Stability of an HTLV-HIV coinfection model with multiple delays and CTL-mediated immunity

Abstract

In the literature, several mathematical models have been formulated and developed to describe the within-host dynamics of either human immunodeficiency virus (HIV) or human T-lymphotropic virus type I (HTLV-I) monoinfections. In this paper, we formulate and analyze a novel within-host dynamics model of HTLV-HIV coinfection taking into consideration the response of cytotoxic T lymphocytes (CTLs). The uninfected $\mathrm{CD}{4}^{+}\mathrm{T}$ cells can be infected via HIV by two mechanisms, free-to-cell and infected-to-cell. On the other hand, the HTLV-I has two modes for transmission, (i) horizontal, via direct infected-to-cell touch, and (ii) vertical, by mitotic division of active HTLV-infected cells. It is well known that the intracellular time delays play an important role in within-host virus dynamics. In this work, we consider six types of distributed-time delays. We investigate the fundamental properties of solutions. Then, we calculate the steady states of the model in terms of threshold parameters. Moreover, we study the global stability of the steady states by using the Lyapunov method. We conduct numerical simulations to illustrate and support our theoretical results. In addition, we discuss the effect of multiple time delays on stability of the steady states of the system.

Introduction

During the past several decades many human viruses and their associated diseases, such as human immunodeficiency virus (HIV), hepatitis C virus (HCV), hepatitis B virus (HBV), dengue virus, human T-lymphotropic virus type I (HTLV-I), and recently coronavirus, have been recognized. Human body can be infected by more that one virus at the same time such as HTLV-HIV, coronavirus/influenza, HCV-HIV, HBV-HIV, HCV-HBV, and malaria-HIV. HTLV and HIV are universal public health matters. HTLV and HIV are two viruses which infect most effective immune cells, $\mathrm{CD}{4}^{+}\mathrm{T}$ cells. Adult T-cell leukemia (ATL) and HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP) are the last stage of HTLV-I infection. Chronic HIV infection leads to acquired immunodeficiency syndrome (AIDS). Both HTLV and HIV have the same ways of transmission such as sharing contaminated needles and unprotected sexual contact with infected partners. Over the last 10 years HTLV-HIV coinfection has been widely documented (see e.g. [1–3], and [4]).

Mathematical models

Mathematical models of HIV and HTLV-I dynamics have become efficient tools to biological and medical scientists. These models can provide a deeper understanding of within-host virus dynamics and assist in predicting the impact of antiviral drug efficacy on viral infection progression (see e.g. [5–16]).

• HIV monoinfection model: The standard HIV dynamics model under the effect of cytotoxic T lymphocytes (CTLs) has been formulated by Nowak and Bangham [17] as follows:

  $$\{\begin{array}{c}\frac{dS(t)}{dt}=\eta -\varrho S(t)-{\vartheta }_{1}S(t)V(t),\hfill \\ \frac{dI(t)}{dt}={\vartheta }_{1}S(t)V(t)-aI(t)-{\mu }_1{C}^I(t)I(t),\hfill \\ \frac{dV(t)}{dt}=bI(t)-\epsilon V(t),\hfill \\ \frac{d{C}^I(t)}{dt}={\sigma }_1{C}^I(t)I(t)-{\pi }_1{C}^I(t),\hfill \end{array}$$ 1

  where $S(t)$, $I(t)$, $V(t)$, and $C^I(t)$ are the concentrations of uninfected $\mathrm{CD}{4}^{+}\mathrm{T}$ cells, active HIV-infected cells, free HIV particles, and HIV-specific CTLs, respectively, and t is the time. η refers to the generation rate of the uninfected $\mathrm{CD}{4}^{+}\mathrm{T}$ cells. The uninfected $\mathrm{CD}{4}^{+}\mathrm{T}$ cells are infected via HIV particles (free-to-cell infection) at rate ${\vartheta }_{1}SV$. The HIV-infected cells produce HIV particles at rate bI. The stimulation rate of effective HIV-specific CTLs due to the presence of HIV-infected cells is defined by ${\sigma }_1{C}^{I}I$. The term ${\mu }_1{C}^{I}I$ accounts for the killing rate of HIV-infected cells due to its specific CTLs. The four compartments S, I, V, and $C^I$ have normal death rates ϱS, aI, εV, and ${\pi }_1{C}^I$, respectively. Several extensions on model (1) have been accomplished (see e.g. [18–20]).
• HTLV-I monoinfection model: The within-host dynamics of HTLV-I has been mathematically modeled in several papers [21–24]. CTL immunity has been included into the HTLV-I dynamics models in many works (see e.g. [25–33]). Lim and Maini [28] have formulated a model for HTLV-I dynamics under the consideration of CTL immunity and mitotic division of active HTLV-infected cells as follows:

  $$\{\begin{array}{c}\frac{dS(t)}{dt}=\eta -\varrho S(t)-{\vartheta }_{3}S(t)Y(t),\hfill \\ \frac{dE(t)}{dt}={\vartheta }_{3}S(t)Y(t)+\mathcal{K}{r}^{\ast }Y(t)-(\psi +\omega )E(t),\hfill \\ \frac{dY(t)}{dt}=\psi E(t)-{\delta }^{\ast }Y(t)-{\mu }_2{C}^Y(t)Y(t),\hfill \\ \frac{d{C}^Y(t)}{dt}={\sigma }_{2}Y(t)-{\pi }_2{C}^Y(t),\hfill \end{array}$$ 2

  where $E(t)$, $Y(t)$, and $C^Y(t)$ are the concentrations of latent HTLV-infected cells, active HTLV-infected cells, and HTLV-specific CTLs at time t, respectively. The term ${\vartheta }_{3}SY$ denotes the infected-to-cell contact rate between HTLV-infected cells and uninfected $\mathrm{CD}{4}^{+}\mathrm{T}$ cells (horizontal transmission). The active HTLV-infected cells transmit vertically to latent compartment at rate $\mathcal{K}{r}^{\ast }Y$ (mitotic transmission), where $\mathcal{K}\in (0,1)$. The HTLV-specific CTLs kill the active HTLV-infected cells at rate ${\mu }_2{C}^{Y}Y$ and are stimulated at rate ${\sigma }_{2}Y$. The term ψE denotes the activation rate of latent HTLV-infected cells. The death rates of E, Y, and $C^Y$ are given by ωE, ${\delta }^{\ast }Y$, and ${\pi }_2{C}^Y$, respectively.
• HTLV-HIV coinfection model: Elaiw and AlShamrani [34] have recently formulated an HTLV-HIV coinfection model as follows:

  $$\{\begin{array}{c}\frac{dS(t)}{dt}=\eta -\varrho S(t)-{\vartheta }_{1}S(t)V(t)-{\vartheta }_{2}S(t)I(t)-{\vartheta }_{3}S(t)Y(t),\hfill \\ \frac{dL(t)}{dt}=(1-\beta )({\vartheta }_{1}S(t)V(t)+{\vartheta }_{2}S(t)I(t))-(\lambda +\gamma )L(t),\hfill \\ \frac{dI(t)}{dt}=\beta ({\vartheta }_{1}S(t)V(t)+{\vartheta }_{2}S(t)I(t))+\lambda L(t)-aI(t)-{\mu }_1{C}^I(t)I(t),\hfill \\ \frac{dE(t)}{dt}=\phi {\vartheta }_{3}S(t)Y(t)-(\psi +\omega )E(t),\hfill \\ \frac{dY(t)}{dt}=\psi E(t)-{\delta }^{\ast }Y(t)-{\mu }_2{C}^Y(t)Y(t),\hfill \\ \frac{dV(t)}{dt}=bI(t)-\epsilon V(t),\hfill \\ \frac{d{C}^I(t)}{dt}={\sigma }_1{C}^I(t)I(t)-{\pi }_1{C}^I(t),\hfill \\ \frac{d{C}^Y(t)}{dt}={\sigma }_2{C}^Y(t)Y(t)-{\pi }_2{C}^Y(t),\hfill \end{array}$$ 3

  where $L(t)$ is the concentration of latent HIV-infected cells. The term ${\vartheta }_{2}SI$ describes the infection rate of uninfected $\mathrm{CD}{4}^{+}\mathrm{T}$ cells by HIV-infected cells. λL and γL are the activation and death rates of latent HIV-infected cells. The parameter $\beta \in (0,1)$ represents the part of newly HIV-infected cells that becomes active, and the other part $1-\beta$ enters a latent stage. The parameter $\phi \in (0,1)$ refers to the part of newly HTLV-infected cells that become latent.

Intracellular delay plays a crucial role in within-host virus dynamics and is defined as the time lapse between viral entry a cell and its production. In case of HIV, it has been estimated that the time between the HIV enters a target cell until producing new HIV particles is about 0.9 days [35]. Time delay has also an important effect in HTLV-I infection. Several works have been devoted to developing mathematical models with time delays to describe the dynamics of HIV (see e.g. [36–43]) and HTLV (see e.g. [44–53]).

Our aim is to take model (3) to further destination by incorporating multiple intracellular time delays and mitotic transmission. We study the fundamental and global properties of the system, then we present numerical simulation. The outcomes of this paper will help clinicians to estimate the suitable time to start the treatment. Our model may be helpful to study different coinfections such as influenza-coronavirus, HCV-HIV, HBV-HIV, and malaria-HIV. It is interesting to note that fractional-order differential equations (FODEs) have been widely studied in several works (see e.g. [54–57]). Modeling and analysis of HIV dynamics with FODEs have been investigated in many papers (see e.g. [58–60]). Clinicians can use the information (in terms of behavior predictions) of fractional-order systems to fit patients’ data with the most appropriate noninteger-order index. As a future work, our coinfection model can be formulated as a system of FODEs.

The multiple delays model

In this section, we extend system (3) by taking under consideration multiple types of distributed-time delays and mitosis of active HTLV-infected cells. We achieve this goal by considering the following system of delay differential equations (DDEs):

$$\{\begin{array}{c}\frac{dS(t)}{dt}=\eta -\varrho S(t)-{\vartheta }_{1}S(t)V(t)-{\vartheta }_{2}S(t)I(t)-{\vartheta }_{3}S(t)Y(t),\hfill \\ \frac{dL(t)}{dt}=(1-\beta ){\int }_0^{{\kappa }_1}{\mathrm{\Lambda }}_1(\mathit{\ell })e^{-{\hslash }_1\mathit{\ell }}S(t-\mathit{\ell })[{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })]d\mathit{\ell }-(\lambda +\gamma )L(t),\hfill \\ \frac{dI(t)}{dt}=\beta {\int }_0^{{\kappa }_2}{\mathrm{\Lambda }}_2(\mathit{\ell })e^{-{\hslash }_2\mathit{\ell }}S(t-\mathit{\ell })[{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })]d\mathit{\ell }\hfill \\ \phantom{\frac{dI(t)}{dt}=}+\lambda {\int }_0^{{\kappa }_3}{\mathrm{\Lambda }}_3(\mathit{\ell })e^{-{\hslash }_3\mathit{\ell }}L(t-\mathit{\ell })d\mathit{\ell }-aI(t)-{\mu }_1{C}^I(t)I(t),\hfill \\ \frac{dE(t)}{dt}=\phi {\vartheta }_3{\int }_0^{{\kappa }_4}{\mathrm{\Lambda }}_4(\mathit{\ell })e^{-{\hslash }_4\mathit{\ell }}S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }+\mathcal{K}{r}^{\ast }Y(t)-(\psi +\omega )E(t),\hfill \\ \frac{dY(t)}{dt}=\psi {\int }_0^{{\kappa }_5}{\mathrm{\Lambda }}_5(\mathit{\ell })e^{-{\hslash }_5\mathit{\ell }}E(t-\mathit{\ell })d\mathit{\ell }+(1-\mathcal{K})r^{\ast }Y(t)-{\delta }^{\ast }Y(t)-{\mu }_2{C}^Y(t)Y(t),\hfill \\ \frac{dV(t)}{dt}=b{\int }_0^{{\kappa }_6}{\mathrm{\Lambda }}_6(\mathit{\ell })e^{-{\hslash }_6\mathit{\ell }}I(t-\mathit{\ell })d\mathit{\ell }-\epsilon V(t),\hfill \\ \frac{d{C}^I(t)}{dt}={\sigma }_1{C}^I(t)I(t)-{\pi }_1{C}^I(t),\hfill \\ \frac{d{C}^Y(t)}{dt}={\sigma }_2{C}^Y(t)Y(t)-{\pi }_2{C}^Y(t).\hfill \end{array}$$ 4

The factor ${\mathrm{\Lambda }}_1(\mathit{\ell })e^{-{\hslash }_1\mathit{\ell }}$ represents the probability that uninfected $\mathrm{CD}{4}^{+}\mathrm{T}$ cells contacted by HIV particles or active HIV-infected cells at time $t-\mathit{\ell }$ survived ℓ time units and become latent infected at time t. The term ${\mathrm{\Lambda }}_2(\mathit{\ell })e^{-{\hslash }_2\mathit{\ell }}$ is the probability that uninfected $\mathrm{CD}{4}^{+}\mathrm{T}$ cells contacted by HIV particles or active HIV-infected cells at time $t-\mathit{\ell }$ survived ℓ time units and become actively infected at time t. The term ${\mathrm{\Lambda }}_3(\mathit{\ell })e^{-{\hslash }_3\mathit{\ell }}$ is the probability that latent HIV-infected $\mathrm{CD}{4}^{+}\mathrm{T}$ cells survived ℓ time units before transmitted to be active at time t. Moreover, the factor ${\mathrm{\Lambda }}_4(\mathit{\ell })e^{-{\hslash }_4\mathit{\ell }}$ demonstrates the probability that the initial infection of uninfected $\mathrm{CD}{4}^{+}\mathrm{T}$ cells and the HTLV-infected cells at time $t-\mathit{\ell }$ completing all the intracellular processes that are required for it to become latent HTLV-infected $\mathrm{CD}{4}^{+}\mathrm{T}$ cells at time t. Further, the probability that latent HTLV-infected $\mathrm{CD}{4}^{+}\mathrm{T}$ cells survived ℓ time units before transmitted to active HTLV-infected cells at time t is given by the factor ${\mathrm{\Lambda }}_5(\mathit{\ell })e^{-{\hslash }_5\mathit{\ell }}$. Furthermore, the term ${\mathrm{\Lambda }}_6(\mathit{\ell })e^{-{\hslash }_6\mathit{\ell }}$ refers to the probability that new immature HIV particles at time $t-\mathit{\ell }$ lost ℓ time units and become mature at time t. Here ${\hslash }_i$, $i=1,2,\dots ,6$, are positive constants. The delay parameter ℓ is randomly taken from a probability distribution function ${\mathrm{\Lambda }}_i(\mathit{\ell })$ over the time interval $[0,{\kappa }_i]$, $i=1,2,\dots ,6$, where ${\kappa }_i$ is the limit superior of this delay period. The function ${\mathrm{\Lambda }}_i(\mathit{\ell })$, $i=1,2,\dots ,6$ satisfies ${\mathrm{\Lambda }}_i(\mathit{\ell })>0$ and

$${\int }_0^{{\kappa }_i}{\mathrm{\Lambda }}_i(\mathit{\ell })d\mathit{\ell }=1\text{and}{\int }_0^{{\kappa }_i}{\mathrm{\Lambda }}_i(\mathit{\ell })e^{-u\mathit{\ell }}d\mathit{\ell }<\mathrm{\infty },$$

where $u>0$. Let us denote

$${\overline{\mathcal{H}}}_i(\mathit{\ell })={\mathrm{\Lambda }}_i(\mathit{\ell })e^{-{\hslash }_i\mathit{\ell }}\text{and}{\mathcal{H}}_i={\int }_0^{{\kappa }_i}{\overline{\mathcal{H}}}_i(\mathit{\ell })d\mathit{\ell },$$

where $i=1,2,\dots ,6$. Thus $0<{\mathcal{H}}_i\le 1$, $i=1,2,\dots 6$.

According to [28], we assume that $r^{\ast }<min\{\varrho ,\omega ,{\delta }^{\ast }\}$. This yields ${\delta }^{\ast }-(1-\mathcal{K})r^{\ast }>0$. Let $r=\mathcal{K}{r}^{\ast }$ and $\delta ={\delta }^{\ast }-(1-\mathcal{K})r^{\ast }$. Then system (4) becomes

$$\{\begin{array}{c}\frac{dS(t)}{dt}=\eta -\varrho S(t)-{\vartheta }_{1}S(t)V(t)-{\vartheta }_{2}S(t)I(t)-{\vartheta }_{3}S(t)Y(t),\hfill \\ \frac{dL(t)}{dt}=(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })[{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })]d\mathit{\ell }-(\lambda +\gamma )L(t),\hfill \\ \frac{dI(t)}{dt}=\beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })[{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })]d\mathit{\ell }\hfill \\ \phantom{\frac{dI(t)}{dt}=}+\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell }-aI(t)-{\mu }_1{C}^I(t)I(t),\hfill \\ \frac{dE(t)}{dt}=\phi {\vartheta }_3{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }+rY(t)-(\psi +\omega )E(t),\hfill \\ \frac{dY(t)}{dt}=\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell }-\delta Y(t)-{\mu }_2{C}^Y(t)Y(t),\hfill \\ \frac{dV(t)}{dt}=b{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })I(t-\mathit{\ell })d\mathit{\ell }-\epsilon V(t),\hfill \\ \frac{d{C}^I(t)}{dt}={\sigma }_1{C}^I(t)I(t)-{\pi }_1{C}^I(t),\hfill \\ \frac{d{C}^Y(t)}{dt}={\sigma }_2{C}^Y(t)Y(t)-{\pi }_2{C}^Y(t).\hfill \end{array}$$ 5

The initial conditions of system (5) are given by:

$$\begin{array}{rl}& S(x)={ϵ}_1(x),L(x)={ϵ}_2(x),I(x)={ϵ}_3(x),E(x)={ϵ}_4(x),\\ & Y(x)={ϵ}_5(x),V(x)={ϵ}_6(x),C^I(x)={ϵ}_7(x),C^Y(x)={ϵ}_8(x),\\ & {ϵ}_j(x)\ge 0,x\in [-\kappa ,0],j=1,2,\dots ,8,\kappa =max\{{\kappa }_1,{\kappa }_2,{\kappa }_3,{\kappa }_4,{\kappa }_5,{\kappa }_6\},\end{array}$$ 6

where ${ϵ}_j(x)\in \mathcal{C}([-\kappa ,0],{\mathbb{R}}_{\ge 0})$, $j=1,2,\dots ,8$, and $\mathcal{C}=\mathcal{C}([-\kappa ,0],{\mathbb{R}}_{\ge 0})$ is the Banach space of continuous functions mapping the interval $[-\kappa ,0]$ into ${\mathbb{R}}_{\ge 0}$ with norm $\parallel {ϵ}_j\parallel ={sup}_{-\kappa \le q\le 0}|{ϵ}_j(q)|$ for ${ϵ}_j\in \mathcal{C}$. Therefore, system (5) with initial conditions (6) has a unique solution by using the standard theory of functional differential equations [61, 62].

Well-posedness of solutions

Proposition 1

All solutions of system (5) with initial conditions (6) are nonnegative and ultimately bounded.

