Dynamics of a cross-superdiffusive SIRS model with delay effects in transmission and treatment

Abstract

We investigate the dynamics of a SIRS epidemiological model taking into account cross-superdiffusion and delays in transmission, Beddington–DeAngelis incidence rate and Holling type II treatment. The superdiffusion is induced by inter-country and inter-urban exchange. The linear stability analysis for the steady-state solutions is performed, and the basic reproductive number is calculated. The sensitivity analysis of the basic reproductive number is presented, and we show that some parameters strongly influence the dynamics of the system. A bifurcation analysis to determine the direction and stability of the model is carried out using the normal form and center manifold theorem. The results reveal a proportionality between the transmission delay and the diffusion rate. The numerical results show the formation of patterns in the model, and their epidemiological implications are discussed.

Introduction

The investigation of diseases has a pragmatic relevance in view of all the implications in human life. Among those implications, we can point out the inherent health problems of a population, economic consequences at all levels of human activity, the psychological conditions derived from changes in the daily routines and many other factors which disrupt the functions of the members of the populations. In the history of the human species, there have been various infections which have affected in different degrees the evolution of humankind. As concrete examples, we can cite the Black Death which killed about half of the European population in the XIV century [1] and more recently various epidemics of influenza, middle-east respiratory syndrome (MERS), Ebola, tuberculosis and the Coronavirus disease also known as COVID-19 [1–3].

In view of the many consequences of diseases in human activities, it is important to propose reliable tools to predict their evolution and to identify measures to control them effectively. From the mathematical point of view, qualitative and quantitative analyses are constantly carried out in order to forecast future trends and to propose an optimal management of human and medical resources [4]. In particular, mathematical modeling has been shown to be an effective way to describe diseases in human populations [5, 6]. Concretely, compartmental systems based on coupled systems of ordinary differential equations, delayed differential equation, stochastic differential equations and partial differential equations have been used to describe the propagation of diseases [7, 8]. Various classical models have been proposed in the form of susceptible–infected (SI) and susceptible–infected–recovered (SIR) models as well as susceptible–infected–recovered–susceptible (SIRS) or susceptible–exposed–infected–recovered (SEIR) models [9–11].

Classical epidemiological models are constantly improved to provide more realistic descriptions. For example, some of those systems consider nonlinear incidence rates in both transmission and treatment [12, 13]. In particular, the Beddington–DeAngelis incidence rate is frequently used to take into account the limitations of disease spreading due to treatment and preventive measures [13]. On the other hand, the Holling type II treatment is capable of accounting for the limitation access to treatment due to financial and economic costs, low test of infected during the disease spreading and unexpected effects of treatment [12]. The analysis of such systems is generally carried out using time series which forecast the epidemiological behavior at a giving time. However, this approach is limited in view that it neglects the spatial structure of disease spreading and can be inapplicable for moving human populations [14]. The spatial structure of epidemics exhibits a heterogeneous macrostructure induced by symmetry breaking process with certain regularity in space known as patterns [2] and constitutes a topic of interest in nonlinear dynamics [2].

Reaction–diffusion in epidemiological systems suppose that the individuals migrate randomly or spread in all directions with the same probability [2]. In real situations, this occurs when individuals move to find resources for their survival [2]. Specifically, superdiffusion arises when populations exchange between cities or countries and the interactions between individuals are allowed to be through long distances [15–17]. Superdiffusion is also known as fractional or anomalous diffusion and corresponds to Lévy flights processes modeled by a fractional Laplacian operator [15]. The cross-diffusion is inherent to biological systems. There are currently studies on cross-superdiffusion in food chain model [18], cross-diffusion and time delay in prey–predator model [19]. In epidemiological dynamics, the cross-diffusion arises when susceptible individuals move in the direction of lower or higher density of infected individuals [20, 21]. In this sense, Chang et al. [22] investigated the dynamics of epidemic cross-diffusive model and revealed that the ability of susceptible individuals to be away of infected individuals can greatly contribute to counter the disease spreading.

In the same direction, Fan [21] improved qualitative analysis of cross-diffusive SIS epidemic model and emphasized the relevance of distancing susceptible individuals from infected populations in the disease controlling. Beside, some studies have been carried out to show that the superdiffusion in epidemiological models can be a way to explain inter-urban and inter-country exchange [15, 17]; however, cross-superdiffusion with time delays has not yet been taking into account in such a process specially in the sense of pattern formation to the best of our knowledge. In the present manuscript, we investigate both analytically and numerically dynamical behavior of a SIRS cross-superdiffusive epidemiological model under the cumulative effect of time delays and saturated incidence rate in both treatment and transmission. All of parameters taking into account in the model are well known to be simultaneously inherent to epidemiological systems. We use in this sens time-delayed Holling type II treatment and Beddington–DeAngelis-type incidence rate in transmission. The model, in the sense of pattern formation, may provide a more realistic description of the epidemiological phenomenon.

The rest of the paper is organized as follows. In Sect. 2, we introduce the mathematical model under investigation in this work together with the model parameters. In Sect. 3, the linear stability analysis of the epidemiological model is presented. We derive the basic reproductive number from the disease-free equilibrium state. In addition, we derive different analytical expressions of the critical delays from the endemic equilibrium. In Sect. 4, we analyze the direction and the stability of the Hopf bifurcation in the system. In Sect. 5, we present various analytical and numerical simulations and provide discussions and interpretations based on those results.

Mathematical model

In this section, we introduce the mathematical model under study in this work. To be more precise, we will consider a time-delayed SIRS epidemic model with space-fractional cross-diffusion, Beddington–DeAngelis-type incidence and Holling type II treatment rates. Throughout, ${\mathbb{R}}_{+}$ represents the set of positive numbers, and ${\overline{\mathbb{R}}}_{+}={\mathbb{R}}_{+}\cup \{0\}$. We suppose that the functions $S,I,R:{\mathbb{R}}^2\times {\overline{\mathbb{R}}}_{+}\to \mathbb{R}$ are sufficiently smooth. From a more physical perspective, $S=S(x,y,t)$, $I=I(x,y,t)$ and $R=R(x,y,t)$ represent, respectively, the numbers of susceptible, infected and recovered individuals at the point $(x,y)\in {\mathbb{R}}^2$ and time $t\ge 0$. The respective population size is denoted by $N=N(x,y,t)$ and it is defined by the identity

$$\begin{array}{c}\hfill N(x,y,t)=S(x,y,t)+I(x,y,t)+R(x,y,t),\end{array}$$ 1

for each $(x,y,t)\in {\mathbb{R}}^2\times {\mathbb{R}}^{+}$. Under these circumstances, the epidemiological model considered in this work is given by the superdiffusive system of partial differential equations

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& \frac{\partial S}{\partial t}=A-\nu S\hfill \\ \hfill & -\frac{\beta S(t-{\tau }_1)I(t-{\tau }_1)}{1+\alpha S(t-{\tau }_1)+\gamma I(t-{\tau }_1)}+\psi R\hfill \\ \hfill & +d_1{\mathrm{\nabla }}^{\delta }S+d_2{\mathrm{\nabla }}^{\delta }I,\hfill \\ \hfill & \frac{\partial I}{\partial t}\hfill \\ \hfill & =\frac{\beta S(t-{\tau }_1)I(t-{\tau }_1)}{1+\alpha S(t-{\tau }_1)+\gamma I(t-{\tau }_1)}-(\nu +d+\zeta )I\hfill \\ \hfill & -\frac{aI(t-{\tau }_2)}{1+bI(t-{\tau }_2)}+d_3{\mathrm{\nabla }}^{\delta }I,\hfill \\ \hfill & \frac{\partial R}{\partial t}\hfill \\ \hfill & =\frac{aI(t-{\tau }_2)}{1+bI(t-{\tau }_2)}+\zeta I-(\nu +\psi )R+d_4{\mathrm{\nabla }}^{\delta }R.\hfill \end{array}\right)\\ \hfill \end{array}$$ 2

In the mathematical model (2), the constant A represents the recruitment rate of the susceptible individuals, $\beta$ denotes the effective contact rate (transmission rate), $\mu$ is the natural death rate of the population, $\psi$ denotes the rate at which recovered individuals lose immunity and return to be susceptible (also called the reinfection force), d is the death rate due to the disease, $\zeta$ is the recovery rate, $\alpha$ is the measure of inhibitions due to social awareness among the susceptible individuals, $\gamma$ is the measure of inhibition taken by infected individuals, a is the cure rate and b represents treatment inhibition due to financial support, low test and unexpected effects of treatment inside the body [13, 23].

The parameter $d_1$ is the susceptible individuals diffusion rate, $d_2$ is the cross-diffusion rate, $d_3$ is the infected individuals diffusion rate and $d_4$ is the recovered individuals diffusion rate. The cross-diffusion coefficient $d_2>0$ denotes the movement of the susceptible in the direction of lower concentration of the infected, while $d_2<0$ means that the susceptible tend to diffuse in the direction of higher concentration of the infected [20, 21]. Meanwhile, ${\tau }_1$ is called the transmission delay and denotes the time delay after which one infected subject effectively becomes infectious. The delay ${\tau }_2$ is the time that the treatment takes before affecting the dynamics of the system, which can be attributed to some biological and chemical process, or some fluctuations due to the organism. Various other physical assumptions are inherent to the mathematical model (2). For example, during the time period $[t-{\tau }_1,t)$, a portion of susceptible subjects equal to

$$\begin{array}{c}\hfill \frac{\beta S(t-{\tau }_1)I(t-{\tau }_1)}{1+\alpha S(t-{\tau }_1)+\gamma I(t-{\tau }_1)}\end{array}$$ 3

becomes infectious at time t [24]. Moreover, we assume that a number of infected individuals equal to

$$\begin{array}{c}\hfill \frac{aI(t-{\tau }_2)}{1+bI(t-{\tau }_2)}\end{array}$$ 4

were treated during the period $[t-{\tau }_2,t)$ and are fully recovered at time t.

