Modeling SARS-CoV-2 and HBV co-dynamics with optimal control

Abstract

Clinical reports have shown that chronic hepatitis B virus (HBV) patients co-infected with SARS-CoV-2 have a higher risk of complications with liver disease than patients without SARS-CoV-2. In this work, a co-dynamical model is designed for SARS-CoV-2 and HBV which incorporates incident infection with the dual diseases. Existence of boundary and co-existence endemic equilibria are proved. The occurrence of backward bifurcation, in the absence and presence of incident co-infection, is investigated through the proposed model. It is noted that in the absence of incident co-infection, backward bifurcation is not observed in the model. However, incident co-infection triggers this phenomenon. For a special case of the study, the disease free and endemic equilibria are shown to be globally asymptotically stable. To contain the spread of both infections in case of an endemic situation, the time dependent controls are incorporated in the model. Also, global sensitivity analysis is carried out by using appropriate ranges of the parameter values which helps to assess their level of sensitivity with reference to the reproduction numbers and the infected components of the model. Finally, numerical assessment of the control system using various intervention strategies is performed, and reached at the conclusion that enhanced preventive efforts against incident co-infection could remarkably control the co-circulation of both SARS-CoV-2 and HBV.

1. Introduction

The Coronavirus disease 2019 (COVID-19) caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) was first discovered in Wuhan, China. The disease has rapidly spread to the entire globe [1]. By September 20, 2022, 600 million confirmed COVID-19 cases with more than 6.5 million deaths worldwide have been estimated by the World Health Organization (WHO) [2]. The WHO also reported an aggregate case-fatality rate of 2.14% [3]. On the other hand, Chronic hepatitis B virus (HBV) infection is a major cause of chronic liver disease worldwide [4]. The global prevalence of HBV is about 3.9% [5], [6]. Generally, SARS-CoV-2 is related to respiratory issues, there has been a strong evidence that shows the link between SARS-CoV-2 infection and liver damage. Though, an exact mechanism is not yet confirmed, it is speculation that the direct attack on hepatocytes or SARS-CoV-2 causing immune-mediated inflammation such as cytokine storm and pneumonia-related hypoxia could lead to liver failure in critically ill SARS-CoV-2 patients [7]. Clinical studies have shown that about 2%–11% of patients with COVID-19 had chronic liver disease [7]. Co-infection between HBV and other viral infections can accelerate liver injury progression which leads to the serious complications [8], [9]. The authors in [10] have observed that patients with SARS-CoV-2 and HBV co-infection with liver injury are more likely to develop serious health problems. Lin et al. [11] has also confirmed that patients with SARS-CoV-2 and HBV co-infection are at risk of greater liver injury. It has also been reported that infection with SARS-CoV-2 could instigate liver injury, and that 14%–53% of COVID-19 patients developed hepatic dysfunction, particularly individuals with severe disease [12]. Infection with SARS-CoV-2 could thus be a major risk factor for critical illness and worse complications among individuals living with HBV [13]. Hence, developing a robust mathematical model to study these interactions is much desirable in making efforts to understand and reduce the incidence of both diseases and their negative consequences.

A lot of mathematical models have been proposed to investigate the dynamics of SARS-CoV-2 infection [14], [15], [16], [17], [18], [19], [20], [21]. The authors in [14] have developed a model to study the dynamics of COVID-19 with main focus on the importance of social distancing, usage of face masks usage and case detection. Wang et al. [15] proposed a mathematical model to study the early trends of COVID-19 in Wuhan. Kucharski et al. [16] used a stochastic dynamical model to study COVID-19. Some important parameters of the model using real data were estimated. Ferguson et al. [17] highlighted the importance of isolation, quarantine and social distancing to bring the death rate due to COVID-19 cases down. Maier et al. [18] proposed a COVID-19 model and pointed out the advantages of containment policies as an adequate means to reduce the spread of the disease. Li et al. [19] developed a robust within-host model to study the interactions between SARS-CoV-2 particles and infected/uninfected epithelial cells. Du and Yuan [20] considered a related model and incorporated the effect of antiviral drugs. Al Agha et al. [21] developed a within-host delay differential equation model to study the impact of SARS-CoV-2 on patients infected with cancer.

Mathematical models to study the co-interactions between SARS-CoV-2 infection and other diseases have attracted the attention of several researchers [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33]. The authors in [22] analyzed a co-dynamical model for dengue and COVID-19, comparing model solutions using various fractional operators. In the same direction, the authors [23] showed that effective chlorine water distribution, sufficient provision of equipments for adequate testing could remarkably curtail the co-circulation of COVID-19 and cholera. Elaiw [24] investigated the global stability of within host model for SARS-CoV-2/HIV incorporating latency. Results obtained in this directions show that HIV patients could suffer worse complications when they get infected with SARS-CoV-2. Also, the authors [25] qualitatively analyzed a within host model for SARS-CoV-2/malaria including antibody immune response. The observed that the shared immune response could reduce the accumulation of SARS-CoV-2 particles and malaria merozoites among co-infected individuals. Pinky and Dobrovolny [26] proposed an in-host model to analyze SARS-CoV-2 co-infection with viruses such as influenza and human rhinovirus.

Optimal control has become an effective tool in modeling the dynamics of different infectious diseases [34], [35], [36], [37], [38], [39], [40], [41]. Khan et al. [34] performed a comprehensive study of a dynamical model for zika virus with optimal control. Jan et al. [35] considered an optimal intervention model for dengue virus. In [36], Ullah et al. analyzed a hepatitis B virus model with optimal control. Also, Bonyah et al. [37] studied an optimal control model for dengue and zika virus. The authors [38] applied optimal control to model the adverse impact of high alcoholic consumption on the dynamics of gonorrhea. Asamoah et al. [39] also used optimal control to analyze a dengue fever dynamical model with asymptomatic and partial immune individuals. Abidemi et al. [40] adopted Lyapunov stability analysis and optimal control measures in mitigating dengue disease transmission. Recently, Ojo et al. [41] comprehensively analyzed a co-dynamical model for COVID-19 and influenza incorporating optimal intervention measures.

These investigations have shown that dual infection with SARS-CoV-2 and HBV could be associated with serious consequences. To the best of our knowledge, there is no robust optimal control model to study the interactions between SARS-CoV-2 and HBV. The aim of this paper is to formulate a comprehensive mathematical model to understand the co-dynamics of both infections.

The remaining part of the paper is organized into different sections described as: In Section 2, a mathematical model for SARS-CoV-2 and HBV is proposed. Basic properties and qualitative analyses of the model without controls are presented in Section 3. The optimal control model is proposed and analyzed in Section 4. Global sensitivity analysis and simulations of the optimal control system to highlight the impact of various intervention measures are carried out in Section 5. Concluding remarks and future directions are given in Section 6.

2. Model formulation

At any given time $t$, underlying population denoted by $\mathcal{N}\left(t\right)$ comprises the different compartments: $\mathcal{S}\left(t\right)$: susceptible or vulnerable individuals, $\mathcal{A}{}_C\left(t\right)$: individuals infected with SARS-CoV-2 in asymptomatic stage, $\mathcal{I}{}_C\left(t\right)$: individuals infected with SARS-CoV-2 in symptomatic stage, $\mathcal{A}{}_H\left(t\right)$: individuals acutely infected with HBV, $\mathcal{I}{}_H\left(t\right)$: individuals with chronic HBV, $\mathcal{I}{}_{CH}\left(t\right)$: individuals infected with both SARS-CoV-2 (in symptomatic stage) and chronic HBV, and $\mathcal{R}\left(t\right)$: individuals who have recovered from or are immune to SARS-CoV-2, HBV or dual infections. In this paper, we shall use the standard incidence rate.

Additional assumptions in the model are stated thus:

• (i.)Susceptible or vulnerable individuals could acquire SARS-CoV-2 infection at the rate $\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}$, where ${\vartheta }_1$ denotes reduced transmissibility rate for asymptomatic individuals relative to symptomatic SARS-CoV-2 -infected individuals [42].
• (ii.)Vulnerable individuals could also acquire HBV at the rate $\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}$, where ${\vartheta }_2$ stands for reduced transmission rate for acutely infected individuals as compared to chronic HBV-infected individuals [43].
• (iii.)Susceptible or vulnerable individuals could also get infected with the dual infections concurrently at the rate $\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}$. There is always a possibility of dual infections [8], [44].
• (iv.)Natural death rate, $\mu$, is uniformly assumed for individuals in every epidemiological state.
• (v.)Individuals with SARS-CoV-2 in symptomatic stage or chronic HBV could get infected with a second infection at the rate ${\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}$ or ${\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}$, where ${\zeta }_1$ and ${\zeta }_2$ are terms denoting vulnerability to a second infection [7].
• (vi.)Disease-induced death is assumed for individuals infected with SARS-CoV-2 (in symptomatic stage), those with chronic HBV and individuals with dual infections at the rates $\eta {}_C,\eta {}_H$ and $\eta {}_{CH}$, separately.
• (vii.)The model assumes recovery from infection at the rates $\xi {}_C,\xi {}_H$ and $\xi {}_{CH}$, for infected with SARS-CoV-2(in symptomatic stage), infected with HBV and infected with dual infections, respectively.

The proposed model did not capture vaccination for both diseases and also did not incorporate re-infection with either or both diseases [45], [46]. By incorporating vaccination and/or re-infection for SARS-CoV-2 and HBV, the proposed model could be extended. Note that the model focuses only on co-infection between symptomatic stage of SARS-CoV-2 infection and chronic HBV infection. The reason behind this is to see how to reduce co-infection among individuals in these critical stages.

All the model parameters are defined in Table 1 whereas the governing system of equations and schematic diagram are given by Eq. (1) and Fig. 1, respectively.

$$\frac{d\mathcal{S}\left(t\right)}{dt}=\Lambda -\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}-\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}-\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}-\mu \mathcal{S},$$$$\frac{d\mathcal{A}{}_C\left(t\right)}{dt}=\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}-({\alpha }_1+\mu )\mathcal{A}{}_C,$$$$\frac{d\mathcal{I}{}_C\left(t\right)}{dt}={\alpha }_1\mathcal{A}{}_C-(\xi {}_C+\eta {}_C+\mu )\mathcal{I}{}_C-{\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C,$$$$\frac{d\mathcal{A}{}_H\left(t\right)}{dt}=\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}-({\alpha }_2+\mu )\mathcal{A}{}_H,$$$$\frac{d\mathcal{I}{}_H\left(t\right)}{dt}={\alpha }_2\mathcal{A}{}_H-(\xi {}_H+\eta {}_H+\mu )\mathcal{I}{}_H-{\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{I}{}_H,$$$$\frac{d\mathcal{I}{}_{CH}\left(t\right)}{dt}=\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}+{\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C+{\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{I}{}_H-(\xi {}_{CH}+\eta {}_{CH}+\mu )\mathcal{I}{}_{CH},$$$$\frac{d\mathcal{R}\left(t\right)}{dt}=\xi {}_C\mathcal{I}{}_C+\xi {}_H\mathcal{I}{}_H+\xi {}_{CH}\mathcal{I}{}_{CH}-\mu \mathcal{R}.$$ (1)

Table 1.

Description of parameters in the model (1).

Parameter	Description	Value	References
$\beta {}_C$	Effective contact rate for SARS-CoV-2 transmission	0.5944/day	[47]
$\beta {}_H$	Effective contact rate for HBV transmission	0.8/day	[43]
$\beta {}_{CH}$	Effective contact rate for dual diseases transmission	0.2/day	Assumed
$\xi {}_C$	SARS-CoV-2 recovery rate	$[\frac{1}{30},\frac{1}{3}]$/day	[42]
$\Lambda$	Human recruitment rate	$\frac{732,500}{76.31\times 365}$/day	[48], [49]
$\xi {}_H$	HBV recovery rate	0.15/day	Assumed
$\xi {}_{CH}$	Co-infection recovery rate	0.15/day	Assumed
$\eta {}_C$	SARS-CoV-2 induced death rate	0.0214/day	[3]
$\eta {}_H$	HBV induced death rate	0.02/day	[43]
$\eta {}_{CH}$	co-infection induced death rate	0.005/day	Assumed
$\mu$	natural death rate	$\frac{1}{76.31\times 365}$/day	[49]
${\zeta }_1,{\zeta }_2$	modification parameter	1.0	[14]
${\vartheta }_1,{\vartheta }_2$	modification parameter	0.5	Assumed
${\alpha }_1,{\alpha }_2$	progression rates to symptomatic stage	$[\frac{1}{14},\frac{1}{3}]$/day	[50]

Fig. 1.

The Model’s schematic diagram.

3. Analysis of the model

In this section, we shall qualitatively analyze the model (1) without controls.

3.1. Basic properties of the model

3.1.1. Non-negativity of the model solutions

For the model (1) to be epidemiologically meaningful, it is appropriate to show that the model solutions are non- negative over time [51]. Let us start with the following:

Theorem 3.1

Given the initial states $\mathcal{S}\left(0\right)\ge 0,\mathcal{A}{}_C\left(0\right)\ge 0,\mathcal{I}{}_C\left(0\right)\ge 0,\mathcal{A}{}_H\left(0\right)\ge 0,\mathcal{I}{}_H\left(0\right)\ge 0,\mathcal{I}{}_{CH}\left(0\right)\ge 0,\mathcal{R}\left(0\right)\ge 0$ .

Then the solutions, $(\mathcal{S}\left(t\right),\mathcal{A}{}_C\left(t\right),\mathcal{I}{}_C\left(t\right),\mathcal{A}{}_H\left(t\right),\mathcal{I}{}_H\left(t\right),\mathcal{I}{}_{CH}\left(t\right),\mathcal{R}\left(t\right))$ of the model (1) are non-negative for all time $t>0$ .

Proof

Let $t_f=sup\{t>0:\mathcal{S}\left(0\right)>0,\mathcal{A}{}_C\left(0\right)>0,\mathcal{I}{}_C\left(0\right)>0,\mathcal{A}{}_H\left(0\right)>0,\mathcal{I}{}_H\left(0\right)>0,\mathcal{I}{}_{CH}\left(0\right)>0,\mathcal{R}\left(0\right)>0\}$.

From the 1st equation of system (1), we have

$$\frac{d\mathcal{S}\left(t\right)}{dt}=\Lambda -(\lambda {}_C\left(t\right)+\lambda {}_H\left(t\right)+\lambda {}_{CH}\left(t\right)+\mu )S$$ (2)

where,

$$\lambda {}_C\left(t\right)=\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C\left(t\right)+\mathcal{I}{}_C\left(t\right))}{\mathcal{N}\left(t\right)},\lambda {}_H\left(t\right)=\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H\left(t\right)+\mathcal{I}{}_H\left(t\right))}{\mathcal{N}\left(t\right)},\lambda {}_{CH}=\frac{\beta {}_{CH}\mathcal{I}{}_{CH}\left(t\right)}{\mathcal{N}\left(t\right)}.$$

Applying the integrating factor method on (2), we obtain that

$$\frac{d}{dt}\left\{S\left(t\right)exp\left[{\int }_0^t(\lambda {}_C\left(u\right)+\lambda {}_H\left(u\right)+\lambda {}_{CH}\left(u\right))du+\mu t\right]\right\}=\Lambda exp\left[{\int }_0^t(\lambda {}_C\left(u\right)+\lambda {}_H\left(u\right)+\lambda {}_{CH}\left(u\right))du+\mu t\right]$$ (3)

Integrating both sides of (3) gives that

$$S\left(t_f\right)exp\left[{\int }_0^{t_f}(\lambda {}_C\left(u\right)+\lambda {}_H\left(u\right)+\lambda {}_{CH}\left(u\right))du+\mu t_f\right]-S\left(0\right)=\Lambda {\int }_0^{t_f}exp\left[{\int }_0^x(\lambda {}_C\left(u\right)+\lambda {}_H\left(u\right)+\lambda {}_{CH}\left(u\right))du+\mu x\right]dx.$$

Thus,

$$S\left(t_f\right)=S\left(0\right)exp\left[-{\int }_0^{t_f}(\lambda {}_C\left(u\right)+\lambda {}_H\left(u\right)+\lambda {}_{CH}\left(u\right))du-\mu t_f\right]+exp\left[-{\int }_0^{t_f}(\lambda {}_C\left(u\right)+\lambda {}_H\left(u\right)+\lambda {}_{CH}\left(u\right))du-\mu t_f\right]\times \Lambda {\int }_0^{t_f}exp\left[{\int }_0^x(\lambda {}_C\left(u\right)+\lambda {}_H\left(u\right)+\lambda {}_{CH}\left(u\right))du+\mu x\right]dx\ge 0.$$

Hence, $S\left(t\right)\ge 0$ for all time $t>0$.