Proof

From the first equation of system (5), we have $\frac{dS(t)}{dt}{|}_{S=0}=\eta >0$, then $S(t)>0$ for all $t\ge 0$. Moreover, the rest of equations of system (5) give us the following:

$$\begin{array}{rl}& \begin{array}{rl}L(t)& ={ϵ}_2(0)e^{-(\lambda +\gamma )t}\\ & +(1-\beta ){\int }_0^t{e}^{-(\lambda +\gamma )(t-\varkappa )}{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(\varkappa -\mathit{\ell })[{\vartheta }_{1}V(\varkappa -\mathit{\ell })+{\vartheta }_{2}I(\varkappa -\mathit{\ell })]d\mathit{\ell }d\varkappa ,\end{array}\\ & \begin{array}{rl}I(t)& ={ϵ}_3(0)e^{-{\int }_0^t(a+{\mu }_1{C}^I(y))dy}\\ & +{\int }_0^t{e}^{-{\int }_{\varkappa }^t(a+{\mu }_1{C}^I(y))dy}[\beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(\varkappa -\mathit{\ell })[{\vartheta }_{1}V(\varkappa -\mathit{\ell })+{\vartheta }_{2}I(\varkappa -\mathit{\ell })]d\mathit{\ell }\\ & +\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(\varkappa -\mathit{\ell })d\mathit{\ell }]d\varkappa ,\end{array}\\ & E(t)={ϵ}_4(0)e^{-(\psi +\omega )t}+{\int }_0^t{e}^{-(\psi +\omega )(t-\varkappa )}[\phi {\vartheta }_3{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(\varkappa -\mathit{\ell })Y(\varkappa -\mathit{\ell })d\mathit{\ell }+rY(\varkappa )]d\varkappa ,\\ & Y(t)={ϵ}_5(0)e^{-{\int }_0^t(\delta +{\mu }_2{C}^Y(y))dy}+\psi {\int }_0^t{e}^{-{\int }_{\varkappa }^t(\delta +{\mu }_2{C}^Y(y))dy}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(\varkappa -\mathit{\ell })d\mathit{\ell }d\varkappa ,\\ & V(t)={ϵ}_6(0)e^{-\epsilon t}+b{\int }_0^t{e}^{-\epsilon (t-\varkappa )}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })I(\varkappa -\mathit{\ell })d\mathit{\ell }d\varkappa ,\\ & C^I(t)={ϵ}_7(0)e^{-{\int }_0^t({\pi }_1-{\sigma }_{1}I(y))dy},\\ & C^Y(t)={ϵ}_8(0)e^{-{\int }_0^t({\pi }_2-{\sigma }_{2}Y(y))dy}.\end{array}$$

Therefore, $L(t),I(t),E(t),Y(t),V(t),C^I(t),C^Y(t)\ge 0$ for all $t\in [0,\kappa ]$. Thus, by a recursive argument, we get $S(t),L(t),I(t),E(t),Y(t),V(t),C^I(t),C^Y(t)\ge 0$ for all $t\ge 0$. Hence, the solutions of system (5) with initial conditions (6) satisfy $(S(t),L(t),I(t),E(t),Y(t),V(t),C^I(t),C^Y(t))\in {\mathbb{R}}_{\ge 0}^8$ for all $t\ge 0$. Next, we establish the boundedness of the model’s solutions. The nonnegativity of the model’s solution implies that ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}S(t)\le \frac{\eta }{\varrho }$. To show the ultimate boundedness of $L(t)$, we let

$${\mathrm{\Psi }}_1(t)=(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }+L(t).$$

Then

$$\begin{array}{rl}\frac{d{\mathrm{\Psi }}_1(t)}{dt}& =(1-\beta )[{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\{\eta -\varrho S(t-\mathit{\ell })\}d\mathit{\ell }-{\vartheta }_3{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }]\\ & -(\lambda +\gamma )L(t)\\ & =(1-\beta )[\eta {\mathcal{H}}_1-\varrho {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }-{\vartheta }_3{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }]\\ & -(\lambda +\gamma )L(t)\\ & \le \eta {\mathcal{H}}_1(1-\beta )-\varrho (1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }-(\lambda +\gamma )L(t)\\ & \le \eta (1-\beta )-{\varphi }_1[(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }+L(t)]=\eta (1-\beta )-{\varphi }_1{\mathrm{\Psi }}_1(t),\end{array}$$

where ${\varphi }_1=min\{\varrho ,\lambda +\gamma \}$. It follows that ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}{\mathrm{\Psi }}_1(t)\le {\mathrm{\Omega }}_1$, where ${\mathrm{\Omega }}_1=\frac{\eta (1-\beta )}{{\varphi }_1}$. Since ${\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }$ and $L(t)$ are nonnegative, then ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}L(t)\le {\mathrm{\Omega }}_1$. Further, we let

$${\mathrm{\Psi }}_2(t)=\beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }+I(t)+\frac{{\mu }_1}{{\sigma }_1}{C}^I(t).$$

Then we obtain

$$\begin{array}{rl}\frac{d{\mathrm{\Psi }}_2(t)}{dt}& =\beta [{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\{\eta -\varrho S(t-\mathit{\ell })\}d\mathit{\ell }-{\vartheta }_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }]\\ & +\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell }-aI(t)-\frac{{\mu }_1{\pi }_1}{{\sigma }_1}{C}^I(t)\\ & =\beta [\eta {\mathcal{H}}_2-\varrho {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }-{\vartheta }_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }]\\ & +\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell }-aI(t)-\frac{{\mu }_1{\pi }_1}{{\sigma }_1}{C}^I(t)\\ & \le \eta \beta {\mathcal{H}}_2+\lambda {\mathrm{\Omega }}_1{\mathcal{H}}_3-\varrho \beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }-aI(t)-\frac{{\mu }_1{\pi }_1}{{\sigma }_1}{C}^I(t)\\ & \le \eta \beta +\lambda {\mathrm{\Omega }}_1-{\varphi }_2[\beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }+I(t)+\frac{{\mu }_1}{{\sigma }_1}{C}^I(t)]\\ & =\eta \beta +\lambda {\mathrm{\Omega }}_1-{\varphi }_2{\mathrm{\Psi }}_2(t),\end{array}$$

where ${\varphi }_2=min\{\varrho ,a,{\pi }_1\}$. It follows that ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}{\mathrm{\Psi }}_2(t)\le {\mathrm{\Omega }}_2$, where ${\mathrm{\Omega }}_2=\frac{\eta \beta +\lambda {\mathrm{\Omega }}_1}{{\varphi }_2}$. Since ${\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }$, $I(t)$ and $C^I(t)$ are nonnegative, then ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}I(t)\le {\mathrm{\Omega }}_2$ and ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}{C}^I(t)\le {\mathrm{\Omega }}_3$, where ${\mathrm{\Omega }}_3=\frac{{\sigma }_1{\mathrm{\Omega }}_2}{{\mu }_1}$. Furthermore, we let

$$\begin{array}{rl}{\mathrm{\Psi }}_3(t)& ={\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }+\frac{1}{\phi }[E(t)+Y(t)]\\ & +\frac{\psi }{\phi }{\int }_0^{{\kappa }_5}{\mathrm{\Lambda }}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^t{e}^{-{\hslash }_5(t-\varkappa )}E(\varkappa )d\varkappa d\mathit{\ell }+\frac{{\mu }_2}{{\sigma }_2\phi }{C}^Y(t).\end{array}$$

Then

$$\begin{array}{rl}\frac{d{\mathrm{\Psi }}_3(t)}{dt}& ={\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[\eta -\varrho S(t-\mathit{\ell })-S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })+{\vartheta }_{3}Y(t-\mathit{\ell })\}]d\mathit{\ell }\\ & +{\vartheta }_3{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }+\frac{r}{\phi }Y(t)-\frac{\psi +\omega }{\phi }E(t)\\ & +\frac{\psi }{\phi }{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell }-\frac{\delta }{\phi }Y(t)\\ & -{\hslash }_5\frac{\psi }{\phi }{\int }_0^{{\kappa }_5}{\mathrm{\Lambda }}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^t{e}^{-{\hslash }_5(t-\varkappa )}E(\varkappa )d\varkappa d\mathit{\ell }\\ & +\frac{\psi }{\phi }E(t){\int }_0^{{\kappa }_5}{\mathrm{\Lambda }}_5(\mathit{\ell })d\mathit{\ell }-\frac{\psi }{\phi }{\int }_0^{{\kappa }_5}{\mathrm{\Lambda }}_5(\mathit{\ell })e^{-{\hslash }_5\mathit{\ell }}E(t-\mathit{\ell })d\mathit{\ell }-\frac{{\mu }_2{\pi }_2}{{\sigma }_2\phi }{C}^Y(t)\\ & \le {\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[\eta -\varrho S(t-\mathit{\ell })]d\mathit{\ell }-\frac{\omega }{\phi }E(t)+(\frac{r}{\phi }-\frac{\delta }{\phi })Y(t)\\ & -{\hslash }_5\frac{\psi }{\phi }{\int }_0^{{\kappa }_5}{\mathrm{\Lambda }}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^t{e}^{-{\hslash }_5(t-\varkappa )}E(\varkappa )d\varkappa d\mathit{\ell }-\frac{{\mu }_2{\pi }_2}{{\sigma }_2\phi }{C}^Y(t)\\ & \le \eta -\varrho {\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }-\frac{\omega }{\phi }E(t)-\frac{\delta -r}{\phi }Y(t)\\ & -{\hslash }_5\frac{\psi }{\phi }{\int }_0^{{\kappa }_5}{\mathrm{\Lambda }}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^t{e}^{-{\hslash }_5(t-\varkappa )}E(\varkappa )d\varkappa d\mathit{\ell }-\frac{{\mu }_2{\pi }_2}{{\sigma }_2\phi }{C}^Y(t).\end{array}$$

Since $\delta -r={\delta }^{\ast }-r^{\ast }>0$, then

$$\begin{array}{rl}\frac{d{\mathrm{\Psi }}_3(t)}{dt}& \le \eta -\varrho {\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }-\frac{\omega }{\phi }E(t)-\frac{{\delta }^{\ast }-r^{\ast }}{\phi }Y(t)\\ & -{\hslash }_5\frac{\psi }{\phi }{\int }_0^{{\kappa }_5}{\mathrm{\Lambda }}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^t{e}^{-{\hslash }_5(t-\varkappa )}E(\varkappa )d\varkappa d\mathit{\ell }-\frac{{\mu }_2{\pi }_2}{{\sigma }_2\phi }{C}^Y(t)\\ & \le \eta -{\varphi }_3[{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }+\frac{1}{\phi }\{E(t)+Y(t)\}\\ & +\frac{\psi }{\phi }{\int }_0^{{\kappa }_5}{\mathrm{\Lambda }}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^t{e}^{-{\hslash }_5(t-\varkappa )}E(\varkappa )d\varkappa d\mathit{\ell }+\frac{{\mu }_2}{{\sigma }_2\phi }{C}^Y(t)]\\ & =\eta -{\varphi }_3{\mathrm{\Psi }}_3(t),\end{array}$$

where ${\varphi }_3=min\{\varrho ,\omega ,{\delta }^{\ast }-r^{\ast },{\hslash }_5,{\pi }_2\}$. It follows that ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}{\mathrm{\Psi }}_3(t)\le \frac{\eta }{{\varphi }_3}$. Since ${\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })d\mathit{\ell }\ge 0$, $E(t)\ge 0$, $Y(t)\ge 0$, and $C^Y(t)\ge 0$, then ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}E(t)\le {\mathrm{\Omega }}_4$, ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}Y(t)\le {\mathrm{\Omega }}_4$, and ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}{C}^Y(t)\le {\mathrm{\Omega }}_5$, where ${\mathrm{\Omega }}_4=\frac{\eta \phi }{{\varphi }_3}$ and ${\mathrm{\Omega }}_5=\frac{\eta {\sigma }_2\phi }{{\mu }_2{\varphi }_3}$. Finally, from the sixth equation of system (5), we have

$$\frac{dV(t)}{dt}=b{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })I(t-\mathit{\ell })d\mathit{\ell }-\epsilon V(t)\le b{\mathcal{H}}_6{\mathrm{\Omega }}_2-\epsilon V(t)\le b{\mathrm{\Omega }}_2-\epsilon V(t).$$

This implies that ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}V(t)\le {\mathrm{\Omega }}_6$, where ${\mathrm{\Omega }}_6=\frac{b{\mathrm{\Omega }}_2}{\epsilon }$. □

According to Proposition 1, we can show that the region

$$\begin{array}{rl}\mathrm{\Theta }& =\{(S,L,I,E,Y,V,C^I,C^Y)\in {\mathcal{C}}_{\ge 0}^8:\parallel S\parallel \le {\mathrm{\Omega }}_1,\parallel L\parallel \le {\mathrm{\Omega }}_1,\parallel I\parallel \le {\mathrm{\Omega }}_2,\\ & \parallel E\parallel \le {\mathrm{\Omega }}_4,\parallel Y\parallel \le {\mathrm{\Omega }}_4,\parallel V\parallel \le {\mathrm{\Omega }}_6,\parallel C^I\parallel \le {\mathrm{\Omega }}_3,\parallel C^Y\parallel \le {\mathrm{\Omega }}_5\}\end{array}$$

is positively invariant with respect to system (5).

Steady states analysis

In this section, we calculate all possible steady states of the model and derive the threshold parameters which guarantee the existence of the steady states. Let us define

$$\mathcal{P}=\lambda {\mathcal{H}}_1{\mathcal{H}}_3(1-\beta )+\beta {\mathcal{H}}_2(\gamma +\lambda ),$$ 7

which will be used throughout the paper. Let $(S,L,I,E,Y,V,C^I,C^Y)$ be any steady state of system (5) satisfying the following equations:

$$\begin{array}{rl}& 0=\eta -\varrho S-{\vartheta }_{1}SV-{\vartheta }_{2}SI-{\vartheta }_{3}SY,\\ & 0={\mathcal{H}}_1(1-\beta )({\vartheta }_{1}SV+{\vartheta }_{2}SI)-(\lambda +\gamma )L,\\ & 0=\beta {\mathcal{H}}_2({\vartheta }_{1}SV+{\vartheta }_{2}SI)+\lambda {\mathcal{H}}_{3}L-aI-{\mu }_1{C}^{I}I,\\ & 0=\phi {\vartheta }_3{\mathcal{H}}_{4}SY+rY-(\psi +\omega )E,\\ & 0=\psi {\mathcal{H}}_{5}E-\delta Y-{\mu }_2{C}^{Y}Y,\\ & 0=b{\mathcal{H}}_{6}I-\epsilon V,\\ & 0=({\sigma }_{1}I-{\pi }_1)C^I,\\ & 0=({\sigma }_{2}Y-{\pi }_2)C^Y.\end{array}$$

We find that system (5) has eight possible steady states.

(i) Infection-free steady state, , where $S_0=\eta /\varrho$. In this case, the body is free from HTLV and HIV.

(ii) Persistent HIV monoinfection steady state with ineffective immune response, , where

$$\begin{array}{rl}& S_1=\frac{S_0}{{\mathrm{\Re }}_1},L_1=\frac{a\epsilon \varrho {\mathcal{H}}_1(1-\beta )}{\mathcal{P}(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}({\mathrm{\Re }}_1-1),\\ & I_1=\frac{\epsilon \varrho }{b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2}({\mathrm{\Re }}_1-1),V_1=\frac{\varrho b{\mathcal{H}}_6}{b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2}({\mathrm{\Re }}_1-1),\end{array}$$ 8

and ${\mathrm{\Re }}_1$ is the basic HIV monoinfection reproduction number for system (5) and is defined as follows:

$${\mathrm{\Re }}_1=\frac{\mathcal{P}{S}_0(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}{a\epsilon (\gamma +\lambda )}={\mathrm{\Re }}_{11}+{\mathrm{\Re }}_{12},$$

where

$${\mathrm{\Re }}_{11}=\frac{\mathcal{P}{S}_{0}b{\vartheta }_1{\mathcal{H}}_6}{a\epsilon (\gamma +\lambda )},{\mathrm{\Re }}_{12}=\frac{\mathcal{P}{S}_0{\vartheta }_2}{a(\gamma +\lambda )}.$$

The parameter ${\mathrm{\Re }}_1$ determines whether or not a persistent HIV infection can be established. In fact, ${\mathrm{\Re }}_{11}$ measures the average number of secondary HIV infected generation caused by an existing free HIV particle due to free-to-cell transmission, while ${\mathrm{\Re }}_{12}$ measures the average numbers of secondary HIV infected generation caused by living active HIV-infected cells due to infected-to-cell transmission. The steady state describes the case of persistent HIV monoinfection without immune response.

(iii) Persistent HTLV monoinfection steady state with ineffective immune response, , where

$$S_2=\frac{S_0}{{\mathrm{\Re }}_2},E_2=\frac{\varrho \delta }{{\vartheta }_3\psi {\mathcal{H}}_5}({\mathrm{\Re }}_2-1),Y_2=\frac{\varrho }{{\vartheta }_3}({\mathrm{\Re }}_2-1),$$

and ${\mathrm{\Re }}_2$ is the basic HTLV monoinfection reproduction number for system (5) and is defined as follows:

$${\mathrm{\Re }}_2=\frac{\phi {\vartheta }_3\psi {\mathcal{H}}_4{\mathcal{H}}_5{S}_0}{(\delta -r{\mathcal{H}}_5)\psi +\delta \omega }.$$

The parameter ${\mathrm{\Re }}_2$ decides whether or not a persistent HTLV infection can be established. The steady state describes a persistent HTLV monoinfection without immune response.

We mention that ${\mathrm{\Re }}_1$ and ${\mathrm{\Re }}_2$ state the threshold dynamics of infection-free equilibrium and can be calculated by different methods such as (a) the next-generation matrix method of van den Driessche and Watmough [63], (b) local stability of the infection-free equilibrium , and (c) the existence of the chronic HIV and HTLV monoinfection equilibria with inactive immune response. In the present paper, we derive ${\mathrm{\Re }}_1$ and ${\mathrm{\Re }}_2$ by method (c).

(iv) Persistent HIV monoinfection steady state with only effective HIV-specific CTL, , where

$$\begin{array}{rl}& S_3=\frac{\epsilon {\sigma }_1\eta }{{\pi }_1(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\varrho \epsilon {\sigma }_1},L_3=\frac{\eta {\pi }_1{\mathcal{H}}_1(1-\beta )(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}{(\gamma +\lambda )[{\pi }_1(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\varrho \epsilon {\sigma }_1]},\\ & I_3=\frac{{\pi }_1}{{\sigma }_1},V_3=\frac{b{\mathcal{H}}_6}{\epsilon }{I}_3=\frac{b{\pi }_1{\mathcal{H}}_6}{\epsilon {\sigma }_1},C_3^I=\frac{a}{{\mu }_1}({\mathrm{\Re }}_3-1),\end{array}$$

and

$${\mathrm{\Re }}_3=\frac{{\sigma }_1\eta \mathcal{P}(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}{a(\gamma +\lambda )[{\pi }_1(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\varrho \epsilon {\sigma }_1]},$$

is the HIV-specific CTL immunity reproduction number in case of HIV monoinfection. The parameter ${\mathrm{\Re }}_3$ determines whether or not the HIV-specific CTL immune response is effective in the absence of HTLV.

(v) Persistent HTLV monoinfection steady state with only effective HTLV-specific CTL, , where

$$\begin{array}{rl}& S_4=\frac{{\sigma }_2\eta }{{\pi }_2{\vartheta }_3+\varrho {\sigma }_2},\\ & Y_4=\frac{{\pi }_2}{{\sigma }_2},\\ & E_4=\frac{{\pi }_2[r({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)+{\vartheta }_3\eta \phi {\sigma }_2{\mathcal{H}}_4]}{{\sigma }_2(\psi +\omega )({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)},\\ & C_4^Y=\frac{(\delta -r{\mathcal{H}}_5)\psi +\delta \omega }{{\mu }_2(\psi +\omega )}({\mathrm{\Re }}_4-1),\end{array}$$

and ${\mathrm{\Re }}_4$ is the HTLV-specific CTL immunity reproduction number in case of HTLV monoinfection and is stated as follows:

$${\mathrm{\Re }}_4=\frac{{\sigma }_2\eta \phi {\vartheta }_3\psi {\mathcal{H}}_4{\mathcal{H}}_5}{({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)[(\delta -r{\mathcal{H}}_5)\psi +\delta \omega ]}.$$

The parameter ${\mathrm{\Re }}_4$ determines whether or not the HTLV-specific CTL immune response is effective in the absence of HIV.

(vi) Persistent HTLV-HIV coinfection steady state with only effective HIV-specific CTL, , where

$$\begin{array}{rl}& S_5=\frac{(\delta -r{\mathcal{H}}_5)\psi +\delta \omega }{\phi {\vartheta }_3\psi {\mathcal{H}}_4{\mathcal{H}}_5}=S_2,\\ & I_5=\frac{{\pi }_1}{{\sigma }_1}=I_3,\\ & V_5=\frac{b{\pi }_1{\mathcal{H}}_6}{\epsilon {\sigma }_1}=V_3,\\ & L_5=\frac{{\pi }_1{\mathcal{H}}_1(1-\beta )(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)[(\delta -r{\mathcal{H}}_5)\psi +\delta \omega ]}{\epsilon {\vartheta }_3{\sigma }_1\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5(\gamma +\lambda )},\\ & E_5=\frac{\delta [{\pi }_1(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\varrho \epsilon {\sigma }_1]}{\epsilon {\vartheta }_3{\sigma }_1\psi {\mathcal{H}}_5}({\mathrm{\Re }}_5-1),\\ & Y_5=\frac{{\pi }_1(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\varrho \epsilon {\sigma }_1}{\epsilon {\vartheta }_3{\sigma }_1}({\mathrm{\Re }}_5-1),\\ & C_5^I=\frac{a}{{\mu }_1}(\frac{{\mathrm{\Re }}_1}{{\mathrm{\Re }}_2}-1),\end{array}$$

where

$${\mathrm{\Re }}_5=\frac{\eta \phi \epsilon {\vartheta }_3{\sigma }_1\psi {\mathcal{H}}_4{\mathcal{H}}_5}{[(\delta -r{\mathcal{H}}_5)\psi +\delta \omega ][{\pi }_1(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\varrho \epsilon {\sigma }_1]}.$$

Here, the parameter ${\mathrm{\Re }}_5$ is the HTLV infection reproduction number in the presence of HIV infection and determines whether or not HIV-infected patients could be dually infected with HTLV.

(vii) Persistent HTLV-HIV coinfection steady state with only effective HTLV-specific CTL, , where

$$\begin{array}{rl}& S_6=\frac{a\epsilon (\gamma +\lambda )}{\mathcal{P}(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}=S_1,\\ & Y_6=\frac{{\pi }_2}{{\sigma }_2}=Y_4,\\ & L_6=\frac{a\epsilon {\mathcal{H}}_1(1-\beta )({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)}{{\sigma }_2\mathcal{P}(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}({\mathrm{\Re }}_6-1),\\ & I_6=\frac{\epsilon ({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)}{{\sigma }_2(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}({\mathrm{\Re }}_6-1),\\ & E_6=\frac{{\pi }_2[r\mathcal{P}(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+a\epsilon {\vartheta }_3\phi {\mathcal{H}}_4(\gamma +\lambda )]}{{\sigma }_2\mathcal{P}(\psi +\omega )(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)},\\ & V_6=\frac{b{\mathcal{H}}_6({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)}{{\sigma }_2(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}({\mathrm{\Re }}_6-1),\\ & C_6^Y=\frac{(\delta -r{\mathcal{H}}_5)\psi +\delta \omega }{{\mu }_2(\psi +\omega )}(\frac{{\mathrm{\Re }}_2}{{\mathrm{\Re }}_1}-1),\end{array}$$

and

$${\mathrm{\Re }}_6=\frac{\eta {\sigma }_2\mathcal{P}(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}{a\epsilon (\gamma +\lambda )({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)},$$

is the HIV infection reproduction number in the presence of HTLV infection. It is clear that ${\mathrm{\Re }}_6$ determines whether or not HTLV-infected patients could be dually infected with HIV.