In this work, the fractional derivative is described by the Weyl fractional operator ${\mathrm{\nabla }}^{\delta }$, which is given by

$$\begin{array}{c}\hfill {\mathrm{\nabla }}^{\delta }=\frac{{\partial }^{\delta }}{\partial X^{\delta }}+\frac{{\partial }^{\delta }}{\partial Y^{\delta }}.\end{array}$$ 5

It is worthwhile to recall that anomalous diffusion holds a power-law scaling relation of the form $⟨x^2(t)⟩\propto t^{\delta }$, which is present in different types of systems. The constant $\delta$ is known as the anomalous diffusion exponent. In one dimension, the Weyl fractional operator is defined by

$$\begin{array}{c}\hfill {\displaystyle {\mathrm{\nabla }}^{\delta }w=-c_{\delta }(D_{-}^{\delta }w+D_{+}^{\delta }w),}\end{array}$$ 6

where w stand for S, I or R. Moreover,

$$\begin{array}{cc}\hfill c_{\delta }& =\frac{1}{2cos(\delta \pi /2)},\hfill \end{array}$$ 7

$$\begin{array}{cc}\hfill D_{+}^{\delta }w& =\frac{1}{\mathrm{\Gamma }(2-\delta )}\frac{d^2}{d{x}^2}{\int }_{-\infty }^x\frac{w(\xi ,t)}{{(x-\xi )}^{{\delta }^{-}1}}d\xi ,\hfill \end{array}$$ 8

$$\begin{array}{cc}\hfill D_{-}^{\delta }w& =\frac{1}{\mathrm{\Gamma }(2-\delta )}\frac{d^2}{d{x}^2}{\int }_x^{+\infty }\frac{w(\xi ,t)}{{(\xi -x)}^{{\delta }^{-}1}}d\xi ,\hfill \end{array}$$ 9

where $\mathrm{\Gamma }$ is the usual Gamma function that generalizes factorials.

It is important to recall that the equivalent definition of the fractional Laplacian in Fourier space allows for a simple generalization of the fractional operator to several spatial variables. Indeed, notice that the anomalous diffusive operator ${(-\mathrm{\Delta })}^{\delta /2}$ is defined through the Fourier transform as

$$\begin{array}{c}\hfill -{(-\mathrm{\Delta })}^{\delta /2}S(x)=-{\mathcal{F}}^{-1}{|k|}^{\delta }\mathcal{F}S(x),\end{array}$$ 10

for each $x\in {\mathbb{R}}^n$ and $\delta >0$. Thus, $-{(-\mathrm{\Delta })}^{\frac{\delta }{2}}$ is frequently denoted by ${\mathrm{\nabla }}^{\delta }$ and satisfies the identity with Fourier transforms

$$\begin{array}{c}\hfill \mathcal{F}({\mathrm{\nabla }}^{\delta }S)=-{|k|}^{\delta }\mathcal{F}(S),\end{array}$$ 11

with $|k|=\sqrt{k_1^2+\dots +k_n^2}$. In various spatial dimensions, the function S is replaced by $S(x_1,x_2,...,x_n,t)$, and the operator ${\mathrm{\nabla }}^{\delta }$ is defined by its Fourier transform, which is actually equal to $-{|k|}^{\delta }\mathcal{F}(S)$ [25].

Steady states and linear stability analysis

In this section, we perform the linear stability analysis of System 2, and provide conditions that result in the formation of patterns. Beforehand it is important to recall that there exist several techniques to investigate the dynamical behavior of nonlinear–linear reaction diffusion systems among which linear stability analysis and nonlinear stability analysis. These techniques consist to perturb the steady state of the system. The nonlinear stability deals with the finite perturbation, and the linear stability analysis deals with infinitesimal perturbation. The linear stability analysis applied in this model is attributed to the highest sensibility of biological systems [26].

Based on this method, we derive the uniform steady states for the system (2) and evaluate their effects in the dynamical behavior of the model trough the linear stability analysis. In epidemiology, this serves to predict the general behavior of the system in terms of stability or instability. The stable domain in this context describes the limited interaction between different model compartments, contrary to the unstable domain which describes the interactions leading to the growth of the disease.

To determine the equilibrium points of system (2), we assume that the solutions S, I and R are constant and solve the algebraic system resulting from that model when the derivatives are all equal to zero [10]. Through the linear stability analysis of this system, we obtain the Jacobian which is the following real matrix of size $3\times 3$:

$$\begin{array}{c}\hfill M=\left(\begin{array}{ccc}{m}_{11}+\beta g_1& m_{12+\beta g_2}& m_{13}\\ m_{21}-\beta g_1& m_{22}-\beta g_2+a{g}_3& m_{23}\\ m_{31}& m_{23}-a{g}_3& m_{33}\end{array}\right).\end{array}$$ 12

We just need to point out here that the constants are defined by ${\mathrm{\Gamma }}_1=\nu +d+\zeta$, ${\mathrm{\Gamma }}_2=\nu +\psi$, $m_{11}=-\nu -d_1{|K|}^{\delta }$, $m_{12}=-d_2{|K|}^{\delta }$, $m_{13}=\psi$, $m_{21}=0$, $m_{22}=-{\mathrm{\Gamma }}_1-d_3{|K|}^{\delta }$, $m_{23}=0$, $m_{31}=0$, $m_{32}=\zeta$, $m_{33}=-{\mathrm{\Gamma }}_2-d_4{|K|}^{\delta }$, where

$$\begin{array}{cc}\hfill g_1& =\frac{\alpha S^{\ast }{I}^{\ast }}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^2}-\frac{I^{\ast }}{1+\alpha S^{\ast }+\gamma I^{\ast }},\hfill \end{array}$$ 13

$$\begin{array}{cc}\hfill g_2& =\frac{\gamma S^{\ast }{I}^{\ast }}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^2}-\frac{S^{\ast }}{1+\alpha S^{\ast }+\gamma I^{\ast }},\hfill \end{array}$$ 14

$$\begin{array}{cc}\hfill g_3& =\frac{b{I}^{\ast }}{{(1+b{I}^{\ast })}^2}-\frac{1}{1+b{I}^{\ast }}.\hfill \end{array}$$ 15

Disease-free equilibrium and basic reproduction number

Notice that the system has two steady-state solutions, namely the disease-free and the endemic equilibria. The disease-free equilibrium is the point $E_0=(\frac{A}{\nu },0,0)$ at which there is no disease. When the disease is present, we are interested in the mechanisms of disease transmission. In particular, we are generally interested in the basic reproductive number, which yields how many people can be infected by a single infected individual [27]. In others words, the basic reproductive number is the average number of secondary cases generated by a single primary case in a fully susceptible population. This indicator is helpful to understand the behavior of the disease spreading and to improve public health strategies [28].

Table 1.

Parameter values of the mathematical model (2)

Parameter	Units
A	Person $·$ day${}^{-1}$
$\alpha$	Person${}^{-1}$
$\beta$	Person${}^{-1}$ $·$ day${}^{-1}$
$\gamma$	Person${}^{-1}$
$\mu$	Day${}^{-1}$
d	Day${}^{-1}$
a	Day${}^{-1}$
b	Person${}^{-1}$
$\psi$	Day${}^{-1}$
${\tau }_1$	Day
${\tau }_2$	Day
$d_1$	$K{m}^2.da{y}^{-1}$
$d_2$	$K{m}^2.da{y}^{-1}$
$d_3$	$K{m}^2.da{y}^{-1}$
$d_4$	$K{m}^2.da{y}^{-1}$

To date, the specialized literature proposes a vast amount of approaches to compute the basic reproductive number. Among other techniques, the next-generation matrix is one of the more widely used in the literature. This methodology biologically groups the compartmental model into two vectors which are denoted by ${\mathcal{F}}_i$ and ${\mathcal{V}}_i$, and represent, respectively, the rate of appearance of new infections in compartment i and the remaining transitional terms (namely births, deaths, disease progression and recovery [12, 27]?). Following [12], we readily obtain that $N\equiv \frac{A}{\nu }=S+R+R$ and, as a consequence, $R=\frac{A}{\nu }-S-I$. Moreover, the following system is obtained:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill \frac{\mathit{dI}}{\mathit{dt}}& =\frac{\beta SI}{1+\alpha S+\gamma I}-{\mathrm{\Gamma }}_{1}I-\frac{\mathit{aI}}{1+bI},\hfill \\ \hfill \frac{\mathit{dS}}{\mathit{dt}}& =A-\nu S-\frac{\beta SI}{1+\alpha S+\gamma I}+\psi \left(\frac{A}{\nu },-,S,-,I\right).\hfill \end{array}\right)\\ \hfill \end{array}$$ 16

For the remainder, let ${\kappa }_0=({\mathrm{\Gamma }}_1+a)(\nu +\psi )$. Using the method of the next-generation matrix, we define the system

$$\begin{array}{c}\hfill \frac{d{x}_i}{\text{d}t}={\mathcal{F}}_i-{\mathcal{V}}_i,\end{array}$$ 17

where

$$\begin{array}{cc}\hfill {\mathcal{F}}_i& =\left(\begin{array}{c}{\displaystyle \frac{\beta SI}{1+\alpha S+\gamma I}}\\ 0\end{array}\right),\hfill \end{array}$$ 18

$$\begin{array}{cc}\hfill {\mathcal{V}}_i& =\left(\begin{array}{c}{\mathrm{\Gamma }}_{1}I+{\displaystyle \frac{\mathit{aI}}{1+bI}}\\ -A+\nu S+{\displaystyle \frac{\beta SI}{1+\alpha S+\gamma I}}-\psi \left({\displaystyle \frac{A}{\nu }},-,S,-,I\right)\end{array}\right).\hfill \end{array}$$ 19

As a consequence, the Jacobian of the system (16) at the disease-free equilibrium solution yields the square matrices

$$\begin{array}{cccc}\hfill F& =\left(\begin{array}{cc}{\displaystyle \frac{\beta A}{\nu +\alpha A}}& 0\\ 0& 0\end{array}\right),\hfill & \hfill V& =\left(\begin{array}{cc}{\mathrm{\Gamma }}_1+a& 0\\ {\displaystyle \frac{\beta A}{\nu +\alpha A}}+\psi & \nu +\psi \end{array}\right).\hfill \end{array}$$ 20

Straightforward algebraic calculations readily show that

$$\begin{array}{c}\hfill F{V}^{-1}=\frac{1}{{\kappa }_0}{\displaystyle \frac{\beta A}{\nu +\alpha A}}\left(\begin{array}{cc}\nu +\psi & -{\displaystyle \frac{\beta A}{\nu +\alpha A}}+\psi \\ 0& 0\end{array}\right).\\ \hfill \end{array}$$ 21

We conclude that the basic reproductive number of our epidemiological system is given by the formula

$$\begin{array}{c}\hfill R_0^2=\frac{\beta A}{({\mathrm{\Gamma }}_1+a)(\nu +\alpha A)}.\end{array}$$ 22