Similarly, it can be shown that:

$\mathcal{A}{}_C\left(t\right)\ge 0,\mathcal{I}{}_C\left(t\right)\ge 0,\mathcal{A}{}_H\left(t\right)\ge 0,\mathcal{I}{}_H\left(t\right)\ge 0,\mathcal{I}{}_{CH}\left(t\right)\ge 0,\mathcal{R}\left(t\right)\ge 0$.

3.1.2. Boundedness

Theorem 3.2

The closed set $\mathcal{D}$ given by

$$\mathcal{D}=.\left\{(\mathcal{S},\mathcal{A}{}_C,\mathcal{I}{}_C,\mathcal{A}{}_H,\mathcal{I}{}_H,\mathcal{I}{}_{CH},\mathcal{R})\in {\mathbb{R}}_{+}^7:\mathcal{S}+\mathcal{A}{}_C+\mathcal{I}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_H+\mathcal{I}{}_{CH}+\mathcal{R}\le \frac{\Lambda }{\mu }.\right\},$$

is positively invariant with respect to the model (1) .

Proof

By adding all equations of system (1), gives

$$\frac{d\mathcal{N}\left(t\right)}{dt}=\Lambda -\mu \mathcal{N}\left(t\right)-[\eta {}_C\mathcal{I}{}_C\left(t\right)+\eta {}_H\mathcal{I}{}_H\left(t\right)+\eta {}_{CH}\mathcal{I}{}_{CH}\left(t\right)].$$ (4)

From (4), we have

$$\frac{d\mathcal{N}\left(t\right)}{dt}\le \Lambda -\mu \mathcal{N}\left(t\right),$$

that is,

$$\frac{d\mathcal{N}\left(t\right)}{dt}+\mu \mathcal{N}\left(t\right)\le \Lambda ,$$ (5)

By applying the integrating factor method to (5) and simplifying, we obtain

$$\mathcal{N}\left(t\right)\le \frac{\Lambda }{\mu }+\left(\mathcal{N}\left(0\right)-\frac{\Lambda }{\mu }\right)e^{-\mu t},$$

which further implies that

$$lim\underset{t\to \infty }{sup}\mathcal{N}\left(t\right)\le \frac{\Lambda }{\mu }.$$ (6)

Thus, $\mathcal{N}\left(t\right)\le \frac{\Lambda }{\mu }$ as $t\to \infty$. Hence, the system (1) has the solution in $\mathcal{D}$. Thus, the given system is positively invariant.

3.2. The basic reproduction number of the model

The disease free equilibrium (DFE) associated with the model (1) is obtained by setting the right-hand sides of the equations in system (1) to zero which is given by

$$\mathcal{Q}{}_0=\left({\mathcal{S}}^0,\mathcal{A}{}_C^0,\mathcal{I}{}_C^0,\mathcal{A}{}_H^0,\mathcal{I}{}_H^0,\mathcal{I}{}_{CH}^0,{\mathcal{R}}^0\right)=\left(\frac{\Lambda }{\mu },0,0,0,0,0,0\right).$$

The basic reproduction number of the model is established by applying the next generation operator method [52] on the system (1). The transfer matrices are given by

$$F=\left(\begin{array}{ccccc}\hfill {\vartheta }_1\beta {}_C\hfill & \hfill \beta {}_C\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\vartheta }_2\beta {}_H\hfill & \hfill \beta {}_H\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \beta {}_{CH}\hfill \end{array}\right),V=\left(\begin{array}{ccccc}\hfill {\mathcal{G}}_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -{\alpha }_1\hfill & \hfill {\mathcal{G}}_2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathcal{G}}_3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -{\alpha }_2\hfill & \hfill {\mathcal{G}}_4\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathcal{G}}_5\hfill \end{array}\right)$$ (7)

where,

$${\mathcal{G}}_1={\alpha }_1+\mu ,{\mathcal{G}}_2=\xi {}_C+\eta {}_C+\mu ,{\mathcal{G}}_3={\alpha }_2+\mu ,{\mathcal{G}}_4=\xi {}_H+\eta {}_H+\mu ,{\mathcal{G}}_5=\xi {}_{CH}+\eta {}_{CH}+\mu$$

Note that ${\mathcal{R}}_0=\rho \left(F{V}^{-1}\right)=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}$, where ${\mathcal{R}}_{0C},{\mathcal{R}}_{0H}$ and ${\mathcal{R}}_{0CH}$ are the associated reproduction numbers for SARS-CoV-2, hepatitis B virus and the co-infection of both diseases, respectively and are given by

$${\mathcal{R}}_{0C}=\frac{\beta {}_C[{\vartheta }_1(\xi {}_C+\eta {}_C+\mu )+{\alpha }_1]}{({\alpha }_1+\mu )(\xi {}_C+\eta {}_C+\mu )},{\mathcal{R}}_{0H}=\frac{\beta {}_H[{\vartheta }_2(\xi {}_H+\eta {}_H+\mu )+{\alpha }_2]}{({\alpha }_2+\mu )(\xi {}_H+\eta {}_H+\mu )},{\mathcal{R}}_{0CH}=\frac{\beta {}_{CH}}{\xi {}_{CH}+\eta {}_{CH}+\mu }.$$

3.3. Local asymptotic stability of the disease free equilibrium (DFE) of the model

Theorem 3.3

The model’s DFE, $\mathcal{Q}{}_0$ is locally asymptotically stable (LAS) if ${\mathcal{R}}_0<1$ , and unstable if ${\mathcal{R}}_0>1$ .

Proof

The model’s local stability can be analyzed using the Jacobian matrix of system (1), when evaluated at the DFE, $\mathcal{Q}{}_0$:

$$\left(\begin{array}{ccccccc}\hfill -\mu \hfill & \hfill -{\vartheta }_1\beta {}_C\hfill & \hfill -\beta {}_C\hfill & \hfill -{\vartheta }_2\beta {}_H\hfill & \hfill -\beta {}_H\hfill & \hfill -\beta {}_{CH}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\vartheta }_1\beta {}_C-{\mathcal{G}}_1\hfill & \hfill \beta {}_C\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\alpha }_1\hfill & \hfill -{\mathcal{G}}_2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\vartheta }_2\beta {}_H-{\mathcal{G}}_3\hfill & \hfill \beta {}_H\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\alpha }_2\hfill & \hfill -{\mathcal{G}}_4\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \beta {}_{CH}-{\mathcal{G}}_5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \xi {}_C\hfill & \hfill 0\hfill & \hfill \xi {}_H\hfill & \hfill \xi {}_{CH}\hfill & \hfill -\mu \hfill \end{array}\right).$$ (8)

The eigenvalues are given by

$${\lambda }_1=-\mu \left(\text{with multiplicity of 2}\right),$$ (9)

and the solutions of the equations

$${\lambda }^2+\left({\mathcal{G}}_1+{\mathcal{G}}_2-\beta {}_C\right)\lambda +\mu {\mathcal{G}}_1{\mathcal{G}}_2(1-{\mathcal{R}}_{0C})=0,$$ (10)

and

$${\lambda }^2+\left({\mathcal{G}}_3+{\mathcal{G}}_4-\beta {}_H\right)\lambda +\mu {\mathcal{G}}_3{\mathcal{G}}_4(1-{\mathcal{R}}_{0H})=0,\text{and}(\lambda +{\mathcal{G}}_5(1-{\mathcal{R}}_{0CH}))=0.$$ (11)

Using Routh Hurwitz stability criterion, the equations in (10), (11) possess all the roots whose real parts are negative if and only if ${\mathcal{R}}_{0C}<1$, ${\mathcal{R}}_{0H}<1$ and ${\mathcal{R}}_{0CH}<1$, respectively. Hence, the DFE, $\mathcal{Q}{}_0$ is locally asymptotically stable if ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}<1$.

Epidemiologically, it can be concluded from Theorem 3.3 that both infections can be eradicated from the population as long as the threshold quantity, ${\mathcal{R}}_0<1$ and if the initial sizes of the sub-populations of the model (1) are within the basin of attraction of the DFE ($\mathcal{Q}{}_0$). Therefore, a little inflow of SARS-CoV-2-infected or HBV-infected persons into the population will not cause large outbreak of the diseases, and both infections will recede over the time.

3.4. Endemic equilibria of the model

In this section, the endemic equilibria (boundary and co-existence) of the model system (1) are obtained.

3.4.1. Boundary endemic equilibria

When the reproduction number ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}>1$, the system (1) is associated with the following boundary endemic equilibria:

• 1.$E_1=({\mathcal{S}}_1^{\ast },\mathcal{A}{}_{C1}^{\ast },\mathcal{I}{}_{C1}^{\ast },0,0,0,{\mathcal{R}}_1^{\ast })$,
• ii.$E_2=({\mathcal{S}}_2^{\ast },0,0,\mathcal{A}{}_{H2}^{\ast },\mathcal{I}{}_{H2}^{\ast },0,{\mathcal{R}}_2^{\ast })$,
• iii.$E_3=({\mathcal{S}}_3^{\ast },0,0,0,0,\mathcal{I}{}_{CH3}^{\ast },{\mathcal{R}}_3^{\ast })$,

where

$${\mathcal{S}}_1^{\ast }=\frac{\Lambda [\mu ({\mathcal{G}}_2+{\alpha }_1)+\xi {}_C{\alpha }_1]}{\mu {\mathcal{G}}_1{\mathcal{G}}_2({\mathcal{R}}_{0C}-1)+\mu [\mu ({\mathcal{G}}_2+{\alpha }_1)+\xi {}_C{\alpha }_1]},\mathcal{A}{}_{C1}^{\ast }=\frac{\Lambda {\mathcal{G}}_2({\mathcal{R}}_{0C}-1)}{[{\mathcal{G}}_1{\mathcal{G}}_2({\mathcal{R}}_{0C}-1)+\mu ({\mathcal{G}}_2+{\alpha }_1)+\xi {}_C{\alpha }_1]},$$$$\mathcal{I}{}_{C1}^{\ast }=\frac{{\alpha }_1\Lambda ({\mathcal{R}}_{0C}-1)}{[{\mathcal{G}}_1{\mathcal{G}}_2({\mathcal{R}}_{0C}-1)+\mu ({\mathcal{G}}_2+{\alpha }_1)+\xi {}_C{\alpha }_1]},{\mathcal{R}}_1^{\ast }=\frac{\xi {}_C{\alpha }_1\Lambda ({\mathcal{R}}_{0C}-1)}{\mu [{\mathcal{G}}_1{\mathcal{G}}_2({\mathcal{R}}_{0C}-1)+\mu ({\mathcal{G}}_2+{\alpha }_1)+\xi {}_C{\alpha }_1]}$$$${\mathcal{S}}_2^{\ast }=\frac{\Lambda [\mu ({\mathcal{G}}_4+{\alpha }_2)+\xi {}_C{\alpha }_2]}{\mu {\mathcal{G}}_3{\mathcal{G}}_4({\mathcal{R}}_{0H}-1)+\mu [\mu ({\mathcal{G}}_4+{\alpha }_2)+\xi {}_H{\alpha }_2]},\mathcal{A}{}_{H2}^{\ast }=\frac{\Lambda {\mathcal{G}}_4({\mathcal{R}}_{0H}-1)}{[{\mathcal{G}}_3{\mathcal{G}}_4({\mathcal{R}}_{0H}-1)+\mu ({\mathcal{G}}_4+{\alpha }_2)+\xi {}_H{\alpha }_2]},$$$$\mathcal{I}{}_{H1}^{\ast }=\frac{{\alpha }_2\Lambda ({\mathcal{R}}_{0H}-1)}{[{\mathcal{G}}_3{\mathcal{G}}_4({\mathcal{R}}_{0H}-1)+\mu ({\mathcal{G}}_4+{\alpha }_2)+\xi {}_H{\alpha }_2]},{\mathcal{R}}_2^{\ast }=\frac{\xi {}_H{\alpha }_2\Lambda ({\mathcal{R}}_{0H}-1)}{\mu [{\mathcal{G}}_3{\mathcal{G}}_4({\mathcal{R}}_{0H}-1)+\mu ({\mathcal{G}}_4+{\alpha }_2)+\xi {}_H{\alpha }_2]}$$$${\mathcal{S}}_3^{\ast }=\frac{\Lambda (\mu +\xi {}_{CH})}{\mu [{\mathcal{G}}_5({\mathcal{R}}_{0CH}-1)+(\mu +\xi {}_{CH})]},{\mathcal{I}}_{CH3}^{\ast }=\frac{\Lambda ({\mathcal{R}}_{0CH}-1)}{{\mathcal{G}}_5({\mathcal{R}}_{0CH}-1)+(\mu +\xi {}_{CH})},{\mathcal{R}}_3^{\ast }=\frac{\Lambda \xi {}_{CH}({\mathcal{R}}_{0CH}-1)}{\mu [{\mathcal{G}}_5({\mathcal{R}}_{0CH}-1)+(\mu +\xi {}_{CH})]}.$$ (12)

3.4.2. Co-existence endemic equilibria

In this subsection, existence of co-existence endemic equilibria of the model (1) is proved for a special case when individuals infected with SARS-CoV-2 (in symptomatic stage) and those infected with chronic HBV do not get additional infection with a different disease (that is, ${\zeta }_1={\zeta }_2=0$). The complete system is not considered analytically for co-existence of endemic equilibria due to its strong nonlinearity.

We now establish the following result.

Theorem 3.4

The special case of system (1) with ${\zeta }_1={\zeta }_2=0$ has a family of co-existence endemic equilibria as long as the associated reproduction numbers are greater than one; that is, when ${\mathcal{R}}_{0C}>1,{\mathcal{R}}_{0H}>1$ , and ${\mathcal{R}}_{0CH}>1$ .

