Global attractivity of the equilibria of the diffusive SIR and SEIR epidemic models with multiple parallel infectious stages and nonlinear incidence mechanism

Abstract

In this paper, we are concerned with the diffusive $SIR$ and $SEIR$ epidemic models with multiple parallel infectious stages and nonlinear incidence mechanism. We first establish a priori $L^{\infty }$-norm estimates for the solutions to the epidemic models, and then derive the global attractivity of disease-free equilibrium and endemic equilibrium for the two models in terms of the basic reproduction number. Our results show that the diversity of infectious stages, the infection mechanism and the diffusion do not affect the global dynamics of the infectious disease.

Introduction

In this paper, we consider the following two diffusive epidemic models with the nonlinear incidence mechanism $v_i{u}^q(q>0)$:

$$\left\{\begin{array}{cc}\frac{\partial u}{\partial t}-d_u\Delta u=\Lambda -\sum _{j=1}^n{\beta }_j{v}_j{u}^q-\mu u,\hfill & x\in \Omega ,t>0,\hfill \\ \frac{\partial v_i}{\partial t}-d_{v_i}\Delta v_i={\theta }_i\sum _{j=1}^n{\beta }_j{v}_j{u}^q-{\eta }_i{v}_i,\hfill & x\in \Omega ,t>0,\hfill \\ \frac{\partial u}{\partial \nu }=\frac{\partial v_i}{\partial \nu }=0,\hfill & x\in \partial \Omega ,t>0,\hfill \\ u(x,0)=u^0\left(x\right)\ge 0,v_i(x,0)=v_i^0\left(x\right)\ge ,⁄\equiv 0,\hfill & x\in \Omega ,\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{cc}\frac{\partial u}{\partial t}-d_u\Delta u=\Lambda -\sum _{j=1}^n{\beta }_j{v}_j{u}^q-\mu u,\hfill & x\in \Omega ,t>0,\hfill \\ \frac{\partial w}{\partial t}-d_w\Delta w=\sum _{j=1}^n{\beta }_j{v}_j{u}^q+\sum _{j=1}^n{a}_j{v}_j-\sigma w,\hfill & x\in \Omega ,t>0,\hfill \\ \frac{\partial v_i}{\partial t}-d_{v_i}\Delta v_i={\gamma }_{i}w-{\eta }_i{v}_i,\hfill & x\in \Omega ,t>0,\hfill \\ \frac{\partial u}{\partial \nu }=\frac{\partial w}{\partial \nu }=\frac{\partial v_i}{\partial \nu }=0,\hfill & x\in \partial \Omega ,t>0,\hfill \\ u(x,0)=u^0\left(x\right),w(x,0)=w^0\left(x\right),v_i(x,0)=v_i^0\left(x\right),\hfill & x\in \Omega ,\hfill \end{array}\right.$$

where $i=1,\dots ,n$; $u(x,t),w(x,t),v_i(x,t)$ stand for, respectively, the densities at location $x$ and time $t$ of susceptible individuals, exposed individuals and infected individuals, and the positive constants $d_u$, $d_w$ and $d_{v_i}$ are their respective diffusion coefficients. The formula $\Lambda -\mu u$ suggests that the susceptible individuals carrying capability are linearly increasing, where the positive constant $\Lambda$ and $\mu$ account for the rates of birth and death of the susceptible individuals, respectively. The positive constant ${\beta }_i$ measures the infection force. The constant ${\theta }_i\in (0,1]$ is the rate for an infected individual to enter the $i$th infective compartment and ${\eta }_i>0$ is the rate of infected individuals of $i$th compartment leaving this compartment. In addition, ${\eta }_i$ is the sum of the recovery rate and the death rate $\mu$ in system (1.1), while it is the sum of the rates of death, recovered and the amelioration $a_i$ of infection individuals in system (1.2) (and hence $a_i<{\eta }_i$). The positive constant $\sigma$ is the rate that exposed hosts leave $w$-compartment $(\sigma >{\sum }_{i=1}^n{\gamma }_i)$. In (1.1), (1.2), $\Omega$ is a bounded smooth domain of ${\mathbb{R}}^N(N\ge 1)$, the homogeneous Neumann boundary conditions imply that there is no population flux across the boundary $\partial \Omega$.