Endemic equilibrium and critical delays

We will investigate now the stability of system (2) around the endemic equilibrium point. For convenience, we will represent this point by $E_e=(S^{\ast },I^{\ast },R^{\ast })$. Evidently,

$$\begin{array}{cc}\hfill S^{\ast }& =\frac{(1+\gamma I^{\ast })(a+{\mathrm{\Gamma }}_1(1+b{I}^{\ast }))}{(\beta (1+b{I}^{\ast })-\alpha (a+{\mathrm{\Gamma }}_1(1+b{I}^{\ast }))},\hfill \end{array}$$ 23

$$\begin{array}{cc}\hfill R^{\ast }& =\frac{a{I}^{\ast }+(1+b{I}^{\ast })\zeta I^{\ast }}{(1+b{I}^{\ast }){\mathrm{\Gamma }}_2}.\hfill \end{array}$$ 24

Meanwhile, the value of $I^{\ast }$ is the solution of the following fourth-order polynomial equation:

$$\begin{array}{c}\hfill C_4{(I^{\ast })}^4+C_3{(I^{\ast })}^3+C_2{(I^{\ast })}^2+C_1{I}^{\ast }+C_0=0.\end{array}$$ 25

The constants $C_1$, $C_2$, $C_3$ and $C_4$ are defined in terms of the parameters in Table 2 in 7 For the sake of physical relevance, the number $I^{\ast }$ must be positive, so we will employ Descartes’ rule of signs to assure the positivity of this constant. According to that rule, there must be at least one sign change in the coefficients of the polynomial equation (25) to guarantee the existence of a least one positive root [29, 30]. Observe that this condition holds, for example, when $C_0<0$, $C_1<0$, $C_2<0$, $C_3<0$ and $C_4>0$. In turn, notice that the characteristic polynomial associated with the Jacobian matrix is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p(\lambda )& ={\lambda }^3+{\delta }_1{\lambda }^2+{\delta }_2\lambda +{\delta }_3\hfill \\ \hfill & +({\rho }_1{\lambda }^2+{\rho }_2\lambda +{\rho }_3)e^{-\lambda {\tau }_1}\hfill \\ \hfill & +({\rho }_4{\lambda }^2+{\rho }_5\lambda +{\rho }_6)e^{-\lambda {\tau }_2}\hfill \\ \hfill & +({\rho }_7\lambda +{\rho }_8)e^{-\lambda ({\tau }_1+{\tau }_2)}.\hfill \end{array}\end{array}$$ 26

Notice that the real part of any root $\lambda$ of $p(\lambda )$ dependents on $\tau$. It follows that the stability of $E^{\ast }$ changes only when the real part of $\lambda$ changes its sign, and this happens when $\tau$ attains a critical value ${\tau }_c$. Moreover, this critical value is readily reached when $\lambda =\pm i{w}_k$, for $w_k>0$ (see [31]).

Table 2.

Table of parameters in the definitions of the constants $C_1$, $C_2$, $C_3$ and $C_4$

${\delta }_1=-m_{11}-m_{22}-m_{33}$

${\delta }_2=m_{11}{m}_{33}+m_{22}{m}_{33}+m_{11}{m}_{22}$

${\delta }_3=-m_{11}{m}_{22}{m}_{33}$

${\rho }_1=(g_2-g_1)\beta$

${\rho }_2=(g_1-g_2)\beta m_{33}+(m_{12}+m_{22})\beta g_1-g_2\beta m_{11}$

${\rho }_3=(m_{13}{m}_{32}-m_{12}{m}_{33}-m_{22}{m}_{33})\beta g_1+m_{11}{m}_{33}\beta g_2$

${\rho }_4=-a{g}_3$

${\rho }_5=(m_{11}+m_{33})a{g}_3$

${\rho }_6=-m_{11}{m}_{33}a{g}_3$

${\rho }_7=a\beta g_1{g}_3$

${\rho }_8=-(m_{13}+m_{33})a\beta g_1{g}_3$

$A_1={\rho }_3+{\rho }_8-{\rho }_1{\omega }_1^2$

$A_2={\rho }_6+{\rho }_8-{\rho }_4{\omega }_2^2$

$B_1=({\rho }_2+{\rho }_7){\omega }_1$

$B_2=({\rho }_5+{\rho }_7){\omega }_2$

$D_1=-({\delta }_2{\omega }_1+{\rho }_5{\omega }_1-{\omega }_1^3)$

$D_2=-(({\delta }_2+{\rho }_2)\omega -{\omega }_2^3))$

$e_1=(-{\delta }_1^2+2{\delta }_2+{\rho }_1^2-2{\delta }_1{\rho }_4-{\rho }_4^2+2{\rho }_5)$

$e_2=2{\delta }_1{\delta }_3-{({\delta }_2-{\rho }_5)}^2-2{\rho }_1({\rho }_3+{\rho }_8)+2{\delta }_3{\rho }_4+2({\delta }_1+{\rho }_4){\rho }_6+{({\rho }_2+{\rho }_7)}^2$

$e_3=-{\delta }_3^2+{\rho }_3^2-2{\delta }_3{\rho }_6-{\rho }_6^2+2{\rho }_3{\rho }_8+{\rho }_8^2$

$e_{11}=2{\delta }_2-{({\delta }_1+{\rho }_1)}^2+2{\rho }_2+{\rho }_4^2$

$e_{33}=-{\delta }_3^2-2{\delta }_3{\rho }_3-{\rho }_3^2+{\rho }_6^2+2{\rho }_6{\rho }_8+{\rho }_8^2$

$e_{22}=2{\delta }_3({\delta }_1+{\rho }_1)-{({\delta }_2+{\rho }_2)}^2+2({\delta }_1+{\rho }_1){\rho }_3-2{\rho }_4({\rho }_6+{\rho }_8)+{({\rho }_5+{\rho }_7)}^2$

In the following, we will study various cases in detail, depending on the values of the parameters ${\tau }_1$ and ${\tau }_2$.

The case with cross-superdiffusion and delay in transmission: ${\tau }_2=0$ and ${\tau }_1\ne 0$.

In this case, the characteristic equation is

$$\begin{array}{cc}\hfill p(\lambda )=& {\lambda }^3+({\delta }_1+{\rho }_4){\lambda }^2+({\delta }_2+{\rho }_5)\lambda +{\delta }_3+{\rho }_6\hfill \\ \hfill & +[{\rho }_1{\lambda }^2+({\rho }_2+{\rho }_7)\lambda +{\rho }_3+{\rho }_8]e^{-\lambda {\tau }_1}.\hfill \\ \hfill \end{array}$$ 27

Letting $\lambda =i\omega$ and separating the real from the imaginary parts, we obtain the following algebraic system:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill A_1({\omega }_1)cos({\omega }_1{\tau }_1)+B_1({\omega }_1)sin({\omega }_1{\tau }_1)& =C_1({\omega }_1),\hfill \\ \hfill B_1({\omega }_1)cos({\omega }_1{\tau }_1)-A_1({\omega }_1)sin({\omega }_1{\tau }_1)& =D_1({\omega }_1).\hfill \end{array}\right)\\ \hfill \end{array}$$ 28

Raising both sides of these equations to the second power and using trigonometric identities, we obtain the following polynomial function in ${\omega }_1$:

$$\begin{array}{c}\hfill {\omega }_1^6-e_1{\omega }_1^4-e_2{\omega }_1^2+e_3=0.\end{array}$$ 29

We are interested in the ordered pair $({\omega }_1,{\tau }_1)$. Observe that $\omega$ can be obtain considering the Pythagorean identity $cos{(\omega {\tau }_1)}^2+sin{(\omega {\tau }_1)}^2=1$. Taking into account only positive values, we obtain that ${\omega }_1={\omega }_{1c}$, where

$$\begin{array}{c}\hfill {\omega }_{1c}=\sqrt{\frac{e_1+\sqrt{e_1^2+3e_2}}{3}}.\end{array}$$ 30

Substituting now into (28), we readily reach that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill cos({\omega }_1{\tau }_1)& =\frac{B_1{D}_1+C_1{A}_1}{A_1^2+B_1^2},\hfill \\ \hfill sin({\omega }_1{\tau }_1)& =\frac{B_1{C}_1-D_1{A}_1}{A_1^2+B_1^2}.\hfill \end{array}\end{array}$$ 31

Using now (31) and following [31], we obtain

$$\begin{array}{c}\hfill {\tau }_{1c}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\omega }_{c1}}}\left[2,\pi ,j,+,arctan,{\displaystyle \frac{T_1}{T_2}}\right],\hfill & \text{if}{T}_1>0,\hfill \\ {\displaystyle \frac{1}{{\omega }_{c1}}}\left[\pi ,+,2,\pi ,j,+,arctan,{\displaystyle \frac{T_1}{T_2}}\right],\hfill & \text{if}{T}_1<0,\hfill \\ {\displaystyle \frac{1}{{\omega }_{c1}}}\left[2,\pi ,+,2,\pi ,j,+,arctan,{\displaystyle \frac{T_1}{T_2}}\right],\hfill & \text{if}{T}_1<0,T_2>0,\hfill \end{array}\right)\\ \hfill \end{array}$$ 32

where $j=0,1...$, $T_1=B_1{C}_1-D_1{A}_1$ and $T_2=B_1{D}_1+C_1{A}_1$. Finally, differentiating now with respect to $\lambda$ and ${\tau }_1$, we obtain the transversality condition

$$\begin{array}{cc}\hfill & {\left[\frac{d\lambda }{d{\tau }_1}\right]}^{-1}\hfill \\ \hfill & =-\frac{3{\lambda }^2+2({\delta }_1+{\rho }_4)\lambda +{\delta }_2+{\rho }_5}{\lambda ({\lambda }^3+({\delta }_1+{\rho }_4){\lambda }^2+({\delta }_2+{\rho }_5)\lambda +{\delta }_3+{\rho }_6)}\hfill \\ \hfill & +\frac{2{\rho }_1\lambda +{\rho }_2+{\rho }_7}{{\tau }_1({\rho }_1{\lambda }^2+({\rho }_2+{\rho }_7)\lambda +{\rho }_3+{\rho }_8)}\hfill \\ \hfill & -\frac{{\tau }_1}{\lambda }\ne 0.\hfill \end{array}$$ 33

The case with cross-superdiffusion and delay in treatment:${\tau }_1=0$ and ${\tau }_2\ne 0$.