Proof

If we let the three forces of infection at steady state be:

$$\lambda {}_C^{\ast }=\frac{\beta {}_C\mathcal{I}{}_C^{\ast }}{{\mathcal{N}}^{\ast }},\lambda {}_H^{\ast }=\frac{\beta {}_H\mathcal{I}{}_H^{\ast }}{{\mathcal{N}}^{\ast }},\lambda {}_{CH}^{\ast }=\frac{\beta {}_{CH}\mathcal{I}{}_{CH}^{\ast }}{{\mathcal{N}}^{\ast }},$$ (13)

the system (1) will possess co-existence endemic equilibria $E^{\ast }=({\mathcal{S}}^{\ast },\mathcal{A}{}_C^{\ast },\mathcal{I}{}_C^{\ast },\mathcal{A}{}_H^{\ast },\mathcal{I}{}_H^{\ast },\mathcal{I}{}_{CH}^{\ast },{\mathcal{R}}^{\ast })$, where

$${\mathcal{S}}^{\ast }=\frac{\Lambda }{({\lambda }_C^{\ast }+{\lambda }_H^{\ast }+\lambda {}_{CH}^{\ast }+\mu )},\mathcal{A}{}_C^{\ast }=\frac{\Lambda \lambda {}_C^{\ast }}{{\mathcal{G}}_1({\lambda }_C^{\ast }+{\lambda }_H^{\ast }+\lambda {}_{CH}^{\ast }+\mu )},\mathcal{I}{}_C^{\ast }=\frac{\Lambda {\alpha }_1\lambda {}_C^{\ast }}{{\mathcal{G}}_1{\mathcal{G}}_2({\lambda }_C^{\ast }+{\lambda }_H^{\ast }+\lambda {}_{CH}^{\ast }+\mu )}$$$$\mathcal{A}{}_H^{\ast }=\frac{\Lambda \lambda {}_H^{\ast }}{{\mathcal{G}}_3({\lambda }_C^{\ast }+{\lambda }_H^{\ast }+\lambda {}_{CH}^{\ast }+\mu )},\mathcal{I}{}_H^{\ast }=\frac{\Lambda {\alpha }_2\lambda {}_H^{\ast }}{{\mathcal{G}}_3{\mathcal{G}}_4({\lambda }_C^{\ast }+{\lambda }_H^{\ast }+\lambda {}_{CH}^{\ast }+\mu )},\mathcal{I}{}_{CH}^{\ast }=\frac{\Lambda \lambda {}_{CH}^{\ast }}{{\mathcal{G}}_5({\lambda }_C^{\ast }+{\lambda }_H^{\ast }+\lambda {}_{CH}^{\ast }+\mu )}$$$${\mathcal{R}}^{\ast }=\frac{\Lambda (\xi {}_C{\alpha }_1{\mathcal{G}}_3{\mathcal{G}}_4{\mathcal{G}}_5\lambda {}_C^{\ast }+\xi {}_H{\alpha }_2{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_5\lambda {}_H^{\ast }+\xi {}_{CH}{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_3{\mathcal{G}}_4\lambda {}_{CH}^{\ast })}{\mu {\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_3{\mathcal{G}}_4{\mathcal{G}}_5({\lambda }_C^{\ast }+{\lambda }_H^{\ast }+\lambda {}_{CH}^{\ast }+\mu )}.$$ (14)

Adding the total populations at steady state, we have

$${\mathcal{N}}^{\ast }=\frac{{\mathcal{G}}_3{\mathcal{G}}_4{\mathcal{G}}_5[\mu ({\mathcal{G}}_2+{\alpha }_1)+\xi {}_C{\alpha }_1]\lambda {}_C^{\ast }+{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_5[\mu ({\mathcal{G}}_4+{\alpha }_2)+\xi {}_H{\alpha }_2]\lambda {}_H^{\ast }+{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_3{\mathcal{G}}_4(\xi {}_{CH}+\mu )\lambda {}_{CH}^{\ast }}{\mu {\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_3{\mathcal{G}}_4{\mathcal{G}}_5(\lambda {}_C^{\ast }+\lambda {}_H^{\ast }+\lambda {}_{CH}^{\ast }+\mu )}.$$

Substituting the steady state solutions into the forces of infection (13) and simplifying, we obtain the following equations:

$${\mathcal{G}}_3{\mathcal{G}}_4{\mathcal{G}}_5[\mu ({\mathcal{G}}_2+{\alpha }_1)+\xi {}_C{\alpha }_1]\lambda {}_C^{\ast }+{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_5[\mu ({\mathcal{G}}_4+{\alpha }_2)+\xi {}_H{\alpha }_2]\lambda {}_H^{\ast }+{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_3{\mathcal{G}}_4(\xi {}_{CH}+\mu )\lambda {}_{CH}^{\ast }=\mu {\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_3{\mathcal{G}}_4{\mathcal{G}}_5\left({\mathcal{R}}_{0C}-1\right),$$$${\mathcal{G}}_3{\mathcal{G}}_4{\mathcal{G}}_5[\mu ({\mathcal{G}}_2+{\alpha }_1)+\xi {}_C{\alpha }_1]\lambda {}_C^{\ast }+{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_5[\mu ({\mathcal{G}}_4+{\alpha }_2)+\xi {}_H{\alpha }_2]\lambda {}_H^{\ast }+{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_3{\mathcal{G}}_4(\xi {}_{CH}+\mu )\lambda {}_{CH}^{\ast }=\mu {\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_3{\mathcal{G}}_4{\mathcal{G}}_5\left({\mathcal{R}}_{0H}-1\right),$$$${\mathcal{G}}_3{\mathcal{G}}_4{\mathcal{G}}_5[\mu ({\mathcal{G}}_2+{\alpha }_1)+\xi {}_C{\alpha }_1]\lambda {}_C^{\ast }+{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_5[\mu ({\mathcal{G}}_4+{\alpha }_2)+\xi {}_H{\alpha }_2]\lambda {}_H^{\ast }+{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_3{\mathcal{G}}_4(\xi {}_{CH}+\mu )\lambda {}_{CH}^{\ast }=\mu {\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{G}}_3{\mathcal{G}}_4{\mathcal{G}}_5\left({\mathcal{R}}_{0CH}-1\right).$$ (15)

Thus, the system will have a co-existence endemic equilibria (the case when SARS-CoV-2, HBV and their co-infection persist in the population) when all associated reproduction numbers are greater than one; that is, when ${\mathcal{R}}_{0C}>1,{\mathcal{R}}_{0H}>1$ and ${\mathcal{R}}_{0CH}>1$.

3.5. Backward bifurcation analysis of the model (1)

The backward bifurcation property that has been investigated in several epidemic models, is typically characterized by the co-existence of a stable disease free equilibrium and a stable endemic equilibrium when the associated reproduction number of the model is less than unity. Epidemiologically, the implication of the backward bifurcation phenomenon of system (1) is; the usual condition of having the model’s reproduction number ${\mathcal{R}}_0$ less than unity is not sufficient for the control of both infections (though necessary). The following result is proved by adopting the approach of Castillo-Chavez and Song [53]. We shall now investigate this important dynamical property for the model (1) in the presence and absence of incident co-infection (the case when a susceptible individual gets infected with both SARS-CoV-2 and HBV concurrently).

3.5.1. Backward bifurcation analysis in the absence of incident co-infection ($\beta {}_{CH}=0$)

For a special case when incident co-infection is not considered in the model (1), we prove the following result:

Theorem 3.5

The model (1) in the absence of incident co-infection, that is, with $\beta {}_{CH}=0$ will not undergo backward bifurcation

Proof

Let us consider the following change of variables:

Let

$$\mathcal{S}=x_1,\mathcal{A}{}_C=x_2,\mathcal{I}{}_C=x_3,\mathcal{A}{}_H=x_4,\mathcal{I}{}_H=x_5,\mathcal{I}{}_{CH}=x_6,\mathcal{R}=x_7,$$

so that system (1) can be re-written in the form

$$\frac{d{x}_1}{dt}=\Lambda -\frac{\beta {}_C({\vartheta }_1{x}_2+x_3)}{\mathcal{N}}{x}_1-\frac{\beta {}_H({\vartheta }_2{x}_4+x_5)}{\mathcal{N}}{x}_1-\mu x_1:f_1,$$$$\frac{d{x}_2}{dt}=\frac{\beta {}_C({\vartheta }_1{x}_2+x_3)}{\mathcal{N}}{x}_1-({\alpha }_1+\mu )x_2:f_2,$$$$\frac{d{x}_3}{dt}={\alpha }_1{x}_2-(\xi {}_C+\eta {}_C+\mu )x_3-{\zeta }_1\frac{\beta {}_H({\vartheta }_2{x}_4+x_5)}{\mathcal{N}}{x}_3:f_3,$$$$\frac{d{x}_4}{dt}=\frac{\beta {}_H({\vartheta }_2{x}_4+x_5)}{\mathcal{N}}{x}_1-({\alpha }_2+\mu )x_4:f_4,$$$$\frac{d{x}_5}{dt}={\alpha }_2{x}_4-(\xi {}_H+\eta {}_H+\mu )x_5-{\zeta }_2\frac{\beta {}_C({\vartheta }_1{x}_2+x_3)}{\mathcal{N}}{x}_5:f_5,$$$$\frac{d{x}_6}{dt}={\zeta }_1\frac{\beta {}_H({\vartheta }_2{x}_4+x_5)}{\mathcal{N}}{x}_3+{\zeta }_2\frac{\beta {}_C({\vartheta }_1{x}_2+x_3)}{\mathcal{N}}{x}_5-(\xi {}_{CH}+\eta {}_{CH}+\mu )x_6:f_6,$$$$\frac{d{x}_7}{dt}=\xi {}_C{x}_3+\xi {}_H{x}_5+\xi {}_{CH}{x}_6-\mu x_7:f_7,$$ (16)

where,

$$\mathcal{N}=\sum _{i=1}^7{x}_i.$$

Consider the case when ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}=1$. In addition, let $\beta {}_C$ be selected as a bifurcation parameter. Solving for $\beta {}_C=\beta {}_C^{\ast }$ from ${\mathcal{R}}_{0C}=1$ gives that

$$\beta {}_C=\beta {}_C^{\ast }=\frac{{\mathcal{G}}_1{\mathcal{G}}_2}{{\vartheta }_1{\mathcal{G}}_2+{\alpha }_1}.$$

If the Jacobian matrix of system (16) is evaluated at the DFE, $J\left({\mathcal{Q}}_0\right)$, then we obtain that

$$\left(\begin{array}{ccccccc}\hfill -\mu \hfill & \hfill -{\vartheta }_1\beta {}_C\hfill & \hfill -\beta {}_C\hfill & \hfill -{\vartheta }_2\beta {}_H\hfill & \hfill -\beta {}_H\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\vartheta }_1\beta {}_C-{\mathcal{G}}_1\hfill & \hfill \beta {}_C\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\alpha }_1\hfill & \hfill -{\mathcal{G}}_2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\vartheta }_2\beta {}_H-{\mathcal{G}}_3\hfill & \hfill \beta {}_H\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\alpha }_2\hfill & \hfill -{\mathcal{G}}_4\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mathcal{G}}_5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \xi {}_C\hfill & \hfill 0\hfill & \hfill \xi {}_H\hfill & \hfill \xi {}_{CH}\hfill & \hfill -\mu \hfill \end{array}\right).$$ (17)

Note that $J\left({\mathcal{Q}}_0\right)$ has a right eigenvector (linked with the zero eigenvalue of $J\left({\mathcal{Q}}_0\right)$) given by $\mathit{\psi }={\left[{\psi }_1,{\psi }_2,{\psi }_3,\dots ,{\psi }_7\right]}^T$ , where the components are

$$\psi {}_1=-\frac{1}{\mu }\left[\beta {}_C({\vartheta }_1{\psi }_2+{\psi }_3)+\beta {}_H({\vartheta }_2{\psi }_4+{\psi }_5)\right]<0,\psi {}_2={\psi }_2>0,$$

$$\psi {}_3=\frac{{\alpha }_1{\psi }_2}{{\mathcal{G}}_2}>0,\psi {}_4=\psi {}_4>0,\psi {}_5=\frac{{\alpha }_2{\psi }_4}{{\mathcal{G}}_4}>0,$$

$$\psi {}_6=0,\psi {}_7=\frac{1}{\mu }\left[\xi {}_C{\psi }_3+\xi {}_H{\psi }_5\right]>0.$$

The components of the left eigenvector of $J\left({\mathcal{Q}}_0\right){|}_{\beta {}_C=\beta {}_C^{\ast }}$, $\mathit{\delta }=[{\delta }_1,{\delta }_2,\dots ,{\delta }_7]$, satisfying $\mathit{\psi .\delta }=1$ are

$$\delta {}_2=\delta {}_2>0,\delta {}_3=\frac{\beta {}_C{\delta }_2}{{\mathcal{G}}_2}>0,\delta {}_4=\delta {}_4>0,\delta {}_5=\frac{\beta {}_H{\delta }_4}{{\mathcal{G}}_4}>0,{\delta }_6=0.$$

By evaluating the non-zero partial derivatives of $f\left(x\right)$ (at the DFE, (${\mathcal{Q}}_0$)), the bifurcation coefficients defined by

$$a=\sum _{k,i,j=1}^7{\delta }_k{\psi }_i{\psi }_j\frac{{\partial }^2{f}_k}{\partial x_i\partial x_j}(0,0)\text{and}b=\sum _{k,i=1}^7{\delta }_k{\psi }_i\frac{{\partial }^2{f}_k}{\partial x_i\partial \beta {}_C^{\ast }}(0,0),$$

are given by

$$a=-\frac{2[\beta {}_C({\vartheta }_1\psi {}_3+\psi {}_2)\delta {}_2+\beta {}_H({\vartheta }_2\psi {}_4+\psi {}_5)\delta {}_4]}{{\mathcal{N}}^{\ast }}(\psi {}_2+\psi {}_3+\psi {}_4+\psi {}_5+\psi {}_7)-\frac{2{\zeta }_1\beta {}_H({\vartheta }_1\psi {}_3+\psi {}_2){\psi }_3}{{\mathcal{N}}^{\ast }}{\delta }_3-\frac{2{\zeta }_2\beta {}_C({\vartheta }_2\psi {}_2+\psi {}_3){\psi }_5}{{\mathcal{N}}^{\ast }}{\delta }_5<0,$$ (18)

and

$$b=\sum _{k,i=1}^7{\delta }_k{\psi }_i\frac{{\partial }^2{f}_k}{\partial x_i\partial \beta {}_C^{\ast }}(0,0)=({\vartheta }_1\psi {}_2+\psi {}_3)\delta {}_2>0.$$

Based on Theorem 4.1 in [53], the model (1), with $\beta {}_{CH}=0$ does not undergo backward bifurcation, since the coefficient $a<0$.

3.5.2. Backward bifurcation analysis in the presence of incident co-infection ($\beta {}_{CH}\ne 0$)

If incident co-infection is assumed in the model (1), then we have the following result:

Theorem 3.6

The model (1) in the presence of incident co-infection, that is, with $\beta {}_{CH}\ne 0$ , undergoes backward bifurcation as long as the coefficient $a$ , given by

$$a=-\frac{2[\beta {}_C({\vartheta }_1\psi {}_3+\psi {}_2)\delta {}_2+\beta {}_H({\vartheta }_2\psi {}_4+\psi {}_5)\delta {}_4+\beta {}_{CH}\psi {}_6\delta {}_6]}{{\mathcal{N}}^{\ast }}(\psi {}_2+\psi {}_3+\psi {}_4+\psi {}_5+\psi {}_6+\psi {}_7)-\frac{2{\zeta }_1\beta {}_H({\vartheta }_1\psi {}_3+\psi {}_2){\psi }_3}{{\mathcal{N}}^{\ast }}({\delta }_3-{\delta }_6)-\frac{2{\zeta }_2\beta {}_C({\vartheta }_2\psi {}_2+\psi {}_3){\psi }_5}{{\mathcal{N}}^{\ast }}({\delta }_5-{\delta }_6),$$

is positive.