By the biological implication, it is natural to assume that the initial data $u^0\left(x\right),w^0\left(x\right),v_i^0\left(x\right)$ are nonnegative on $\overline{\Omega }$. To ensure the local existence and uniqueness of solutions of (1.2), we additionally assume that $u^0\left(x\right),w^0\left(x\right),v_i^0\left(x\right)$ are Hölder continuous on $\overline{\Omega }$, and $u^0\left(x\right)>0$ on $\overline{\Omega }$ when $0<q<1$. Thus, the standard existence theory of parabolic equations shows that for such initial data, (1.2) admits a classical solution $(u(x,t),w(x,t),v_i(x,t))$. It further follows from the parabolic comparison principle that $u(x,t)>0,w(x,t)>0$ and $v_i(x,t)\ge 0$ for all $x\in \overline{\Omega }$ and $t>0$. Furthermore, if $v_i^0\left(x\right)\ge ,⁄\equiv 0$ for all $x\in \overline{\Omega }$, then $v_i(x,t)>0$ for all $x\in \overline{\Omega }$ and $t>0$. A similar assumption is imposed for the initial data of the system (1.1), and the same conclusion also holds for (1.1).

Nowadays, people have recognized that individual motility are significant factors in the study of the spread of infectious diseases, and many reaction–diffusion epidemic models have been developed to investigate the dynamics of disease transmissions. These models are derived from the ODE compartmental epidemic models by introducing random diffusion terms to describe the movement of individuals; see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], to list a few.

The corresponding ODE systems of (1.1), (1.2) with the standard bilinear incidence mechanism ${\sum }_{j=1}^n{\beta }_j{v}_{j}u$ were studied in [16]. Korobeinikov [16] found that the levels of contagiousness and the lengths of the infectious periods did not always the same by studying the Hepatitis B. Then they proposed SIR and SEIR systems with several parallel infectious stages (there has $n$ alternative infectious pathways and $n$ noninteracting infective subclasses $v_i,i=1,2,\dots ,n$) and the bilinear incidence mechanism $v_{i}u$. This kind of incidence mechanism was first introduced by Kermack and McKendrick [17] to describe the plague epidemic in Bombay. Since then, such a bilinear infection mechanism has been used in many models, for instance, [4], [18], [19], [20], [21]. However, as mentioned by [22], [23], [24], this kind of infection mechanism has some shortcomings and may require modifications in certain situations to better model the disease transmission. Hence, many different nonlinear infection mechanisms have been proposed. Among them, the nonlinear incidence form $u^q{v}^p$ ($0<p\le 1,q>0$) has been used to describe the transmission of infectious disease, for example, in [22], [24], [25], [26], [27], [28]. Especially, the nonlinear incidence mechanism $u^{q}v$ ($q>0$) was investigated in this article.

For the corresponding ODE systems of (1.1), (1.2) with the nonlinear incidence mechanism ${\sum }_{j=1}^n{\beta }_j{v}_j{u}^q$ is replaced by ${\sum }_{j=1}^n{\beta }_j{v}_{j}u$, Korobeinikov [16] analysed the global stability of the disease-free equilibrium and endemic equilibrium by the Lyapunov method. His results implies that the classic compartment models of infectious diseases are staunch and that the diversity of the infection level and the lengths of the infectious period do not affect the basic properties of these models. For more studies on multistage or multigroup models of infectious diseases, one may refer to [29], [30], [31], [32], [33] and the references therein.

In general, the global stability of systems (1.1), (1.2) is determined by the basic reproduction number. Next, we define the basic reproduction number [34] for systems (1.1), (1.2) as follows:

$$R_0=\sum _{j=1}^n\frac{{\beta }_j{\theta }_j}{{\eta }_j}{\left(\frac{\Lambda }{\mu }\right)}^q\text{and}{\mathcal{R}}_0=\sum _{j=1}^n\frac{{\beta }_j{\gamma }_j}{\sigma {\eta }_j}{\left(\frac{\Lambda }{\mu }\right)}^q+\sum _{j=1}^n\frac{a_j{\gamma }_j}{\sigma {\eta }_j}.$$