In this case, the characteristic equation is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p(\lambda )& ={\lambda }^3+({\delta }_1+{\rho }_1){\lambda }^2+({\delta }_2+{\rho }_2)\lambda +{\delta }_3+{\rho }_3\hfill \\ \hfill & +({\rho }_4{\lambda }^2+({\rho }_5+{\rho }_7)\lambda +{\rho }_6+{\rho }_8)e^{-\lambda {\tau }_2}.\hfill \end{array}\\ \hfill \end{array}$$ 34

Proceeding as in the previous case, we may reach the system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill A_2({\omega }_2)cos({\omega }_2{\tau }_2)+B_2({\omega }_2)sin({\omega }_2{\tau }_2)& =C_2({\omega }_2),\hfill \\ \hfill B_2({\omega }_2)cos({\omega }_2{\tau }_2)-A_2({\omega }_2)sin({\omega }_2{\tau }_2)& =D_2({\omega }_2).\hfill \end{array}\right)\\ \hfill \end{array}$$ 35

Using the Pythagorean identity, squaring both sides on both equations of system (35) and adding both equations together, we easily obtain that

$$\begin{array}{c}\hfill {\omega }_2^6-e_{11}{\omega }_2^4-e_{22}{\omega }_2^2+e_{33}=0.\end{array}$$ 36

A simple substitution leads then to the system

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill cos({\omega }_2{\tau }_2)& =\frac{B_2{D}_2+C_2{A}_2}{A_2^2+B_2^2},\hfill \\ \hfill sin({\omega }_2{\tau }_2)& =\frac{B_2{C}_2-D_2{A}_2}{A_2^2+B_2^2}.\hfill \end{array}\end{array}$$ 37

As a consequence, we obtain that

$$\begin{array}{c}\hfill {\tau }_{2c}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\omega }_{2c}}}\left[2,\pi ,j,+,arctan,{\displaystyle \frac{T_3}{T_4}}\right],\hfill & \text{if}{T}_3>0,\hfill \\ {\displaystyle \frac{1}{{\omega }_{2c}}}\left[\pi ,+,2,\pi ,j,+,arctan,{\displaystyle \frac{T_3}{T_4}}\right],\hfill & \text{if}{T}_4<0,\hfill \\ {\displaystyle \frac{1}{{\omega }_{2c}}}\left[2,\pi ,+,2,\pi ,j,+,arctan,{\displaystyle \frac{T_3}{T_4}}\right]+,\hfill & \text{if}{T}_3>0,T_4<0.\hfill \end{array}\right)\\ \hfill \end{array}$$ 38

where $T_3=B_2{C}_2-D_2{A}_2$ and $T_4=B_2{D}_2+C_2{A}_2$. Finally, we may obtain the following identity by differentiating with respect to $\lambda$:

$$\begin{array}{cc}\hfill & {\left[\frac{d\lambda }{d{\tau }_2}\right]}^{-1}\hfill \\ \hfill & =-\frac{3{\lambda }^2+2({\delta }_1+{\rho }_1)\lambda +{\delta }_2+{\rho }_2}{\lambda ({\lambda }^3+({\delta }_1+{\rho }_1){\lambda }^2+({\delta }_2+{\rho }_2)\lambda +{\delta }_3+{\rho }_3)}\hfill \\ \hfill & +\frac{2{\rho }_4\lambda +{\rho }_5+{\rho }_7}{\lambda ({\rho }_4{\lambda }^2+({\rho }_5+{\rho }_7)\lambda +{\rho }_6+{\rho }_8)}\hfill \\ \hfill & -\frac{{\tau }_2}{\lambda }\ne 0.\hfill \\ \hfill \end{array}$$ 39

The case with cross-superdiffusion, delays in transmission and treatment:${\tau }_1\ne 0$ and ${\tau }_2\ne 0$.

The characteristic equation in this case is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill p(\lambda )& ={\lambda }^3+{\delta }_1{\lambda }^2+{\delta }_2\lambda +{\delta }_3\hfill \\ \hfill & +({\rho }_1{\lambda }^2+{\rho }_2\lambda +{\rho }_3)e^{-\lambda {\tau }_1}\hfill \\ \hfill & +({\rho }_4{\lambda }^2+{\rho }_5\lambda +{\rho }_6)e^{-\lambda {\tau }_2}\hfill \\ \hfill & +({\rho }_7\lambda +{\rho }_8)e^{-\lambda ({\tau }_1+{\tau }_2)}.\hfill \end{array}\end{array}$$ 40

Using a similar approach, we can reach the algebraic system of equations

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill M_1(\omega )cos(\omega {\tau }_1)+M_2(\omega )sin(\omega {\tau }_1)& =M_3(\omega ),\hfill \\ \hfill M_2(\omega )cos(\omega {\tau }_1)-M_1(\omega )sin(\omega {\tau }_1)& =M_4(\omega ).\hfill \end{array}\right)\\ \hfill \end{array}$$ 41

As a consequence,

$$\begin{array}{c}\hfill {\tau }_c=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\omega }_c}}\left[2,\pi ,j,+,arctan,{\displaystyle \frac{T_5}{T_6}}\right],\hfill & \text{if}{T}_5>0,\hfill \\ {\displaystyle \frac{1}{{\omega }_c}}\left[\pi ,+,2,\pi ,j,+,arctan,{\displaystyle \frac{T_5}{T_6}}\right],\hfill & \text{if}{T}_6<0,\hfill \\ {\displaystyle \frac{1}{{\omega }_c}}\left[2,\pi ,+,2,\pi ,j,+,arctan,{\displaystyle \frac{T_5}{T_6}}\right],\hfill & \text{if}{T}_5>0,T_6<0,\hfill \end{array}\right)\\ \hfill \end{array}$$ 42

where $T_5=M_3(\omega )M_2(\omega )-M_1(\omega )M_4(\omega )$, $T_6=M_2(\omega )M_4(\omega )+M_3(\omega )M_1(\omega )$,

$$\begin{array}{cc}\hfill M_1(\omega )=& {\rho }_3-{\rho }_1{\omega }^2+{\rho }_{8}cos(\omega {\tau }_2)+{\rho }_7\omega sin(\omega {\tau }_2),\hfill \\ \hfill M_2(\omega )=& {\rho }_2\omega +{\rho }_7\omega cos(\omega {\tau }_2)-{\rho }_{8}sin(\omega {\tau }_2),\hfill \\ \hfill M_3(\omega )=& -{\delta }_3+{\delta }_1{\omega }^2-({\rho }_6-{\rho }_4{\omega }^2)cos(\omega {\tau }_2),\hfill \\ \hfill M_4(\omega )=& {\omega }^3-{\delta }_2\omega -({\rho }_5+{\rho }_4\omega )\omega sin(\omega {\tau }_2).\hfill \end{array}$$ 43

Finally, differentiate the characteristic polynomial with respect ${\tau }_1$ and obtain the following condition for Hopf bifurcation:

$$\begin{array}{c}\hfill {\left[\frac{d\lambda }{d{\tau }_1}\right]}^{-1}=\frac{{\sigma }_1+{\sigma }_2}{{\sigma }_3}\ne 0.\end{array}$$ 44

In this expression,

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill {\sigma }_1& =3{\lambda }^2+2{\delta }_1\lambda +{\delta }_2+(2{\rho }_1\lambda +{\rho }_2)e^{-\lambda {\tau }_1}\hfill \\ \hfill & -{\tau }_1[{\rho }_1{\lambda }^2+{\rho }_2\lambda +{\rho }_3]e^{-\lambda {\tau }_1},\hfill \end{array}\hfill \end{array}$$ 45

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill {\sigma }_2& =(2{\rho }_4\lambda +{\rho }_5)e^{-\lambda {\tau }_1}\hfill \\ \hfill & -{\tau }_2({\rho }_4{\lambda }^2+{\rho }_5\lambda +{\rho }_6)e^{-\lambda {\tau }_2}\hfill \\ \hfill & +({\rho }_7-({\tau }_1+{\tau }_2)({\rho }_7\lambda +{\rho }_8))e^{-\lambda ({\tau }_1+{\tau }_2)},\hfill \end{array}\hfill \end{array}$$ 46

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill {\sigma }_3& =\lambda ({\rho }_1{\lambda }^2+{\rho }_2\lambda +{\rho }_3+{\rho }_3)e^{-\lambda {\tau }_1}\hfill \\ \hfill & +\lambda ({\rho }_7\lambda +{\rho }_8)e^{-\lambda ({\tau }_1+{\tau }_2)}.\hfill \end{array}\hfill \end{array}$$ 47

Direction and stability of Hoph bifurcation

We perform the Hoph bifurcation analysis under the case that the transmission delay ${\tau }_1$ is a bifurcation parameter. So, we study the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. The analysis is carried out using the normal form theory and the center manifold theorem [11, 32, 33].

Let $t=\frac{t}{{\tau }_1}$, $u_1({\tau }_{1}t)=S(t)-S^{\ast }$, $u_2({\tau }_{1}t)=I(t)-I^{\ast }$ and $u_3({\tau }_{1}t)=R(t)-R^{\ast }$. Define ${\tau }_1={\tau }_1+\eta$, where $\eta \in \mathbb{R}$. Obviously, $\eta =0$ is the Hopf bifurcation of (2) for the new bifurcation parameter [34]. Moreover, the nonlinear system (2) is transformed into a functional differential equation in $\mathcal{C}=\mathcal{C}([-1,0],{\mathbb{R}}^3)$ as

$$\begin{array}{c}\hfill \dot{u}(t)=L_{\eta }(u_t)+f(\eta ,u_t),\end{array}$$ 48

where $u(t)={(u_1(t),u_2(t),u_3(t))}^T\in {\mathbb{R}}^3$, and $f:\mathbb{R}\times \mathbb{C}$ $\to$ ${\mathbb{R}}^3$ and $L_{\eta }:\mathbb{C}\to {\mathbb{R}}^3$ are given by $f(\eta ,u_t)=({\tau }_1^{\ast }+\eta ){(f_1,f_2,f_3)}^T$ and

$$\begin{array}{cc}& L_{\eta }(\varphi )=({\tau }_1^{\ast }+\eta )\hfill \\ \hfill & \left[M,\varphi ,(0),+,B_1,\varphi ,\left(-,\frac{{\tau }_2^{\ast }}{{\tau }_1^{\ast }}\right),+,B_2,\varphi ,(-1)\right].\hfill \end{array}$$ 49