Proof

If

$$\mathcal{S}=x_1,\mathcal{A}{}_C=x_2,\mathcal{I}{}_C=x_3,\mathcal{A}{}_H=x_4,\mathcal{I}{}_H=x_5,\mathcal{I}{}_{CH}=x_6,\mathcal{R}=x_7,$$

then the model (1) can be re-presented in the following form

$$\frac{d{x}_1}{dt}=\Lambda -\frac{\beta {}_C({\vartheta }_1{x}_2+x_3)}{\mathcal{N}}{x}_1-\frac{\beta {}_H({\vartheta }_2{x}_4+x_5)}{\mathcal{N}}{x}_1-\frac{\beta {}_{CH}{x}_6}{\mathcal{N}}{x}_1-\mu x_1:f_1,$$$$\frac{d{x}_2}{dt}=\frac{\beta {}_C({\vartheta }_1{x}_2+x_3)}{\mathcal{N}}{x}_1-({\alpha }_1+\mu )x_2:f_2,$$$$\frac{d{x}_3}{dt}={\alpha }_1{x}_2-(\xi {}_C+\eta {}_C+\mu )x_3-{\zeta }_1\frac{\beta {}_H({\vartheta }_2{x}_4+x_5)}{\mathcal{N}}{x}_3:f_3,$$$$\frac{d{x}_4}{dt}=\frac{\beta {}_H({\vartheta }_2{x}_4+x_5)}{\mathcal{N}}{x}_1-({\alpha }_2+\mu )x_4:f_4,$$$$\frac{d{x}_5}{dt}={\alpha }_2{x}_4-(\xi {}_H+\eta {}_H+\mu )x_5-{\zeta }_2\frac{\beta {}_C({\vartheta }_1{x}_2+x_3)}{\mathcal{N}}{x}_5:f_5,$$$$\frac{d{x}_6}{dt}=\frac{\beta {}_{CH}{x}_6}{\mathcal{N}}{x}_1+{\zeta }_1\frac{\beta {}_H({\vartheta }_2{x}_4+x_5)}{\mathcal{N}}{x}_3+{\zeta }_2\frac{\beta {}_C({\vartheta }_1{x}_2+x_3)}{\mathcal{N}}{x}_5-(\xi {}_{CH}+\eta {}_{CH}+\mu )x_6:f_6,$$$$\frac{d{x}_7}{dt}=\xi {}_C{x}_3+\xi {}_H{x}_5+\xi {}_{CH}{x}_6-\mu x_7:f_7,$$ (19)

where,

$$\mathcal{N}=\sum _{i=1}^7{x}_i.$$

Consider the case when ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}=1$. In addition, let $\beta {}_{CH}$ be selected as a bifurcation parameter. Solving for $\beta {}_{CH}=\beta {}_{CH}^{\ast }$ from ${\mathcal{R}}_{0CH}=1$ gives that

$$\beta {}_{CH}=\beta {}_{CH}^{\ast }={\mathcal{G}}_5.$$

If the Jacobian matrix of system (19) is evaluated at the DFE, $J\left({\mathcal{Q}}_0\right)$, then $J\left({\mathcal{Q}}_0\right)$ would possess the right eigenvector $\mathit{\psi }={\left[{\psi }_1,{\psi }_2,{\psi }_3,\dots ,{\psi }_7\right]}^T$, where its components are given by:

$$\psi {}_1=-\frac{1}{\mu }\left[\beta {}_C({\vartheta }_1{\psi }_2+{\psi }_3)+\beta {}_H({\vartheta }_2{\psi }_4+{\psi }_5)+\beta {}_{CH}{\psi }_6\right]<0,\psi {}_2={\psi }_2>0,$$

$$\psi {}_3=\frac{{\alpha }_1{\psi }_2}{{\mathcal{G}}_2}>0,\psi {}_4=\psi {}_4>0,\psi {}_5=\frac{{\alpha }_2{\psi }_4}{{\mathcal{G}}_4}>0,$$

$$\psi {}_6=\psi {}_6>0,\psi {}_7=\frac{1}{\mu }\left[\xi {}_C{\psi }_3+\xi {}_H{\psi }_5+\xi {}_{CH}{\psi }_6\right]>0.$$

The left eigenvector components of $J\left({\mathcal{Q}}_0\right){|}_{\beta {}_{CH}=\beta {}_{CH}^{\ast }}$, $\mathit{\delta }=[{\delta }_1,{\delta }_2,\dots ,{\delta }_7]$, satisfying $\mathit{\psi .\delta }=1$ are

$$\delta {}_2=\delta {}_2>0,\delta {}_3=\frac{\beta {}_C{\delta }_2}{{\mathcal{G}}_2}>0,\delta {}_4=\delta {}_4>0,\delta {}_5=\frac{\beta {}_H{\delta }_4}{{\mathcal{G}}_4}>0,{\delta }_6={\delta }_6>0.$$

By evaluating non-zero partial derivatives of $f\left(x\right)$ (at the DFE, (${\mathcal{Q}}_0$)), the bifurcation coefficients defined by

$$a=\sum _{k,i,j=1}^7{\delta }_k{\psi }_i{\psi }_j\frac{{\partial }^2{f}_k}{\partial x_i\partial x_j}(0,0)\text{and}b=\sum _{k,i=1}^7{\delta }_k{\psi }_i\frac{{\partial }^2{f}_k}{\partial x_i\partial \beta {}_{CH}^{\ast }}(0,0),$$

are given by

$$a=-\frac{2[\beta {}_C({\vartheta }_1\psi {}_3+\psi {}_2)\delta {}_2+\beta {}_H({\vartheta }_2\psi {}_4+\psi {}_5)\delta {}_4+\beta {}_{CH}\psi {}_6\delta {}_6]}{{\mathcal{N}}^{\ast }}(\psi {}_2+\psi {}_3+\psi {}_4+\psi {}_5+\psi {}_6+\psi {}_7)-\frac{2{\zeta }_1\beta {}_H({\vartheta }_1\psi {}_3+\psi {}_2){\psi }_3}{{\mathcal{N}}^{\ast }}({\delta }_3-{\delta }_6)-\frac{2{\zeta }_2\beta {}_C({\vartheta }_2\psi {}_2+\psi {}_3){\psi }_5}{{\mathcal{N}}^{\ast }}({\delta }_5-{\delta }_6),$$ (20)

and

$$b=\sum _{k,i=1}^7{\delta }_k{\psi }_i\frac{{\partial }^2{f}_k}{\partial x_i\partial \beta {}_{CH}^{\ast }}(0,0)=\psi {}_6\delta {}_6>0.$$

Based on Theorem 4.1 in [53], the model (1) with incident co-infection $\beta {}_{CH}\ne 0$, undergoes backward bifurcation when $a>0$. Thus, the incident co-infection with both SARS-CoV-2 and HBV could trigger the phenomenon of backward bifurcation in the model (1). Also, it is worthmentioning that with the incident co-infection in place, setting the modification parameters responsible for infection with a second disease: ${\zeta }_1={\zeta }_2=0$, the bifurcation coefficient $a$ becomes negative which rules out the occurrence of a backward bifurcation.

In the following subsections, we shall establish the global asymptotic stability of both the infection-free and endemic equilibria of the model (1), for a special case when ${\zeta }_1={\zeta }_2=0$ (the cause of backward bifurcation is removed).

3.6. Global asymptotic stability (GAS) of disease free and endemic equilibria of the model (1)

In order to establish the global stability of an epidemic system, one of the most effective methods is the direct Lyapunov method [54] which requires an auxiliary function $\mathcal{L}\left(X\right)$ (say), with $X\in {\mathbb{R}}^n$, defined on a neighborhood $U$ of the origin and satisfies the following properties:

• (i.)$\mathcal{L}\left(X\right)>0$, for all $X\in U\setminus \left\{\mathbf{0}\right\}$,
• (ii.)$\mathcal{L}\left(\mathbf{0}\right)=0$,
• (iii.)$\frac{d\mathcal{L}}{dt}\le 0$, for all $X\in U$.

3.6.1. GAS of disease free equilibrium (DFE)

The following result deals with global asymptotic stability of DFE.

Theorem 3.7

In the absence of additional infection by individuals singly infected with SARS-CoV-2 (in symptomatic stage) or chronic HBV (i.e. ${\zeta }_1={\zeta }_2=0$ ), the DFE of the model (1) , given by ${\mathcal{Q}}_0$ , is GAS in $\mathcal{D}$ whenever ${\mathcal{R}}_0\le 1$ .

Proof

Consider the following function as a candidate for Lyapunov function

$${\mathcal{L}}_1=\frac{({\mathcal{G}}_2+{\alpha }_1)}{{\mathcal{G}}_1{\mathcal{G}}_2}\mathcal{A}{}_C+\frac{1}{{\mathcal{G}}_1}\mathcal{I}{}_C+\frac{({\mathcal{G}}_4+{\alpha }_2)}{{\mathcal{G}}_3{\mathcal{G}}_4}\mathcal{A}{}_H+\frac{1}{{\mathcal{G}}_3}\mathcal{I}{}_H+\frac{1}{{\mathcal{G}}_5}\mathcal{I}{}_{CH}$$

with,

$${\dot{\mathcal{L}}}_1=\frac{({\mathcal{G}}_2+{\alpha }_1)}{{\mathcal{G}}_1{\mathcal{G}}_2}\left(\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}-({\alpha }_1+\mu )\mathcal{A}{}_C\right)+\frac{1}{{\mathcal{G}}_1}\left({\alpha }_1\mathcal{A}{}_C-(\xi {}_C+\eta {}_C+\mu )\mathcal{I}{}_C-{\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C\right)+\frac{({\mathcal{G}}_4+{\alpha }_2)}{{\mathcal{G}}_3{\mathcal{G}}_4}\left(\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}-({\alpha }_2+\mu )\mathcal{A}{}_H\right)+\frac{1}{{\mathcal{G}}_3}\left({\alpha }_2\mathcal{A}{}_H-(\xi {}_H+\eta {}_H+\mu )\mathcal{I}{}_H\right)+\frac{1}{{\mathcal{G}}_5}\left(\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}-(\xi {}_{CH}+\eta {}_{CH}+\mu )\mathcal{I}{}_{CH}\right),$$

which can be further simplified into

$${\dot{\mathcal{L}}}_1=\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)({\vartheta }_1{\mathcal{G}}_2+{\alpha }_1)}{{\mathcal{G}}_1{\mathcal{G}}_2\mathcal{N}}S-{\vartheta }_1\mathcal{A}{}_C-\mathcal{I}{}_C+\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)({\vartheta }_2{\mathcal{G}}_4+{\alpha }_2)}{{\mathcal{G}}_3{\mathcal{G}}_4\mathcal{N}}S-{\vartheta }_2\mathcal{A}{}_H-\mathcal{I}{}_H+\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{{\mathcal{G}}_5\mathcal{N}}\mathcal{S}-\mathcal{I}{}_{CH}.$$

Simplifying further (noting that, $\frac{\mathcal{S}}{\mathcal{N}}<1$), we have

$${\dot{\mathcal{L}}}_1\le \left(\frac{\beta {}_C({\vartheta }_1{\mathcal{G}}_2+{\alpha }_1)}{{\mathcal{G}}_1{\mathcal{G}}_2}-1\right)\left({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C\right)+\left(\frac{\beta {}_H({\vartheta }_2{\mathcal{G}}_4+{\alpha }_2)}{{\mathcal{G}}_3{\mathcal{G}}_4}-1\right)\left({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H\right)+\left(\frac{\beta {}_{CH}}{{\mathcal{G}}_5}-1\right)\mathcal{I}{}_{CH}=\left({\mathcal{R}}_{0C}-1\right)\left({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C\right)+\left({\mathcal{R}}_{0H}-1\right)\left({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H\right)+\left({\mathcal{R}}_{0CH}-1\right)\mathcal{I}{}_{CH}.$$

Note that the model parameters and variables are non-negative, it is observed that ${\dot{\mathcal{L}}}_1\le 0$ for ${\mathcal{R}}_0\le 1$, and ${\dot{\mathcal{L}}}_1=0$ if and only if $\mathcal{A}{}_C=\mathcal{I}{}_{HF}=\mathcal{A}{}_H=\mathcal{I}{}_H=\mathcal{I}{}_{CH}=0$. Hence, ${\mathcal{L}}_1$ is a candidate for a Lyapunov function on $\mathcal{D}$. By La Salle’s Invariance Principle [54], $\mathcal{A}{}_C\to 0,\mathcal{I}{}_C\to 0,\mathcal{A}{}_H\to 0,\mathcal{I}{}_H\to 0\text{and}\mathcal{I}{}_{CH}\to 0$ as $t\to \infty$. Substituting $\mathcal{A}{}_C=\mathcal{I}{}_C=\mathcal{A}{}_H=\mathcal{I}{}_H=\mathcal{I}{}_{CH}=0$ in (1) shows that $\mathcal{R}\to 0,\mathcal{S}\to {\mathcal{S}}^{\ast }$, as $t\to \infty$. Thus, every solution to the system of Eqs. (1) with ${\zeta }_1={\zeta }_2=0$, and with initial conditions in $\mathcal{D}$, converges to the DFE as $t\to \infty$ whenever ${\mathcal{R}}_0\le 1$. The epidemiological consequence of the above theorem is that if individuals in symptomatic stage of SARS-CoV-2 infection and those infected with chronic HBV do not get additional infection with a different disease, then both infections can be eliminated as long as the threshold, ${\mathcal{R}}_0\le 1$, regardless of the initial population sizes.

3.6.2. GAS of endemic equilibrium (EE)

The global asymptotic stability of EE of the model is discussed in the following result.

Theorem 3.8

In the absence of additional infection by individuals singly infected with SARS-CoV-2 (in symptomatic stage) or chronic HBV (i.e. ${\zeta }_1={\zeta }_2=0$ ), the endemic equilibrium, ${\mathcal{Q}}_e$ , of the model (1) , with negligible disease-induced death rates (that is, $\eta {}_C=\eta {}_H=\eta {}_{CH}=0$ ) is globally asymptotically stable (GAS) in $\mathcal{D}⧵{\mathcal{D}}_0$ whenever ${\overline{\mathcal{R}}}_0={\mathcal{R}}_0{|}_{\eta {}_C=\eta {}_H=\eta {}_{CH}=0}>1$ , where

$${\mathcal{D}}_0=.\left\{(\mathcal{S},\mathcal{A}{}_C,\mathcal{I}{}_C,\mathcal{A}{}_H,\mathcal{I}{}_H,\mathcal{I}{}_{CH},\mathcal{R})\in \mathcal{D}:\mathcal{A}{}_C=\mathcal{I}{}_C=\mathcal{A}{}_H=\mathcal{I}{}_H=\mathcal{I}{}_{CH}=0.\right\}.$$

Proof

Consider the model (1) with $\eta {}_C=\eta {}_H=\eta {}_{CH}=0(\text{so that}\mathcal{N}=\frac{\Lambda }{\mu },\overline{\beta }{}_C=\frac{\mu \beta {}_C}{\Lambda },\overline{\beta }{}_H=\frac{\mu \beta {}_H}{\Lambda },\overline{\beta }{}_{CH}=\frac{\mu \beta {}_{CH}}{\Lambda })$  and ${\mathcal{R}}_0>1$.