The main purpose of this paper is to discuss the global stability of the disease-free equilibrium (DFE) and the endemic equilibrium (EE) of systems (1.1), (1.2). For any $q>0$, it can be easily shown that the system (1.1) has two equilibria: a DFE $(\hat{u},\widehat{v_i})=(\frac{\Lambda }{\mu },0)$, and a unique EE $(u^{\ast },{v_i}^{\ast })$ with $u^{\ast },v_i^{\ast }$ such that $\Lambda -{\sum }_{j=1}^n{\beta }_j{v}_j^{\ast }{\left(u^{\ast }\right)}^q-\mu u^{\ast }=0$, ${\theta }_i{\sum }_{j=1}^n{\beta }_j{v}_j^{\ast }{\left(u^{\ast }\right)}^q-{\eta }_i{v_i}^{\ast }=0$ where

$$u^{\ast }={\left(\sum _{j=1}^n\frac{{\beta }_j{\theta }_j}{{\eta }_j}\right)}^{-\frac{1}{q}}>0,v_i^{\ast }=\frac{(\Lambda -\mu u^{\ast }){\theta }_i}{{\eta }_i}=\left(R_0^{\frac{1}{q}}-1\right)\frac{\mu u^{\ast }{\theta }_i}{{\eta }_i}>0$$

provided that $R_0>1$.

For the system (1.1), we can state the following result on the global stability of the disease-free equilibrium and the endemic equilibrium.

Theorem 1.1

The following assertions hold for (1.1).

• If $R_0\le 1$, then the disease-free equilibrium is globally attractive.

• If $R_0>1$, then the unique endemic equilibrium is globally attractive.

We now turn to the system (1.2). For any $q>0$, the system (1.2) has DFE $(\tilde{u},\widetilde{v_i},\tilde{w})=(\frac{\Lambda }{\mu },0,0)$. Next, we need to determine the existence of endemic equilibrium. The EE of (1.2) satisfies $\Lambda -{\sum }_{j=1}^n{\beta }_j{v}_j{u}^q-\mu u=0$, ${\sum }_{j=1}^n{\beta }_j{v}_j{u}^q+{\sum }_{j=1}^n{a}_j{v}_j-\sigma w=0$, ${\gamma }_{i}w-{\eta }_i{v}_i=0$.

For any $q>0$, we have $v_i=\frac{{\gamma }_i}{{\eta }_i}w$. Inserting this into the second equation, we can get $u^{\ast \ast }={\left(\frac{\sigma -K_2}{K_1}\right)}^{\frac{1}{q}}$ with $K_1≔{\sum }_{j=1}^n\frac{{\beta }_j{\gamma }_j}{{\eta }_j}$ and $K_2≔{\sum }_{j=1}^n\frac{a_j{\gamma }_j}{{\eta }_j}$. It follows from $\sigma >{\sum }_{i=1}^n{\gamma }_i$ and $a_i<{\eta }_i$ that $\sigma -K_2=\sigma -{\sum }_{j=1}^n\frac{a_j{\gamma }_j}{{\eta }_j}$ is a positive constant. Adding the first and second equation, it is easily seen that (1.2) has a unique positive solution $(u^{\ast \ast },w^{\ast \ast },v_i^{\ast \ast })$ with

$$w^{\ast \ast }=\frac{\Lambda -\mu u^{\ast \ast }}{\sigma -K_2}=\frac{\mu \left[{(\sigma {\mathcal{R}}_0-K_2)}^{1/q}-{(\sigma -K_2)}^{1/q}\right]}{K_1^{1/q}(\sigma -K_2)},v_i^{\ast \ast }=\frac{{\gamma }_i}{{\eta }_i}{w}^{\ast \ast }=\frac{{\gamma }_i\mu \left[{(\sigma {\mathcal{R}}_0-K_2)}^{1/q}-{(\sigma -K_2)}^{1/q}\right]}{{\eta }_i{K}_1^{1/q}(\sigma -K_2)}$$

provided that ${\mathcal{R}}_0>1$.

For the system (1.2), we have the following result on the global stability of the disease-free equilibrium and the endemic equilibrium.

Theorem 1.2

The following assertions hold for the system (1.2).

• If ${\mathcal{R}}_0\le 1$, then the disease-free equilibrium is globally attractive.

• If ${\mathcal{R}}_0>1$, then the endemic equilibrium is globally attractive.

To prove Theorem 1.1, Theorem 1.2, as in [16], we shall employ the Lyapunov function approach. However, it is worth mentioning that the construction of the Lyapunov function in the case of $q=1$ is rather different from that for the case of $q\ne 1$; in the former case, we just need to modify the ones used in [16], but in the latter case we have to use some new types of Lyapunov function. Moreover, in order to derive the desired results, we need to establish the uniform bounds of solutions to (1.1), (1.2).

The rest of this paper is devoted to the uniform bounds of solutions to (1.1), (1.2) and the proof of the global attractivity of the disease-free equilibrium and the endemic equilibrium of (1.1), (1.2).