Note that $\varphi (\theta )={({\varphi }_1(\theta ),{\varphi }_2(\theta ),{\varphi }_3(\theta ))}^T\in \mathcal{C}$, and we used the matrix

$$\begin{array}{c}\hfill M=\left(\begin{array}{ccc}{m}_{11}& m_{12}& m_{13}\\ m_{21}& m_{22}& m_{23}\\ m_{31}& m_{23}& m_{33}\end{array}\right).\end{array}$$ 50

Here, $m_{11}=-\nu -d_1{|k|}^{\delta }$, $m_{12}=-d_2{|k|}^{\delta }$, $m_{13}=\psi$, $m_{21}=m_{23}=m_{31}=0$, $m_{22}=-{\mathrm{\Gamma }}_1-d_3{|k|}^{\delta }$, $m_{32}=\zeta$, and $m_{33}=-{\mathrm{\Gamma }}_2-d_4{|k|}^{\delta }$. Moreover,

$$\begin{array}{c}\hfill B_1=\left(\begin{array}{ccc}{b}_{11}& b_{12}& 0\\ b_{21}& b_{22}& 0\\ 0& 0& 0\end{array}\right),\end{array}$$ 51

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill b_{11}& =-\frac{\beta I^{\ast }(1+\gamma I^{\ast })}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^2},\hfill \\ \hfill b_{12}& =-\frac{\beta S^{\ast }(1+\alpha S^{\ast })}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^2},\hfill \\ \hfill b_{21}& =\frac{\beta I^{\ast }(1+\gamma I^{\ast })}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^2},\hfill \\ \hfill b_{22}& =\frac{\beta S^{\ast }(1+\alpha S^{\ast })}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^2}.\hfill \end{array}\end{array}$$ 52

Moreover, we let $b_4=-a\mathrm{\Omega }{(\mathrm{\Omega }+b{I}^{\ast })}^{-1}$ and $b_5=-b_4$, and set

$$\begin{array}{c}\hfill B_2=\left(\begin{array}{ccc}0& 0& 0\\ 0& b_4& 0\\ 0& b_5& 0\end{array}\right).\end{array}$$ 53

In the following, we define the constants

$$\begin{array}{cc}\hfill f_1=& -[a_{1}I{(t)}^2+a_{2}S(t)I(t)+a_{3}S{(t)}^2]\hfill \\ \hfill & -[a_{4}I{(t)}^3+a_{5}S(t)I{(t)}^2\hfill \\ \hfill & +a_{6}I(t)S{(t)}^2+a_{7}S{(t)}^3],\hfill \end{array}$$ 54

$$\begin{array}{cc}\hfill f_2=& -a_{8}I{(t)}^2-a_{9}I{(t)}^3+[a_1{I}^2+a_{2}S(t)I(t)\hfill \\ \hfill & +a_{3}S{(t)}^2]+[a_{4}I{(t)}^3+a_{5}S(t)I{(t)}^2\hfill \\ \hfill & +a_{6}I(t)S{(t)}^2+a_{7}S{(t)}^3],\hfill \end{array}$$ 55

$$\begin{array}{cc}\hfill f_3=& a_{8}I{(t)}^2+a_{9}I{(t)}^3,\hfill \end{array}$$ 56

where

$$\begin{array}{cc}\hfill a_1=& -\frac{\alpha \beta I^{\ast }(1+\gamma I^{\ast })}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^3},\hfill \\ \hfill a_2=& -\frac{\beta (1+\alpha S^{\ast }+\gamma I^{\ast }+2\alpha \gamma S^{\ast }{I}^{\ast })}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^3},\hfill \\ \hfill a_3=& \frac{\beta \gamma S^{\ast }(1+\alpha S^{\ast })}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^3},a_4=\frac{\beta {\alpha }^2{I}^{\ast }(1+\gamma I^{\ast })}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^4},\hfill \\ \hfill a_5=& \frac{\alpha \beta ({\gamma }^2{I^{\ast }}^2-\alpha S^{\ast }-2\alpha \gamma S^{\ast }{I}^{\ast }-1)}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^4},\hfill \\ \hfill a_6=& \frac{\beta \gamma ({\alpha }^2{S^{\ast }}^2-\gamma I^{\ast }-2\alpha \gamma S^{\ast }{I}^{\ast }-1)}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^4},\hfill \\ \hfill a_7=& \frac{\beta {\gamma }^2{S}^{\ast }(1+\alpha S^{\ast })}{{(1+\alpha S^{\ast }+\gamma I^{\ast })}^4},\hfill \\ \hfill a_8=& \frac{-ab\mathrm{\Omega }}{{(\mathrm{\Omega }+b{I}^{\ast })}^3},a_9=\frac{a\mathrm{\Omega }{b}^2}{{(\mathrm{\Omega }+b{I}^{\ast })}^4}.\hfill \\ \hfill \end{array}$$ 57

For the remainder, we set $a_{11}=b_{11}{K}_1+m_{11}$, $a_{12}=b_{12}{K}_1+m_{12}$, $a_{13}=m_{13}$, $a_{21}=b_{21}{K}_1$, $a_{22}=b_{22}{K}_1+b_4{K}_2+m_{22}$, $a_{23}=a_{31}=0$, $a_{32}=b_5{K}_2+m_{32}$ and $a_{33}=m_{33}$. Using the same approach as in [35], we obtain the following matrix:

$$\begin{array}{c}\hfill A=\left(\begin{array}{ccc}{a}_{11}-i\omega & a_{12}& a_{13}\\ a_{21}& a_{22}-i\omega & a_{23}\\ a_{31}& a_{32}& a_{33}-i\omega \end{array}\right).\end{array}$$ 58

It is known that $\pm i\omega$ is eigenvalue for A and $A^{\ast }$. The corresponding eigenvectors are $q(\theta )=(1,q_1,$$q_2{)}^T{e}^{i\omega {\tau }_1\theta }$ and $q^{\ast }(s)=D{(1,q_1^{\ast },q_2^{\ast })}^T{e}^{i\omega {\tau }_{1}s}$, where

$$\begin{array}{cc}\hfill q_1=& \frac{(a_{11}-i\omega )a_{23}-a_{21}{a}_{13}}{(a_{22}-i\omega )a_{13}-a_{12}{a}_{23}},\hfill \\ \hfill q_2=& \frac{(a_{11}-i\omega )a_{32}-a_{31}{a}_{12}}{(a_{33}-i\omega )a_{12}-a_{13}{a}_{32}}.\hfill \end{array}$$ 59

Moreover,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \overline{D}& =[1+q_1{q}_1^{\ast }+q_2{q}_2^{\ast }+(q_1^{\ast }{b}_4+q_2^{\ast }{b}_5)q_2{\tau }_2^{\ast }{e}^{-i{\omega }_2{\tau }_2}\hfill \\ \hfill & +(b_{11}+q_1^{\ast }{b}_{21}+(b_{12}+q_1^{\ast }{b}_{22})q_1){\tau }_1^{\ast }{e}^{-i{\omega }_1{\tau }_1}{]}^{-1}.\hfill \end{array}\\ \hfill \end{array}$$ 60

We compute the coordinate to describe the center manifold ${\mathcal{C}}_0$ at $\eta =0$. After some algebraic calculations and substitutions, we can check that

$$\begin{array}{cc}\hfill g(z,\overline{z})=& \overline{q^{\ast }}(0)f_0(z,\overline{z})\hfill \\ \hfill =& P_1{Z}^2+P_{2}Z\overline{Z}+P_3{\overline{Z}}^2+P_4{Z}^2\overline{Z}+\dots \hfill \\ \hfill \end{array}$$ 61

where

$$\begin{array}{cc}\hfill P_1=& \overline{D}{\tau }_1^{\ast }\left[-,(a_1{q}_1^2+a_2{q}_1{q}_0+a_3^2{q}_0^2),e^{-2i{\omega }_{{\tau }_1}{\tau }_1^{\ast }}\right)\hfill \\ \hfill & \left(+,q_2^{\ast },a_8,q_1^2,e^{-2i{\omega }_{{\tau }_2}\frac{{\tau }_1^{\ast }}{{\tau }_2}},+,q_1^{\ast },\left((,a_1,q_1^2,+,a_2,q_1,q_0\right)\right)\hfill \\ \hfill & \left(\left(+,a_3^2,q_0^2,),e^{-2i{\omega }_{{\tau }_1}{\tau }_1^{\ast }},-,a_8,q_1^2,e^{-2i{\omega }_{{\tau }_2}\frac{{\tau }_1^{\ast }}{{\tau }_2}}\right)\right],\hfill \end{array}$$ 62

$$\begin{array}{cc}\hfill P_2=& 2\overline{D}{\tau }_1^{\ast }\left[-,(,a_1,|,q_1,{|}^2,+,a_2,\text{Re},q_1,+,a_3,|,q_0,{|}^2,)\right)\hfill \\ \hfill & \left(+,q_2^{\ast },a_8,{|q_1|}^2,+,q_1^{\ast }\right)\hfill \\ \hfill & \left((,a_1,|,q_1,{|}^2,-,a_8,|,q_1,{|}^2,+,a_2,\text{Re},q_1,+,a_3,|,q_0,{|}^2,)\right],\hfill \end{array}$$ 63

$$\begin{array}{cc}\hfill P_3=& \overline{D}{\tau }_1^{\ast }\left[-,(a_1{\overline{q}}_1^2+a_2{\overline{q}}_1\overline{q_0}+a_3^2),e^{2i{\omega }_{{\tau }_1}{\tau }_1^{\ast }}\right)\hfill \\ \hfill & \left(+,q_2^{\ast },a_8,{\overline{q}}_1^2,e^{2i{\omega }_{{\tau }_2}\frac{{\tau }_1^{\ast }}{{\tau }_2}},+,q_1^{\ast },((,a_1,{\overline{q}}_1^2,+,a_2,{\overline{q}}_1,\overline{q_0}\right)\hfill \\ \hfill & \left(+,a_3^2,),e^{2i{\omega }_{{\tau }_1}{\tau }_1^{\ast }},-,a_8,{\overline{q}}_1^2,e^{2i{\omega }_{{\tau }_2}\frac{{\tau }_1^{\ast }}{{\tau }_2}},)\right],\hfill \end{array}$$ 64