(viii) Persistent HTLV-HIV coinfection steady state with effective HIV-specific CTL and HTLV-specific CTL, , where

$$\begin{array}{rl}& S_7=\frac{\epsilon {\sigma }_1{\sigma }_2\eta }{{\pi }_1{\sigma }_2(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\epsilon {\sigma }_1({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)},\\ & L_7=\frac{{\pi }_1{\sigma }_2\eta {\mathcal{H}}_1(1-\beta )(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}{(\gamma +\lambda )[{\pi }_1{\sigma }_2(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\epsilon {\sigma }_1({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)]},\\ & E_7=\frac{{\pi }_2[{\vartheta }_3\epsilon {\sigma }_1{\sigma }_2\eta \phi {\mathcal{H}}_4+r\{{\pi }_1{\sigma }_2(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\epsilon {\sigma }_1({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)\}]}{{\sigma }_2(\psi +\omega )[{\pi }_1{\sigma }_2(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\epsilon {\sigma }_1({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)]},\\ & I_7=\frac{{\pi }_1}{{\sigma }_1}=I_3=I_5,\\ & Y_7=\frac{{\pi }_2}{{\sigma }_2}=Y_4=Y_6,\\ & V_7=\frac{b{\pi }_1{\mathcal{H}}_6}{\epsilon {\sigma }_1}=V_3=V_5,\\ & C_7^I=\frac{a}{{\mu }_1}({\mathrm{\Re }}_7-1),\\ & C_7^Y=\frac{(\delta -r{\mathcal{H}}_5)\psi +\delta \omega }{{\mu }_2(\psi +\omega )}({\mathrm{\Re }}_8-1),\end{array}$$

and

$$\begin{array}{rl}& {\mathrm{\Re }}_7=\frac{{\sigma }_1{\sigma }_2\eta \mathcal{P}(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}{a(\gamma +\lambda )[{\pi }_1{\sigma }_2(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\epsilon {\sigma }_1({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)]},\\ & {\mathrm{\Re }}_8=\frac{\epsilon {\vartheta }_3\eta {\sigma }_1{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{[{\pi }_1{\sigma }_2(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\epsilon {\sigma }_1({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)][(\delta -r{\mathcal{H}}_5)\psi +\delta \omega ]}.\end{array}$$

The parameter ${\mathrm{\Re }}_7$ is the competed HIV-specific CTL immunity reproduction number in case of HTLV-HIV coinfection. The parameter ${\mathrm{\Re }}_8$ is the competed HTLV-specific CTL immunity reproduction number in case of HTLV-HIV coinfection. Clearly, exists when ${\mathrm{\Re }}_7>1$ and ${\mathrm{\Re }}_8>1$.

Global stability analysis

In this section, we use the Lyapunov method to show the global asymptotic stability of the model’s steady states. For formation of Lyapunov functionals, we follow the works [64, 65]. Denote $U=U(t)$, where $U\in (S,L,I,E,Y,V,C^I,C^Y)$.

Let a function ${\mathrm{\Phi }}_j(S,L,I,E,Y,V,C^I,C^Y)$ and ${\mathrm{\Upsilon }}_j^{\prime }$ be the largest invariant subset of

$${\mathrm{\Upsilon }}_j=\{(S,L,I,E,Y,V,C^I,C^Y):\frac{d{\mathrm{\Phi }}_j}{dt}=0\},j=0,1,2,\dots ,7.$$

We define a function $Ϝ(x)=x-1-lnx$.

Theorem 1

If ${\mathrm{\Re }}_1\le 1$ and ${\mathrm{\Re }}_2\le 1$, then is globally asymptotically stable (GAS).

Proof

We define a Lyapunov functional as follows:

$$\begin{array}{rl}{\mathrm{\Phi }}_0& =\mathcal{P}{S}_0Ϝ\left(\frac{S}{S_0}\right)+\lambda {\mathcal{H}}_{3}L+(\gamma +\lambda )I+\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}E+\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y+\frac{\mathcal{P}{\vartheta }_1{S}_0}{\epsilon }V\\ & +\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}{C}^I+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y+\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}S(\varkappa )\\ & \times [{\vartheta }_{1}V(\varkappa )+{\vartheta }_{2}I(\varkappa )]d\varkappa d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}S(\varkappa )[{\vartheta }_{1}V(\varkappa )+{\vartheta }_{2}I(\varkappa )]d\varkappa d\mathit{\ell }\\ & +\lambda (\gamma +\lambda ){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}L(\varkappa )d\varkappa d\mathit{\ell }+\frac{\mathcal{P}{\vartheta }_3}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}S(\varkappa )Y(\varkappa )d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}E(\varkappa )d\varkappa d\mathit{\ell }+\frac{b\mathcal{P}{\vartheta }_1{S}_0}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}I(\varkappa )d\varkappa d\mathit{\ell }.\end{array}$$

Clearly, ${\mathrm{\Phi }}_0(S,L,I,E,Y,V,C^I,C^Y)>0$ for all $S,L,I,E,Y,V,C^I,C^Y>0$, and ${\mathrm{\Phi }}_0(S_0,0,0,0,0,0,0,0)=0$. We calculate $\frac{d{\mathrm{\Phi }}_0}{dt}$ along the solutions of model (5) as follows:

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_0}{dt}& =\mathcal{P}(1-\frac{S_0}{S})(\eta -\varrho S-{\vartheta }_{1}SV-{\vartheta }_{2}SI-{\vartheta }_{3}SY)+\lambda {\mathcal{H}}_3[(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })\\ & \times \{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }-(\lambda +\gamma )L]+(\gamma +\lambda )[\beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })\\ & \times \{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }+\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell }-aI-{\mu }_1{C}^{I}I]\\ & +\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}[\phi {\vartheta }_3{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }+rY-(\psi +\omega )E]\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}[\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell }-\delta Y-{\mu }_2{C}^{Y}Y]\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_0}{\epsilon }[b{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })I(t-\mathit{\ell })d\mathit{\ell }-\epsilon V]+\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}({\sigma }_1{C}^{I}I-{\pi }_1{C}^I)\\ & +\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}({\sigma }_2{C}^{Y}Y-{\pi }_2{C}^Y)+\mathcal{P}({\vartheta }_{1}SV+{\vartheta }_{2}SI)-\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\\ & \times S(t-\mathit{\ell })[{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })]d\mathit{\ell }-\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })\\ & \times [{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })]d\mathit{\ell }+\lambda (\gamma +\lambda ){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })[L-L(t-\mathit{\ell })]d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[SY-S(t-\mathit{\ell })Y(t-\mathit{\ell })]d\mathit{\ell }+\frac{\mathcal{P}(\psi +\omega )}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\\ & \times [E-E(t-\mathit{\ell })]d\mathit{\ell }+\frac{b\mathcal{P}{\vartheta }_1{S}_0}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[I-I(t-\mathit{\ell })]d\mathit{\ell }.\end{array}$$ 9

Summing the terms of Eq. (9), we obtain

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_0}{dt}& =\mathcal{P}(1-\frac{S_0}{S})(\eta -\varrho S)+\mathcal{P}{\vartheta }_2{S}_{0}I-a(\lambda +\gamma )I+\frac{b\mathcal{P}{\vartheta }_1{\mathcal{H}}_6{S}_0}{\epsilon }I+\mathcal{P}{\vartheta }_3{S}_{0}Y\\ & -\frac{\mathcal{P}[(\delta -r{\mathcal{H}}_5)\psi +\delta \omega ]}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y-\frac{{\mu }_1{\pi }_1(\gamma +\lambda )}{{\sigma }_1}{C}^I-\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y.\end{array}$$

Using $S_0=\eta /\varrho$, we obtain

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_0}{dt}& =-\varrho \mathcal{P}\frac{{(S-S_0)}^2}{S}+a(\lambda +\gamma )({\mathrm{\Re }}_1-1)I+\frac{\mathcal{P}[(\delta -r{\mathcal{H}}_5)\psi +\delta \omega ]}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}({\mathrm{\Re }}_2-1)Y\\ & -\frac{{\mu }_1{\pi }_1(\gamma +\lambda )}{{\sigma }_1}{C}^I-\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y.\end{array}$$

Since $r<\delta$ and $0<{\mathcal{H}}_5\le 1$, then $\delta -r{\mathcal{H}}_5>0$. Therefore, $\frac{d{\mathrm{\Phi }}_0}{dt}\le 0$ for all $S,I,Y,C^I,C^Y>0$; moreover, $\frac{d{\mathrm{\Phi }}_0}{dt}=0$ when $(S(t),I(t),Y(t),C^I(t),C^Y(t))=(S_0,0,0,0,0)$. The solutions of system (5) converge to ${\mathrm{\Upsilon }}_0^{\prime }$. The set ${\mathrm{\Upsilon }}_0^{\prime }$ includes elements with $(S(t),I(t),Y(t),C^I(t),C^Y(t))=(S_0,0,0,0,0)$. Then $\frac{dS(t)}{dt}=\frac{dY(t)}{dt}=0$ and the first and fifth equations of system (5) become

$$\begin{array}{rl}& 0=\frac{dS(t)}{dt}=\eta -\varrho S_0-{\vartheta }_1{S}_{0}V(t),\\ & 0=\frac{dY(t)}{dt}=\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell },\end{array}$$

which give $V(t)=E(t)=0$ for all t. In addition, we have $\frac{dI(t)}{dt}=0$, and from the third equation of system (5) we have

$$0=\frac{dI(t)}{dt}=\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell },$$

which yields $L(t)=0$ for all t and hence . Applying Lyapunov–LaSalle asymptotic stability (LLAS) theorem [66–68], we get that is GAS. □

The following equalities are needed in the next theorems:

$$\begin{array}{rl}& ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)=[ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_n}{S_n{V}_{n}L}\right)+ln\left(\frac{S_n}{S}\right)+ln\left(\frac{V_{n}L}{V{L}_n}\right)],\\ & ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)=[ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_n}{S_n{V}_{n}I}\right)+ln\left(\frac{S_n}{S}\right)+ln\left(\frac{V_{n}I}{V{I}_n}\right)],\\ & ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)=[ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_n}{S_n{I}_{n}L}\right)+ln\left(\frac{S_n}{S}\right)+ln\left(\frac{I_{n}L}{I{L}_n}\right)],\\ & ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)=[ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{n}I}\right)+ln\left(\frac{S_n}{S}\right)],\\ & ln\left(\frac{L(t-\mathit{\ell })}{L}\right)=ln\left(\frac{L(t-\mathit{\ell })I_n}{L_{n}I}\right)+ln\left(\frac{L_{n}I}{L{I}_n}\right),\\ & ln\left(\frac{I(t-\mathit{\ell })}{I}\right)=ln\left(\frac{I(t-\mathit{\ell })V_n}{I_{n}V}\right)+ln\left(\frac{I_{n}V}{I{V}_n}\right),\text{where }n=1,3,5,6,7.\end{array}$$ 10

Further,

$$\begin{array}{rl}& ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)=ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_m}{S_m{Y}_{m}E}\right)+ln\left(\frac{S_m}{S}\right)+ln\left(\frac{Y_{m}E}{Y{E}_m}\right),\\ & ln\left(\frac{E(t-\mathit{\ell })}{E}\right)=ln\left(\frac{E(t-\mathit{\ell })Y_m}{E_{m}Y}\right)+ln\left(\frac{Y{E}_m}{Y_{m}E}\right),\text{where }m=2,4,5,6,7.\end{array}$$ 11

Theorem 2

Let ${\mathrm{\Re }}_1>1$, ${\mathrm{\Re }}_2/{\mathrm{\Re }}_1\le 1$, and ${\mathrm{\Re }}_3\le 1$, then is GAS.

Proof

Define a functional as follows:

$$\begin{array}{rl}{\mathrm{\Phi }}_1& =\mathcal{P}{S}_1Ϝ\left(\frac{S}{S_1}\right)+\lambda {\mathcal{H}}_3{L}_1Ϝ\left(\frac{L}{L_1}\right)+(\gamma +\lambda )I_1Ϝ\left(\frac{I}{I_1}\right)+\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}E+\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_1}{\epsilon }{V}_1Ϝ\left(\frac{V}{V_1}\right)+\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}{C}^I+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y\\ & +{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta )S_1{V}_1{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )V(\varkappa )}{S_1{V}_1}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta )S_1{I}_1{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )I(\varkappa )}{S_1{I}_1}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_1\beta (\gamma +\lambda )S_1{V}_1{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )V(\varkappa )}{S_1{V}_1}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_2\beta (\gamma +\lambda )S_1{I}_1{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )I(\varkappa )}{S_1{I}_1}\right)d\varkappa d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_1{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{L(\varkappa )}{L_1}\right)d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}S(\varkappa )Y(\varkappa )d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}E(\varkappa )d\varkappa d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_1{I}_1}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{I(\varkappa )}{I_1}\right)d\varkappa d\mathit{\ell }.\end{array}$$

Calculate $\frac{d{\mathrm{\Phi }}_1}{dt}$ as follows:

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_1}{dt}& =\mathcal{P}(1-\frac{S_1}{S})(\eta -\varrho S-{\vartheta }_{1}SV-{\vartheta }_{2}SI-{\vartheta }_{3}SY)+\lambda {\mathcal{H}}_3(1-\frac{L_1}{L})\\ & \times [(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }-(\lambda +\gamma )L]\\ & +(\gamma +\lambda )(1-\frac{I_1}{I})[\beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }\\ & +\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell }-aI-{\mu }_1{C}^{I}I]\\ & +\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}[\phi {\vartheta }_3{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }\\ & +rY-(\psi +\omega )E]+\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}[\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell }-\delta Y-{\mu }_2{C}^{Y}Y]\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_1}{\epsilon }(1-\frac{V_1}{V})[b{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })I(t-\mathit{\ell })d\mathit{\ell }-\epsilon V]\\ & +\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}({\sigma }_1{C}^{I}I-{\pi }_1{C}^I)+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}({\sigma }_2{C}^{Y}Y-{\pi }_2{C}^Y)\\ & +{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta )S_1{V}_1\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{SV}{S_1{V}_1}-\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{S_1{V}_1}+ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)]d\mathit{\ell }\\ & +{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta )S_1{I}_1\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{SI}{S_1{I}_1}-\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_1{I}_1}+ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)]d\mathit{\ell }\\ & +{\vartheta }_1\beta (\gamma +\lambda )S_1{V}_1\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{SV}{S_1{V}_1}-\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{S_1{V}_1}+ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)]d\mathit{\ell }\\ & +{\vartheta }_2\beta (\gamma +\lambda )S_1{I}_1\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{SI}{S_1{I}_1}-\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_1{I}_1}+ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)]d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_1{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })[\frac{L}{L_1}-\frac{L(t-\mathit{\ell })}{L_1}+ln\left(\frac{L(t-\mathit{\ell })}{L}\right)]d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[SY-S(t-\mathit{\ell })Y(t-\mathit{\ell })]d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })[E-E(t-\mathit{\ell })]d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_1{I}_1}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[\frac{I}{I_1}-\frac{I(t-\mathit{\ell })}{I_1}+ln\left(\frac{I(t-\mathit{\ell })}{I}\right)]d\mathit{\ell }.\end{array}$$ 12

Summing the terms of Eq. (12), we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_1}{dt}& =\mathcal{P}(1-\frac{S_1}{S})(\eta -\varrho S)+\mathcal{P}{\vartheta }_2{S}_{1}I+\mathcal{P}{\vartheta }_3{S}_{1}Y\\ & -{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_1}{L}d\mathit{\ell }\\ & -{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_1}{L}d\mathit{\ell }+\lambda {\mathcal{H}}_3(\lambda +\gamma )L_1-a(\lambda +\gamma )I\\ & -{\vartheta }_1\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_1}{I}d\mathit{\ell }\\ & -{\vartheta }_2\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })I_1}{I}d\mathit{\ell }\\ & -\lambda (\lambda +\gamma ){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })\frac{L(t-\mathit{\ell })I_1}{I}d\mathit{\ell }+a(\lambda +\gamma )I_1\\ & +{\mu }_1(\lambda +\gamma )C^I{I}_1+\frac{\mathcal{P}r}{\phi {\mathcal{H}}_4}Y-\frac{\mathcal{P}\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y\\ & -\frac{\mathcal{P}b{\vartheta }_1{S}_1}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })\frac{I(t-\mathit{\ell })V_1}{V}d\mathit{\ell }+\mathcal{P}{\vartheta }_1{S}_1{V}_1\\ & -\frac{{\mu }_1{\pi }_1(\lambda +\gamma )}{{\sigma }_1}{C}^I-\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_1{V}_1{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_1{I}_1{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_1{S}_1{V}_1{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_2{S}_1{I}_1{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_1{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })ln\left(\frac{L(t-\mathit{\ell })}{L}\right)d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_1{\mathcal{H}}_6}{\epsilon }I+\frac{\mathcal{P}b{\vartheta }_1{S}_1{I}_1}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })ln\left(\frac{I(t-\mathit{\ell })}{I}\right)d\mathit{\ell }.\end{array}$$

The steady state conditions for are given by

$$\begin{array}{rl}& \eta =\varrho S_1+{\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1,{\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1)=(\lambda +\gamma )L_1,\\ & \beta {\mathcal{H}}_2({\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1)+\lambda {\mathcal{H}}_3{L}_1=a{I}_1{V}_1=\frac{b{\mathcal{H}}_6{I}_1}{\epsilon }.\end{array}$$

Then we get

$$\mathcal{P}({\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1)=a(\lambda +\gamma )I_1.$$

Further, we obtain

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_1}{dt}& =\mathcal{P}(1-\frac{S_1}{S})(\varrho S_1-\varrho S)+\mathcal{P}({\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1)(1-\frac{S_1}{S})+\mathcal{P}{\vartheta }_3{S}_{1}Y\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_1{V}_1{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_1}{S_1{V}_{1}L}d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_1{I}_1{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_1}{S_1{I}_{1}L}d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_1{\mathcal{H}}_3(1-\beta )({\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1)\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_1{V}_1{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_1}{S_1{V}_{1}I}d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_1{I}_1{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{1}I}d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })\frac{L(t-\mathit{\ell })I_1}{L_{1}I}d\mathit{\ell }\\ & +\mathcal{P}({\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1)+{\mu }_1(\lambda +\gamma )C^I{I}_1-\frac{\mathcal{P}[(\delta -r{\mathcal{H}}_5)\psi +\delta \omega ]}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y\\ & -\frac{\mathcal{P}{\vartheta }_1{S}_1{V}_1}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })\frac{I(t-\mathit{\ell })V_1}{I_{1}V}d\mathit{\ell }\\ & +\mathcal{P}{\vartheta }_1{S}_1{V}_1-\frac{{\mu }_1{\pi }_1(\lambda +\gamma )}{{\sigma }_1}{C}^I-\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_1{V}_1{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_1{I}_1{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_1{S}_1{V}_1{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_2{S}_1{I}_1{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })ln\left(\frac{L(t-\mathit{\ell })}{L}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_1{V}_1}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })ln\left(\frac{I(t-\mathit{\ell })}{I}\right)d\mathit{\ell }.\end{array}$$

Using the equalities given by (10) in case of $n=1$, we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_1}{dt}& =-\varrho \mathcal{P}\frac{{(S-S_1)}^2}{S}-\mathcal{P}({\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1)[\frac{S_1}{S}-1-ln\left(\frac{S_1}{S}\right)]\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_1{V}_1\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_1}{S_1{V}_{1}L}-1-ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_1}{S_1{V}_{1}L}\right)]d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_1{I}_1\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_1}{S_1{I}_{1}L}-1-ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_1}{S_1{I}_{1}L}\right)]d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_1{V}_1\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_1}{S_1{V}_{1}I}-1-ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_1}{S_1{V}_{1}I}\right)]d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_1{I}_1{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{1}I}-1-ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{1}I}\right)]d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1)\\ & \times {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })[\frac{L(t-\mathit{\ell })I_1}{L_{1}I}-1-ln\left(\frac{L(t-\mathit{\ell })I_1}{L_{1}I}\right)]d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_1{S}_1{V}_1}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[\frac{I(t-\mathit{\ell })V_1}{I_{1}V}-1-ln\left(\frac{I(t-\mathit{\ell })V_1}{I_{1}V}\right)]d\mathit{\ell }\\ & +\frac{\mathcal{P}[(\delta -r{\mathcal{H}}_5)\psi +\delta \omega ]}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(\frac{{\vartheta }_3\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5{S}_1}{(\delta -r{\mathcal{H}}_5)\psi +\delta \omega }-1)Y\\ & +{\mu }_1(\lambda +\gamma )(I_1-\frac{{\pi }_1}{{\sigma }_1})C^I-\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y.\end{array}$$ 13

Therefore, Eq. (13) becomes

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_1}{dt}& =-\varrho \mathcal{P}\frac{{(S-S_1)}^2}{S}-\mathcal{P}({\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1)Ϝ\left(\frac{S_1}{S}\right)\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_1{V}_1{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_1}{S_1{V}_{1}L}\right)d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_1{I}_1{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_1}{S_1{I}_{1}L}\right)d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_1{V}_1{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_1}{S_1{V}_{1}I}\right)d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_1{I}_1{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{1}I}\right)d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_1{V}_1+{\vartheta }_2{S}_1{I}_1){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })Ϝ\left(\frac{L(t-\mathit{\ell })I_1}{L_{1}I}\right)d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_1{S}_1{V}_1}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })Ϝ\left(\frac{I(t-\mathit{\ell })V_1}{I_{1}V}\right)d\mathit{\ell }+\frac{\mathcal{P}[(\delta -r{\mathcal{H}}_5)\psi +\delta \omega ]}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(\frac{{\mathrm{\Re }}_2}{{\mathrm{\Re }}_1}-1)Y\\ & +\frac{{\mu }_1(\gamma +\lambda )[{\pi }_1(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\varrho \epsilon {\sigma }_1]}{{\sigma }_1(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}({\mathrm{\Re }}_3-1)C^I-\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y.\end{array}$$

Since ${\mathrm{\Re }}_2/{\mathrm{\Re }}_1\le 1$ and ${\mathrm{\Re }}_3\le 1$, then $\frac{d{\mathrm{\Phi }}_1}{dt}\le 0$ for all $S,L,I,Y,V,C^I,C^Y>0$. Moreover, $\frac{d{\mathrm{\Phi }}_1}{dt}=0$ when $S=S_1$ and $Y=C^I=C^Y=Ϝ=0$. The solutions of system (5) converge to ${\mathrm{\Upsilon }}_1^{\prime }$ which includes elements that satisfy $S(t)=S_1$ and $Ϝ=0$ i.e.

$$\begin{array}{rl}\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_1}{S_1{V}_{1}L}& =\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_1}{S_1{I}_{1}L}=\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_1}{S_1{V}_{1}I}\\ & =\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{1}I}=\frac{L(t-\mathit{\ell })I_1}{L_{1}I}=\frac{I(t-\mathit{\ell })V_1}{I_{1}V}=1\end{array}$$ 14

for all $t\in [0,\kappa ]$. If $S(t)=S_1$, then from Eq. (14) we get $L(t)=L_1$, $I(t)=I_1$, and $V(t)=V_1$ for all t. Further, for each element of ${\mathrm{\Upsilon }}_1^{\prime }$, we have $Y(t)=0$ and then $\frac{dY(t)}{dt}=0$. The fifth equation of system (5) becomes

$$0=\frac{dY(t)}{dt}=\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell },$$

which provides $E(t)=0$ for all t, and hence . Therefore, using LLAS theorem we get that is GAS. □

Theorem 3

If ${\mathrm{\Re }}_2>1$, ${\mathrm{\Re }}_1/{\mathrm{\Re }}_2\le 1$, and ${\mathrm{\Re }}_4\le 1$, then is GAS.