Define the following function as a candidate for a Lyapunov function:

$${\mathcal{L}}_2=\mathcal{S}-{\mathcal{S}}^{\ast }-{\mathcal{S}}^{\ast }ln\left(\frac{\mathcal{S}}{{\mathcal{S}}^{\ast }}\right)+\mathcal{A}{}_C-\mathcal{A}{}_C^{\ast }-\mathcal{A}{}_C^{\ast }ln\left(\frac{\mathcal{A}{}_C}{\mathcal{A}{}_C^{\ast }}\right)+\mathcal{A}{}_H-\mathcal{A}{}_H^{\ast }-\mathcal{A}{}_H^{\ast }ln\left(\frac{\mathcal{A}{}_H}{\mathcal{A}{}_H^{\ast }}\right)+\frac{\overline{\beta }{}_C{\mathcal{S}}^{\ast }}{{\mathcal{G}}_2}\left[\mathcal{I}{}_C-\mathcal{I}{}_C^{\ast }-\mathcal{I}{}_C^{\ast }ln\left(\frac{\mathcal{I}{}_C}{\mathcal{I}{}_C^{\ast }}\right)\right]+\frac{\overline{\beta }{}_H{\mathcal{S}}^{\ast }}{{\mathcal{G}}_4}\left[\mathcal{I}{}_H-\mathcal{I}{}_H^{\ast }-\mathcal{I}{}_H^{\ast }ln\left(\frac{\mathcal{I}{}_H}{\mathcal{I}{}_H^{\ast }}\right)\right]+\left[\mathcal{I}{}_{CH}-\mathcal{I}{}_{CH}^{\ast }-\mathcal{I}{}_{CH}^{\ast }ln\left(\frac{\mathcal{I}{}_{CH}}{\mathcal{I}{}_{CH}^{\ast }}\right)\right],$$

wirh time derivative is given by,

$${\dot{\mathcal{L}}}_2=\left(1-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}\right)\dot{\mathcal{S}}+\left(1-\frac{\mathcal{A}{}_C^{\ast }}{\mathcal{A}{}_C}\right)\dot{\mathcal{A}}{}_C+\left(1-\frac{\mathcal{A}{}_H^{\ast }}{\mathcal{A}{}_H}\right)\dot{\mathcal{A}}{}_H+\frac{\overline{\beta }{}_C{\mathcal{S}}^{\ast }}{{\mathcal{G}}_2}\left(1-\frac{\mathcal{I}{}_C^{\ast }}{\mathcal{I}{}_C}\right)\dot{\mathcal{I}}{}_C+\frac{\overline{\beta }{}_H{\mathcal{S}}^{\ast }}{{\mathcal{G}}_4}\left(1-\frac{\mathcal{I}{}_H^{\ast }}{\mathcal{I}{}_H}\right)\dot{\mathcal{I}}{}_H+\left(1-\frac{\mathcal{I}{}_{CH}^{\ast }}{\mathcal{I}{}_{CH}}\right)\dot{\mathcal{I}}{}_{CH}.$$ (21)

If the derivatives in system (1) are substituted into ${\dot{\mathcal{L}}}_2$, then it becomes

$${\dot{\mathcal{L}}}_2=\left(1-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}\right)\left[\Lambda -\overline{\beta }{}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{S}-\overline{\beta }{}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{S}-\overline{\beta }{}_{CH}\mathcal{I}{}_{CH}\mathcal{S}-\mu \mathcal{S}\right]+\left(1-\frac{\mathcal{A}{}_C^{\ast }}{\mathcal{A}{}_C}\right)\left[\overline{\beta }{}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{S}-{\mathcal{G}}_1\mathcal{A}{}_C\right]+\left(1-\frac{\mathcal{A}{}_H^{\ast }}{\mathcal{A}{}_H}\right)\left[\overline{\beta }{}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{S}-{\mathcal{G}}_3\mathcal{A}{}_H\right]+\frac{\overline{\beta }{}_C{\mathcal{S}}^{\ast }}{{\mathcal{G}}_2}\left(1-\frac{\mathcal{I}{}_C^{\ast }}{\mathcal{I}{}_C}\right)\left[{\alpha }_1\mathcal{A}{}_C-{\mathcal{G}}_2\mathcal{I}{}_C\right]+\frac{\overline{\beta }{}_H{\mathcal{S}}^{\ast }}{{\mathcal{G}}_4}\left(1-\frac{\mathcal{I}{}_H^{\ast }}{\mathcal{I}{}_H}\right)\left[{\alpha }_2\mathcal{A}{}_H-{\mathcal{G}}_4\mathcal{I}{}_H\right]+\left(1-\frac{\mathcal{I}{}_{CH}^{\ast }}{\mathcal{I}{}_{CH}}\right)\left[\overline{\beta }{}_{CH}\mathcal{I}{}_{CH}\mathcal{S}-{\mathcal{G}}_5\mathcal{I}{}_{CH}\right].$$ (22)

It is observed from the system (1) at steady state that

$$\Lambda =\overline{\beta }{}_C\text{(}{\vartheta }_1\mathcal{A}{}_C^{\ast }+\mathcal{I}{}_C^{\ast }\text{)}{\mathcal{S}}^{\ast }+\overline{\beta }{}_H({\vartheta }_2\mathcal{A}{}_H^{\ast }+\mathcal{I}{}_H^{\ast }){\mathcal{S}}^{\ast }+\overline{\beta }{}_{CH}\mathcal{I}{}_{CH}^{\ast }{\mathcal{S}}^{\ast }+\mu {\mathcal{S}}^{\ast },$$$$\overline{\beta }{}_C({\vartheta }_1\mathcal{A}{}_C^{\ast }+\mathcal{I}{}_C^{\ast }){\mathcal{S}}^{\ast }={\mathcal{G}}_1\mathcal{A}{}_C^{\ast },$$$${\alpha }_1\mathcal{A}{}_C^{\ast }={\mathcal{G}}_2\mathcal{I}{}_C^{\ast },$$$$\overline{\beta }{}_H({\vartheta }_2\mathcal{A}{}_H^{\ast }+\mathcal{I}{}_H^{\ast }){\mathcal{S}}^{\ast }={\mathcal{G}}_3\mathcal{A}{}_H^{\ast },$$$${\alpha }_2\mathcal{A}{}_H^{\ast }={\mathcal{G}}_4\mathcal{I}{}_H^{\ast }$$$$\overline{\beta }{}_{CH}\mathcal{I}{}_{CH}^{\ast }{\mathcal{S}}^{\ast }={\mathcal{G}}_5\mathcal{I}{}_{CH}^{\ast },$$$$\xi {}_C\mathcal{I}{}_C^{\ast }+\xi {}_H\mathcal{I}{}_H^{\ast }+\xi {}_{CH}\mathcal{I}{}_{CH}^{\ast }=\mu {\mathcal{R}}^{\ast }.$$ (23)

If the expressions in (23) is substituted into (22), then we have

$${\dot{\mathcal{L}}}_2=\left(1-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}\right)\left[\overline{\beta }{}_C({\vartheta }_1\mathcal{A}{}_C^{\ast }+\mathcal{I}{}_C^{\ast }){\mathcal{S}}^{\ast }+\overline{\beta }{}_H({\vartheta }_2\mathcal{A}{}_H^{\ast }+\mathcal{I}{}_H^{\ast }){\mathcal{S}}^{\ast }+\overline{\beta }{}_{CH}\mathcal{I}{}_{CH}^{\ast }{\mathcal{S}}^{\ast }+\mu {\mathcal{S}}^{\ast }-\overline{\beta }{}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{S}\right)\left(-\overline{\beta }{}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{S}-\overline{\beta }{}_{CH}\mathcal{I}{}_{CH}\mathcal{S}-\mu \mathcal{S}\right]+\left(1-\frac{\mathcal{A}{}_C^{\ast }}{\mathcal{A}{}_C}\right)\left[\overline{\beta }{}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{S}-{\mathcal{G}}_1\mathcal{A}{}_C\right]+\left(1-\frac{\mathcal{A}{}_H^{\ast }}{\mathcal{A}{}_H}\right)\left[\overline{\beta }{}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{S}-{\mathcal{G}}_3\mathcal{A}{}_H\right]+\frac{\overline{\beta }{}_C{\mathcal{S}}^{\ast }}{{\mathcal{G}}_2}\left(1-\frac{\mathcal{I}{}_C^{\ast }}{\mathcal{I}{}_C}\right)\left[{\alpha }_1\mathcal{A}{}_C-{\mathcal{G}}_2\mathcal{I}{}_C\right]+\frac{\overline{\beta }{}_H{\mathcal{S}}^{\ast }}{{\mathcal{G}}_4}\left(1-\frac{\mathcal{I}{}_H^{\ast }}{\mathcal{I}{}_H}\right)\left[{\alpha }_2\mathcal{A}{}_H-{\mathcal{G}}_4\mathcal{I}{}_H\right]+\left(1-\frac{\mathcal{I}{}_{CH}^{\ast }}{\mathcal{I}{}_{CH}}\right)\left[\overline{\beta }{}_{CH}\mathcal{I}{}_{CH}\mathcal{S}-{\mathcal{G}}_5\mathcal{I}{}_{CH}\right],$$ (24)

The Eq. (24) now reduces to

$${\dot{\mathcal{L}}}_2=\mu {\mathcal{S}}^{\ast }\left(2-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}-\frac{\mathcal{S}}{{\mathcal{S}}^{\ast }}\right)+\overline{\beta }{}_C({\vartheta }_1\mathcal{A}{}_C^{\ast }+\mathcal{I}{}_C^{\ast }){\mathcal{S}}^{\ast }+\overline{\beta }{}_H({\vartheta }_2\mathcal{A}{}_H^{\ast }+\mathcal{I}{}_H^{\ast }){\mathcal{S}}^{\ast }+\overline{\beta }{}_{CH}\mathcal{I}{}_{CH}^{\ast }{\mathcal{S}}^{\ast }-\frac{\overline{\beta }{}_C({\vartheta }_1\mathcal{A}{}_C^{\ast }+\mathcal{I}{}_C^{\ast }){\mathcal{S}}^{\ast 2}}{\mathcal{S}}-\frac{\overline{\beta }{}_H({\vartheta }_2\mathcal{A}{}_H^{\ast }+\mathcal{I}{}_H^{\ast }){\mathcal{S}}^{\ast 2}}{\mathcal{S}}-\frac{\overline{\beta }{}_{CH}\mathcal{I}{}_{CH}^{\ast }{\mathcal{S}}^{\ast 2}}{\mathcal{S}}+\overline{\beta }{}_C({\vartheta }_1\mathcal{A}{}_C^{\ast }+\mathcal{I}{}_C^{\ast }){\mathcal{S}}^{\ast }-\frac{\overline{\beta }{}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{S}\mathcal{A}{}_C^{\ast }}{\mathcal{A}{}_C}-\frac{\overline{\beta }{}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{S}\mathcal{A}{}_H^{\ast }}{\mathcal{A}{}_H}+\overline{\beta }{}_H({\vartheta }_2\mathcal{A}{}_H^{\ast }+\mathcal{I}{}_H^{\ast }){\mathcal{S}}^{\ast }-\overline{\beta }{}_{CH}\mathcal{I}{}_{CH}^{\ast }\mathcal{S}+\overline{\beta }{}_{CH}\mathcal{I}{}_{CH}^{\ast }{\mathcal{S}}^{\ast }-\frac{\overline{\beta }{}_C\mathcal{I}{}_C^{\ast 2}\mathcal{A}{}_C{\mathcal{S}}^{\ast }}{\mathcal{I}{}_C\mathcal{A}{}_C^{\ast }}+\overline{\beta }{}_C\mathcal{I}{}_C^{\ast }{\mathcal{S}}^{\ast }-\frac{\overline{\beta }{}_H\mathcal{I}{}_H^{\ast 2}\mathcal{A}{}_H{\mathcal{S}}^{\ast }}{\mathcal{I}{}_H\mathcal{A}{}_H^{\ast }}+\overline{\beta }{}_H\mathcal{I}{}_H^{\ast }{\mathcal{S}}^{\ast }.$$ (25)

Simplifying further, we obtain that

$${\dot{\mathcal{L}}}_2=\mu {\mathcal{S}}^{\ast }\left(2-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}-\frac{\mathcal{S}}{{\mathcal{S}}^{\ast }}\right)+\overline{\beta }{}_C{\vartheta }_1\mathcal{A}{}_C^{\ast }{\mathcal{S}}^{\ast }\left(2-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}-\frac{\mathcal{S}}{{\mathcal{S}}^{\ast }}\right)+\overline{\beta }{}_H{\vartheta }_2\mathcal{A}{}_H^{\ast }{\mathcal{S}}^{\ast }\left(2-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}-\frac{\mathcal{S}}{{\mathcal{S}}^{\ast }}\right)+\overline{\beta }{}_C\mathcal{I}{}_C^{\ast }{\mathcal{S}}^{\ast }\left(3-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}-\frac{\mathcal{I}{}_C\mathcal{A}{}_C^{\ast }\mathcal{S}}{\mathcal{I}{}_C^{\ast }\mathcal{A}{}_C{\mathcal{S}}^{\ast }}-\frac{\mathcal{I}{}_C^{\ast }\mathcal{A}{}_C}{\mathcal{I}{}_C\mathcal{A}{}_C^{\ast }}\right)+\overline{\beta }{}_H\mathcal{I}{}_H^{\ast }{\mathcal{S}}^{\ast }\left(3-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}-\frac{\mathcal{I}{}_H\mathcal{A}{}_H^{\ast }\mathcal{S}}{\mathcal{I}{}_H^{\ast }\mathcal{A}{}_H{\mathcal{S}}^{\ast }}-\frac{\mathcal{I}{}_H^{\ast }\mathcal{A}{}_H}{\mathcal{I}{}_H\mathcal{A}{}_H^{\ast }}\right)+\overline{\beta }{}_{CH}\mathcal{I}{}_{CH}^{\ast }{\mathcal{S}}^{\ast }\left(2-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}-\frac{\mathcal{S}}{{\mathcal{S}}^{\ast }}\right).$$ (26)

As the geometric mean is always less than the arithmetic mean, it follows from (26) that

$$\left(2-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}-\frac{\mathcal{S}}{{\mathcal{S}}^{\ast }}\right)\le 0,\left(3-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}-\frac{\mathcal{I}{}_C\mathcal{A}{}_C^{\ast }\mathcal{S}}{\mathcal{I}{}_C^{\ast }\mathcal{A}{}_C{\mathcal{S}}^{\ast }}-\frac{\mathcal{I}{}_C^{\ast }\mathcal{A}{}_C}{\mathcal{I}{}_C\mathcal{A}{}_C^{\ast }}\right)\le 0,\left(3-\frac{{\mathcal{S}}^{\ast }}{\mathcal{S}}-\frac{\mathcal{I}{}_H\mathcal{A}{}_H^{\ast }\mathcal{S}}{\mathcal{I}{}_H^{\ast }\mathcal{A}{}_H{\mathcal{S}}^{\ast }}-\frac{\mathcal{I}{}_H^{\ast }\mathcal{A}{}_H}{\mathcal{I}{}_H\mathcal{A}{}_H^{\ast }}\right)\le 0.$$

Consequently, we have ${\dot{\mathcal{L}}}_2\le 0$ whenever ${\overline{\mathcal{R}}}_0>1$. As a result, ${\mathcal{L}}_2$ is a well defined candidate for a Lyapunov function in $\mathcal{D}$ and based on La Salle’s Invariance principle [54], it is concluded that the endemic equilibrium is globally asymptotically stable for ${\overline{\mathcal{R}}}_0>1$.

This implies that every solution to the equations of the model (1) with ${\zeta }_1={\zeta }_2=\eta {}_C=\eta {}_H=\eta {}_{CH}=0$, and initial conditions in $\mathcal{D}⧵{\mathcal{D}}_0$ converges to the respective unique endemic equilibrium ${\mathcal{Q}}_e$, of the model (1) as $t\to \infty$ for ${\overline{\mathcal{R}}}_0>1$. The epidemiological significance of Theorem 3.8 is that, if individuals in symptomatic stage of SARS-CoV-2 infection and those infected with chronic HBV do not get additional infection with a different disease, and if diseases induced death is insignificant, then both infections will persist within the population as long as ${\overline{\mathcal{R}}}_0>1$.

4. Optimal control analysis

In this section, the model (1) is considered with time dependent controls ${\theta }_1\left(t\right)$, ${\theta }_2\left(t\right),{\theta }_3\left(t\right)$ and ${\theta }_4\left(t\right)$ to obtain the optimal intervention strategies for curtailing the spread of both diseases. The controls: ${\theta }_1\left(t\right),{\theta }_2\left(t\right),{\theta }_3\left(t\right)$ and ${\theta }_4\left(t\right)$ are bounded, “Lebesgue integrable” functions which are defined below:

• (i.)${\theta }_1\left(t\right)$: SARS-CoV-2 prevention (in the form of vaccination, social distancing, use of face-mask in the public places, use of personal protective equipment (PPE) and hand gloves by health workers and so on),
• (ii.)${\theta }_2\left(t\right)$: HBV prevention (in the form of vaccination, condom use by sexually active individuals, abstinence and so on),
• (iii.)${\theta }_3\left(t\right)$: efforts against incident co-infection with both SARS-CoV-2 and HBV (combined SARS-CoV-2 and HBV preventive measures highlighted in (i.) and (ii.) above),
• (iv.)${\theta }_4\left(t\right)$: efforts against second infection by individuals singly infected with SARS-CoV-2 (in symptomatic stage) or chronic HBV (in the form of intensified case detection and isolation of infected individuals, quarantine and so on).