$$\begin{array}{cc}\hfill P_4=& \overline{D}{\tau }_1^{\ast }\left[-,2,a_1,\left(\frac{W_{20}^{(2)}(-1)}{2},\overline{q_1},+,W_{11}^{(2)},(-1),q_1\right)\right)\hfill \\ \hfill & -a_2\left(\frac{W_{20}^{(1)}(-1)}{2},\overline{q_1},+,\frac{W_{20}^{(2)}(-1)}{2},\overline{q_0},+,W_{11}^{(1)},(-1),q_1\right)\hfill \\ \hfill & \left(+,W_{11}^{(2)},(-1),q_0\right)-2a_3\left(\frac{W_{20}^{(1)}(-1)}{2},\overline{q_0}\right)\hfill \\ \hfill & \left(+,W_{11}^{(1)},(-1),q_0\right)-3a_4{q}_1^2\overline{q_1}-a_5(\overline{q_0}{q}_1^2\hfill \\ \hfill & \left(+,2,q_0,|,q_1,{|}^2,)-,a_6,(,q_0^2,\overline{q_1}+2|,q_0,{|}^2{q}_1),-,3,a_7,q_1^2,\overline{q_1}\right]\hfill \\ \hfill & +\overline{D}{\tau }_2^{\ast }{q}_1^{\ast }[2a_1\left(\frac{W_{20}^{(2)}(-1)}{2},\overline{q_1},+,W_{11}^{(2)},(-1),q_1\right)\hfill \\ \hfill & +3a_7{q}_1^2\overline{q_1}+3a_4{q}_1^2\overline{q_1}+a_2(\frac{W_{20}^{(1)}(-1)}{2}\overline{q_1}\hfill \\ \hfill & +\frac{W_{20}^{(2)}(-1)}{2}\overline{q_0}+W_{11}^{(1)}(-1)q_1+W_{11}^{(2)}(-1)q_0)\hfill \\ \hfill & +2a_3\left(\frac{W_{20}^{(1)}(-1)}{2},\overline{q_0},+,W_{11}^{(1)},(-1),q_0\right)\hfill \\ \hfill & +a_6(q_0^2\overline{q_1}+2|q_0{|}^2{q}_1)\hfill \\ \hfill & -a_8\left(\frac{W_{20}^{(1)}(-\frac{{\tau }_1^{\ast }}{{\tau }_2^{\ast }})}{2},\overline{q_1},+,W_{11}^{(1)},(-\frac{{\tau }_1^{\ast }}{{\tau }_2^{\ast }}),q_1\right)\hfill \\ \hfill & \left(-,3,a_9,q_1^2,\overline{q_1},+,a_5,(\overline{q_0},q_1^2,+,2,q_0,|,q_1,{|}^2)\right]\hfill \\ \hfill & +\overline{D}{\tau }_2^{\ast }{q}_2^{\ast }\left[a_8,\left(\frac{W_{20}^{(1)}(-\frac{{\tau }_1^{\ast }}{{\tau }_2^{\ast }})}{2},\overline{q_1},+,W_{11}^{(1)},(-\frac{{\tau }_1^{\ast }}{{\tau }_2^{\ast }}),q_1\right)\right)\hfill \\ \hfill & +3a_9{q}_1^2\overline{q_1}].\hfill \end{array}$$ 65

Moreover,

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill W_{20}(\theta )& =\frac{i{g}_{20}}{{\omega }^{\ast }{\tau }^{\ast }}q(0)e^{i{\omega }^{\ast }{\tau }^{\ast }\theta }\hfill \\ \hfill & +\frac{i{\overline{g}}_{20}}{3{\omega }^{\ast }{\tau }^{\ast }}\overline{q}(0)e^{-i{\omega }^{\ast }{\tau }^{\ast }\theta }+E_1{e}^{2i{\omega }^{\ast }\theta }\hfill \end{array}\hfill \end{array}$$ 66

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill W_{11}(\theta )& =-\frac{i{g}_{11}}{{\omega }^{\ast }{\tau }^{\ast }}q(0)e^{i{\omega }^{\ast }{\tau }^{\ast }\theta }\hfill \\ \hfill & +\frac{i{\overline{g}}_{11}}{{\omega }^{\ast }{\tau }^{\ast }}\overline{q}(0)e^{-i{\omega }^{\ast }{\tau }^{\ast }\theta }+E_2.\hfill \end{array}\hfill \end{array}$$ 67

It is worthwhile to notice that

$$\begin{array}{c}\hfill \begin{array}{cc}& \left(\begin{array}{ccc}2i\omega -a_{11}& -a_{12}& -a_{13}\\ -a_{21}& 2i\omega -a_{22}& -a_{23}\\ a_{31}& a_{32}& 2i\omega -a_{33}\end{array}\right)E_1\hfill \\ \hfill & =2\left(\begin{array}{c}-(a_1{q}_1^2+a_2{q}_1{q}_0+a_3^2{q}_0^2)e^{-2i{\omega }_{{\tau }_1}{\tau }_1^{\ast }}\\ (a_1{q}_1^2+a_2{q}_1{q}_0+a_3^2{q}_0^2)e^{-2i{\omega }_{{\tau }_1}{\tau }_1^{\ast }}-a_8{q}_1^2{e}^{-2i{\omega }_{{\tau }_2}\frac{{\tau }_1^{\ast }}{{\tau }_2}}\\ a_8{q}_1^2{e}^{-2i{\omega }_{{\tau }_2}\frac{{\tau }_1^{\ast }}{{\tau }_2}}\end{array}\right)\hfill \end{array}\\ \hfill \end{array}$$ 68

and

$$\begin{array}{c}\hfill \begin{array}{cc}& \left(\begin{array}{ccc}-a_{11}& -a_{12}& -a_{13}\\ -a_{21}& -a_{22}& -a_{23}\\ a_{31}& a_{32}& -a_{33}\end{array}\right)\hfill \\ \hfill & E_2=\left(\begin{array}{c}-(a_1|q_1{|}^2+a_2\text{Re}{q}_1+a_3|q_0{|}^2)\\ a_1|q_1{|}^2-a_8|q_1{|}^2+a_2\text{Re}{q}_1+a_3{|q_0|}^2\\ a_8{|q_1|}^2\end{array}\right),\hfill \end{array}\\ \hfill \end{array}$$ 69

where

$$\begin{array}{cc}\hfill & C_1(0)=\frac{i}{2{\tau }^{\ast }{\omega }^{\ast }}\left(g_{11},g_{20},-,2,{|g_{11}|}^2,-,\frac{|g_{02}{|}^2}{3}\right)+\frac{g_{21}}{2},\hfill \\ \hfill & {\mu }_2=\frac{\text{Re}[C_1(0)]}{\text{Re}[{\lambda }^{\prime }({\tau }^{\ast })]},\hfill \\ \hfill & T_2=-\frac{\text{Im}[C_1(0)]+{\mu }_2\text{Im}[{\lambda }^{\prime }({\tau }^{\ast })]}{{\omega }^{\ast }{\tau }^{\ast }},\hfill \\ \hfill & {\beta }_2=2\text{Re}{C}_1(0).\hfill \end{array}$$ 70

Moreover, $g_{20}(\theta )=2p_1$, $g_{11}(\theta )=p_2$, $g_{02}(\theta )=2p_3$ and $g_{21}(\theta )=2p_4$. Notice that the sign of ${\mu }_2$ determines the direction of the bifurcation. Meanwhile, the sign of ${\beta }_2$ determines the stability of the bifurcating periodic solution and $T_2$ provides the period of the bifurcating solutions.

Results

Analytical results

Here, we present the analytical results obtained based on the aforementioned techniques. The parameters values used in the analysis were obtained from various reports in the literature for validation purposes [3, 23]. Figure 1 shows the effects of treatment on the basic reproduction number. We observe that $R_0$ decreases as the treatment rate increases. Figure 2 provides a sensitivity analysis of the basic reproduction number using a partial rank correlation coefficient (PRCC) between the values of the parameters and the value of the response function [36]. The sensibility analysis globally exhibits two ranges of behavior, namely upwards and downwards. Those directed upwards and downwards are, respectively, proportional and inversely proportionally to $R_0$. As a consequence, it is observed that the transmission rate $\beta$ is directed upwards. We observe there that the death rate $\mu$ and the inhibition taken by susceptible $\alpha$ decrease the basic reproductive number more than the treatment rate.

Fig. 1.

Basic reproductive number versus treatment rate using $A=15$, $d=0.005$, $\alpha =0.002$, $\beta =0.02$, $\nu =0.00002$, $\zeta =0.026$

Fig. 2.

Sensitivity analysis of the basic reproduction number: $A=15$, $d=0.005$, $\alpha =0.002$, $\beta =0.02$, $\nu =0.00002$, $\zeta =0.026$, $a=0.02$

The general behavior of susceptible and infected compartments at the endemic equilibrium is depicted in Fig. 3. It is observed in this figure the global instability of both compartments around the steady state. This validates the possible action of treatment in the system, confirming that the treatment can destabilize the endemic equilibrium [37]. Figure 4 shows the behavior of the transmission delay versus diffusion rates, while Fig. 5 provides graphs of transmission and reinfection force. The susceptible diffusion rate depicted in Fig. 4(i), the recovered diffusion rate in Fig. 4(iv) and the reinfection force in Fig. 5(ii) are proportional to the critical transmission delay. Meanwhile, the cross-diffusion rate presented in Fig. 4(ii) and the infected diffusion rate shown in Fig. 4(iii) are inversely proportional to the critical transmission delay. As a consequence, it can be deduced that those parameters which are proportional to the critical delay tend to stabilize the system at certain range of their values. Thus, those parameters are not favorable for the disease spreading. On the contrary, those which are inversely proportional tend to destabilize the system by increasing the spreading.

Fig. 3.

Phase plane($S^{\ast }$,$I^{\ast }$): $b=0.33$, $A=15$, $d=0.005$, $\alpha =0.002$, $\beta =0.11$, $\gamma =0.004$, $\zeta =0.026$, $\nu =0.00002$, $\psi =0.0009$

Fig. 4.

Transmission delay versus different diffusion coefficient: $A=15$, $\nu =0.00002$, $\beta =0.11$, $\alpha =0.002$, $\gamma =0.003$, $b=0.33$, $\psi =0.0009$, $\zeta =0.026$, $a=0.02$

Fig. 5.