Proof

Define

$$\begin{array}{rl}{\mathrm{\Phi }}_2& =\mathcal{P}{S}_2Ϝ\left(\frac{S}{S_2}\right)+\lambda {\mathcal{H}}_{3}L+(\gamma +\lambda )I+\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}{E}_2Ϝ\left(\frac{E}{E_2}\right)+\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_2Ϝ\left(\frac{Y}{Y_2}\right)\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_2}{\epsilon }V+\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}{C}^I+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y+\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}S(\varkappa )\\ & \times [{\vartheta }_{1}V(\varkappa )+{\vartheta }_{2}I(\varkappa )]d\varkappa d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}S(\varkappa )[{\vartheta }_{1}V(\varkappa )+{\vartheta }_{2}I(\varkappa )]d\varkappa d\mathit{\ell }\\ & +\lambda (\gamma +\lambda ){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}L(\varkappa )d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_2{Y}_2}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )Y(\varkappa )}{S_2{Y}_2}\right)d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )E_2}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{E(\varkappa )}{E_2}\right)d\varkappa d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_2}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}I(\varkappa )d\varkappa d\mathit{\ell }.\end{array}$$

We calculate $\frac{d{\mathrm{\Phi }}_2}{dt}$ as follows:

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_2}{dt}& =\mathcal{P}(1-\frac{S_2}{S})(\eta -\varrho S-{\vartheta }_{1}SV-{\vartheta }_{2}SI-{\vartheta }_{3}SY)\\ & +\lambda {\mathcal{H}}_3[(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }-(\lambda +\gamma )L]\\ & +(\gamma +\lambda )[\beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }\\ & +\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell }-aI-{\mu }_1{C}^{I}I]\\ & +\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}(1-\frac{E_2}{E})[\phi {\vartheta }_3{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }+rY-(\psi +\omega )E]\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(1-\frac{Y_2}{Y})[\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell }-\delta Y-{\mu }_2{C}^{Y}Y]+\frac{\mathcal{P}{\vartheta }_1{S}_2}{\epsilon }\\ & \times [b{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })I(t-\mathit{\ell })d\mathit{\ell }-\epsilon V]+\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}({\sigma }_1{C}^{I}I-{\pi }_1{C}^I)+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}\\ & \times ({\sigma }_2{C}^{Y}Y-{\pi }_2{C}^Y)+\mathcal{P}({\vartheta }_{1}SV+{\vartheta }_{2}SI)-\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })\\ & \times [{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })]d\mathit{\ell }-\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })[{\vartheta }_{1}V(t-\mathit{\ell })\\ & +{\vartheta }_{2}I(t-\mathit{\ell })]d\mathit{\ell }+\lambda (\gamma +\lambda ){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })[L-L(t-\mathit{\ell })]d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_2{Y}_2}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[\frac{SY}{S_2{Y}_2}\\ & -\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{S_2{Y}_2}+ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)]d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )E_2}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })[\frac{E}{E_2}-\frac{E(t-\mathit{\ell })}{E_2}+ln\left(\frac{E(t-\mathit{\ell })}{E}\right)]d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_2}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[I-I(t-\mathit{\ell })]d\mathit{\ell }.\end{array}$$ 15

By collecting the terms of Eq. (15), we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_2}{dt}& =\mathcal{P}[(1-\frac{S_2}{S})(\eta -\varrho S)+{\vartheta }_2{S}_{2}I+{\vartheta }_3{S}_{2}Y-\frac{a(\lambda +\gamma )}{\mathcal{P}}I+\frac{r}{\phi {\mathcal{H}}_4}Y\\ & -\frac{{\vartheta }_3}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_2}{E}d\mathit{\ell }-\frac{r}{\phi {\mathcal{H}}_4}\frac{Y{E}_2}{E}+\frac{\psi +\omega }{\phi {\mathcal{H}}_4}{E}_2-\frac{\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y\\ & -\frac{\psi +\omega }{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\frac{E(t-\mathit{\ell })Y_2}{Y}d\mathit{\ell }+\frac{\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_2+\frac{{\mu }_2(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y{Y}_2\\ & -\frac{{\mu }_1{\pi }_1(\gamma +\lambda )}{{\sigma }_1\mathcal{P}}{C}^I-\frac{{\mu }_2{\pi }_2(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y\\ & +\frac{{\vartheta }_3{S}_2{Y}_2}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)d\mathit{\ell }\\ & +\frac{(\psi +\omega )E_2}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })ln\left(\frac{E(t-\mathit{\ell })}{E}\right)d\mathit{\ell }+\frac{b{\vartheta }_1{S}_2{\mathcal{H}}_6}{\epsilon }I].\end{array}$$

Using the steady state conditions for

$$\eta =\varrho S_2+{\vartheta }_3{S}_2{Y}_2,{\vartheta }_3{S}_2{Y}_2+\frac{r{Y}_2}{\phi {\mathcal{H}}_4}=\frac{(\psi +\omega )E_2}{\phi {\mathcal{H}}_4}=\frac{\delta (\psi +\omega )Y_2}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5},$$

we obtain

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_2}{dt}& =\mathcal{P}[(1-\frac{S_2}{S})(\varrho S_2-\varrho S)+{\vartheta }_3{S}_2{Y}_2(1-\frac{S_2}{S})\\ & -\frac{{\vartheta }_3{S}_2{Y}_2}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_2}{S_2{Y}_{2}E}d\mathit{\ell }-\frac{r{Y}_2}{\phi {\mathcal{H}}_4}\frac{Y{E}_2}{Y_{2}E}\\ & +{\vartheta }_3{S}_2{Y}_2+\frac{r{Y}_2}{\phi {\mathcal{H}}_4}-(\frac{{\vartheta }_3{S}_2{Y}_2}{{\mathcal{H}}_5}+\frac{r{Y}_2}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\frac{E(t-\mathit{\ell })Y_2}{E_{2}Y}d\mathit{\ell }\\ & +{\vartheta }_3{S}_2{Y}_2+\frac{r{Y}_2}{\phi {\mathcal{H}}_4}+\frac{{\mu }_2(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y{Y}_2-\frac{{\mu }_1{\pi }_1(\gamma +\lambda )}{{\sigma }_1\mathcal{P}}{C}^I-\frac{{\mu }_2{\pi }_2(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y\\ & +\frac{{\vartheta }_3{S}_2{Y}_2}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)d\mathit{\ell }\\ & +(\frac{{\vartheta }_3{S}_2{Y}_2}{{\mathcal{H}}_5}+\frac{r{Y}_2}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })ln\left(\frac{E(t-\mathit{\ell })}{E}\right)d\mathit{\ell }\\ & +\frac{a(\lambda +\gamma )}{\mathcal{P}}\{\frac{\mathcal{P}{S}_2(\epsilon {\vartheta }_2+b{\vartheta }_1{\mathcal{H}}_6)}{a\epsilon (\lambda +\gamma )}-1\}I].\end{array}$$

Using the equalities given by (11) in case of $m=2$, we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_2}{dt}& =-\mathcal{P}[\varrho \frac{{(S-S_2)}^2}{S}+{\vartheta }_3{S}_2{Y}_2\{\frac{S_2}{S}-1-ln\left(\frac{S_2}{S}\right)\}+\frac{r{Y}_2}{\phi {\mathcal{H}}_4}\{\frac{Y{E}_2}{Y_{2}E}-1-ln\left(\frac{Y{E}_2}{Y_{2}E}\right)\}\\ & +\frac{{\vartheta }_3{S}_2{Y}_2}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\{\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_2}{S_2{Y}_{2}E}-1-ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_2}{S_2{Y}_{2}E}\right)\}d\mathit{\ell }\\ & +(\frac{{\vartheta }_3{S}_2{Y}_2}{{\mathcal{H}}_5}+\frac{r{Y}_2}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\{\frac{E(t-\mathit{\ell })Y_2}{E_{2}Y}-1-ln\left(\frac{E(t-\mathit{\ell })Y_2}{E_{2}Y}\right)\}d\mathit{\ell }\\ & -\frac{{\mu }_1{\pi }_1(\gamma +\lambda )}{{\sigma }_1\mathcal{P}}{C}^I+\frac{{\mu }_2(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(Y_2-\frac{{\pi }_2}{{\sigma }_2})C^Y]\\ & +a(\lambda +\gamma )(\frac{\mathcal{P}{S}_2(\epsilon {\vartheta }_2+b{\vartheta }_1{\mathcal{H}}_6)}{a\epsilon (\lambda +\gamma )}-1)I.\end{array}$$ 16

Therefore, Eq. (16) becomes

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_2}{dt}& =-\mathcal{P}[\varrho \frac{{(S-S_2)}^2}{S}+\frac{r{Y}_2}{\phi {\mathcal{H}}_4}Ϝ\left(\frac{Y{E}_2}{Y_{2}E}\right)\\ & +\frac{{\vartheta }_3{S}_2{Y}_2}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\{Ϝ\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_2}{S_2{Y}_{2}E}\right)+Ϝ\left(\frac{S_2}{S}\right)\}d\mathit{\ell }\\ & +(\frac{{\vartheta }_3{S}_2{Y}_2}{{\mathcal{H}}_5}+\frac{r{Y}_2}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })Ϝ\left(\frac{E(t-\mathit{\ell })Y_2}{E_{2}Y}\right)d\mathit{\ell }\\ & -\frac{{\mu }_1{\pi }_1(\gamma +\lambda )}{{\sigma }_1\mathcal{P}}{C}^I+\frac{{\mu }_2(\psi +\omega )(\varrho {\sigma }_2+{\vartheta }_3{\pi }_2)}{{\vartheta }_3{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}({\mathrm{\Re }}_4-1)C^Y]\\ & +a(\lambda +\gamma )(\frac{{\mathrm{\Re }}_1}{{\mathrm{\Re }}_2}-1)I.\end{array}$$

Thus, if ${\mathrm{\Re }}_1/{\mathrm{\Re }}_2\le 1$ and ${\mathrm{\Re }}_4\le 1$, then $\frac{d{\mathrm{\Phi }}_2}{dt}\le 0$ for all $S,I,E,Y,C^I,C^Y>0$. Moreover, $\frac{d{\mathrm{\Phi }}_2}{dt}=0$ when $(S,E,Y,I,C^I,C^Y)=(S_2,E_2,Y_2,0,0,0)$. The solutions of system (5) converge to ${\mathrm{\Upsilon }}_2^{\prime }$ which includes elements with $(S(t),E(t),Y(t),I(t),C^I(t),C^Y(t))=(S_2,E_2,Y_2,0,0,0)$. Then we have $\frac{dS(t)}{dt}=0$, and the first equation of system (5) becomes

$$0=\frac{dS(t)}{dt}=\eta -\varrho S_2-{\vartheta }_1{S}_{2}V(t)-{\vartheta }_3{S}_2{Y}_2,$$

which yields $V(t)=0$ for all t. Moreover, we have $\frac{dI(t)}{dt}=0$ and from the third equation of system (5) we get

$$0=\frac{dI(t)}{dt}=\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell },$$

which implies that $L(t)=0$ for all t. Therefore, . Applying LLAS theorem, we get is GAS. □

Theorem 4

Let ${\mathrm{\Re }}_3>1$ and ${\mathrm{\Re }}_5\le 1$, then is GAS.

Proof

Define a functional as follows:

$$\begin{array}{rl}{\mathrm{\Phi }}_3& =\mathcal{P}{S}_3Ϝ\left(\frac{S}{S_3}\right)+\lambda {\mathcal{H}}_3{L}_3Ϝ\left(\frac{L}{L_3}\right)+(\gamma +\lambda )I_3Ϝ\left(\frac{I}{I_3}\right)+\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}E+\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_3}{\epsilon }{V}_3Ϝ\left(\frac{V}{V_3}\right)+\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}{C}_3^IϜ\left(\frac{C^I}{C_3^I}\right)+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y\\ & +{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta )S_3{V}_3{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )V(\varkappa )}{S_3{V}_3}\right)d\varkappa d\mathit{\ell }+{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta )S_3{I}_3\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )I(\varkappa )}{S_3{I}_3}\right)d\varkappa d\mathit{\ell }+{\vartheta }_1\beta (\gamma +\lambda )S_3{V}_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\\ & \times {\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )V(\varkappa )}{S_3{V}_3}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_2\beta (\gamma +\lambda )S_3{I}_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )I(\varkappa )}{S_3{I}_3}\right)d\varkappa d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_3{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{L(\varkappa )}{L_3}\right)d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}S(\varkappa )Y(\varkappa )d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}E(\varkappa )d\varkappa d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_3{I}_3}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{I(\varkappa )}{I_3}\right)d\varkappa d\mathit{\ell }.\end{array}$$ 17

We calculate $\frac{d{\mathrm{\Phi }}_3}{dt}$ as follows:

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_3}{dt}& =\mathcal{P}(1-\frac{S_3}{S})(\eta -\varrho S-{\vartheta }_{1}SV-{\vartheta }_{2}SI-{\vartheta }_{3}SY)+\lambda {\mathcal{H}}_3(1-\frac{L_3}{L})\\ & \times [(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }-(\lambda +\gamma )L]\\ & +(\gamma +\lambda )(1-\frac{I_3}{I})[\beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }\\ & +\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell }-aI-{\mu }_1{C}^{I}I]\\ & +\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}[\phi {\vartheta }_3{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }+rY-(\psi +\omega )E]\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}[\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell }-\delta Y-{\mu }_2{C}^{Y}Y]\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_3}{\epsilon }(1-\frac{V_3}{V})[b{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })I(t-\mathit{\ell })d\mathit{\ell }-\epsilon V]+\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}\\ & \times (1-\frac{C_3^I}{C^I})({\sigma }_1{C}^{I}I-{\pi }_1{C}^I)+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}({\sigma }_2{C}^{Y}Y-{\pi }_2{C}^Y)\\ & +{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta )S_3{V}_3\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{SV}{S_3{V}_3}-\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{S_3{V}_3}+ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)]d\mathit{\ell }\\ & +{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta )S_3{I}_3\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{SI}{S_3{I}_3}-\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_3{I}_3}+ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)]d\mathit{\ell }\\ & +{\vartheta }_1\beta (\gamma +\lambda )S_3{V}_3\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{SV}{S_3{V}_3}-\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{S_3{V}_3}+ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)]d\mathit{\ell }\\ & +{\vartheta }_2\beta (\gamma +\lambda )S_3{I}_3\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{SI}{S_3{I}_3}-\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_3{I}_3}+ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)]d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_3{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })[\frac{L}{L_3}-\frac{L(t-\mathit{\ell })}{L_3}+ln\left(\frac{L(t-\mathit{\ell })}{L}\right)]d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[SY-S(t-\mathit{\ell })Y(t-\mathit{\ell })]d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })[E-E(t-\mathit{\ell })]d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_3{I}_3}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[\frac{I}{I_3}-\frac{I(t-\mathit{\ell })}{I_3}+ln\left(\frac{I(t-\mathit{\ell })}{I}\right)]d\mathit{\ell }.\end{array}$$ 18

Collecting the terms of Eq. (18), we derive

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_3}{dt}& =\mathcal{P}(1-\frac{S_3}{S})(\eta -\varrho S)+\mathcal{P}{\vartheta }_2{S}_{3}I+\mathcal{P}{\vartheta }_3{S}_{3}Y-{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\\ & \times \frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_3}{L}d\mathit{\ell }-{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_3}{L}d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(\lambda +\gamma )L_3-a(\lambda +\gamma )I-{\vartheta }_1\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_3}{I}d\mathit{\ell }\\ & -{\vartheta }_2\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })I_3}{I}d\mathit{\ell }\\ & -\lambda (\lambda +\gamma ){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })\frac{L(t-\mathit{\ell })I_3}{I}d\mathit{\ell }+a(\lambda +\gamma )I_3+{\mu }_1(\lambda +\gamma )C^I{I}_3\\ & +\frac{\mathcal{P}r}{\phi {\mathcal{H}}_4}Y-\frac{\mathcal{P}\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y-\frac{\mathcal{P}b{\vartheta }_1{S}_3}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })\frac{I(t-\mathit{\ell })V_3}{V}d\mathit{\ell }\\ & +\mathcal{P}{\vartheta }_1{S}_3{V}_3-\frac{{\mu }_1{\pi }_1(\lambda +\gamma )}{{\sigma }_1}{C}^I-{\mu }_1(\lambda +\gamma )C_3^{I}I\\ & +\frac{{\mu }_1{\pi }_1(\lambda +\gamma )}{{\sigma }_1}{C}_3^I-\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_3{V}_3{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_3{I}_3{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_1{S}_3{V}_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_2{S}_3{I}_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_3{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })ln\left(\frac{L(t-\mathit{\ell })}{L}\right)d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_3{\mathcal{H}}_6}{\epsilon }I+\frac{b\mathcal{P}{\vartheta }_1{S}_3{I}_3}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })ln\left(\frac{I(t-\mathit{\ell })}{I}\right)d\mathit{\ell }.\end{array}$$

Using the steady state conditions for

$$\begin{array}{rl}& \eta =\varrho S_3+{\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3,{\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3)=(\lambda +\gamma )L_3,\\ & \beta {\mathcal{H}}_2({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3)+\lambda {\mathcal{H}}_3{L}_3=(a+{\mu }_1{C}_3^I)I_3,I_3=\frac{{\pi }_1}{{\sigma }_1},V_3=\frac{b{\mathcal{H}}_6}{\epsilon }{I}_3,\end{array}$$

we get

$$\mathcal{P}({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3)=(\lambda +\gamma )(a+{\mu }_1{C}_3^I)I_3.$$

Further, we obtain

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_3}{dt}& =\mathcal{P}(1-\frac{S_3}{S})(\varrho S_3-\varrho S)+\mathcal{P}({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3)(1-\frac{S_3}{S})+\mathcal{P}{\vartheta }_3{S}_{3}Y\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_3{V}_3{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_3}{S_3{V}_{3}L}d\mathit{\ell }-\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_3{I}_3\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_3}{S_3{I}_{3}L}d\mathit{\ell }+\lambda {\mathcal{H}}_1{\mathcal{H}}_3(1-\beta )({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3)\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_3{V}_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_3}{S_3{V}_{3}I}d\mathit{\ell }-\beta (\gamma +\lambda ){\vartheta }_2{S}_3{I}_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\\ & \times \frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{3}I}d\mathit{\ell }-\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })\frac{L(t-\mathit{\ell })I_3}{L_{3}I}d\mathit{\ell }\\ & +\mathcal{P}({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3)-\frac{\mathcal{P}[(\delta -r{\mathcal{H}}_5)\psi +\delta \omega ]}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y-\frac{\mathcal{P}{\vartheta }_1{S}_3{V}_3}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })\\ & \times \frac{I(t-\mathit{\ell })V_3}{I_{3}V}d\mathit{\ell }+\mathcal{P}{\vartheta }_1{S}_3{V}_3-\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y+\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_3{V}_3\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_3{I}_3{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_1{S}_3{V}_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_2{S}_3{I}_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })ln\left(\frac{L(t-\mathit{\ell })}{L}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_3{V}_3}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })ln\left(\frac{I(t-\mathit{\ell })}{I}\right)d\mathit{\ell }.\end{array}$$

Using the equalities given by (10) in case of $n=3$, we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_3}{dt}& =-\varrho \mathcal{P}\frac{{(S-S_3)}^2}{S}-\mathcal{P}({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3)[\frac{S_3}{S}-1-ln\left(\frac{S_3}{S}\right)]\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_3{V}_3\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_3}{S_3{V}_{3}L}-1-ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_3}{S_3{V}_{3}L}\right)]d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_3{I}_3\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_3}{S_3{I}_{3}L}-1-ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_3}{S_3{I}_{3}L}\right)]d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_3{V}_3\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_3}{S_3{V}_{3}I}-1-ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_3}{S_3{V}_{3}I}\right)]d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_3{I}_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{3}I}-1-ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{3}I}\right)]d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3)\\ & \times {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })[\frac{L(t-\mathit{\ell })I_3}{L_{3}I}-1-ln\left(\frac{L(t-\mathit{\ell })I_3}{L_{3}I}\right)]d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_1{S}_3{V}_3}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[\frac{I(t-\mathit{\ell })V_3}{I_{3}V}-1-ln\left(\frac{I(t-\mathit{\ell })V_3}{I_{3}V}\right)]d\mathit{\ell }\\ & +\mathcal{P}{\vartheta }_3(S_3-\frac{(\delta -r{\mathcal{H}}_5)\psi +\delta \omega }{{\vartheta }_3\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5})Y-\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y.\end{array}$$ 19

Therefore, Eq. (19) becomes

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_3}{dt}& =-\varrho \mathcal{P}\frac{{(S-S_3)}^2}{S}-\mathcal{P}({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3)Ϝ\left(\frac{S_3}{S}\right)\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_3{V}_3{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_3}{S_3{V}_{3}L}\right)d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_3{I}_3{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_3}{S_3{I}_{3}L}\right)d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_3{V}_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_3}{S_3{V}_{3}I}\right)d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_3{I}_3{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{3}I}\right)d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })Ϝ\left(\frac{L(t-\mathit{\ell })I_3}{L_{3}I}\right)d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_1{S}_3{V}_3}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })Ϝ\left(\frac{I(t-\mathit{\ell })V_3}{I_{3}V}\right)d\mathit{\ell }\\ & +\mathcal{P}{\vartheta }_3(S_3-S_5)Y-\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y.\end{array}$$

Hence, if ${\mathrm{\Re }}_5\le 1$, then does not exist since $E_5\le 0$ and $Y_5\le 0$. In this case

$$\begin{array}{rl}& \frac{dE(t)}{dt}=\phi {\vartheta }_3{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }+rY(t)-(\psi +\omega )E(t)\le 0,\\ & \frac{dY(t)}{dt}=\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell }-\delta Y(t)-{\mu }_2{C}^{Y}Y\le 0.\end{array}$$

Now we want to find the value S̄ such that, for all $0<S(t)\le \overline{S}$, we get $\frac{dE(t)}{dt}\le 0$ and $\frac{dY(t)}{dt}\le 0$. Let us consider

$$\begin{array}{rl}& \frac{d}{dt}[\frac{1}{{\mathcal{H}}_4}E+\frac{\psi +\omega }{\psi {\mathcal{H}}_4{\mathcal{H}}_5}Y+\frac{\phi {\vartheta }_3}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}S(\varkappa )Y(\varkappa )d\varkappa d\mathit{\ell }\\ & +\frac{\psi +\omega }{{\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}E(\varkappa )d\varkappa d\mathit{\ell }]\\ & =\phi {\vartheta }_{3}SY-\frac{(\delta -r{\mathcal{H}}_5)\psi +\delta \omega }{\psi {\mathcal{H}}_4{\mathcal{H}}_5}Y-\frac{{\mu }_2(\psi +\omega )}{\psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^{Y}Y\\ & =\phi {\vartheta }_3(S-\frac{(\delta -r{\mathcal{H}}_5)\psi +\delta \omega }{\phi {\vartheta }_3\psi {\mathcal{H}}_4{\mathcal{H}}_5})Y-\frac{{\mu }_2(\psi +\omega )}{\psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^{Y}Y\le 0\text{for all }{C}^Y,Y>0.\end{array}$$

This happens when $S_3\le \overline{S}=\frac{(\delta -r{\mathcal{H}}_5)\psi +\delta \omega }{{\vartheta }_3\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}=S_5$. Clearly, $\frac{d{\mathrm{\Phi }}_3}{dt}\le 0$ for all $S,L,I,Y,V,C^Y>0$, where $\frac{d{\mathrm{\Phi }}_3}{dt}=0$ occurs at $S=S_3$ and $Y=C^Y=0$. The solutions of system (5) converge to ${\mathrm{\Upsilon }}_3^{\prime }$ which includes elements satisfying $S(t)=S_3$ and $Ϝ=0$ i.e.

$$\begin{array}{rl}\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_3}{S_3{V}_{3}L}& =\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_3}{S_3{I}_{3}L}=\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_3}{S_3{V}_{3}I}\\ & =\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{3}I}=\frac{L(t-\mathit{\ell })I_3}{L_{3}I}=\frac{I(t-\mathit{\ell })V_3}{I_{3}V}=1\end{array}$$ 20

for all $t\in [0,\kappa ]$. If $S(t)=S_3$, then from Eq. (20) we get $L(t)=L_3$, $I(t)=I_3$, and $V(t)=V_3$ for all t. Thus, ${\mathrm{\Upsilon }}_3^{\prime }$ contains elements with $I(t)=I_3$, $V(t)=V_3$, $Y(t)=0$, and then $\frac{dI(t)}{dt}=0$, $\frac{dY(t)}{dt}=0$. The third and fifth equations of system (5) become

$$\begin{array}{rl}& 0=\frac{dI(t)}{dt}=\beta {\mathcal{H}}_2({\vartheta }_1{S}_3{V}_3+{\vartheta }_2{S}_3{I}_3)+\lambda {\mathcal{H}}_3{L}_3-a{I}_3-{\mu }_1{C}^I(t)I_3,\\ & 0=\frac{dY(t)}{dt}=\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_{5}E(t-\mathit{\ell })d\mathit{\ell },\end{array}$$

which yield $C^I(t)=C_3^I$ and $E(t)=0$ for all t. Therefore, . Applying LLAS theorem, we get is GAS. □

Theorem 5

If ${\mathrm{\Re }}_4>1$ and ${\mathrm{\Re }}_6\le 1$, then is GAS.