Also, assume that $0<{\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4\le 1$. The optimal control model is now given by:

$$\frac{d\mathcal{S}\left(t\right)}{dt}=\Lambda -(1-{\theta }_1)\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}-(1-{\theta }_2)\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}-(1-{\theta }_3)\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}-\mu \mathcal{S},$$$$\frac{d\mathcal{A}{}_C\left(t\right)}{dt}=(1-{\theta }_1)\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}-({\alpha }_1+\mu )\mathcal{A}{}_C,$$$$\frac{d\mathcal{I}{}_C\left(t\right)}{dt}={\alpha }_1\mathcal{A}{}_C-(\xi {}_C+\eta {}_C+\mu )\mathcal{I}{}_C-(1-{\theta }_4){\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C$$$$\frac{d\mathcal{A}{}_H\left(t\right)}{dt}=(1-{\theta }_2)\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}-({\alpha }_2+\mu )\mathcal{A}{}_H,$$$$\frac{d\mathcal{I}{}_H\left(t\right)}{dt}={\alpha }_2\mathcal{A}{}_H-(\xi {}_H+\eta {}_H+\mu )\mathcal{I}{}_H-(1-{\theta }_4){\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{I}{}_H,$$$$\frac{d\mathcal{I}{}_{CH}\left(t\right)}{dt}=(1-{\theta }_3)\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}+(1-{\theta }_4){\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C+(1-{\theta }_4){\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{I}{}_H-(\xi {}_{CH}+\eta {}_{CH}+\mu )\mathcal{I}{}_{CH},$$$$\frac{d\mathcal{R}\left(t\right)}{dt}=\xi {}_C\mathcal{I}{}_C+\xi {}_H\mathcal{I}{}_H+\xi {}_{CH}\mathcal{I}{}_{CH}-\mu \mathcal{R}.$$ (27)

subject to the initial conditions

$${\mathcal{S}}_0=\mathcal{S}\left(0\right),{\mathcal{A}}_{C0}={\mathcal{A}}_C\left(0\right),{\mathcal{I}}_{C0}={\mathcal{I}}_C\left(0\right),{\mathcal{A}}_{H0}={\mathcal{A}}_H\left(0\right),{\mathcal{I}}_{H0}={\mathcal{I}}_H\left(0\right),{\mathcal{I}}_{CH0}={\mathcal{I}}_{CH}\left(0\right),{\mathcal{R}}_0=\mathcal{R}\left(0\right).$$ (28)

Let us consider the following cost functional

$$J\left[\theta {}_1,\theta {}_2,\theta {}_3,\theta {}_4,\theta {}_4\right]={\int }_0^T\left[\mathcal{A}{}_C\left(t\right)+\mathcal{I}{}_C\left(t\right)+\mathcal{A}{}_H\left(t\right)+\mathcal{I}{}_H\left(t\right)+\mathcal{I}{}_{CH}\left(t\right)+\frac{{\varsigma }_1}{2}\theta {}_1^2+\frac{{\varsigma }_2}{2}\theta {}_2^2+\frac{{\varsigma }_3}{2}\theta {}_3^2+\frac{{\varsigma }_4}{2}\theta {}_4^2\right]dt,$$ (29)

where $T$ is the final time.

An optimal control, $\theta {}_1^{\ast },\theta {}_2^{\ast },\theta {}_3^{\ast },\theta {}_4^{\ast }$, is now evaluated such that

$$J(\theta {}_1^{\ast },\theta {}_2^{\ast },\theta {}_3^{\ast },\theta {}_4^{\ast })=min\left\{J(\theta {}_1,\theta {}_2,\theta {}_3,\theta {}_4)\right|\theta {}_1,\theta {}_2,\theta {}_3,\theta {}_4\in U\},$$ (30)

where $U=\left\{(\theta {}_1,\theta {}_2,\theta {}_3,\theta {}_4)\right\}$ is a control set and $\theta {}_1^{\ast },\theta {}_2^{\ast },\theta {}_3^{\ast },\theta {}_4^{\ast }$ are measurable with $0\le \theta {}_1^{\ast },\theta {}_2^{\ast },\theta {}_3^{\ast },\theta {}_4^{\ast }\le 1$ for $t\in [0,T]$. The Hamiltonian is given by:

$$\mathcal{X}=\mathcal{A}{}_C\left(t\right)+\mathcal{I}{}_C\left(t\right)+\mathcal{A}{}_H\left(t\right)+\mathcal{I}{}_H\left(t\right)+\mathcal{I}{}_{CH}\left(t\right)+\frac{{\varsigma }_1}{2}\theta {}_1^2+\frac{{\varsigma }_2}{2}\theta {}_2^2+\frac{{\varsigma }_3}{2}\theta {}_3^2+\frac{{\varsigma }_4}{2}\theta {}_4^2+{\varrho }_1\left(\Lambda -(1-{\theta }_1)\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}-(1-{\theta }_2)\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}-(1-{\theta }_3)\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}-\mu \mathcal{S}\right)+{\varrho }_2\left((1-{\theta }_1)\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}-({\alpha }_1+\mu )\mathcal{A}{}_C\right)+{\varrho }_3\left({\alpha }_1\mathcal{A}{}_C-(\xi {}_C+\eta {}_C+\mu )\mathcal{I}{}_C-(1-{\theta }_4){\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C\right)+{\varrho }_4\left((1-{\theta }_2)\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}-({\alpha }_2+\mu )\mathcal{A}{}_H\right)+{\varrho }_5\left({\alpha }_2\mathcal{A}{}_H-(\xi {}_H+\eta {}_H+\mu )\mathcal{I}{}_H-(1-{\theta }_4){\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{I}{}_H\right)+{\varrho }_6\left((1-{\theta }_3)\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}+(1-{\theta }_4){\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C+(1-{\theta }_4){\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{I}{}_H-(\xi {}_{CH}+\eta {}_{CH}+\mu )\mathcal{I}{}_{CH}\right)+{\varrho }_7\left(\xi {}_C\mathcal{I}{}_C+\xi {}_H\mathcal{I}{}_H+\xi {}_{CH}\mathcal{I}{}_{CH}-\mu \mathcal{R}\right).$$ (31)

4.1. Existence

We shall now prove the existence of solution for the optimal control system (27) which minimizes the objective functional $J$.

Theorem 4.1

Let$J$be defined on control set$U$, subject to system(27)with non-negative initial conditions, then an optimal control quadruple$u^{\ast }=({\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4)$exists such that$J\left(u^{\ast }\right)=min\left\{J({\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4)|{\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4\in U\right\}$, if following conditions hold [55] :

• (i.) The admissible control set U is convex and closed.
• (ii.) The state system is bounded by a linear function in state and control variables.
• (iii.) The integrand of the objective functional in (29) is convex with respect to the controls.
• (iv.) The Lagrangian is bounded below by $\varpi {}_1|\theta {|}^{\varpi {}_3}-\varpi {}_2$ , where, $\varpi {}_1>0,\varpi {}_2>0,\varpi {}_3>1$ .

Proof

Let $U={[0,{\theta }_{max}]}^4$ be the control set, $\theta =({\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4)\in U$, $x=(\mathcal{S},\mathcal{A}{}_C,\mathcal{I}{}_C,\mathcal{A}{}_H,\mathcal{I}{}_H,\mathcal{I}{}_{CH},\mathcal{R})$ and $f(t,x,\theta )$ be the right hand of (27), that is

$$f(t,x,\theta )=\left(\begin{array}{c}\hfill \Lambda -(1-{\theta }_1)\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}-(1-{\theta }_2)\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}-(1-{\theta }_3)\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}-\mu \mathcal{S}\hfill \\ \hfill (1-{\theta }_1)\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}-({\alpha }_1+\mu )\mathcal{A}{}_C\hfill \\ \hfill {\alpha }_1\mathcal{A}{}_C-(\xi {}_C+\eta {}_C+\mu )\mathcal{I}{}_C-(1-{\theta }_4){\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C\hfill \\ \hfill (1-{\theta }_2)\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}-({\alpha }_2+\mu )\mathcal{A}{}_H\hfill \\ \hfill {\alpha }_2\mathcal{A}{}_H-(\xi {}_H+\eta {}_H+\mu )\mathcal{I}{}_H-(1-{\theta }_4){\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{I}{}_H\hfill \\ \hfill (1-{\theta }_3)\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}+(1-{\theta }_4){\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C+(1-{\theta }_4){\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{I}{}_H-(\xi {}_{CH}+\eta {}_{CH}+\mu )\mathcal{I}{}_{CH}\hfill \\ \hfill \xi {}_C\mathcal{I}{}_C+\xi {}_H\mathcal{I}{}_H+\xi {}_{CH}\mathcal{I}{}_{CH}-\mu \mathcal{R}\hfill \end{array}\right).$$ (32)

The proof of Theorem 4.1 is proceeded as below:

• (i.)
  Given the control set $U$; by definition, it is closed. Suppose that $\phi ,\sigma \in U$, where $\phi =({\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4)$, and $\sigma =({\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4)$. Then, note that [56],

  $$\lambda \phi +(1-\lambda )\sigma \in {[0,{\theta }_{max}]}^4,\forall \lambda \in [0,1],$$

  and hence, $U$ is convex.
• (ii.)
  We shall verify the second condition of Theorem 4.1 following a similar approach given in [57], [58]. The system (27) can be re-written as a linear function of the control variables $({\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4)$ with the coefficients as functions of time and state variables:

  $$f(t,x,\theta )=\gamma (t,x)+\zeta (t,x)\theta$$

  with

  $$\gamma (t,x)=\left(\begin{array}{c}\hfill \Lambda -\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}-\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}-\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}-\mu \mathcal{S}\hfill \\ \hfill \frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}-({\alpha }_1+\mu )\mathcal{A}{}_C\hfill \\ \hfill {\alpha }_1\mathcal{A}{}_C-(\xi {}_C+\eta {}_C+\mu )\mathcal{I}{}_C-{\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C\hfill \\ \hfill \frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}-({\alpha }_2+\mu )\mathcal{A}{}_H\hfill \\ \hfill {\alpha }_2\mathcal{A}{}_H-(\xi {}_H+\eta {}_H+\mu )\mathcal{I}{}_H-{\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{I}{}_H\hfill \\ \hfill \frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}+{\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C+{\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{I}{}_H-(\xi {}_{CH}+\eta {}_{CH}+\mu )\mathcal{I}{}_{CH}\hfill \\ \hfill \xi {}_C\mathcal{I}{}_C+\xi {}_H\mathcal{I}{}_H+\xi {}_{CH}\mathcal{I}{}_{CH}-\mu \mathcal{R}\hfill \end{array}\right),$$

  $$\zeta (t,x)=\left(\begin{array}{cccc}\hfill \frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}\hfill & \hfill \frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}\hfill & \hfill \frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}\hfill & \hfill 0\hfill \\ \hfill -\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{S}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C\hfill \\ \hfill 0\hfill & \hfill -\frac{\beta {}_C({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{S}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{I}{}_H\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\frac{\beta {}_{CH}\mathcal{I}{}_{CH}}{\mathcal{N}}\mathcal{S}\hfill & \hfill -{\zeta }_1\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)}{\mathcal{N}}\mathcal{I}{}_C-{\zeta }_2\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)}{\mathcal{N}}\mathcal{I}{}_H\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

  Also, we have

  $$\Vert f(t,x,\theta )\Vert \le \Vert \gamma (t,x)\Vert +\Vert \zeta (t,x)\Vert \Vert \theta \Vert \le a+b\Vert \theta \Vert ,wherea>0,b>0.$$

  with,

  $$a\le \sqrt{max\{c_1,c_2,c_3\}{\left(3\Lambda \right)}^2},$$

  $$b\le \sqrt{max\{d_1,d_2,d_3\}{\left(3\Lambda \right)}^2},$$

  where,

  $$c_1=\frac{2\beta {}_C^2({\vartheta }_1^2+1)({\zeta }_2^2+1)}{{\mu }^2},c_2=\frac{2\beta {}_H^2({\vartheta }_2^2+1)({\zeta }_1^2+1)}{{\mu }^2},$$

  $$c_3=\frac{2\beta {}_{CH}^2+5{\mu }^2+2({\alpha }_1^2+{\alpha }_2^2)+2({\alpha }_1+{\alpha }_2)\mu +(\xi {}_C^2+\xi {}_H^2+\xi {}_{CH}^2)+{(\xi {}_C+\eta {}_C+\mu )}^2+{(\xi {}_H+\eta {}_H+\mu )}^2+{(\xi {}_{CH}+\eta {}_{CH}+\mu )}^2}{{\mu }^2},$$

  $$d_1=\frac{2\beta {}_C^2({\vartheta }_1^2+1)({\zeta }_2^2+1)}{{\mu }^2},d_2=\frac{2\beta {}_H^2({\vartheta }_2^2+1)({\zeta }_1^2+1)}{{\mu }^2},d_3=\frac{2\beta {}_{CH}^2}{{\mu }^2}.$$
• (iii.)
  Lagrangian of the optimal control system is given by

  $$\mathcal{L}=\mathcal{A}{}_C\left(t\right)+\mathcal{I}{}_C\left(t\right)+\mathcal{A}{}_H\left(t\right)+\mathcal{I}{}_H\left(t\right)+\mathcal{I}{}_{CH}\left(t\right)+\frac{1}{2}\sum _{i=1}^4{\varsigma }_i{\theta }_i^2.$$ (33)

  Let $\phi ,\sigma \in U$, with $\phi =({\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4),\sigma =({\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4)$. and $\lambda \in [0,1]$. We now show that,

  $$\mathcal{L}[t,x,(1-\lambda )\phi +\lambda \sigma ]\le (1-\lambda )\mathcal{L}(t,x,\phi )+\lambda \mathcal{L}(t,x,\sigma ).$$

  From (33), we have

  $$\mathcal{L}[t,x,(1-\lambda )\phi +\lambda \sigma ]=\mathcal{A}{}_C\left(t\right)+\mathcal{I}{}_C\left(t\right)+\mathcal{A}{}_H\left(t\right)+\mathcal{I}{}_H\left(t\right)+\mathcal{I}{}_{CH}\left(t\right)+\frac{1}{2}\sum _{i=1}^4{\varsigma }_i{[(1-\lambda ){\theta }_i+\lambda {\sigma }_i]}^2,(1-\lambda )\mathcal{L}(t,x,\phi )+\lambda \mathcal{L}(t,x,\sigma )=\mathcal{A}{}_C\left(t\right)+\mathcal{I}{}_C\left(t\right)+\mathcal{A}{}_H\left(t\right)+\mathcal{I}{}_H\left(t\right)+\mathcal{I}{}_{CH}\left(t\right)+\frac{1}{2}(1-\lambda )\sum _{i=1}^4{\varsigma }_i{\phi }_i^2+\frac{1}{2}\lambda \sum _{i=1}^4{\varsigma }_i{\sigma }_i^2.$$ (34)

  By taking the difference of the above two equations in (34), we have

  $$\mathcal{L}[t,x,(1-\lambda )\phi +\lambda \sigma ]-[(1-\lambda )\mathcal{L}(t,x,\phi )+\lambda \mathcal{L}(t,x,\sigma )]=\frac{1}{2}({\lambda }^2-\lambda )\sum _{i=1}^4{\varsigma }_i{({\phi }_i-{\sigma }_i)}^2\le 0,since\lambda \in [0,1].$$ (35)