Transmission delay under the effect of infection rate and reinfection rate: $A=15$; $\nu =0.00002$, $\beta =0.11$, $\alpha =0.002$, $\gamma =0.004$, $b=0.33$, $a=0.02$ and $\zeta =0.026$, $d_1=0.6$, $d_2=0.25$, $d_3=0.75$, $d_4=0$

Figure 4(i) shows that the susceptible diffusion rate does not behave monotonically. The transmission delay decreases at a certain range of the diffusion rate and increases for others. This illustrates how the diffusion rate for susceptible individuals can be used to control the disease spreading. Globally, those parameters proportional to the critical delay can be useful tools in achieving herd immunity as they delayed the occurrence of the instabilities in the system by slowing down the outbreak of the disease. Figure 5(i) shows the behavior of the critical transmission delay versus the infection rate, and it evidences that critical delay does not behave monotonically. This behavior can be interpreted as the capacity of transmission rate to induce instabilities in the system.

Figure 6(i) exhibits the global behavior of the direction of bifurcation. From Theorem 3.1 in [11], it is specifically observed that the increasing values of the cross-diffusion rate ensure the transition from subcritical to supercritical bifurcation. In Fig. 6(ii), we studied the stability of the bifurcation under the cumulative effects of fractional order and cross-diffusion rate. It is observed that the increasing values of the fractional order significantly reduce the stability of the bifurcation but do not change the nature of the dynamical behavior. Globally, it is observed in both figures that the fractional order and cross-diffusion affect the direction of the bifurcation but do not change the nature of the stability behavior. Moreover, we notice that the minimum values of both bifurcation parameters are obtained for the negatives values of cross-diffusion rate which arise in real world when susceptible individuals move in the same direction with infected individuals. When behaving in that form, people contribute to keep the state then limiting the transition from one state to another. This ensures the disease release.

Fig. 6.

Bifurcation parameters analysis (${\mu }_2$,${\beta }_2$) under the effects of fractional order $\delta$ and cross-diffusion rate $d_2$: $A=15$, $\nu =0.00002$, ${\tau }_1=0.36$, ${\tau }_2=0.001$, $\beta =0.02$, $\alpha =0.002$, $\gamma =0.004$, $b=0.33$, $a=0.02$, $\zeta =0.026$, $d_1=0.6$, $d_2=0.25$, $d_3=0.8$, $d_4=0$

Computational results

In this section, we will provide numerical simulations to approximate the solutions of the nonlinear model (2). To that end, we will use a fully explicit finite difference scheme. It is worth pointing out that there exist other numerical methods available in the literature with higher orders of consistency. However, we have chosen the present numerical method in view that it is relatively faster than other computational approaches. This is an important feature in view that all the numerical methods for fractional systems require a substantial amount of computer time and resources [38]. Moreover, the finite difference scheme is well known to better preserve qualitatively some properties of the model than other methods which hold only for small nodes [39, 40].

The spatial domain for the system (2) is confined to the two-dimensional rectangle $[0,L]\times [0,L]$ and $s=\frac{d_i\mathrm{\Delta }t}{\mathrm{\Delta }{x}^2}=\frac{d_i\mathrm{\Delta }t}{\mathrm{\Delta }{y}^2}<\frac{1}{2}$ for all of computations [40], where $L=100$ and $d_i$ is the diffusion coefficient. $\mathrm{\Delta }t$, $\mathrm{\Delta }x$, $\mathrm{\Delta }y$ are the time and space step, respectively, and have been fixed for convenience. In the numerical simulation, we use the variable T which indicates the final time of simulation. Further, the model is analyzed under the zero flux boundary conditions and the following initial conditions:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S(x,y,0)& =S^{\ast }+\delta (x,y),\hfill \\ \hfill I(x,y,0)& =I^{\ast }+\delta (x,y),\hfill \\ \hfill R(x,y,0)& =R^{\ast }+\delta (x,y).\hfill \\ \hfill \end{array}\end{array}$$ 71

where $S^{\ast }=22$, $I^{\ast }=2686$, $R^{\ast }=75958$ and $\delta (x,y)=0.02sin(\frac{x}{2})sin(\frac{y}{2})$, where $\delta (x,y)$ is a slight perturbation around the steady state and T the time of simulation.

As a result, from our simulations, Fig. 7 shows different infected population patterns under the effect of treatment. Figure 7(i) and (ii) shows that the infected population initially increases in the presence of treatment. Then they transit from high density to low density of infected in Fig. 7(iii) before significantly decrease in Fig. 7(iv). This clearly exhibits the cleaning process of disease by treatment which consists in initially lowering down the density of infected before reducing their number. The increasing number of infected individuals at the beginning of the treatment effects can be due to some interaction between virus population and immune defense of the organisms which induce delay in treatment. Figure 8(i) and (ii) shows that increasing the recovered diffusion rate ensures the transition from low and stationary density of susceptible individuals to traveling pattern with the significant decreasing of the number of the population. In addition, Fig. 9 shows that the increasing values of the reinfection forces reduce the number of susceptible population, and transits it from stationary to traveling wave patterns, similarly as with the recovered diffusion rate.

Fig. 7.

Infected patterns under the effects of treatment rate: $b=0.33$, $A=15$, $d=0.005$, $\alpha =0.002$, $\beta =0.02$, $\gamma =0.004$, $\zeta =0.026$, $\nu =0.00002$, $\psi =0.0009$, ${\tau }_2=0.001$, ${\tau }_1=0.1$, $d_1=0.6$, $d_2=0.25$, $d_3=0.75$, $d_4=0$, $T=1000$

Fig. 8.

Susceptible pattern under the effect of recovered diffusion rate: $b=0.33$, $A=15$,$a=0.08$, $d=0.005$, $\alpha =0.002$, $\beta =0.02$, $\gamma =0.004$, $\zeta =0.026$, $\nu =0.00002$, $\psi =0.0009$, ${\tau }_2=0.001$, ${\tau }_1=0.39$, $d_1=0.6$, $d_2=0.25$, $d_3=0.75$, $T=1000$

Fig. 9.

Dynamical behavior of the infected class under the effect of reinfection force: $\psi$: $a=0.02$, $b=0.33$, $A=15$, $d=0.005$, $\alpha =0.002$, $d_1=0.6$, $d_2=0.25$, $d_3=0.75$, $d_4=0$, $\beta =0.02$, $\gamma =0.004$, $\zeta =0.026$, $\psi =0.0009$, $\nu =0.00002$, ${\tau }_1=0.3$, ${\tau }_2=0.5$, $T=1000$

Results from Figs. 8 and 9 validate the analytical predictions obtained from Figs. 4 and 5. As predicted, an important number of recovered individuals from the disease return to the susceptible compartment and contribute to build herd immunity. It is important to clarify that the reinfection force in this case does not necessarily lead to a reinfection of those individuals leaving the recovered class. There is a possibility of reinfection or immunization. The results show that the increasing rate of recovered individuals in the susceptible class can contribute to immunize susceptible subjects. However, Fig. 11 shows that increasing the value of transmission delay ensures the transition for infected individuals from traveling wave to chaotic patterns, which favors the disease spreading. This result clearly shows how transmission delay can change the dynamics of the system and leads to a new outbreak. Moreover, we also observe that the behavior of the system under the effects of treatment and a normal diffusion differs with the case of superdiffusion, as shown in Fig. 10. This figure shows that the number of infected population significantly increases, despite the increasing value of treatment rate. This shows the limitation of the treatment in the case of superdiffusion. The low densities of infected population observed in this case are not those cured by treatment but rather new infected individuals. In view of this remark, Fig. 12 shows the effects of critical transmission delays in the delayed cross-diffusive system. It is observed that the critical delay increases the infected population and ensures the transition of the cross-diffusive system from the stable domain (predominance of low density) to unstable domain (predominance of high density).

Fig. 11.

Infected patterns under the effects of treatment rate and transmission delay: $b=0.33$, $A=15$, $d=0.005$, $\alpha =0.002$, $\beta =0.02$, $\gamma =0.004$, $\zeta =0.026$, $\nu =0.00002$, $\psi =0.0009$, ${\tau }_2=0.001$, $d_1=0.6$, $d_2=0.25$, $d_3=0.75$, $d_4=0$, $T=1000$

Fig. 10.

Dynamical behavior of infected patterns under the effects of treatment in the case of superdiffusion: $\delta =1.5$, $b=0.33$, $A=15$,$a=0.08$, $d=0.005$, $\alpha =0.002$, $\beta =0.02$, $\gamma =0.004$, $\zeta =0.026$, $\nu =0.00002$, $\psi =0.0009$, ${\tau }_2=0.001$, ${\tau }_1=0.1$, $d_1=0.6$, $d_2=0.25$, $d_3=0.75$, $d_4=0$, $T=1000$

Fig. 12.

Infected patterns around critical point of transmission delay: $a=0.02$, $b=0.33$, $A=15$, $d=0.005$, $\alpha =0.002$, $\beta =0.11$, $\gamma =0.004$, $\zeta =0.026$, $\nu =0.00002$, $\psi =0.0009$, ${\tau }_2=0.5$, $d_1=0.6$, $d_2=0.25$, $d_3=0.75$, $d_4=0$, $T=500$

As noted previously in [9, 11], if treatment delay is considered, then the system is globally stable before the critical value of the transmission delay and globally unstable after the critical value. These two facts are confirmed by Figs. 13 and 14, respectively. It is worth pointing out that the number of infected individuals reduces under the presence of treatment when individuals move in the direction of low density of infected subjects. This fact is observed in Figs. 15 and 16, which show that the decreasing negative values of cross-diffusion rate increase the number of infected individuals, resulting in a maximum density of infected. Figure 16 shows that the increasing positive values of cross-diffusion rate decrease the number of infected and lead to their minimum density. Thus, cross-diffusion rate is an important tool for policy makers during the disease spreading as long as it can be considered in the case of partial lockdown when directing population in their partial movement. Figure 17 shows that the number of infected individuals increases as the fractional order increases. This exhibits the contribution of long-distance interaction in the disease spreading. These interactions occur when the population exhibits inter-urban or inter-country migrations or, in low-income countries, when people living in the country side come to supply those living in town with food.

Fig. 13.

Infected patterns before critical point of transmission delay: $b=0.33$, $A=15$, $d=0.005$, $\alpha =0.002$, $\beta =0.02$, $\gamma =0.004$, $\mu =0.05$, $d_1=0.6$, $d_2=0.25$, $d_3=0.75$, ${\tau }_1=0.30$, $T=500$

Fig. 14.

Infected patterns after critical point of transmission delay: $a=0.02$, $b=0.33$, $A=15$, $d=0.005$, $\alpha =0.002$, $d_1=0.6$, $d_2=0.25$, $d_3=0.75$, $d_4=0$, ${\tau }_1=0.39$, $\beta =0.11$, $\gamma =0.004$, $\zeta =0.026$, $\nu =0.00002$, $\psi =0.0009$, ${\tau }_1=0.39$, $T=500$

Fig. 15.