Proof

Let

$$\begin{array}{rl}{\mathrm{\Phi }}_4& =\mathcal{P}{S}_4Ϝ\left(\frac{S}{S_4}\right)+\lambda {\mathcal{H}}_{3}L+(\gamma +\lambda )I+\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}{E}_4Ϝ\left(\frac{E}{E_4}\right)+\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_4Ϝ\left(\frac{Y}{Y_4}\right)\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_4}{\epsilon }V+\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}{C}^I+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}_4^YϜ\left(\frac{C^Y}{C_4^Y}\right)\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}S(\varkappa )[{\vartheta }_{1}V(\varkappa )+{\vartheta }_{2}I(\varkappa )]d\varkappa d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}S(\varkappa )[{\vartheta }_{1}V(\varkappa )+{\vartheta }_{2}I(\varkappa )]d\varkappa d\mathit{\ell }\\ & +\lambda (\gamma +\lambda ){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}L(\varkappa )d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_4{Y}_4}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )Y(\varkappa )}{S_4{Y}_4}\right)d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )E_4}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{E(\varkappa )}{E_4}\right)d\varkappa d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_4}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell }){\int }_{t-\mathit{\ell }}^{t}I(\varkappa )d\varkappa d\mathit{\ell }.\end{array}$$

Calculate $\frac{d{\mathrm{\Phi }}_4}{dt}$ as follows:

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_4}{dt}& =\mathcal{P}(1-\frac{S_4}{S})(\eta -\varrho S-{\vartheta }_{1}SV-{\vartheta }_{2}SI-{\vartheta }_{3}SY)\\ & +\lambda {\mathcal{H}}_3[(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }-(\lambda +\gamma )L]\\ & +(\gamma +\lambda )[\beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }\\ & +\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell }-aI-{\mu }_1{C}^{I}I]\\ & +\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}(1-\frac{E_4}{E})[\phi {\vartheta }_3{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }+rY-(\psi +\omega )E]\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(1-\frac{Y_4}{Y})[\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell }-\delta Y-{\mu }_2{C}^{Y}Y]\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_4}{\epsilon }[b{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })I(t-\mathit{\ell })d\mathit{\ell }-\epsilon V]+\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}({\sigma }_1{C}^{I}I-{\pi }_1{C}^I)\\ & +\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(1-\frac{C_4^Y}{C^Y})({\sigma }_2{C}^{Y}Y-{\pi }_2{C}^Y)+\mathcal{P}({\vartheta }_{1}SV+{\vartheta }_{2}SI)\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })[{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })]d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })[{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })]d\mathit{\ell }\\ & +\lambda (\gamma +\lambda ){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })[L-L(t-\mathit{\ell })]d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_4{Y}_4}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[\frac{SY}{S_4{Y}_4}-\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{S_4{Y}_4}\\ & +ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)]d\mathit{\ell }+\frac{\mathcal{P}(\psi +\omega )E_4}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })[\frac{E}{E_4}-\frac{E(t-\mathit{\ell })}{E_4}\\ & +ln\left(\frac{E(t-\mathit{\ell })}{E}\right)]d\mathit{\ell }+\frac{b\mathcal{P}{\vartheta }_1{S}_4}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[I-I(t-\mathit{\ell })]d\mathit{\ell }.\end{array}$$ 21

Summing the terms of Eq. (21), we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_4}{dt}& =\mathcal{P}[(1-\frac{S_4}{S})(\eta -\varrho S)+{\vartheta }_2{S}_{4}I+{\vartheta }_3{S}_{4}Y-\frac{a(\lambda +\gamma )}{\mathcal{P}}I+\frac{r}{\phi {\mathcal{H}}_4}Y\\ & -\frac{{\vartheta }_3}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_4}{E}d\mathit{\ell }-\frac{r}{\phi {\mathcal{H}}_4}\frac{Y{E}_4}{E}+\frac{\psi +\omega }{\phi {\mathcal{H}}_4}{E}_4\\ & -\frac{\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y-\frac{\psi +\omega }{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\frac{E(t-\mathit{\ell })Y_4}{Y}d\mathit{\ell }+\frac{\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_4\\ & +\frac{{\mu }_2(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y{Y}_4-\frac{{\mu }_1{\pi }_1(\gamma +\lambda )}{{\sigma }_1\mathcal{P}}{C}^I-\frac{{\mu }_2{\pi }_2(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y-\frac{{\mu }_2(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}_4^{Y}Y\\ & +\frac{{\mu }_2{\pi }_2(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}_4^Y+\frac{{\vartheta }_3{S}_4{Y}_4}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)d\mathit{\ell }\\ & +\frac{(\psi +\omega )E_4}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })ln\left(\frac{E(t-\mathit{\ell })}{E}\right)d\mathit{\ell }+\frac{b{\vartheta }_1{S}_4{\mathcal{H}}_6}{\epsilon }I].\end{array}$$

Using the steady state conditions for

$$\begin{array}{rl}& \eta =\varrho S_4+{\vartheta }_3{S}_4{Y}_4,Y_4=\frac{{\pi }_2}{{\sigma }_2},\\ & {\vartheta }_3{S}_4{Y}_4+\frac{r{Y}_4}{\phi {\mathcal{H}}_4}=\frac{(\psi +\omega )E_4}{\phi {\mathcal{H}}_4}=\frac{\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_4+\frac{{\mu }_2(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}_4^Y{Y}_4,\end{array}$$

we obtain

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_4}{dt}& =\mathcal{P}[(1-\frac{S_4}{S})(\varrho S_4-\varrho S)+{\vartheta }_3{S}_4{Y}_4(1-\frac{S_4}{S})\\ & -\frac{{\vartheta }_3{S}_4{Y}_4}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_4}{S_4{Y}_{4}E}d\mathit{\ell }\\ & -\frac{r{Y}_4}{\phi {\mathcal{H}}_4}\frac{Y{E}_4}{Y_{4}E}+{\vartheta }_3{S}_4{Y}_4+\frac{r{Y}_4}{\phi {\mathcal{H}}_4}\\ & -(\frac{{\vartheta }_3{S}_4{Y}_4}{{\mathcal{H}}_5}+\frac{r{Y}_4}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\frac{E(t-\mathit{\ell })Y_4}{E_{4}Y}d\mathit{\ell }\\ & +{\vartheta }_3{S}_4{Y}_4+\frac{r{Y}_4}{\phi {\mathcal{H}}_4}-\frac{{\mu }_1{\pi }_1(\gamma +\lambda )}{{\sigma }_1\mathcal{P}}{C}^I\\ & +\frac{{\vartheta }_3{S}_4{Y}_4}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)d\mathit{\ell }\\ & +(\frac{{\vartheta }_3{S}_4{Y}_4}{{\mathcal{H}}_5}+\frac{r{Y}_4}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })ln\left(\frac{E(t-\mathit{\ell })}{E}\right)d\mathit{\ell }\\ & +\frac{a(\lambda +\gamma )}{\mathcal{P}}\{\frac{\mathcal{P}{S}_4(\epsilon {\vartheta }_2+b{\vartheta }_1{\mathcal{H}}_6)}{a\epsilon (\lambda +\gamma )}-1\}I].\end{array}$$

Using the equalities given by (11) in case of $m=4$, we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_4}{dt}& =-\mathcal{P}[\varrho \frac{{(S-S_4)}^2}{S}+{\vartheta }_3{S}_4{Y}_4\{\frac{S_4}{S}-1-ln\left(\frac{S_4}{S}\right)\}+\frac{r{Y}_4}{\phi {\mathcal{H}}_4}\{\frac{Y{E}_4}{Y_{4}E}-1-ln\left(\frac{Y{E}_4}{Y_{4}E}\right)\}\\ & +\frac{{\vartheta }_3{S}_4{Y}_4}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\{\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_4}{S_4{Y}_{4}E}-1-ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_4}{S_4{Y}_{4}E}\right)\}d\mathit{\ell }\\ & +(\frac{{\vartheta }_3{S}_4{Y}_4}{{\mathcal{H}}_5}+\frac{r{Y}_4}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\{\frac{E(t-\mathit{\ell })Y_4}{E_{4}Y}-1-ln\left(\frac{E(t-\mathit{\ell })Y_4}{E_{4}Y}\right)\}d\mathit{\ell }\\ & +\frac{a(\lambda +\gamma )}{\mathcal{P}}\{\frac{\mathcal{P}{S}_4(\epsilon {\vartheta }_2+b{\vartheta }_1{\mathcal{H}}_6)}{a\epsilon (\lambda +\gamma )}-1\}I-\frac{{\mu }_1{\pi }_1(\gamma +\lambda )}{{\sigma }_1\mathcal{P}}{C}^I].\end{array}$$ 22

Therefore, Eq. (22) becomes

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_4}{dt}& =-\mathcal{P}[\varrho \frac{{(S-S_4)}^2}{S}+\frac{r{Y}_4}{\phi {\mathcal{H}}_4}Ϝ\left(\frac{Y{E}_4}{Y_{4}E}\right)\\ & +\frac{{\vartheta }_3{S}_4{Y}_4}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\{Ϝ\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_4}{S_4{Y}_{4}E}\right)+Ϝ\left(\frac{S_4}{S}\right)\}d\mathit{\ell }\\ & +(\frac{{\vartheta }_3{S}_4{Y}_4}{{\mathcal{H}}_5}+\frac{r{Y}_4}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })Ϝ\left(\frac{E(t-\mathit{\ell })Y_4}{E_{4}Y}\right)d\mathit{\ell }\\ & +\frac{a(\lambda +\gamma )}{\mathcal{P}}({\mathrm{\Re }}_6-1)I-\frac{{\mu }_1{\pi }_1(\gamma +\lambda )}{{\sigma }_1\mathcal{P}}{C}^I].\end{array}$$

Hence, if ${\mathrm{\Re }}_6\le 1$, then $\frac{d{\mathrm{\Phi }}_4}{dt}\le 0$ for all $S,I,E,Y,V,C^I>0$, where $\frac{d{\mathrm{\Phi }}_4}{dt}=0$ occurs at $S=S_4$, $E=E_4$, $Y=Y_4$, and $I=C^I=0$. The trajectories of system (5) converge to ${\mathrm{\Upsilon }}_4^{\prime }$ which includes elements with $S(t)=S_4$, $E(t)=E_4$, $Y(t)=Y_4$, and then $\frac{dS(t)}{dt}=\frac{dY(t)}{dt}=0$. The first and fifth equations of system (5) become

$$\begin{array}{rl}& 0=\frac{dS(t)}{dt}=\eta -\varrho S_4-{\vartheta }_1{S}_{4}V(t)-{\vartheta }_3{S}_4{Y}_4,\\ & 0=\frac{dY(t)}{dt}=\psi {\mathcal{H}}_5{E}_4-\delta Y_4-{\mu }_2{C}^Y(t)Y_4,\end{array}$$

which imply that $V(t)=0$ and $C^Y(t)=C_4^Y$ for all t. Moreover, we have $\frac{dI(t)}{dt}=0$, then the third equation of system (5) becomes

$$0=\frac{dI(t)}{dt}=\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell },$$

which yields $L(t)=0$ for all t, and then . Applying LLAS theorem, we get is GAS. □

Theorem 6

If ${\mathrm{\Re }}_5>1$, ${\mathrm{\Re }}_8\le 1$, and ${\mathrm{\Re }}_1/{\mathrm{\Re }}_2>1$, then is GAS.

Proof

Define

$$\begin{array}{rl}{\mathrm{\Phi }}_5& =\mathcal{P}{S}_5Ϝ\left(\frac{S}{S_5}\right)+\lambda {\mathcal{H}}_3{L}_5Ϝ\left(\frac{L}{L_5}\right)+(\gamma +\lambda )I_5Ϝ\left(\frac{I}{I_5}\right)+\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}{E}_5Ϝ\left(\frac{E}{E_5}\right)\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_5Ϝ\left(\frac{Y}{Y_5}\right)+\frac{\mathcal{P}{\vartheta }_1{S}_5}{\epsilon }{V}_5Ϝ\left(\frac{V}{V_5}\right)+\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}{C}_5^IϜ\left(\frac{C^I}{C_5^I}\right)\\ & +\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y+{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta )S_5{V}_5{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )V(\varkappa )}{S_5{V}_5}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta )S_5{I}_5{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )I(\varkappa )}{S_5{I}_5}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_1\beta (\gamma +\lambda )S_5{V}_5{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )V(\varkappa )}{S_5{V}_5}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_2\beta (\gamma +\lambda )S_5{I}_5{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )I(\varkappa )}{S_5{I}_5}\right)d\varkappa d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_5{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{L(\varkappa )}{L_5}\right)d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_5{Y}_5}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )Y(\varkappa )}{S_5{Y}_5}\right)d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )E_5}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{E(\varkappa )}{E_5}\right)d\varkappa d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_5{I}_5}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{I(\varkappa )}{I_5}\right)d\varkappa d\mathit{\ell }.\end{array}$$

Calculate $\frac{d{\mathrm{\Phi }}_5}{dt}$ as follows:

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_5}{dt}& =\mathcal{P}(1-\frac{S_5}{S})(\eta -\varrho S-{\vartheta }_{1}SV-{\vartheta }_{2}SI-{\vartheta }_{3}SY)+\lambda {\mathcal{H}}_3(1-\frac{L_5}{L})\\ & \times [(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }-(\lambda +\gamma )L]\\ & +(\gamma +\lambda )(1-\frac{I_5}{I})[\beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }\\ & +\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell }-aI-{\mu }_1{C}^{I}I]\\ & +\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}(1-\frac{E_5}{E})[\phi {\vartheta }_3{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }+rY-(\psi +\omega )E]\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(1-\frac{Y_5}{Y})[\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell }-\delta Y-{\mu }_2{C}^{Y}Y]\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_5}{\epsilon }(1-\frac{V_5}{V})[b{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })I(t-\mathit{\ell })d\mathit{\ell }-\epsilon V]\\ & +\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}(1-\frac{C_5^I}{C^I})({\sigma }_1{C}^{I}I-{\pi }_1{C}^I)+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}({\sigma }_2{C}^{Y}Y-{\pi }_2{C}^Y)\\ & +{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta )S_5{V}_5\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{SV}{S_5{V}_5}-\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{S_5{V}_5}+ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)]d\mathit{\ell }\\ & +{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta )S_5{I}_5\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{SI}{S_5{I}_5}-\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_5{I}_5}+ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)]d\mathit{\ell }\\ & +{\vartheta }_1\beta (\gamma +\lambda )S_5{V}_5\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{SV}{S_5{V}_5}-\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{S_5{V}_5}+ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)]d\mathit{\ell }\\ & +{\vartheta }_2\beta (\gamma +\lambda )S_5{I}_5\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{SI}{S_5{I}_5}-\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_5{I}_5}+ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)]d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_5{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })[\frac{L}{L_5}-\frac{L(t-\mathit{\ell })}{L_5}+ln\left(\frac{L(t-\mathit{\ell })}{L}\right)]d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_5{Y}_5}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[\frac{SY}{S_5{Y}_5}-\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{S_5{Y}_5}+ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)]d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )E_5}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })[\frac{E}{E_5}-\frac{E(t-\mathit{\ell })}{E_5}+ln\left(\frac{E(t-\mathit{\ell })}{E}\right)]d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_5{I}_5}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[\frac{I}{I_5}-\frac{I(t-\mathit{\ell })}{I_5}+ln\left(\frac{I(t-\mathit{\ell })}{I}\right)]d\mathit{\ell }.\end{array}$$ 23

Summing the terms of Eq. (23), we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_5}{dt}& =\mathcal{P}(1-\frac{S_5}{S})(\eta -\varrho S)+\mathcal{P}{\vartheta }_2{S}_{5}I+\mathcal{P}{\vartheta }_3{S}_{5}Y-{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\\ & \times \frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_5}{L}d\mathit{\ell }-{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_5}{L}d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(\lambda +\gamma )L_5-a(\lambda +\gamma )I-{\vartheta }_1\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_5}{I}d\mathit{\ell }\\ & -{\vartheta }_2\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })I_5}{I}d\mathit{\ell }\\ & -\lambda (\lambda +\gamma ){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })\frac{L(t-\mathit{\ell })I_5}{I}d\mathit{\ell }+a(\lambda +\gamma )I_5+{\mu }_1(\lambda +\gamma )C^I{I}_5\\ & +\frac{\mathcal{P}r}{\phi {\mathcal{H}}_4}Y-\frac{\mathcal{P}{\vartheta }_3}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_5}{E}d\mathit{\ell }\\ & -\frac{\mathcal{P}r}{\phi {\mathcal{H}}_4}\frac{Y{E}_5}{E}+\frac{\mathcal{P}(\psi +\omega )}{\phi {\mathcal{H}}_4}{E}_5-\frac{\mathcal{P}\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y\\ & -\frac{\mathcal{P}(\psi +\omega )}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\frac{E(t-\mathit{\ell })Y_5}{Y}d\mathit{\ell }+\frac{\mathcal{P}\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_5\\ & +\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y{Y}_5-\frac{b\mathcal{P}{\vartheta }_1{S}_5}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })\frac{I(t-\mathit{\ell })V_5}{V}d\mathit{\ell }+\mathcal{P}{\vartheta }_1{S}_5{V}_5\\ & -\frac{{\mu }_1{\pi }_1(\gamma +\lambda )}{{\sigma }_1}{C}^I-{\mu }_1(\lambda +\gamma )C_5^{I}I+\frac{{\mu }_1{\pi }_1(\lambda +\gamma )}{{\sigma }_1}{C}_5^I-\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_5{V}_5{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_5{I}_5{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_1{S}_5{V}_5{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_2{S}_5{I}_5{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_5{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })ln\left(\frac{L(t-\mathit{\ell })}{L}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_5{Y}_5}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )E_5}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })ln\left(\frac{E(t-\mathit{\ell })}{E}\right)d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_5{\mathcal{H}}_6}{\epsilon }I+\frac{b\mathcal{P}{\vartheta }_1{S}_5{I}_5}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })ln\left(\frac{I(t-\mathit{\ell })}{I}\right)d\mathit{\ell }.\end{array}$$

Using the steady state conditions for

$$\begin{array}{rl}& \eta =\varrho S_5+{\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5+{\vartheta }_3{S}_5{Y}_5,{\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5)=(\lambda +\gamma )L_5,\\ & \beta {\mathcal{H}}_2({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5)+\lambda {\mathcal{H}}_3{L}_5=(a+{\mu }_1{C}_5^I)I_5{I}_5=\frac{{\pi }_1}{{\sigma }_1},V_5=\frac{b{\mathcal{H}}_6}{\epsilon }{I}_5,\\ & {\vartheta }_3{S}_5{Y}_5+\frac{r{Y}_5}{\phi {\mathcal{H}}_4}=\frac{(\psi +\omega )E_5}{\phi {\mathcal{H}}_4}=\frac{\delta (\psi +\omega )Y_5}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5},\end{array}$$

we obtain

$$\mathcal{P}({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5)=(\lambda +\gamma )(a+{\mu }_1{C}_5^I)I_5.$$