  Therefore, we have

  $$\mathcal{L}[t,x,(1-\lambda )\phi +\lambda \sigma ]\le [(1-\lambda )\mathcal{L}(t,x,\phi )+\lambda \mathcal{L}(t,x,\sigma )],$$

  and hence

  $$\mathcal{L}[t,x,(1-\lambda )\phi +\lambda \sigma ]-[(1-\lambda )\mathcal{L}(t,x,\phi )+\lambda \mathcal{L}(t,x,\sigma )]\le 0.$$

  Thus $\mathcal{L}$ is convex.
• (iv.)
  We now show that there exists some constants $\varpi {}_1,\varpi {}_2$ and $\varpi {}_3$ such that $\mathcal{L}\ge \varpi {}_1{\left|\theta \right|}^{\varpi {}_3}-\varpi {}_2$, where, $\varpi {}_1>0$, $\varpi {}_2>0$, $\varpi {}_3>1$. Note that $\varsigma {}_4{\theta }_4^2\le \varsigma {}_4$ since ${\theta }_4\in [0,1]$, and $\frac{1}{2}\varsigma {}_4{\theta }_4^2\le \frac{1}{2}\varsigma {}_4$. Now,

  $$\mathcal{L}>\frac{{\varsigma }_1}{2}\theta {}_1^2+\frac{{\varsigma }_2}{2}\theta {}_2^2+\frac{{\varsigma }_3}{2}\theta {}_3^2+\frac{{\varsigma }_4}{2}\theta {}_4^2\ge \frac{{\varsigma }_1}{2}\theta {}_1^2+\frac{{\varsigma }_2}{2}\theta {}_2^2+\frac{{\varsigma }_3}{2}\theta {}_3^2+\frac{{\varsigma }_4}{2}\theta {}_4^2-\frac{{\varsigma }_4}{2}\ge min\left(\frac{{\varsigma }_1}{2},\frac{{\varsigma }_2}{2},\frac{{\varsigma }_3}{2},\frac{{\varsigma }_4}{2}\right)\left({\theta }_1^2+{\theta }_2^2+{\theta }_3^2+{\theta }_4^2\right)-\frac{{\varsigma }_4}{2}\ge min\left(\frac{{\varsigma }_1}{2},\frac{{\varsigma }_2}{2},\frac{{\varsigma }_3}{2},\frac{{\varsigma }_4}{2}\right){|{\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4|}^2-\frac{{\varsigma }_4}{2}$$

  Hence,

  $$\mathcal{L}\ge \varpi {}_1{\left|\theta \right|}^{\varpi {}_3}-\varpi {}_2,\text{where},\varpi {}_1=min\left(\frac{{\varsigma }_1}{2},\frac{{\varsigma }_2}{2},\frac{{\varsigma }_3}{2},\frac{{\varsigma }_4}{2}\right)>0,\varpi {}_2=\frac{{\varsigma }_4}{2}>0\text{and}\varpi {}_3=2>1.\square$$

Theorem 4.2

Suppose the set $\theta =\{{\theta }_1^{\ast },{\theta }_2^{\ast },{\theta }_3^{\ast },{\theta }_4^{\ast }\}$ minimizes $J$ over $U$ , then there exist adjoint variables, ${\varrho }_1,{\varrho }_2,\dots ,{\varrho }_7$ , satisfying the adjoint equations

$$-\frac{\partial {\varrho }_i}{\partial t}=\frac{\partial \mathcal{X}}{\partial i},$$

with transversality condition (with adjoint system provided in the Appendix )

$${\varrho }_i\left(t_f\right)=0,\text{where},i=\mathcal{S},\mathcal{A}{}_C,\mathcal{I}{}_C,\mathcal{A}{}_H,\mathcal{I}{}_H,\mathcal{I}{}_{CH},\mathcal{R}.$$ (36)

Furthermore,

$${\theta }_1^{\ast }=min\left\{1,max\left(0,\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{S}({\varrho }_2-{\varrho }_1)}{{\varsigma }_1\mathcal{N}}\right)\right\},$$$${\theta }_2^{\ast }=min\left\{1,max\left(0,\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{S}({\varrho }_4-{\varrho }_1)}{{\varsigma }_2\mathcal{N}}\right)\right\},$$$${\theta }_3^{\ast }=min\left\{1,max\left(0,\frac{\beta {}_{CH}\mathcal{I}{}_{CH}\mathcal{S}({\varrho }_6-{\varrho }_1)}{{\varsigma }_3\mathcal{N}}\right)\right\},$$$${\theta }_4^{\ast }=min\left\{1,max\left(0,\frac{{\zeta }_1\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{I}{}_C({\varrho }_6-{\varrho }_3)+{\zeta }_2\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{I}{}_H({\varrho }_6-{\varrho }_5)}{{\varsigma }_4\mathcal{N}}\right)\right\}.$$ (37)

Proof of Theorem 4.2

Let $U^{\ast }=({\theta }_1^{\ast },{\theta }_2^{\ast },{\theta }_3^{\ast },{\theta }_4^{\ast })$ and ${\mathcal{S}}^{\ast },\mathcal{A}{}_C^{\ast },\mathcal{I}{}_C^{\ast },\mathcal{A}{}_H^{\ast },\mathcal{I}{}_H^{\ast },\mathcal{I}{}_{CH}^{\ast },{\mathcal{R}}^{\ast }$ be the relevant solutions. With the help of the Pontryagin’s Maximum Principle [59], there exist adjoint variables which fulfills:

$$-\frac{d{\varrho }_1}{dt}=\frac{\partial \mathcal{X}}{\partial \mathcal{S}},{\varrho }_1\left(t_f\right)=0,-\frac{d{\varrho }_2}{dt}=\frac{\partial \mathcal{X}}{\partial \mathcal{A}{}_C},{\varrho }_2\left(t_f\right)=0,-\frac{d{\varrho }_3}{dt}=\frac{\partial \mathcal{X}}{\partial \mathcal{I}{}_C},{\varrho }_3\left(t_f\right)=0,-\frac{d{\varrho }_4}{dt}=\frac{\partial \mathcal{X}}{\partial \mathcal{A}{}_H},{\varrho }_4\left(t_f\right)=0,-\frac{d{\varrho }_5}{dt}=\frac{\partial \mathcal{X}}{\partial \mathcal{I}{}_H},{\varrho }_5\left(t_f\right)=0,-\frac{d{\varrho }_6}{dt}=\frac{\partial \mathcal{X}}{\partial \mathcal{I}{}_{CH}},{\varrho }_6\left(t_f\right)=0,-\frac{d{\varrho }_7}{dt}=\frac{\partial \mathcal{X}}{\partial \mathcal{R}},{\varrho }_7\left(t_f\right)=0.$$ (38)

On the interior of the set, where $0<{\theta }_j<1$   $\forall$ ($j=1,\dots ,4$), we have

$$0=\frac{\partial \mathcal{X}}{\partial {\theta }_1}={\varsigma }_1\mathcal{N}{\theta }_1^{\ast }-\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{S}({\varrho }_2-{\varrho }_1),$$$$0=\frac{\partial \mathcal{X}}{\partial {\theta }_2}={\varsigma }_2\mathcal{N}{\theta }_2^{\ast }-\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{S}({\varrho }_4-{\varrho }_1),$$$$0=\frac{\partial \mathcal{X}}{\partial {\theta }_3}={\varsigma }_3\mathcal{N}{\theta }_3^{\ast }-\beta {}_{CH}\mathcal{I}{}_{CH}\mathcal{S}({\varrho }_6-{\varrho }_1),$$$$0=\frac{\partial \mathcal{X}}{\partial {\theta }_4}={\varsigma }_4\mathcal{N}{\theta }_4^{\ast }-\left[{\zeta }_1\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{I}{}_C({\varrho }_6-{\varrho }_3)+{\zeta }_2\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{I}{}_H({\varrho }_6-{\varrho }_5)\right].$$ (39)

Therefore,

$${\theta }_1^{\ast }=\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{S}({\varrho }_2-{\varrho }_1)}{{\varsigma }_1\mathcal{N}},$$$${\theta }_2^{\ast }=\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{S}({\varrho }_4-{\varrho }_1)}{{\varsigma }_2\mathcal{N}},$$$${\theta }_3^{\ast }=\frac{\beta {}_{CH}\mathcal{I}{}_{CH}\mathcal{S}({\varrho }_6-{\varrho }_1)}{{\varsigma }_3\mathcal{N}},$$$${\theta }_4^{\ast }=\frac{{\zeta }_1\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{I}{}_C({\varrho }_6-{\varrho }_3)+{\zeta }_2\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{I}{}_H({\varrho }_6-{\varrho }_5)}{{\varsigma }_4\mathcal{N}}.$$ (40)

$${\theta }_1^{\ast }=min\left\{1,max\left(0,\frac{\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{S}({\varrho }_2-{\varrho }_1)}{{\varsigma }_1\mathcal{N}}\right)\right\},$$$${\theta }_2^{\ast }=min\left\{1,max\left(0,\frac{\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{S}({\varrho }_4-{\varrho }_1)}{{\varsigma }_2\mathcal{N}}\right)\right\},$$$${\theta }_3^{\ast }=min\left\{1,max\left(0,\frac{\beta {}_{CH}\mathcal{I}{}_{CH}\mathcal{S}({\varrho }_6-{\varrho }_1)}{{\varsigma }_3\mathcal{N}}\right)\right\},$$$${\theta }_4^{\ast }=min\left\{1,max\left(0,\frac{{\zeta }_1\beta {}_H({\vartheta }_2\mathcal{A}{}_H+\mathcal{I}{}_H)\mathcal{I}{}_C({\varrho }_6-{\varrho }_3)+{\zeta }_2\beta {}_C({\vartheta }_1\mathcal{A}{}_C+\mathcal{I}{}_C)\mathcal{I}{}_H({\varrho }_6-{\varrho }_5)}{{\varsigma }_4\mathcal{N}}\right)\right\}.$$ (41)

5. Numerical simulations

Simulations conducted on the optimal system (27), adjoint Eqs. (38) and the control characterizations (37) are implemented in MATLAB by adopting the forward–backward sweep method proposed by Lenhart and Workman [60]. The scheme combines the forward application of First order Runge–Kutta (RK) method to the state system (27) together with the backward application of the same method to the adjoint system (given in the Appendix). The stability and accuracy of the RK methods have been extensively established by Butcher [61], [62]. Specifically, to minimize the error as much as possible, we set the tolerance, $\delta =0.0001$. This approach requires an initial value of the optimal control $U^{\ast }$. The initial conditions given in (28) are used to solve the state system forward in time with the help of MATLAB built-in routine of ODE45. Subsequently, the adjoint system given in the Appendix is solved by using the transversality condition (36) and the approximate solution obtained for the state system. Then, the control variables are computed using the characterizations given in (37), and the control set $U^{\ast }$ is updated through a convex combination of the previous as well as current values of the control characterizations. This process is repeated until the state and adjoint variables as well as control values attain convergence. The quadratic cost functions $\frac{1}{2}{\varsigma }_1\theta {}_1^2,\frac{1}{2}{\varsigma }_2\theta {}_2^2,\frac{1}{2}{\varsigma }_3\theta {}_3^2$ and $\frac{1}{2}{\varsigma }_4\theta {}_4^2$ are adopted, over the time. The weight parameters are assumed as: ${\varsigma }_1=0.5,{\varsigma }_2=0.7,{\varsigma }_3=0.9$ and ${\varsigma }_4=0.9$. It is important to note that, the choice of the weight parameters are of theoretical sense just to highlight the control measures implemented in this paper.

5.1. Uncertainty and sensitivity analysis

Due to uncertainties which may arise in parameters estimation, sensitivity analysis shall be carried out in this section, using the method in [63]. A Latin Hypercube Sampling (LHS) is performed on the model parameters. For the sensitivity analysis, a Partial Rank Correlation Coefficient (PRCC) was calculated between values of the parameters in the response function and the values of the response function derived from the sensitivity analysis. A total of 1000 simulations (of the model (1)) per LHS run were implemented. The PRCC values are presented in Table 2. PRCC values given in bold fonts are for those parameters which are highly sensitive with respect to each of the response functions. As depicted in Fig. 2(a), using the SARS-CoV-2 only reproduction number ${\mathcal{R}}_{0C}$ input, the dominant terms are: the contact rate for SARS-CoV-2 transmission, ($\beta {}_C$, positively correlated) and recovery rate ($\xi {}_C$, negatively correlated). Thus, in order to effectively bring down the burden of SARS-CoV-2, more efforts should be put in place to reduce disease spread and also to provide desired treatment for infected individuals. Similar conclusions can be made when the HBV and co-infection associated reproduction numbers are used as inputs. These can be observed in Fig. 2, Fig. 2. If the class of individuals infected with SARS-CoV-2 in asymptomatic stage $\mathcal{A}{}_C$ (Fig. 3(a)) is used as input, the most important parameters driving the infection are the effective contact rates for the transmission of SARS-CoV-2: $\beta {}_C$ (positively correlated) and HBV: ${\beta }_H$ (negatively correlated) as well as the modification parameter for transmissibility of acutely infected individuals: ${\vartheta }_1$ (positively correlated). If the class of symptomatic infectious individuals with SARS-CoV-2, $\mathcal{I}{}_C$ is used as response function (as observed in Fig. 3(b)), the dominant parameters are the contact rate for the transmission of SARS-CoV-2: ${\beta }_C$ (positively correlated), contact rate for HBV transmission: ${\beta }_H$ (negatively correlated), progression rates to symptomatic stage: ${\alpha }_1$ (positively correlated), the modification parameter: ${\vartheta }_1$ (positively correlated) and the recovery rate ${\xi }_C$ (negatively correlated). In a similar manner, when the class of individuals acutely infected with HBV and those with chronic HBV are used as response functions, as presented in Fig. 3, Fig. 3, the most sensitive parameters are the contact rate for HBV transmission (${\beta }_H$), SARS-CoV-2 transmission rate (${\beta }_C$), progression rate to chronic HBV stage (${\alpha }_2$) and the modification parameter ${\vartheta }_2$. Also, the recovery rate ${\xi }_H$ influences the dynamics of the disease in both stages. Furthermore, using the co-infection class $\mathcal{I}{}_{CH}$ as input (as depicted by Fig. 3(e)), the most sensitive parameters are the contact rate for dual diseases transmission: $\beta {}_{CH}$ and the recovery rate $\xi {}_{CH}$. This shows that to avert high incidence of co-infection cases in the population, serious efforts must be put in place to reduce the spread of dual infections, and also provide effective treatment for those already infected with both infections. When the total infected population is used as the response function, as can be observed in Fig. 3(f), the most sensitive parameters are the SARS-CoV-2, HBV and co-infection transmission rates.

Table 2.

PRCC values for the model (1) parameters using the infected classes and the related reproduction numbers as response functions.

Parameters	$A{}_C$	$I{}_C$	$A{}_H$	$I{}_H$	$I{}_{CH}$	Total infected	${\mathcal{R}}_{0C}$	${\mathcal{R}}_{0H}$	${\mathcal{R}}_{0CH}$
$\mu$	0.0278	−0.0528	0.0148	−0.0349	0.0441	0.0343	0.0312	0.0126	0.0128
$\beta {}_C$	0.9498	0.9301	−0.4623	−0.4883	0.2203	0.5141	0.9608	–	–
$\beta {}_H$	−0.4595	−0.4807	0.9531	0.9256	0.1712	0.5582	–	0.9548	–
$\beta {}_{CH}$	−0.1831	−0.0738	−0.2289	−0.0413	0.8029	0.4035	–	–	0.8428
$\vartheta {}_1$	0.7028	0.6509	−0.2570	−0.2739	0.0846	0.2887	0.1627	–	–
$\vartheta {}_2$	−0.2632	−0.2954	0.7076	0.6321	0.0352	0.3087	–	0.1410	–
$\alpha {}_1$	−0.2972	0.6521	−0.0967	−0.0973	0.0655	0.0142	−0.2573	–	–
$\alpha {}_2$	−0.0189	−0.0570	−0.3965	0.6389	0.0996	0.0098	–	−0.2208	–
$\eta {}_C$	0.0006963	−0.0302	0.0473	−0.0594	−0.0153	0.0061	−0.2014	–	–
$\eta {}_H$	−0.0093	−0.0330	−0.0056	−0.0602	−0.0095	−0.0449	–	−0.1751	–
$\eta {}_{CH}$	0.0482	0.0078	0.1369	−0.0325	−0.0194	−0.0183	–	–	−0.1117
$\zeta {}_1$	−0.0624	−0.0560	−0.0637	−0.0300	0.0635	−0.0086	–	–	–
$\zeta {}_2$	0.0124	−0.0191	−0.0106	−0.0397	0.0378	−0.0049	–	–	–
$\xi {}_C$	−0.3890	−0.5487	0.0173	0.0940	−0.0462	−0.1662	−0.8890	–	–
$\xi {}_H$	0.0600	0.1423	−0.4101	−0.5330	−0.1175	−0.1976	–	−0.8846	–
$\xi {}_{CH}$	0.0560	0.0389	0.0616	−0.0149	−0.4173	−0.2731	–	–	−0.9037

Fig. 2.