Infected pattern under the effects of negative values of cross-diffusion rate: $a=0.02$, $b=0.33$, $A=15$, $d=0.005$, $\alpha =0.002$, $d_1=0.6$,$d_3=0.8$,$d_4=0$, $\beta =0.11$, $\gamma =0.04$, $\zeta =0.026$, $\psi =0.0009$, $\nu =0.00002$, ${\tau }_1=0.4$, ${\tau }_2=0.5$, $T=1000$

Fig. 16.

Infected pattern under the effects of positive values of cross-diffusion rate: $a=0.02$, $b=0.33$, $A=15$, $d=0.005$, $\alpha =0.002$, $d_1=0.6$, $d_3=0.75$, $d_4=0$, $\beta =0.11$, $\gamma =0.04$, $\zeta =0.026$, $\nu =0.00002$, $\psi =0.0009$, ${\tau }_1=0.4$, ${\tau }_2=0.5$, $d_4=0$, $T=1000$

Fig. 17.

Infected pattern under the effects of fractional order, case of superdiffusion: $a=0.02$, $b=0.33$, $A=15$, $d=0.005$, $\alpha =0.002$, $d_1=0.6$, $d_3=0.75$, $d_4=0$, $\beta =0.11$, $\gamma =0.04$, $\zeta =0.026$, $\nu =0.00002$, $\psi =0.0009$, ${\tau }_1=0.6$, ${\tau }_2=0.5$, $T=1000$

Generally, the population feels psychologically secure and safe when using treatment. In those cases, individuals seldom respect any restriction measures, and so, they travel freely. The later result shows how this behavior can highly contributed to the disease resurgence. These results are in agreement with those reported in the literature [15, 16]. Finally, Fig. 18 shows that increasing the value of the transmission time delay in the superdiffusive leads to an exponential growth of the infected population. Globally, it is observed that transmission delay and superdiffusion both destabilize the system and sufficiently limit the action of treatment. At certain range of these parameters, the treatment does not affect the dynamics of the disease. Then the cumulative effect of the superdiffusion and time delay is found very favorable to the development of the infectious disease in the population.

Fig. 18.

Infected pattern under the effects of transmission time delay in the super diffusive domain: $a=0.02$, $b=0.33$, $A=15$, $d=0.005$, $\alpha =0.002$, $d_1=0.6$, $d2=0.25$, $d_3=0.75$, $d_4=0$, $\beta =0.11$, $\gamma =0.04$, $\zeta =0.026$, $\psi =0.0009$, $\delta =1.5$, $\nu =0.00002$, ${\tau }_2=0.5$, $T=1000$

Conclusion

We investigated analytically and numerically a SIRS epidemiological model with fractional cross-diffusion of the Weyl type, and temporal delays in the transmission and treatment. Beddington–DeAngelis-type incidence and Holling type II treatment rates were considered in the model. We studied the general impact of the treatment in the dynamics of the disease spreading subjected to the general population behavior. The analytical study established the effects of the diffusion rates and the reinfection force. A bifurcation analysis was carried out to elucidate the general effect of the cross-diffusion and fractional order as well as the stability of the bifurcating system. Our numerical simulations exhibited the presence of various patterns. In particular, we observed that the treatment rate decreases the value of the basic reproduction number and that the social awareness among the susceptible individuals decreases even more dramatically its value. This fact validates the commonsense idea that prevention is better than cure. Moreover, these results show that the effects of treatment are better when the system is stable.

To obtain the best effects of the treatment, our simulations established that it is necessary to take into account the impact of different parameters able to affect the dynamics of the system. In some cases, it is necessary to stabilize the system before starting the treatment. This stability is attained when we limit the long-distance interaction, or when we let the population move in the direction of low density of infected instead of in the direction of high density. Globally, We show that such a model can be a good continuous epidemiological model of a random walk describing the behavior of individuals probability to perform a long-distance interaction during the disease spreading. The results obtained in this paper can be a good improvement for understanding the dynamical behavior of epidemiological system under cross-superdiffusion.

Finally, it is important to point out that one of the reviewers suggested that the analytical instabilities could be caused by the numerical instability of the scheme. To check that this was not the case, the method was fully analyzed for the stability. The theoretical results (not included here in view of the length of the proofs and the fact that the journal is not a forum for numerical analysis) showed that the scheme is conditionally stable in time. However, the temporal time step was chosen to guarantee the stability of the approximations. We chose to use an explicit scheme in view of the expensive nature of an implicit scheme (which would possibly be unconditionally stable). Moreover, it is well known that numerical instabilities lead to the blow up of solutions in finite time, a feature which was not witnessed in our simulations. As a conclusion, the patterns shown in the simulations correspond to analytical features of the mathematical model, and not to numerical characteristics in our discretization.

Appendix

$$\begin{array}{cc}\hfill C_0=& aA\alpha {\mathrm{\Gamma }}_2-A\beta {\mathrm{\Gamma }}_2+A\alpha {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2\hfill \\ \hfill & +a{\mathrm{\Gamma }}_2\nu +{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2\nu ,\hfill \end{array}$$ 72

$$\begin{array}{cc}\hfill C_1=& -a^2\alpha {\mathrm{\Gamma }}_2+2aAb\alpha {\mathrm{\Gamma }}_2+a\beta {\mathrm{\Gamma }}_2-3Ab\beta {\mathrm{\Gamma }}_2\hfill \\ \hfill & +\alpha {\mathrm{\Gamma }}_1\zeta \mathrm{\Psi }-\beta \zeta \mathrm{\Psi }-a\beta \mathrm{\Psi }\hfill \\ \hfill & -2a\alpha {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2+3Ab\alpha {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2+\beta {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2\hfill \\ \hfill & -\alpha {\mathrm{\Gamma }}_1^2{\mathrm{\Gamma }}_2+a\alpha \zeta \mathrm{\Psi }+a^2\alpha \mathrm{\Psi }\hfill \\ \hfill & +2ab{\mathrm{\Gamma }}_2\nu +a\gamma {\mathrm{\Gamma }}_2\nu +3b{\mathrm{\Gamma }}_1\hfill \\ \hfill & {\mathrm{\Gamma }}_2\nu +\gamma {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2\nu +a\alpha {\mathrm{\Gamma }}_1\mathrm{\Psi },\hfill \end{array}$$ 73

$$\begin{array}{cc}\hfill C_2=& -a^{2}b\alpha {\mathrm{\Gamma }}_2+aA{b}^2\alpha {\mathrm{\Gamma }}_2+ab\beta {\mathrm{\Gamma }}_2-3A{b}^2\beta {\mathrm{\Gamma }}_2\hfill \\ \hfill & -b(\beta -\alpha {\mathrm{\Gamma }}_1)\zeta \mathrm{\Psi }\hfill \\ \hfill & -3ab\alpha {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2+3A{b}^2\alpha {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2\hfill \\ \hfill & +2b\beta {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2+2b\alpha {\mathrm{\Gamma }}_1\zeta \mathrm{\Psi }\hfill \\ \hfill & -2b\alpha {\mathrm{\Gamma }}_1^2{\mathrm{\Gamma }}_2+ab(\beta -\alpha {\mathrm{\Gamma }}_1){\mathrm{\Gamma }}_2\hfill \\ \hfill & -2b\beta \zeta \mathrm{\Psi }+2ab\alpha \zeta \mathrm{\Psi }\hfill \\ \hfill & +b{\mathrm{\Gamma }}_1(\beta -\alpha {\mathrm{\Gamma }}_1){\mathrm{\Gamma }}_2+a{b}^2{\mathrm{\Gamma }}_2\nu \hfill \\ \hfill & +2ab\gamma {\mathrm{\Gamma }}_2\nu -ab(\beta -\alpha {\mathrm{\Gamma }}_1)\mathrm{\Psi }\hfill \\ \hfill & +3b^2{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2\nu +3b\gamma {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2\nu \hfill \\ \hfill & +a^{2}b\alpha \mathrm{\Psi }-ab\beta \mathrm{\Psi }+ab\alpha {\mathrm{\Gamma }}_1\mathrm{\Psi },\hfill \end{array}$$ 74

$$\begin{array}{cc}\hfill C_3=& -A{b}^3\beta {\mathrm{\Gamma }}_2-a{b}^2\alpha {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2+A{b}^3\alpha {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2\hfill \\ \hfill & +b^2\beta {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2-2b^2(\beta -\alpha {\mathrm{\Gamma }}_1)\zeta \mathrm{\Psi }\hfill \\ \hfill & -b^2\alpha {\mathrm{\Gamma }}_1^2{\mathrm{\Gamma }}_2+a{b}^2(\beta -\alpha {\mathrm{\Gamma }}_1){\mathrm{\Gamma }}_2\hfill \\ \hfill & -b^2\beta \zeta \mathrm{\Psi }+b^2\alpha {\mathrm{\Gamma }}_1\zeta \mathrm{\Psi }\hfill \\ \hfill & +2b^2{\mathrm{\Gamma }}_1(\beta -\alpha {\mathrm{\Gamma }}_1){\mathrm{\Gamma }}_2+a{b}^2\gamma {\mathrm{\Gamma }}_2\nu \hfill \\ \hfill & +b^3{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2\nu +a{b}^2\alpha \zeta \mathrm{\Psi }\hfill \\ \hfill & +3b^2\gamma {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2\nu \hfill \\ \hfill & -a{b}^2(\beta -\alpha {\mathrm{\Gamma }}_1)\mathrm{\Psi },\hfill \end{array}$$ 75

and

$$\begin{array}{cc}\hfill C_4=& b^3{\mathrm{\Gamma }}_1(\beta -\alpha {\mathrm{\Gamma }}_1){\mathrm{\Gamma }}_2\hfill \\ \hfill & +b^3\gamma {\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2\nu -b^3(\beta -\alpha {\mathrm{\Gamma }}_1)\zeta \mathrm{\Psi }.\hfill \end{array}$$ 76

Author Contributions

All authors contributed equally in this study. All authors read and approved the final manuscript

Funding

The corresponding author (J.E.M.-D.) was funded by the National Council of Science and Technology of Mexico (CONACYT) through grant A1-S-45928.

Data Availability

The datasets generated during and/or analyzed during the current study are not publicly available but are available from the corresponding author (J.E.M.-D.) on reasonable request.

Declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.