Moreover, we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_5}{dt}& =\mathcal{P}(1-\frac{S_5}{S})(\varrho S_5-\varrho S)+\mathcal{P}({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5+{\vartheta }_3{S}_5{Y}_5)(1-\frac{S_5}{S})\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_5{V}_5{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_5}{S_5{V}_{5}L}d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_5{I}_5{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_5}{S_5{I}_{5}L}d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_1{\mathcal{H}}_3(1-\beta )({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5)\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_5{V}_5{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_5}{S_5{V}_{5}I}d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_5{I}_5{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{5}I}d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })\frac{L(t-\mathit{\ell })I_5}{L_{5}I}d\mathit{\ell }\\ & +\mathcal{P}({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5)-\frac{\mathcal{P}{\vartheta }_3{S}_5{Y}_5}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_5}{S_5{Y}_{5}E}d\mathit{\ell }\\ & -\frac{\mathcal{P}r{Y}_5}{\phi {\mathcal{H}}_4}\frac{Y{E}_5}{Y_{5}E}+\mathcal{P}{\vartheta }_3{S}_5{Y}_5+\frac{\mathcal{P}r{Y}_5}{\phi {\mathcal{H}}_4}\\ & -(\frac{\mathcal{P}{\vartheta }_3{S}_5{Y}_5}{{\mathcal{H}}_5}+\frac{\mathcal{P}r{Y}_5}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\frac{E(t-\mathit{\ell })Y_5}{E_{5}Y}d\mathit{\ell }+\mathcal{P}{\vartheta }_3{S}_5{Y}_5+\frac{\mathcal{P}r{Y}_5}{\phi {\mathcal{H}}_4}\\ & -\frac{\mathcal{P}{\vartheta }_1{S}_5{V}_5}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })\frac{I(t-\mathit{\ell })V_5}{I_{5}V}d\mathit{\ell }+\mathcal{P}{\vartheta }_1{S}_5{V}_5\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_5{V}_5{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_5{I}_5{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_1{S}_5{V}_5{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_2{S}_5{I}_5{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })ln\left(\frac{L(t-\mathit{\ell })}{L}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_5{Y}_5}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)d\mathit{\ell }\\ & +(\frac{\mathcal{P}{\vartheta }_3{S}_5{Y}_5}{{\mathcal{H}}_5}+\frac{\mathcal{P}r{Y}_5}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })ln\left(\frac{E(t-\mathit{\ell })}{E}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_5{V}_5}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })ln\left(\frac{I(t-\mathit{\ell })}{I}\right)d\mathit{\ell }+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(Y_5-\frac{{\pi }_2}{{\sigma }_2})C^Y.\end{array}$$

Using the equalities given by (10) and (11) in case of $n=m=5$, we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_5}{dt}& =-\varrho \mathcal{P}\frac{{(S-S_5)}^2}{S}-\mathcal{P}({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5+{\vartheta }_3{S}_5{Y}_5)[\frac{S_5}{S}-1-ln\left(\frac{S_5}{S}\right)]\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_5{V}_5\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_5}{S_5{V}_{5}L}-1-ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_5}{S_5{V}_{5}L}\right)]d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_5{I}_5\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_5}{S_5{I}_{5}L}-1-ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_5}{S_5{I}_{5}L}\right)]d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_5{V}_5\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_5}{S_5{V}_{5}I}-1-ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_5}{S_5{V}_{5}I}\right)]d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_5{I}_5{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{5}I}-1-ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{5}I}\right)]d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5)\\ & \times {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })[\frac{L(t-\mathit{\ell })I_5}{L_{5}I}-1-ln\left(\frac{L(t-\mathit{\ell })I_5}{L_{5}I}\right)]d\mathit{\ell }\\ & -\frac{\mathcal{P}r{Y}_5}{\phi {\mathcal{H}}_4}[\frac{Y{E}_5}{Y_{5}E}-1-ln\left(\frac{Y{E}_5}{Y_{5}E}\right)]\\ & -\frac{\mathcal{P}{\vartheta }_3{S}_5{Y}_5}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_5}{S_5{Y}_{5}E}-1-ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_5}{S_5{Y}_{5}E}\right)]d\mathit{\ell }\\ & -(\frac{\mathcal{P}{\vartheta }_3{S}_5{Y}_5}{{\mathcal{H}}_5}+\frac{\mathcal{P}r{Y}_5}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })[\frac{E(t-\mathit{\ell })Y_5}{E_{5}Y}-1-ln\left(\frac{E(t-\mathit{\ell })Y_5}{E_{5}Y}\right)]d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_1{S}_5{V}_5}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[\frac{I(t-\mathit{\ell })V_5}{I_{5}V}-1-ln\left(\frac{I(t-\mathit{\ell })V_5}{I_{5}V}\right)]d\mathit{\ell }\\ & +\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(Y_5-\frac{{\pi }_2}{{\sigma }_2})C^Y.\end{array}$$ 24

Therefore, Eq. (24) becomes

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_5}{dt}& =-\varrho \mathcal{P}\frac{{(S-S_5)}^2}{S}-\mathcal{P}({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5+{\vartheta }_3{S}_5{Y}_5)Ϝ\left(\frac{S_5}{S}\right)\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_5{V}_5{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_5}{S_5{V}_{5}L}\right)d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_5{I}_5{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_5}{S_5{I}_{5}L}\right)d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_5{V}_5{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_5}{S_5{V}_{5}I}\right)d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_5{I}_5{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{5}I}\right)d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })Ϝ\left(\frac{L(t-\mathit{\ell })I_5}{L_{5}I}\right)d\mathit{\ell }\\ & -\frac{\mathcal{P}r{Y}_5}{\phi {\mathcal{H}}_4}Ϝ\left(\frac{Y{E}_5}{Y_{5}E}\right)-\frac{\mathcal{P}{\vartheta }_3{S}_5{Y}_5}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_5}{S_5{Y}_{5}E}\right)d\mathit{\ell }\\ & -(\frac{\mathcal{P}{\vartheta }_3{S}_5{Y}_5}{{\mathcal{H}}_5}+\frac{\mathcal{P}r{Y}_5}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })Ϝ\left(\frac{E(t-\mathit{\ell })Y_5}{E_{5}Y}\right)d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_1{S}_5{V}_5}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })Ϝ\left(\frac{I(t-\mathit{\ell })V_5}{I_{5}V}\right)d\mathit{\ell }\\ & +\frac{{\mu }_2\mathcal{P}(\psi +\omega )[{\pi }_1{\sigma }_2(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\epsilon {\sigma }_1({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)]}{\phi \psi {\vartheta }_3\epsilon {\sigma }_1{\sigma }_2{\mathcal{H}}_4{\mathcal{H}}_5}({\mathrm{\Re }}_8-1)C^Y.\end{array}$$

Hence, if ${\mathrm{\Re }}_8\le 1$, then $\frac{d{\mathrm{\Phi }}_5}{dt}\le 0$ for all $S,L,I,E,Y,V,C^Y>0$. One can show that $\frac{d{\mathrm{\Phi }}_5}{dt}=0$ when $(S,L,I,E,Y,V,C^Y)=(S_5,L_5,I_5,E_5,Y_5,V_5,0)$. The solutions of model (5) tend to ${\mathrm{\Upsilon }}_5^{\prime }$ which includes elements with $(S(t),L(t),I(t),V(t))=(S_5,L_5,I_5,V_5)$, and then $\frac{dI(t)}{dt}=0$. The third equation of system (5) becomes

$$0=\frac{dI(t)}{dt}=\beta {\mathcal{H}}_2({\vartheta }_1{S}_5{V}_5+{\vartheta }_2{S}_5{I}_5)+\lambda {\mathcal{H}}_3{L}_5-a{I}_5-{\mu }_1{C}^I(t)I_5,$$

which yields $C^I(t)=C_5^I$ for all t, and hence . Applying LLAS theorem, we get is GAS. □

Theorem 7

If ${\mathrm{\Re }}_6>1$, ${\mathrm{\Re }}_7\le 1$, and ${\mathrm{\Re }}_2/{\mathrm{\Re }}_1>1$, then is GAS.

Proof

Define

$$\begin{array}{rl}{\mathrm{\Phi }}_6& =\mathcal{P}{S}_6Ϝ\left(\frac{S}{S_6}\right)+\lambda {\mathcal{H}}_3{L}_6Ϝ\left(\frac{L}{L_6}\right)+(\gamma +\lambda )I_6Ϝ\left(\frac{I}{I_6}\right)+\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}{E}_6Ϝ\left(\frac{E}{E_6}\right)\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_6Ϝ\left(\frac{Y}{Y_6}\right)+\frac{\mathcal{P}{\vartheta }_1{S}_6}{\epsilon }{V}_6Ϝ\left(\frac{V}{V_6}\right)+\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}{C}^I\\ & +\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}_6^YϜ\left(\frac{C^Y}{C_6^Y}\right)\\ & +{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta )S_6{V}_6{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )V(\varkappa )}{S_6{V}_6}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta )S_6{I}_6{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )I(\varkappa )}{S_6{I}_6}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_1\beta (\gamma +\lambda )S_6{V}_6{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )V(\varkappa )}{S_6{V}_6}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_2\beta (\gamma +\lambda )S_6{I}_6{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )I(\varkappa )}{S_6{I}_6}\right)d\varkappa d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_6{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{L(\varkappa )}{L_6}\right)d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_6{Y}_6}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )Y(\varkappa )}{S_6{Y}_6}\right)d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )E_6}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{E(\varkappa )}{E_6}\right)d\varkappa d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_6{I}_6}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{I(\varkappa )}{I_6}\right)d\varkappa d\mathit{\ell }.\end{array}$$

Calculate $\frac{d{\mathrm{\Phi }}_6}{dt}$ as follows:

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_6}{dt}& =\mathcal{P}(1-\frac{S_6}{S})(\eta -\varrho S-{\vartheta }_{1}SV-{\vartheta }_{2}SI-{\vartheta }_{3}SY)+\lambda {\mathcal{H}}_3(1-\frac{L_6}{L})\\ & \times [(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }-(\lambda +\gamma )L]\\ & +(\gamma +\lambda )(1-\frac{I_6}{I})[\beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }\\ & +\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell }-aI-{\mu }_1{C}^{I}I]\\ & +\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}(1-\frac{E_6}{E})[\phi {\vartheta }_3{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }+rY-(\psi +\omega )E]\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(1-\frac{Y_6}{Y})[\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell }-\delta Y-{\mu }_2{C}^{Y}Y]\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_6}{\epsilon }(1-\frac{V_6}{V})[b{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })I(t-\mathit{\ell })d\mathit{\ell }-\epsilon V]\\ & +\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}({\sigma }_1{C}^{I}I-{\pi }_1{C}^I)+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(1-\frac{C_6^Y}{C^Y})({\sigma }_2{C}^{Y}Y-{\pi }_2{C}^Y)\\ & +{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta )S_6{V}_6\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{SV}{S_6{V}_6}-\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{S_6{V}_6}+ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)]d\mathit{\ell }\\ & +{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta )S_6{I}_6\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{SI}{S_6{I}_6}-\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_6{I}_6}+ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)]d\mathit{\ell }\\ & +{\vartheta }_1\beta (\gamma +\lambda )S_6{V}_6\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{SV}{S_6{V}_6}-\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{S_6{V}_6}+ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)]d\mathit{\ell }\\ & +{\vartheta }_2\beta (\gamma +\lambda )S_6{I}_6\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{SI}{S_6{I}_6}-\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_6{I}_6}+ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)]d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_6{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })[\frac{L}{L_6}-\frac{L(t-\mathit{\ell })}{L_6}+ln\left(\frac{L(t-\mathit{\ell })}{L}\right)]d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_6{Y}_6}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[\frac{SY}{S_6{Y}_6}-\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{S_6{Y}_6}+ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)]d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )E_6}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })[\frac{E}{E_6}-\frac{E(t-\mathit{\ell })}{E_6}+ln\left(\frac{E(t-\mathit{\ell })}{E}\right)]d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_6{I}_6}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[\frac{I}{I_6}-\frac{I(t-\mathit{\ell })}{I_6}+ln\left(\frac{I(t-\mathit{\ell })}{I}\right)]d\mathit{\ell }.\end{array}$$ 25

Collecting the terms of Eq. (25), we obtain

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_6}{dt}& =\mathcal{P}(1-\frac{S_6}{S})(\eta -\varrho S)+\mathcal{P}{\vartheta }_2{S}_{6}I+\mathcal{P}{\vartheta }_3{S}_{6}Y-{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\\ & \times \frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_6}{L}d\mathit{\ell }-{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_6}{L}d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(\lambda +\gamma )L_6-a(\lambda +\gamma )I-{\vartheta }_1\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_6}{I}d\mathit{\ell }\\ & -{\vartheta }_2\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })I_6}{I}d\mathit{\ell }\\ & -\lambda (\lambda +\gamma ){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })\frac{L(t-\mathit{\ell })I_6}{I}d\mathit{\ell }+a(\lambda +\gamma )I_6+{\mu }_1(\lambda +\gamma )C^I{I}_6\\ & +\frac{\mathcal{P}r}{\phi {\mathcal{H}}_4}Y-\frac{\mathcal{P}{\vartheta }_3}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_6}{E}d\mathit{\ell }\\ & -\frac{\mathcal{P}r}{\phi {\mathcal{H}}_4}\frac{Y{E}_6}{E}+\frac{\mathcal{P}(\psi +\omega )}{\phi {\mathcal{H}}_4}{E}_6-\frac{\mathcal{P}\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y\\ & -\frac{\mathcal{P}(\psi +\omega )}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\frac{E(t-\mathit{\ell })Y_6}{Y}d\mathit{\ell }+\frac{\mathcal{P}\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_6+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y{Y}_6\\ & -\frac{b\mathcal{P}{\vartheta }_1{S}_6}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })\frac{I(t-\mathit{\ell })V_6}{V}d\mathit{\ell }+\mathcal{P}{\vartheta }_1{S}_6{V}_6-\frac{{\mu }_1{\pi }_1(\lambda +\gamma )}{{\sigma }_1}{C}^I\\ & -\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y-\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}_6^{Y}Y+\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}_6^Y\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_6{V}_6{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_6{I}_6{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_1{S}_6{V}_6{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_2{S}_6{I}_6{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_6{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })ln\left(\frac{L(t-\mathit{\ell })}{L}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_6{Y}_6}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )E_6}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })ln\left(\frac{E(t-\mathit{\ell })}{E}\right)d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_6{\mathcal{H}}_6}{\epsilon }I+\frac{b\mathcal{P}{\vartheta }_1{S}_6{I}_6}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })ln\left(\frac{I(t-\mathit{\ell })}{I}\right)d\mathit{\ell }.\end{array}$$

Using the steady state conditions for

$$\begin{array}{rl}& \eta =\varrho S_6+{\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6+{\vartheta }_3{S}_6{Y}_6,{\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6)=(\lambda +\gamma )L_6,\\ & \beta {\mathcal{H}}_2({\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6)+\lambda {\mathcal{H}}_3{L}_6=a{I}_6,Y_6=\frac{{\pi }_2}{{\sigma }_2},V_6=\frac{b{\mathcal{H}}_6{I}_6}{\epsilon },\\ & {\vartheta }_3{S}_6{Y}_6+\frac{r{Y}_6}{\phi {\mathcal{H}}_4}=\frac{(\psi +\omega )E_6}{\phi {\mathcal{H}}_4}=\frac{\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_6+\frac{{\mu }_2(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}_6^Y{Y}_6,\end{array}$$

we get

$$\mathcal{P}({\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6)=a(\lambda +\gamma )I_6.$$

Moreover, we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_6}{dt}& =\mathcal{P}(1-\frac{S_6}{S})(\varrho S_6-\varrho S)+\mathcal{P}({\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6+{\vartheta }_3{S}_6{Y}_6)(1-\frac{S_6}{S})\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_6{V}_6{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_6}{S_6{V}_{6}L}d\mathit{\ell }-\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_6{I}_6\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_6}{S_6{I}_{6}L}d\mathit{\ell }+\lambda {\mathcal{H}}_1{\mathcal{H}}_3(1-\beta )({\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6)\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_6{V}_6{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_6}{S_6{V}_{6}I}d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_6{I}_6{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{6}I}d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })\frac{L(t-\mathit{\ell })I_6}{L_{6}I}d\mathit{\ell }\\ & +\mathcal{P}({\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6)-\frac{\mathcal{P}{\vartheta }_3{S}_6{Y}_6}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_6}{S_6{Y}_{6}E}d\mathit{\ell }\\ & -\frac{\mathcal{P}r{Y}_6}{\phi {\mathcal{H}}_4}\frac{Y{E}_6}{Y_{6}E}+\mathcal{P}{\vartheta }_3{S}_6{Y}_6+\frac{\mathcal{P}r{Y}_6}{\phi {\mathcal{H}}_4}\\ & -(\frac{\mathcal{P}{\vartheta }_3{S}_6{Y}_6}{{\mathcal{H}}_5}+\frac{\mathcal{P}r{Y}_6}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\frac{E(t-\mathit{\ell })Y_6}{E_{6}Y}d\mathit{\ell }\\ & +\mathcal{P}{\vartheta }_3{S}_6{Y}_6+\frac{\mathcal{P}r{Y}_6}{\phi {\mathcal{H}}_4}-\frac{\mathcal{P}{\vartheta }_1{S}_6{V}_6}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })\frac{I(t-\mathit{\ell })V_6}{I_{6}V}d\mathit{\ell }+\mathcal{P}{\vartheta }_1{S}_6{V}_6\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_6{V}_6{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_6{I}_6{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_1{S}_6{V}_6{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_2{S}_6{I}_6{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })ln\left(\frac{L(t-\mathit{\ell })}{L}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_6{Y}_6}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)d\mathit{\ell }\\ & +(\frac{\mathcal{P}{\vartheta }_3{S}_6{Y}_6}{{\mathcal{H}}_5}+\frac{\mathcal{P}r{Y}_6}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })ln\left(\frac{E(t-\mathit{\ell })}{E}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_6{V}_6}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })ln\left(\frac{I(t-\mathit{\ell })}{I}\right)d\mathit{\ell }+{\mu }_1(\lambda +\gamma )(I_6-\frac{{\pi }_1}{{\sigma }_1})C^I.\end{array}$$

Using the equalities given by (10) and (11) in case of $n=m=6$, we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_6}{dt}& =-\varrho \mathcal{P}\frac{{(S-S_6)}^2}{S}-\mathcal{P}({\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6+{\vartheta }_3{S}_6{Y}_6)[\frac{S_6}{S}-1-ln\left(\frac{S_6}{S}\right)]\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_6{V}_6\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_6}{S_6{V}_{6}L}-1-ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_6}{S_6{V}_{6}L}\right)]d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_6{I}_6\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_6}{S_6{I}_{6}L}-1-ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_6}{S_6{I}_{6}L}\right)]d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_6{V}_6\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_6}{S_6{V}_{6}I}-1-ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_6}{S_6{V}_{6}I}\right)]d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_6{I}_6{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{6}I}-1-ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{6}I}\right)]d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6)\\ & \times {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell }))[\frac{L(t-\mathit{\ell })I_6}{L_{6}I}-1-ln\left(\frac{L(t-\mathit{\ell })I_6}{L_{6}I}\right)]d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_3{S}_6{Y}_6}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_6}{S_6{Y}_{6}E}-1-ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_6}{S_6{Y}_{6}E}\right)]d\mathit{\ell }\\ & -\frac{\mathcal{P}r{Y}_6}{\phi {\mathcal{H}}_4}[\frac{Y{E}_6}{Y_{6}E}-1-ln\left(\frac{Y{E}_6}{Y_{6}E}\right)]\\ & -(\frac{\mathcal{P}{\vartheta }_3{S}_6{Y}_6}{{\mathcal{H}}_5}+\frac{\mathcal{P}r{Y}_6}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })[\frac{E(t-\mathit{\ell })Y_6}{E_{6}Y}-1-ln\left(\frac{E(t-\mathit{\ell })Y_6}{E_{6}Y}\right)]d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_1{S}_6{V}_6}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[\frac{I(t-\mathit{\ell })V_6}{I_{6}V}-1-ln\left(\frac{I(t-\mathit{\ell })V_6}{I_{6}V}\right)]d\mathit{\ell }\\ & +{\mu }_1(\lambda +\gamma )(I_6-\frac{{\pi }_1}{{\sigma }_1})C^I.\end{array}$$ 26

Therefore, Eq. (26) becomes

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_6}{dt}& =-\varrho \mathcal{P}\frac{{(S-S_6)}^2}{S}-\mathcal{P}({\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6+{\vartheta }_3{S}_6{Y}_6)+Ϝ\left(\frac{S_6}{S}\right)\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_6{V}_6{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_6}{S_6{V}_{6}L}\right)d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_6{I}_6{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_6}{S_6{I}_{6}L}\right)d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_6{V}_6{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_6}{S_6{V}_{6}I}\right)d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_6{I}_6{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{6}I}\right)d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_6{V}_6+{\vartheta }_2{S}_6{I}_6){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell }))Ϝ\left(\frac{L(t-\mathit{\ell })I_6}{L_{6}I}\right)d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_3{S}_6{Y}_6}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_6}{S_6{Y}_{6}E}\right)d\mathit{\ell }-\frac{\mathcal{P}r{Y}_6}{\phi {\mathcal{H}}_4}Ϝ\left(\frac{Y{E}_6}{Y_{6}E}\right)\\ & -(\frac{\mathcal{P}{\vartheta }_3{S}_6{Y}_6}{{\mathcal{H}}_5}+\frac{\mathcal{P}r{Y}_6}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })Ϝ\left(\frac{E(t-\mathit{\ell })Y_6}{E_{6}Y}\right)d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_1{S}_6{V}_6}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })Ϝ\left(\frac{I(t-\mathit{\ell })V_6}{I_{6}V}\right)d\mathit{\ell }\\ & +\frac{{\mu }_1(\gamma +\lambda )[{\pi }_1{\sigma }_2(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)+\epsilon {\sigma }_1({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)]}{{\sigma }_1{\sigma }_2(b{\vartheta }_1{\mathcal{H}}_6+\epsilon {\vartheta }_2)}({\mathrm{\Re }}_7-1)C^I.\end{array}$$

Hence, if ${\mathrm{\Re }}_7\le 1$, then $\frac{d{\mathrm{\Phi }}_6}{dt}\le 0$ for all $S,L,I,E,Y,V,C^I>0$. Similar to the previous theorems, one can show that $\frac{d{\mathrm{\Phi }}_6}{dt}=0$ at $(S,L,I,E,Y,V,C^I)=(S_6,L_6,I_6,E_6,Y_6,V_6,0)$. The solutions of system (5) reach ${\mathrm{\Upsilon }}_6^{\prime }$ which contains elements with $E(t)=E_6$, $Y(t)=Y_6$, and then $\frac{dY(t)}{dt}=0$. The fifth equation of system (5) becomes

$$0=\frac{dY(t)}{dt}=\psi {\mathcal{H}}_5{E}_6-\delta Y_6-{\mu }_2{C}^Y(t)Y_6,$$

which yields $C^Y(t)=C_6^Y$ for all t, and hence . Applying LLAS theorem, we get is GAS. □

Theorem 8

If ${\mathrm{\Re }}_7>1$ and ${\mathrm{\Re }}_8>1$, then is GAS.