Graphical results of the sensitivity analysis with the SARS-CoV-2, HBV, co-infection related reproduction numbers as response functions. Reasonable ranges of the parameter values given in Table 1 are used.

Fig. 3.

Graphical results of the sensitivity analysis with different epidemiological states and total infected population as response functions. Reasonable ranges of the parameter values given in Table 1 are used.

In the proceeding sections, the impact of some control strategies shall be investigated. One of the limitations of this work is that, it is not a case study, due to insufficient data on SARS-CoV-2 and HBV co-infection. It was difficult to obtain certain information about the co-interaction of the two diseases. Most parameter values used for the simulations were obtained from relevant literatures, while others were reasonably assumed. However, as regards the recruitment rate, $\Lambda$ and natural death rate $\mu$, their values were estimated from the total population and life expectancy for the district of Jinshan in Shanghai Province, China (a region that has high endemicity of both diseases). Thus, the two parameters $\Lambda$ and $\mu$ are approximately taken as 732,500 and 76.31, respectively [48], [49]. The initial conditions that were used for the numerical experiment are assumed thus: $\mathcal{S}\left(0\right)=700000;\mathcal{A}{}_C\left(0\right)=500;\mathcal{I}{}_C\left(0\right)=300;\mathcal{A}{}_H\left(0\right)=500;\mathcal{I}{}_H\left(0\right)=500;\mathcal{I}{}_{CH}\left(0\right)=20;\mathcal{R}\left(0\right)=100$.

5.2. Strategy A: SARS-CoV-2 prevention ($u_1\ne 0$)

The optimal control system (27) is simulated using the strategy against incident SARS-CoV-2 infection ($u_1\ne 0$). As depicted in Figs. 4(a), 4(b), 4(c), and using the transmission rates: $\beta {}_C=0.15,\beta {}_H=0.15,\beta {}_{CH}=0.2$, so that ${\mathcal{R}}_0=1.2900>1$, a remarkable reduction in the number of infected persons with SARS-CoV-2 (in asymptomatic and symptomatic stages), is observed. Interestingly, this strategy also has remarkable impact on co-infected cases. It is observed that, a remarkable number of new co-infected cases of SARS-CoV-2 and chronic HBV infections are also averted by this intervention strategy (as can be seen in Fig. 4(c)). The control profile for this scenario is depicted in Fig. 4(d), where it is observed that the control was at its peak for more about 120 days from the onset of the simulation before it starts to drop steadily.

Fig. 4.

Impact of SARS-CoV-2 prevention control. Here, $\beta {}_C=0.15,\beta {}_H=0.15,\beta {}_{CH}=0.2,{\eta }_C=0.05,{\eta }_H=0.05,{\eta }_{CH}=0.005$, so that ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}=1.2900>1$.

5.3. Strategy B: HBV prevention ($u_2\ne 0$)

Results of simulating the optimal control system (27) when the strategy that averts incident HBV infection ($u_2\ne 0$) is implemented, are depicted by Figs. 5(a), 5(b), and 5(c). Upon implementing this intervention measure, for $\beta {}_C=0.15,\beta {}_H=0.15,\beta {}_{CH}=0.2$, so that ${\mathcal{R}}_0=1.2900>1$, reveals a remarkable decrease in the total number of individuals acutely infected with HBV and those with chronic HBV, as expected (shown in Fig. 5, Fig. 5). Also, this intervention measure positively impact the co-infected individuals, as observed in Fig. 5(c). It is noticed that, remarkable co-infected cases of SARS-CoV-2 and chronic HBV are averted by this intervention strategy. Thus, averting incident HBV infection is key to averting co-infection cases. Control profile associated with this intervention measure, depicted in Fig. 5(d) shows that this strategy has very high impact in averting new cases of HBV in the population.

Fig. 5.

Impact of HBV prevention control. Here, $\beta {}_C=0.15,\beta {}_H=0.15,\beta {}_{CH}=0.2,{\eta }_C=0.05,{\eta }_H=0.05,{\eta }_{CH}=0.005$, so that ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}=1.2900>1$.

5.4. Strategy C: Efforts against co-infection with both SARS-CoV-2 and HBV ($u_3\ne 0,u_4\ne 0$)

Result of Simulating system (27) when the intervention measure that averts co-infection with both SARS-CoV-2 and HBV ($u_3\ne 0$) is implemented, is shown in Fig. 6(a). Upon administering this intervention measure, using the transmission rates: $\beta {}_C=0.15,\beta {}_H=0.15,\beta {}_{CH}=0.2$, so that ${\mathcal{R}}_0=1.2900>1$, shows a remarkable reduction in co-infected cases. Thus, in order to curtail the spread of both diseases, efforts must also be geared towards preventing individuals from acquiring concurrent infections. In addition, the result of simulating system (27) when the intervention strategy that averts additional infection by those already infected with SARS-CoV-2 (in symptomatic stage) or chronic HBV, is presented in Fig. 6(b). We noticed a remarkable decrease in co-infection population. Control profiles describing this scenario are depicted by Fig. 6, Fig. 6. It can be observed that, the control profile denoting efforts against concurrent dual infections reaches its maximum value early enough during the simulation and sustains this value for the remaining simulation period. However, the control profile describing efforts against additional infection with a different disease, was its peak value for the entire simulation period.

Fig. 6.

Impact of control against co-infection with both diseases. Here, $\beta {}_C=0.15,\beta {}_H=0.15,\beta {}_{CH}=0.2,{\eta }_C=0.05,{\eta }_H=0.05,{\eta }_{CH}=0.005$, so that ${\mathcal{R}}_0=max\{{\mathcal{R}}_{0C},{\mathcal{R}}_{0H},{\mathcal{R}}_{0CH}\}=1.2900>1$.

6. Conclusion

In this work, a co-dynamical model for SARS-CoV-2 and Hepatitis B Virus incorporating optimal control was proposed and analyzed. The model captures incident co-infection with both viruses. Boundary and co-existence endemic equilibria were investigated for the model. The dynamical property of backward bifurcation was investigated for the model. It has been shown that in the absence of incident co-infection, backward bifurcation does not occur whereas presence of incident co-infection give rise backward bifurcation . The infection free and endemic equilibria were shown to be asymptotically stable in global sense, when individuals are in asymptomatic stage of SARS-CoV-2 infection and those with chronic HBV do not get additional infection with a different disease. Global sensitivity analysis was performed using appropriate ranges for the model parameters in order to ascertain which of them are most sensitive with reference to the reproduction numbers and infected components of the model. For instance, when the total infected population is used as response function (as can be observed in Table 2 and Fig. 3(f)), the most sensitive parameters driving the dynamics of both infections are the contact rates for SARS-CoV-2, HBV and co-infection transmissions. Thus, to reduce the spread of the dual infections within the population in an effective way, the model is incorporated with time-variant controls which comprises prevention efforts against SARS-CoV-2, HBV and their co-infection.

Highlights from the numerical assessment of the control system (27) include:

• (i.)SARS-CoV-2 only prevention effort could also cause great decrease in the co-infection cases (as observed in Fig. 4(c)).
• (ii.)HBV only prevention efforts also has a positive population level impact on the cases of co-infection (as shown in Fig. 5(c)).
• (iii.)Preventive effort against co-infection with dual infections was most significant in reducing co-infection cases (as can be seen in Fig. 6(a)).

The proposed model in this work did not capture the impact of vaccination for both diseases and re-infection with either or both diseases. If vaccination and or re-infection for SARS-CoV-2 and HBV is incorporated in the model, this could be possible extension to the model. Also, the model could consider separate recovery classes for SARS-CoV-2, HBV and co-infection. In addition, the mutations of several variants of SARS-CoV-2 demands more studies on their co-infection with other diseases. We could thus, model the dynamics of multiple variants of SARS-CoV-2 and co-infection with other emerging diseases.

CRediT authorship contribution statement

Andrew Omame: Formal analysis, Methodology, Writing – original draft, Editing, Software. Mujahid Abbas: Conceptualization, Validation, Writing – review & editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix. Adjoint system

$${\varrho }_1^{\prime }={\varrho }_1\left(\mu -\frac{\beta {}_C\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}-\frac{\beta {}_H\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}\right)\left(-\frac{\mathcal{I}{}_{CH}\beta {}_{CH}\left({\theta }_3-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}+\frac{\mathcal{S}\beta {}_C\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(+\frac{\mathcal{S}\beta {}_H\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(+\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-{\varrho }_6\left(\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(-\frac{\mathcal{I}{}_{CH}\beta {}_{CH}\left({\theta }_3-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}+\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(+\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)+{\varrho }_2\left(\frac{\beta {}_C\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}-\frac{\mathcal{S}\beta {}_C\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)+{\varrho }_4\left(\frac{\beta {}_H\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}-\frac{\mathcal{S}\beta {}_H\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)+\frac{\mathcal{I}{}_H\beta {}_C{\varrho }_5{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}+\frac{\mathcal{I}{}_C\beta {}_H{\varrho }_3{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}$$

$${\varrho }_2^{\prime }={\varrho }_1\left(\frac{\mathcal{S}\beta {}_C\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-\frac{\mathcal{S}\beta {}_C{\vartheta }_1\left({\theta }_1-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}\right)\left(+\frac{\mathcal{S}\beta {}_H\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}+\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-{\varrho }_3\left({\alpha }_1-\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-{\varrho }_6\left(\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(-\frac{\mathcal{I}{}_H\beta {}_C{\vartheta }_1{\zeta }_2\left({\theta }_4-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}+\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(+\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)+{\varrho }_2\left({\alpha }_1+\mu +\frac{\mathcal{S}\beta {}_C{\vartheta }_1\left({\theta }_1-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}-\frac{\mathcal{S}\beta {}_C\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-{\varrho }_5\left(\frac{\mathcal{I}{}_H\beta {}_C{\vartheta }_1{\zeta }_2\left({\theta }_4-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}-\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-\frac{\mathcal{S}\beta {}_H{\varrho }_4\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-1$$

$${\varrho }_3^{\prime }={\varrho }_2\left(\frac{\mathcal{S}\beta {}_C\left({\theta }_1-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}-\frac{\mathcal{S}\beta {}_C\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-{\varrho }_5\left(\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left({\theta }_4-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}-\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-{\varrho }_7\xi {}_C-{\varrho }_6\left(\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-\frac{\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}\right)\left(-\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left({\theta }_4-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}+\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)$$

$$\left(+\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)+{\varrho }_1\left(\frac{\mathcal{S}\beta {}_C\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-\frac{\mathcal{S}\beta {}_C\left({\theta }_1-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}\right)\left(+\frac{\mathcal{S}\beta {}_H\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}+\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)+{\varrho }_3\left(\eta {}_C+\mu +\xi {}_C-\frac{\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}+\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-\frac{\mathcal{S}\beta {}_H{\varrho }_4\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-1$$

$${\varrho }_4^{\prime }={\varrho }_1\left(\frac{\mathcal{S}\beta {}_C\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-\frac{\mathcal{S}\beta {}_H{\vartheta }_2\left({\theta }_2-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}\right)\left(+\frac{\mathcal{S}\beta {}_H\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}+\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-{\varrho }_5\left({\alpha }_2-\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-{\varrho }_6\left(\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(-\frac{\mathcal{I}{}_C\beta {}_H{\vartheta }_2{\zeta }_1\left({\theta }_4-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}+\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(+\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)+{\varrho }_4\left({\alpha }_2+\mu +\frac{\mathcal{S}\beta {}_H{\vartheta }_2\left({\theta }_2-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}-\frac{\mathcal{S}\beta {}_H\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-{\varrho }_3\left(\frac{\mathcal{I}{}_C\beta {}_H{\vartheta }_2{\zeta }_1\left({\theta }_4-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}-\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-\frac{\mathcal{S}\beta {}_C{\varrho }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-1$$

$${\varrho }_5^{\prime }={\varrho }_4\left(\frac{\mathcal{S}\beta {}_H\left({\theta }_2-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}-\frac{\mathcal{S}\beta {}_H\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-{\varrho }_3\left(\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left({\theta }_4-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}-\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-{\varrho }_7\xi {}_H-{\varrho }_6\left(\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-\frac{\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}\right)\left(-\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left({\theta }_4-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}+\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(+\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)+{\varrho }_1\left(\frac{\mathcal{S}\beta {}_C\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-\frac{\mathcal{S}\beta {}_H\left({\theta }_2-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}\right)\left(+\frac{\mathcal{S}\beta {}_H\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(+\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)+{\varrho }_5\left(\eta {}_H+\mu +\xi {}_H-\frac{\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}\right)\left(+\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-\frac{\mathcal{S}\beta {}_C{\varrho }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-1$$

$${\varrho }_6^{\prime }={\varrho }_1\left(\frac{\mathcal{S}\beta {}_C\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-\frac{\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}\right)\left(+\frac{\mathcal{S}\beta {}_H\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)$$

$$\left(+\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-{\varrho }_7\xi {}_{CH}+{\varrho }_6\left(\eta {}_{CH}+\mu +\xi {}_{CH}\right)\left(+\frac{\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}\right)\left(-\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(-\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-\frac{\mathcal{S}\beta {}_C{\varrho }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-\frac{\mathcal{S}\beta {}_H{\varrho }_4\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}+\frac{\mathcal{I}{}_H\beta {}_C{\varrho }_5{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}+\frac{\mathcal{I}{}_C\beta {}_H{\varrho }_3{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-1$$

$${\varrho }_7^{\prime }={\varrho }_7\mu -{\varrho }_6\left(\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}+\frac{\mathcal{I}{}_H\beta {}_C{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(+\frac{\mathcal{I}{}_C\beta {}_H{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)+{\varrho }_1\left(\frac{\mathcal{S}\beta {}_C\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)\left(+\frac{\mathcal{S}\beta {}_H\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}+\frac{\mathcal{I}{}_{CH}\mathcal{S}\beta {}_{CH}\left({\theta }_3-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}\right)-\frac{\mathcal{S}\beta {}_C{\varrho }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_1-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}-\frac{\mathcal{S}\beta {}_H{\varrho }_4\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_2-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}+\frac{\mathcal{I}{}_H\beta {}_C{\varrho }_5{\zeta }_2\left(\mathcal{I}{}_C+\mathcal{A}{}_C{\vartheta }_1\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}+\frac{\mathcal{I}{}_C\beta {}_H{\varrho }_3{\zeta }_1\left(\mathcal{I}{}_H+\mathcal{A}{}_H{\vartheta }_2\right)\left({\theta }_4-1\right)}{{\left(\mathcal{A}{}_C+\mathcal{A}{}_H+\mathcal{I}{}_C+\mathcal{I}{}_{CH}+\mathcal{I}{}_H+\mathcal{R}+\mathcal{S}\right)}^2}$$

Data availability

Data will be made available on request.