Proof

Consider

$$\begin{array}{rl}{\mathrm{\Phi }}_7& =\mathcal{P}{S}_7Ϝ\left(\frac{S}{S_7}\right)+\lambda {\mathcal{H}}_3{L}_7Ϝ\left(\frac{L}{L_7}\right)+(\gamma +\lambda )I_7Ϝ\left(\frac{I}{I_7}\right)+\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}{E}_7Ϝ\left(\frac{E}{E_7}\right)\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_7Ϝ\left(\frac{Y}{Y_7}\right)+\frac{\mathcal{P}{\vartheta }_1{S}_7}{\epsilon }{V}_7Ϝ\left(\frac{V}{V_7}\right)+\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}{C}_7^IϜ\left(\frac{C^I}{C_7^I}\right)\\ & +\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}_7^YϜ\left(\frac{C^Y}{C_7^Y}\right)\\ & +{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta )S_7{V}_7{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )V(\varkappa )}{S_7{V}_7}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta )S_7{I}_7{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )I(\varkappa )}{S_7{I}_7}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_1\beta (\gamma +\lambda )S_7{V}_7{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )V(\varkappa )}{S_7{V}_7}\right)d\varkappa d\mathit{\ell }\\ & +{\vartheta }_2\beta (\gamma +\lambda )S_7{I}_7{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )I(\varkappa )}{S_7{I}_7}\right)d\varkappa d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_7{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{L(\varkappa )}{L_7}\right)d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_7{Y}_7}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{S(\varkappa )Y(\varkappa )}{S_7{Y}_7}\right)d\varkappa d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )E_7}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{E(\varkappa )}{E_7}\right)d\varkappa d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_7{I}_7}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell }){\int }_{t-\mathit{\ell }}^tϜ\left(\frac{I(\varkappa )}{I_7}\right)d\varkappa d\mathit{\ell }.\end{array}$$

Calculate $\frac{d{\mathrm{\Phi }}_7}{dt}$ as follows:

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_7}{dt}& =\mathcal{P}(1-\frac{S_7}{S})(\eta -\varrho S-{\vartheta }_{1}SV-{\vartheta }_{2}SI-{\vartheta }_{3}SY)+\lambda {\mathcal{H}}_3(1-\frac{L_7}{L})\\ & \times [(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }-(\lambda +\gamma )L]\\ & +(\gamma +\lambda )(1-\frac{I_7}{I})[\beta {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })S(t-\mathit{\ell })\{{\vartheta }_{1}V(t-\mathit{\ell })+{\vartheta }_{2}I(t-\mathit{\ell })\}d\mathit{\ell }\\ & +\lambda {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })L(t-\mathit{\ell })d\mathit{\ell }-aI-{\mu }_1{C}^{I}I]\\ & +\frac{\mathcal{P}}{\phi {\mathcal{H}}_4}(1-\frac{E_7}{E})[\phi {\vartheta }_3{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })S(t-\mathit{\ell })Y(t-\mathit{\ell })d\mathit{\ell }+rY-(\psi +\omega )E]\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(1-\frac{Y_7}{Y})[\psi {\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })E(t-\mathit{\ell })d\mathit{\ell }-\delta Y-{\mu }_2{C}^{Y}Y]\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_7}{\epsilon }(1-\frac{V_7}{V})[b{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })I(t-\mathit{\ell })d\mathit{\ell }-\epsilon V]\\ & +\frac{{\mu }_1(\gamma +\lambda )}{{\sigma }_1}(1-\frac{C_7^I}{C^I})({\sigma }_1{C}^{I}I-{\pi }_1{C}^I)\\ & +\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}(1-\frac{C_7^Y}{C^Y})({\sigma }_2{C}^{Y}Y-{\pi }_2{C}^Y)\\ & +{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta )S_7{V}_7\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{SV}{S_7{V}_7}-\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{S_7{V}_7}+ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)]d\mathit{\ell }\\ & +{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta )S_7{I}_7\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{SI}{S_7{I}_7}-\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_7{I}_7}+ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)]d\mathit{\ell }\\ & +{\vartheta }_1\beta (\gamma +\lambda )S_7{V}_7\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{SV}{S_7{V}_7}-\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{S_7{V}_7}+ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)]d\mathit{\ell }\\ & +{\vartheta }_2\beta (\gamma +\lambda )S_7{I}_7\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{SI}{S_7{I}_7}-\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_7{I}_7}+ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)]d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_7{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })[\frac{L}{L_7}-\frac{L(t-\mathit{\ell })}{L_7}+ln\left(\frac{L(t-\mathit{\ell })}{L}\right)]d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_7{Y}_7}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[\frac{SY}{S_7{Y}_7}-\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{S_7{Y}_7}+ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)]d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )E_7}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })[\frac{E}{E_7}-\frac{E(t-\mathit{\ell })}{E_7}+ln\left(\frac{E(t-\mathit{\ell })}{E}\right)]d\mathit{\ell }\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_7{I}_7}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[\frac{I}{I_7}-\frac{I(t-\mathit{\ell })}{I_7}+ln\left(\frac{I(t-\mathit{\ell })}{I}\right)]d\mathit{\ell }.\end{array}$$ 27

Summing the terms of Eq. (27), we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_7}{dt}& =\mathcal{P}(1-\frac{S_7}{S})(\eta -\varrho S)+\mathcal{P}{\vartheta }_2{S}_{7}I+\mathcal{P}{\vartheta }_3{S}_{7}Y-{\vartheta }_1\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\\ & \times \frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_7}{L}d\mathit{\ell }-{\vartheta }_2\lambda {\mathcal{H}}_3(1-\beta ){\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_7}{L}d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(\lambda +\gamma )L_7-a(\lambda +\gamma )I-{\vartheta }_1\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_7}{I}d\mathit{\ell }\\ & -{\vartheta }_2\beta (\gamma +\lambda ){\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })I_7}{I}d\mathit{\ell }\\ & -\lambda (\lambda +\gamma ){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })\frac{L(t-\mathit{\ell })I_7}{I}d\mathit{\ell }+a(\lambda +\gamma )I_7+{\mu }_1(\lambda +\gamma )C^I{I}_7\\ & +\frac{\mathcal{P}r}{\phi {\mathcal{H}}_4}Y-\frac{\mathcal{P}{\vartheta }_3}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_7}{E}d\mathit{\ell }-\frac{\mathcal{P}r}{\phi {\mathcal{H}}_4}\frac{Y{E}_7}{E}\\ & +\frac{\mathcal{P}(\psi +\omega )}{\phi {\mathcal{H}}_4}{E}_7-\frac{\mathcal{P}\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}Y-\frac{\mathcal{P}(\psi +\omega )}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\frac{E(t-\mathit{\ell })Y_7}{Y}d\mathit{\ell }\\ & +\frac{\mathcal{P}\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_7+\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y{Y}_7-\frac{b\mathcal{P}{\vartheta }_1{S}_7}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })\frac{I(t-\mathit{\ell })V_7}{V}d\mathit{\ell }\\ & +\mathcal{P}{\vartheta }_1{S}_7{V}_7-\frac{{\mu }_1{\pi }_1(\lambda +\gamma )}{{\sigma }_1}{C}^I-{\mu }_1(\lambda +\gamma )C_7^{I}I+\frac{{\mu }_1{\pi }_1(\lambda +\gamma )}{{\sigma }_1}{C}_7^I\\ & -\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}^Y-\frac{{\mu }_2\mathcal{P}(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}_7^{Y}Y+\frac{{\mu }_2{\pi }_2\mathcal{P}(\psi +\omega )}{{\sigma }_2\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}_7^Y\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_7{V}_7{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_7{I}_7{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_1{S}_7{V}_7{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_2{S}_7{I}_7{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\lambda (\gamma +\lambda )L_7{\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })ln\left(\frac{L(t-\mathit{\ell })}{L}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_7{Y}_7}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}(\psi +\omega )E_7}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}{\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })ln\left(\frac{E(t-\mathit{\ell })}{E}\right)d\mathit{\ell }+\frac{b\mathcal{P}{\vartheta }_1{S}_7{\mathcal{H}}_6}{\epsilon }I\\ & +\frac{b\mathcal{P}{\vartheta }_1{S}_7{I}_7}{\epsilon }{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })ln\left(\frac{I(t-\mathit{\ell })}{I}\right)d\mathit{\ell }.\end{array}$$

Using the steady state conditions for

$$\begin{array}{rl}& \eta =\varrho S_7+{\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7+{\vartheta }_3{S}_7{Y}_7,\\ & {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7)=(\lambda +\gamma )L_7,\\ & \beta {\mathcal{H}}_2({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7)+\lambda {\mathcal{H}}_3{L}_7=(a+{\mu }_1{C}_7^I)I_7,\\ & I_7=\frac{{\pi }_1}{{\sigma }_1},Y_7=\frac{{\pi }_2}{{\sigma }_2},V_7=\frac{b{\mathcal{H}}_6{I}_7}{\epsilon },\\ & {\vartheta }_3{S}_7{Y}_7+\frac{r{Y}_7}{\phi {\mathcal{H}}_4}=\frac{(\psi +\omega )E_7}{\phi {\mathcal{H}}_4}=\frac{\delta (\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{Y}_7+\frac{{\mu }_2(\psi +\omega )}{\phi \psi {\mathcal{H}}_4{\mathcal{H}}_5}{C}_7^Y{Y}_7,\end{array}$$

we get

$$\mathcal{P}({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7)=(\lambda +\gamma )(a+{\mu }_1{C}_7^I)I_7.$$

Moreover, we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_7}{dt}& =\mathcal{P}(1-\frac{S_7}{S})(\varrho S_7-\varrho S)+\mathcal{P}({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7+{\vartheta }_3{S}_7{Y}_7)(1-\frac{S_7}{S})\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_7{V}_7{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_7}{S_7{V}_{7}L}d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_7{I}_7{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_7}{S_7{I}_{7}L}d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_1{\mathcal{H}}_3(1-\beta )({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7)\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_7{V}_7{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_7}{S_7{V}_{7}I}d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_7{I}_7{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{7}I}d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })\frac{L(t-\mathit{\ell })I_7}{L_{7}I}d\mathit{\ell }\\ & +\mathcal{P}({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7)-\frac{\mathcal{P}{\vartheta }_3{S}_7{Y}_7}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_7}{S_7{Y}_{7}E}d\mathit{\ell }\\ & -\frac{\mathcal{P}r{Y}_7}{\phi {\mathcal{H}}_4}\frac{Y{E}_7}{Y_{7}E}+\mathcal{P}{\vartheta }_3{S}_7{Y}_7+\frac{\mathcal{P}r{Y}_7}{\phi {\mathcal{H}}_4}\\ & -(\frac{\mathcal{P}{\vartheta }_3{S}_7{Y}_7}{{\mathcal{H}}_5}+\frac{\mathcal{P}r{Y}_7}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })\frac{E(t-\mathit{\ell })Y_7}{E_{7}Y}d\mathit{\ell }\\ & +\mathcal{P}{\vartheta }_3{S}_7{Y}_7+\frac{\mathcal{P}r{Y}_7}{\phi {\mathcal{H}}_4}{Y}_7-\frac{\mathcal{P}{\vartheta }_1{S}_7{V}_7}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })\frac{I(t-\mathit{\ell })V_7}{I_{7}V}d\mathit{\ell }\\ & +\mathcal{P}{\vartheta }_1{S}_7{V}_7+\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_7{V}_7{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_7{I}_7{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_1{S}_7{V}_7{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })}{SV}\right)d\mathit{\ell }\\ & +\beta (\gamma +\lambda ){\vartheta }_2{S}_7{I}_7{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{SI}\right)d\mathit{\ell }\\ & +\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell })ln\left(\frac{L(t-\mathit{\ell })}{L}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_3{S}_7{Y}_7}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })}{SY}\right)d\mathit{\ell }\\ & +(\frac{\mathcal{P}{\vartheta }_3{S}_7{Y}_7}{{\mathcal{H}}_5}+\frac{\mathcal{P}r{Y}_7}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })ln\left(\frac{E(t-\mathit{\ell })}{E}\right)d\mathit{\ell }\\ & +\frac{\mathcal{P}{\vartheta }_1{S}_7{V}_7}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })ln\left(\frac{I(t-\mathit{\ell })}{I}\right)d\mathit{\ell }.\end{array}$$

Using the equalities given by (10) and (11) in case of $n=m=7$, we get

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_7}{dt}& =-\varrho \mathcal{P}\frac{{(S-S_7)}^2}{S}-\mathcal{P}({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7+{\vartheta }_3{S}_7{Y}_7)[\frac{S_7}{S}-1-ln\left(\frac{S_7}{S}\right)]\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_7{V}_7\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_7}{S_7{V}_{7}L}-1-ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_7}{S_7{V}_{7}L}\right)]d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_7{I}_7\\ & \times {\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })[\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_7}{S_7{I}_{7}L}-1-ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_7}{S_7{I}_{7}L}\right)]d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_7{V}_7\\ & \times {\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_7}{S_7{V}_{7}I}-1-ln\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_7}{S_7{V}_{7}I}\right)]d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_7{I}_7{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })[\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{7}I}-1-ln\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{7}I}\right)]d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7)\\ & \times {\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell }))[\frac{L(t-\mathit{\ell })I_7}{L_{7}I}-1-ln\left(\frac{L(t-\mathit{\ell })I_7}{L_{7}I}\right)]d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_3{S}_7{Y}_7}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })[\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_7}{S_7{Y}_{7}E}-1-ln\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_7}{S_7{Y}_{7}E}\right)]d\mathit{\ell }\\ & -\frac{\mathcal{P}r{Y}_7}{\phi {\mathcal{H}}_4}[\frac{Y{E}_7}{Y_{7}E}-1-ln\left(\frac{Y{E}_7}{Y_{7}E}\right)]\\ & -(\frac{\mathcal{P}{\vartheta }_3{S}_7{Y}_7}{{\mathcal{H}}_5}+\frac{\mathcal{P}r{Y}_7}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })[\frac{E(t-\mathit{\ell })Y_7}{E_{7}Y}-1-ln\left(\frac{E(t-\mathit{\ell })Y_7}{E_{7}Y}\right)]d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_1{S}_7{V}_7}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })[\frac{I(t-\mathit{\ell })V_7}{I_{7}V}-1-ln\left(\frac{I(t-\mathit{\ell })V_7}{I_{7}V}\right)]d\mathit{\ell }.\end{array}$$ 28

Therefore, Eq. (28) becomes

$$\begin{array}{rl}\frac{d{\mathrm{\Phi }}_7}{dt}& =-\varrho \mathcal{P}\frac{{(S-S_7)}^2}{S}-\mathcal{P}({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7+{\vartheta }_3{S}_7{Y}_7)Ϝ\left(\frac{S_7}{S}\right)\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_1{S}_7{V}_7{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })L_7}{S_7{V}_{7}L}\right)d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_3(1-\beta ){\vartheta }_2{S}_7{I}_7{\int }_0^{{\kappa }_1}{\overline{\mathcal{H}}}_1(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })L_7}{S_7{I}_{7}L}\right)d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_1{S}_7{V}_7{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })V(t-\mathit{\ell })I_7}{S_7{V}_{7}I}\right)d\mathit{\ell }\\ & -\beta (\gamma +\lambda ){\vartheta }_2{S}_7{I}_7{\int }_0^{{\kappa }_2}{\overline{\mathcal{H}}}_2(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })I(t-\mathit{\ell })}{S_{7}I}\right)d\mathit{\ell }\\ & -\lambda {\mathcal{H}}_1(1-\beta )({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7){\int }_0^{{\kappa }_3}{\overline{\mathcal{H}}}_3(\mathit{\ell }))Ϝ\left(\frac{L(t-\mathit{\ell })I_7}{L_{7}I}\right)d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_3{S}_7{Y}_7}{{\mathcal{H}}_4}{\int }_0^{{\kappa }_4}{\overline{\mathcal{H}}}_4(\mathit{\ell })Ϝ\left(\frac{S(t-\mathit{\ell })Y(t-\mathit{\ell })E_7}{S_7{Y}_{7}E}\right)d\mathit{\ell }-\frac{\mathcal{P}r{Y}_7}{\phi {\mathcal{H}}_4}Ϝ\left(\frac{Y{E}_7}{Y_{7}E}\right)\\ & -(\frac{\mathcal{P}{\vartheta }_3{S}_7{Y}_7}{{\mathcal{H}}_5}+\frac{\mathcal{P}r{Y}_7}{\phi {\mathcal{H}}_4{\mathcal{H}}_5}){\int }_0^{{\kappa }_5}{\overline{\mathcal{H}}}_5(\mathit{\ell })Ϝ\left(\frac{E(t-\mathit{\ell })Y_7}{E_{7}Y}\right)d\mathit{\ell }\\ & -\frac{\mathcal{P}{\vartheta }_1{S}_7{V}_7}{{\mathcal{H}}_6}{\int }_0^{{\kappa }_6}{\overline{\mathcal{H}}}_6(\mathit{\ell })Ϝ\left(\frac{I(t-\mathit{\ell })V_7}{I_{7}V}\right)d\mathit{\ell }.\end{array}$$

Hence, $\frac{d{\mathrm{\Phi }}_7}{dt}\le 0$ for all $S,L,I,E,Y,V>0$. Similar to the previous theorems, one can show that $\frac{d{\mathrm{\Phi }}_7}{dt}=0$ when $(S,L,I,E,Y,V)=(S_7,L_7,I_7,E_7,Y_7,V_7)$. The solutions of system (5) converge to ${\mathrm{\Upsilon }}_7^{\prime }$ which includes elements with $(S,L,I,E,Y,V)(t)=(S_7,L_7,I_7,E_7,Y_7,V_7)$. Then $\frac{dI(t)}{dt}=\frac{dY(t)}{dt}=0$. The third and fifth equations of system (5) become

$$\begin{array}{rl}& 0=\frac{dI(t)}{dt}=\beta {\mathcal{H}}_2({\vartheta }_1{S}_7{V}_7+{\vartheta }_2{S}_7{I}_7)+\lambda {\mathcal{H}}_3{L}_7-a{I}_7-{\mu }_1{C}^I(t)I_7,\\ & 0=\frac{dY(t)}{dt}=\psi {\mathcal{H}}_5{E}_7-\delta Y_7-{\mu }_2{C}^Y(t)Y_7,\end{array}$$

which yield $C^I(t)=C_7^I$ and $C^Y(t)=C_7^Y$ for all t, and hence . Applying LLAS theorem, we get is GAS. □

Numerical simulations

In this section, we perform numerical simulations to illustrate the results of Theorems 1–8. Moreover, we study the influence of time delays on the dynamical behavior of the system. Let us choose a Dirac delta function $D(\cdot )$ as a special form of the kernel ${\mathrm{\Lambda }}_i(\cdot )$ as follows:

$${\mathrm{\Lambda }}_i(x)=D(x-{\mathit{\ell }}_i),{\mathit{\ell }}_i\in [0,{\kappa }_i],i=1,2,\dots ,6.$$

Let ${\kappa }_i\to \mathrm{\infty }$, then we get

$$\begin{array}{c}{\int }_0^{\mathrm{\infty }}{\mathrm{\Lambda }}_j(\varsigma )d\varsigma =1,\hfill \\ {\mathcal{H}}_j={\int }_0^{\mathrm{\infty }}D(\varsigma -{\mathit{\ell }}_j)e^{-{\hslash }_j\varsigma }d\varsigma =e^{-{\hslash }_j{\mathit{\ell }}_j},j=1,2,\dots ,6.\hfill \end{array}$$

Thus, model (4) reduces to

$$\{\begin{array}{c}\frac{dS(t)}{dt}=\eta -\varrho S(t)-{\vartheta }_{1}S(t)V(t)-{\vartheta }_{2}S(t)I(t)-{\vartheta }_{3}S(t)Y(t),\hfill \\ \frac{dL(t)}{dt}=(1-\beta )e^{-{\hslash }_1{\mathit{\ell }}_1}S(t-{\mathit{\ell }}_1)[{\vartheta }_{1}V(t-{\mathit{\ell }}_1)+{\vartheta }_{2}I(t-{\mathit{\ell }}_1)]-(\lambda +\gamma )L(t),\hfill \\ \frac{dI(t)}{dt}=\beta e^{-{\hslash }_2{\mathit{\ell }}_2}S(t-{\mathit{\ell }}_2)[{\vartheta }_{1}V(t-{\mathit{\ell }}_2)+{\vartheta }_{2}I(t-{\mathit{\ell }}_2)]\hfill \\ \phantom{\frac{dI(t)}{dt}=}+\lambda e^{-{\hslash }_3{\mathit{\ell }}_3}L(t-{\mathit{\ell }}_3)-aI(t)-{\mu }_1{C}^I(t)I(t),\hfill \\ \frac{dE(t)}{dt}=\phi {\vartheta }_3{e}^{-{\hslash }_4{\mathit{\ell }}_4}S(t-{\mathit{\ell }}_4)Y(t-{\mathit{\ell }}_4)+\mathcal{K}{r}^{\ast }Y(t)-(\psi +\omega )E(t),\hfill \\ \frac{dY(t)}{dt}=\psi e^{-{\hslash }_5{\mathit{\ell }}_5}E(t-{\mathit{\ell }}_5)+(1-\mathcal{K})r^{\ast }Y(t)-{\delta }^{\ast }Y(t)-{\mu }_2{C}^Y(t)Y(t),\hfill \\ \frac{dV(t)}{dt}=b{e}^{-{\hslash }_6{\mathit{\ell }}_6}I(t-{\mathit{\ell }}_6)-\epsilon V(t),\hfill \\ \frac{d{C}^I(t)}{dt}={\sigma }_1{C}^I(t)I(t)-{\pi }_1{C}^I(t),\hfill \\ \frac{d{C}^Y(t)}{dt}={\sigma }_2{C}^Y(t)Y(t)-{\pi }_2{C}^Y(t).\hfill \end{array}$$ 29

For model (29), the threshold parameters are given by

$$\begin{array}{rl}& {\mathrm{\Re }}_1=\frac{\mathcal{P}{S}_0(b{\vartheta }_1{e}^{-{\hslash }_6{\mathit{\ell }}_6}+\epsilon {\vartheta }_2)}{a\epsilon (\gamma +\lambda )},{\mathrm{\Re }}_2=\frac{\phi {\vartheta }_3\psi e^{-({\hslash }_4{\mathit{\ell }}_4+{\hslash }_5{\mathit{\ell }}_5)}{S}_0}{(\delta -r{e}^{-{\hslash }_5{\mathit{\ell }}_5})\psi +\delta \omega },\\ & {\mathrm{\Re }}_3=\frac{{\sigma }_1\eta \mathcal{P}(b{\vartheta }_1{e}^{-{\hslash }_6{\mathit{\ell }}_6}+\epsilon {\vartheta }_2)}{a(\gamma +\lambda )[{\pi }_1(b{\vartheta }_1{e}^{-{\hslash }_6{\mathit{\ell }}_6}+\epsilon {\vartheta }_2)+\varrho \epsilon {\sigma }_1]},\\ & {\mathrm{\Re }}_4=\frac{{\sigma }_2\eta \phi {\vartheta }_3\psi e^{-({\hslash }_4{\mathit{\ell }}_4+{\hslash }_5{\mathit{\ell }}_5)}}{({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)[(\delta -r{e}^{-{\hslash }_5{\mathit{\ell }}_5})\psi +\delta \omega ]},\\ & {\mathrm{\Re }}_5=\frac{\eta \phi \epsilon {\vartheta }_3{\sigma }_1\psi e^{-({\hslash }_4{\mathit{\ell }}_4+{\hslash }_5{\mathit{\ell }}_5)}}{[(\delta -r{e}^{-{\hslash }_5{\mathit{\ell }}_5})\psi +\delta \omega ][{\pi }_1(b{\vartheta }_1{e}^{-{\hslash }_6{\mathit{\ell }}_6}+\epsilon {\vartheta }_2)+\varrho \epsilon {\sigma }_1]},\\ & {\mathrm{\Re }}_6=\frac{\eta {\sigma }_2\mathcal{P}(b{\vartheta }_1{e}^{-{\hslash }_6{\mathit{\ell }}_6}+\epsilon {\vartheta }_2)}{a\epsilon (\gamma +\lambda )({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)},\\ & {\mathrm{\Re }}_7=\frac{{\sigma }_1{\sigma }_2\eta \mathcal{P}(b{\vartheta }_1{e}^{-{\hslash }_6{\mathit{\ell }}_6}+\epsilon {\vartheta }_2)}{a(\gamma +\lambda )[{\pi }_1{\sigma }_2(b{\vartheta }_1{e}^{-{\hslash }_6{\mathit{\ell }}_6}+\epsilon {\vartheta }_2)+\epsilon {\sigma }_1({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)]},\\ & {\mathrm{\Re }}_8=\frac{\epsilon {\vartheta }_3\eta {\sigma }_1{\sigma }_2\phi \psi e^{-({\hslash }_4{\mathit{\ell }}_4+{\hslash }_5{\mathit{\ell }}_5)}}{[{\pi }_1{\sigma }_2(b{\vartheta }_1{e}^{-{\hslash }_6{\mathit{\ell }}_6}+\epsilon {\vartheta }_2)+\epsilon {\sigma }_1({\pi }_2{\vartheta }_3+\varrho {\sigma }_2)][(\delta -r{e}^{-{\hslash }_5{\mathit{\ell }}_5})\psi +\delta \omega ]},\end{array}$$ 30

where

$$\mathcal{P}=\lambda e^{-({\hslash }_1{\mathit{\ell }}_1+{\hslash }_3{\mathit{\ell }}_3)}(1-\beta )+\beta e^{-{\hslash }_2{\mathit{\ell }}_2}(\gamma +\lambda ).$$ 31

To solve system (29) numerically, we fix the values of some parameters (see Table 1) and the others will be varied.

Table 1.

The data of model (29)

Parameter	Value	Parameter	Value	Parameter	Value
η	10	b	5	ω	0.03
ϱ	0.01	${\pi }_1$	0.1	ψ	0.003
${\vartheta }_i$, i = 1,2,3	Varied	${\pi }_2$	0.1	${\hslash }_1$	0.2
β	0.7	${\mu }_1$	0.2	${\hslash }_2$	0.3
a	0.5	${\mu }_2$	0.2	${\hslash }_3$	0.4
φ	0.2	ε	2	${\hslash }_4$	0.5
$\mathcal{K}$	0.9	γ	0.1	${\hslash }_5$	0.6
$r^{\ast }$	0.008	${\sigma }_i$, i = 1,2	Varied	${\hslash }_6$	0.9
${\delta }^{\ast }$	0.05	λ	0.2	${\xi }_i$, i = 1,2,…,6	Varied

Stability of the steady states

In this subsection, we select the delay parameters as ${\mathit{\ell }}_1=1$, ${\mathit{\ell }}_2=0.8$, ${\mathit{\ell }}_3=0.6$, ${\mathit{\ell }}_4=0.4$, ${\mathit{\ell }}_5=0.2$, ${\mathit{\ell }}_6=0.1$. Besides, we choose the following three different initial conditions for system (29):

Initial-1: $(S(\mathit{\ell }),L(\mathit{\ell }),I(\mathit{\ell }),E(\mathit{\ell }),Y(\mathit{\ell }),V(\mathit{\ell }),C^I(\mathit{\ell }),C^Y(\mathit{\ell }))=(800,0.5,0.7,12,0.51.52.5,0.3)$,

Initial-2: $(S(\mathit{\ell }),L(\mathit{\ell }),I(\mathit{\ell }),E(\mathit{\ell }),Y(\mathit{\ell }),V(\mathit{\ell }),C^I(\mathit{\ell }),C^Y(\mathit{\ell }))=(700,0.8,1,8,0.3,2.5,1.5,0.2)$,

Initial-3: $(S(\mathit{\ell }),L(\mathit{\ell }),I(\mathit{\ell }),E(\mathit{\ell }),Y(\mathit{\ell }),V(\mathit{\ell }),C^I(\mathit{\ell }),C^Y(\mathit{\ell }))=(600,1.31.2,10,0.4,3.5,0.5,0.1)$, where $\mathit{\ell }\in [-1,0]$.

Choosing different values of ${\vartheta }_1$, ${\vartheta }_2$, ${\vartheta }_3$, ${\vartheta }_4$, ${\sigma }_1$, and ${\sigma }_2$ under the above initial conditions leads to the following sets:

Set 1 (Stability of ): ${\vartheta }_1=0.0002$, ${\vartheta }_2=0.0001$, ${\vartheta }_3=0.001$, ${\sigma }_1=0.3$, and ${\sigma }_2=0.5$. For this set of parameters, we have ${\mathrm{\Re }}_1=0.76<1$ and ${\mathrm{\Re }}_2=0.27<1$. Figure 1 illustrates that the trajectories starting different initials converge to the steady state . This supports the global stability result of Theorem 1. Here, a healthy state will be reached where both viruses are absent.

Figure 1.

Solutions of system (29) when ${\mathrm{\Re }}_1\le 1$ and ${\mathrm{\Re }}_2\le 1$

Set 2 (Stability of ): ${\vartheta }_1=0.0005$, ${\vartheta }_2=0.0003$, ${\vartheta }_3=0.0007$, ${\sigma }_1=0.003$, and ${\sigma }_2=0.2$. With such a choice we get ${\mathrm{\Re }}_2=0.19<1<1.96={\mathrm{\Re }}_1$, ${\mathrm{\Re }}_3=0.34<1$, and hence ${\mathrm{\Re }}_2/{\mathrm{\Re }}_1=0.1<1$. The steady state exists with . The stability of given in Theorem 2 is shown in Fig. 2. This leads to the case where HIV monoinfection is chronic but with an ineffective CTL immunity.

Figure 2.

Solutions of system (29) when ${\mathrm{\Re }}_1>1$, ${\mathrm{\Re }}_2/{\mathrm{\Re }}_1\le 1$, and ${\mathrm{\Re }}_3\le 1$

Set 3 (Stability of ): ${\vartheta }_1=0.0001$, ${\vartheta }_2=0.0003$, ${\vartheta }_3=0.005$, ${\sigma }_1=0.001$, and ${\sigma }_2=0.05$. Then we calculate ${\mathrm{\Re }}_1=0.72<1<1.34={\mathrm{\Re }}_2$, ${\mathrm{\Re }}_4=0.67<1$, and then ${\mathrm{\Re }}_1/{\mathrm{\Re }}_2=0.54<1$ and . We can see from Fig. 3 that the system’s solutions tend to , which is compatible with Theorem 3. This case means that an HTLV monoinfection is chronic with an ineffective CTL immunity.

Figure 3.

Solutions of system (29) when ${\mathrm{\Re }}_2>1$, ${\mathrm{\Re }}_1/{\mathrm{\Re }}_2\le 1$, and ${\mathrm{\Re }}_4\le 1$

Set 4 (Stability of ): ${\vartheta }_1=0.0005$, ${\vartheta }_2=0.0003$, ${\vartheta }_3=0.002$, ${\sigma }_1=0.05$, and ${\sigma }_2=0.005$. Then we calculate ${\mathrm{\Re }}_3=1.52>1$ and ${\mathrm{\Re }}_5=0.42<1$. Figure 4 shows that the trajectories starting with different states tend to . Therefore, is GAS, and this supports Theorem 4. Hence, an HIV monoinfection is chronic with effective HIV-specific CTL immunity.

Figure 4.

Solutions of system (29) when ${\mathrm{\Re }}_3>1$ and ${\mathrm{\Re }}_5\le 1$

Set 5 (Stability of ): ${\vartheta }_1=0.0002$, ${\vartheta }_2=0.0002$, ${\vartheta }_3=0.02$, ${\sigma }_1=0.07$, and ${\sigma }_2=0.4$. Then we calculate ${\mathrm{\Re }}_4=3.57>1$ and ${\mathrm{\Re }}_6=0.60<1$, and exists with . We observe from Fig. 5 that the system’s trajectories tend to and it is GAS. Here, an HTLV monoinfection is chronic with effective HTLV-specific CTL immunity.

Figure 5.

Solutions of system (29) when ${\mathrm{\Re }}_4>1$ and ${\mathrm{\Re }}_6\le 1$

Set 6 (Stability of ): ${\vartheta }_1=0.001$, ${\vartheta }_2=0.0001$, ${\vartheta }_3=0.005$, ${\sigma }_1=0.15$, and ${\sigma }_2=0.01$. Then we calculate ${\mathrm{\Re }}_5=1.15>1$, ${\mathrm{\Re }}_8=0.22<1$, and ${\mathrm{\Re }}_1/{\mathrm{\Re }}_2=2.42>1$. The numerical solutions of the system drawn in Fig. 6 confirm that exists and is GAS. This case leads to a chronic coinfection with HTLV and HIV where the HIV-specific CTL immunity is effective while the HTLV-specific CTL immunity is ineffective.

Figure 6.

Solutions of system (29) when ${\mathrm{\Re }}_5>1$, ${\mathrm{\Re }}_8\le 1$, and ${\mathrm{\Re }}_1/{\mathrm{\Re }}_2>1$

Set 7 (Stability of ): ${\vartheta }_1=0.0004$, ${\vartheta }_2=0.0002$, ${\vartheta }_3=0.01$, ${\sigma }_1=0.007$, and ${\sigma }_2=0.7$. We compute ${\mathrm{\Re }}_6=1.32>1$, ${\mathrm{\Re }}_7=0.55<1$, and ${\mathrm{\Re }}_2/{\mathrm{\Re }}_1=1.77>1$. According to these values, we obtain that exists. The numerical solutions of our system plotted in Fig. 7 show that is GAS (Theorem 7). This case leads to a chronic coinfection with HTLV and HIV where the HTLV-specific CTL immunity is effective and the HIV-specific CTL immunity is not working.

Figure 7.

Solutions of system (29) when ${\mathrm{\Re }}_6>1$, ${\mathrm{\Re }}_7\le 1$, and ${\mathrm{\Re }}_2/{\mathrm{\Re }}_1>1$

Set 8 (Stability of ): ${\vartheta }_1=0.0005$, ${\vartheta }_2=0.0003$, ${\vartheta }_3=0.01$, ${\sigma }_1=0.1$, and ${\sigma }_2=0.3$. These data give ${\mathrm{\Re }}_7=1.33>1$ and ${\mathrm{\Re }}_8=1.81>1$. According to these values, the steady state exists. Figure 8 illustrates that the solutions of the system initiating with three different states tend to . In this case, a chronic coinfection with HTLV and HIV is reached where both immune responses are well working.

Figure 8.

Solutions of system (29) when ${\mathrm{\Re }}_7>1$ and ${\mathrm{\Re }}_8>1$

Effect of time delays on the HTLV-HIV dynamics

In this part we vary the delay parameters ${\mathit{\ell }}_i$, $i=1,2,\dots ,6$, and fix the parameters ${\vartheta }_1=0.0005$, ${\vartheta }_2=0.0003$, ${\vartheta }_3=0.01$, ${\sigma }_1=0.04$, and ${\sigma }_2=0.7$. Since ${\mathrm{\Re }}_1$ and ${\mathrm{\Re }}_2$ given by Eqs. (30) and (31) depend on ${\mathit{\ell }}_i$, $i=1,2,\dots ,6$, then changing the parameters ${\mathit{\ell }}_i$ will change the stability of steady states. Let us consider the following situations:

(D.P.S1)

${\mathit{\ell }}_1={\mathit{\ell }}_2={\mathit{\ell }}_3={\mathit{\ell }}_4={\mathit{\ell }}_5={\mathit{\ell }}_6=0$,

(D.P.S2)

${\mathit{\ell }}_1=0.4$, ${\mathit{\ell }}_2=0.5$, ${\mathit{\ell }}_3=0.6$, ${\mathit{\ell }}_4=0.7$, ${\mathit{\ell }}_5=0.8$, and ${\mathit{\ell }}_6=0.9$,

(D.P.S3)

${\mathit{\ell }}_1=0.6$, ${\mathit{\ell }}_2=0.7$, ${\mathit{\ell }}_3=0.8$, ${\mathit{\ell }}_4=0.9$, ${\mathit{\ell }}_5=1$, and ${\mathit{\ell }}_6=1.2$,

(D.P.S4)

${\mathit{\ell }}_1=10$, ${\mathit{\ell }}_2=11$, ${\mathit{\ell }}_3=12$, ${\mathit{\ell }}_4=13$, ${\mathit{\ell }}_5=14$, and ${\mathit{\ell }}_6=15$.

With these values we solve system (29) under the following initial condition:

Initial-4: $(S(\mathit{\ell }),L(\mathit{\ell }),I(\mathit{\ell }),E(\mathit{\ell }),Y(\mathit{\ell }),V(\mathit{\ell }),C^I(\mathit{\ell }),C^Y(\mathit{\ell }))=(800,1,2,4,0.14,3,1,0.1)$, where $\mathit{\ell }\in [-max{\mathit{\ell }}_i,0]$, $i=1,2,\dots ,6$.

From Fig. 9 we observe that the presence of time delays can increase the number of uninfected CD4⁺ T cells and decrease the number of other compartments. Table 2 presents the values ${\mathrm{\Re }}_1$ and ${\mathrm{\Re }}_2$ for selected values of ${\mathit{\ell }}_i$, $i=1,2,\dots ,6$. It is clear that ${\mathrm{\Re }}_1$ and ${\mathrm{\Re }}_2$ are decreased when ${\mathit{\ell }}_i$ are increased, and thus the stability of can be changed. Let us calculate the critical value of the time delay that changes the stability of . Without loss of generality, we let the parameters $\mathit{\ell }={\mathit{\ell }}_1={\mathit{\ell }}_2={\mathit{\ell }}_3$ and fix ${\mathit{\ell }}_j$, $j=5,6$, and write ${\mathrm{\Re }}_1$ and ${\mathrm{\Re }}_2$ as functions of ℓ and ${\mathit{\ell }}_4$, respectively, as follows:

$$\begin{array}{rl}& {\mathrm{\Re }}_1(\mathit{\ell })=\frac{[\lambda e^{-\mathit{\ell }({\hslash }_1+{\hslash }_3)}(1-\beta )+\beta e^{-{\hslash }_2\mathit{\ell }}(\gamma +\lambda )]S_0(b{\vartheta }_1{e}^{-{\hslash }_6{\mathit{\ell }}_6}+\epsilon {\vartheta }_2)}{a\epsilon (\gamma +\lambda )},\\ & {\mathrm{\Re }}_2({\mathit{\ell }}_4)=\frac{\phi {\vartheta }_3\psi e^{-({\hslash }_4{\mathit{\ell }}_4+{\hslash }_5{\mathit{\ell }}_5)}{S}_0}{(\delta -r{e}^{-{\hslash }_5{\mathit{\ell }}_5})\psi +\delta \omega }.\end{array}$$

To force the threshold parameters ${\mathrm{\Re }}_1$ and ${\mathrm{\Re }}_2$ to satisfy ${\mathrm{\Re }}_1(\mathit{\ell })\le 1$ and ${\mathrm{\Re }}_2({\mathit{\ell }}_4)\le 1$, we choose $\mathit{\ell }\ge {\mathit{\ell }}^{min}$, where ${\mathit{\ell }}^{min}$ is the solution of

$$\frac{[\lambda e^{-{\mathit{\ell }}^{min}({\hslash }_1+{\hslash }_3)}(1-\beta )+\beta e^{-{\hslash }_2{\mathit{\ell }}^{min}}(\gamma +\lambda )]S_0(b{\vartheta }_1{e}^{-{\hslash }_6{\mathit{\ell }}_6}+\epsilon {\vartheta }_2)}{a\epsilon (\gamma +\lambda )}=1,$$

and

$${\mathit{\ell }}_4\ge {\mathit{\ell }}_4^{min},\text{where }{\mathit{\ell }}_4^{min}=max\{0,\frac{1}{{\hslash }_4}ln\frac{\phi {\vartheta }_3\psi e^{-{\hslash }_5{\mathit{\ell }}_5}{S}_0}{(\delta -r{e}^{-{\hslash }_5{\mathit{\ell }}_5})\psi +\delta \omega }\}.$$

Therefore, if $\mathit{\ell }\ge {\mathit{\ell }}^{min}$ and ${\mathit{\ell }}_4\ge {\mathit{\ell }}_4^{min}$, then is GAS. Let us choose the value ${\mathit{\ell }}_5=0.2$ and ${\mathit{\ell }}_6=0.1$ and compute ${\mathit{\ell }}^{min}$, ${\mathit{\ell }}_4^{min}$ as ${\mathit{\ell }}^{min}=2.73728$, ${\mathit{\ell }}_4^{min}=2.36794$. It follows that:

Figure 9.

Impact of delay parameters ${\ell }_i$, $i=1,2,\dots ,6$, on the behavior of solution trajectories of system (29)

Table 2.

The variation of ${\mathrm{\Re }}_1$ and ${\mathrm{\Re }}_2$ with respect to the delay parameters

Delay parameters	${\mathrm{\Re }}_1$	${\mathrm{\Re }}_2$
${\ell }_1={\ell }_2={\ell }_3={\ell }_4={\ell }_5={\ell }_6=0$	2.790	3.690
${\ell }_1=0.3$, ${\ell }_2=0.4$, ${\ell }_3=0.5$, ${\ell }_4=0.6$, ${\ell }_5=0.7$, and ${\ell }_6=0.8$	1.408	1.787
${\ell }_1=0.4$, ${\ell }_2=0.5$, ${\ell }_3=0.6$, ${\ell }_4=0.7$, ${\ell }_5=0.8$, and ${\ell }_6=0.9$	1.280	1.600
${\ell }_1=0.6$, ${\ell }_2=0.7$, ${\ell }_3=0.8$, ${\ell }_4=0.9$, ${\ell }_5=1$, and ${\ell }_6=1.2$	1.009	1.283
${\ell }_1=1$, ${\ell }_2=1.5$, ${\ell }_3=2$, ${\ell }_4=2.5$, ${\ell }_5=3$, and ${\ell }_6=3.5$	0.368	0.173
${\ell }_1=2$, ${\ell }_2=3$, ${\ell }_3=4$, ${\ell }_4=5$, ${\ell }_5=6$, and ${\ell }_6=7$	0.188	0.008
${\ell }_1=3$, ${\ell }_2=4$, ${\ell }_3=5$, ${\ell }_4=6$, ${\ell }_5=7$, and ${\ell }_6=8$	0.136	0.003
${\ell }_1=4$, ${\ell }_2=6$, ${\ell }_3=8$, ${\ell }_4=9$, ${\ell }_5=10$, and ${\ell }_6=11$	0.072	0.1 × 10−3
${\ell }_1=6$, ${\ell }_2=7$, ${\ell }_3=9$, ${\ell }_4=10$, ${\ell }_5=11$, and ${\ell }_6=12$	0.052	0.3 × 10−4
${\ell }_1=10$, ${\ell }_2=11$, ${\ell }_3=12$, ${\ell }_4=13$, ${\ell }_5=14$, and ${\ell }_6=15$	0.016	1.2 × 10−6

(i) If $\mathit{\ell }\ge 2.73728$ and ${\mathit{\ell }}_4\ge 2.36794$, then ${\mathrm{\Re }}_1(\mathit{\ell })\le 1$, ${\mathrm{\Re }}_2({\mathit{\ell }}_4)\le 1$ and is GAS.

(ii) If $\mathit{\ell }<2.73728$ or ${\mathit{\ell }}_4<2.36794$, then ${\mathrm{\Re }}_1(\mathit{\ell })>1$ or ${\mathrm{\Re }}_2({\mathit{\ell }}_4)>1$ and will lose its stability.

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