A note on the dynamics analysis of a diffusive cholera epidemic model with nonlinear incidence rate

Abstract

This paper is concerned with a diffusive cholera epidemic model with nonlinear incidence rate. The global stability of the cholera-free steady state(CFSS) of the model is studied when the basic reproduction number ${\mathcal{R}}_0=1$, which can be regarded as a continuation work of Wang et al. (2020). The theoretical result is also supported with numerical simulations.

Introduction

It is known that Cholera, a water-borne disease caused by the bacterium Vibrio cholerae, continues to represent a significant public health burden in developing countries [1], [2]. In order to better understand the fundamental mechanism of transmission for cholera, many types of cholera epidemic model and their modifications have been proposed and extensively studied in the literature due to their theoretical and practical importance (see, for example,[3], [4], [5], [6], [7] and the references therein).

Very recently, when the diffusion effect and spatial heterogeneity are taken into account, Wang et al. in [8] proposed and studied the following diffusive cholera epidemic model with nonlinear incidence rate:

$$\left\{\begin{array}{cc}{S}_t(t,x)=\hfill & \nabla \cdot (D_1\left(x\right)\nabla S)+b\left(x\right)-a\left(x\right)S-f(x,S,I)-g(x,S,P),x\in \Omega ,t>0,\hfill \\ I_t(t,x)=\hfill & \nabla \cdot (D_2\left(x\right)\nabla I)+f(x,S,I)+g(x,S,P)-(a\left(x\right)+c\left(x\right))I,x\in \Omega ,t>0,\hfill \\ P_t(t,x)=\hfill & \nabla \cdot (D_3\left(x\right)\nabla P)+c\left(x\right)I-m\left(x\right)P,x\in \Omega ,t>0,\hfill \\ \frac{\partial S(t,x)}{\partial \upsilon }=\hfill & \frac{\partial I(t,x)}{\partial \upsilon }=\frac{\partial P(t,x)}{\partial \upsilon }=0,x\in \partial \Omega ,t>0,\hfill \end{array}\right.$$

with

$$S(0,x)=S_0\left(x\right)\ge 0,I(0,x)=I_0\left(x\right)\ge 0,P(0,x)=P_0\left(x\right)\ge 0,\mathrm{for}x\in \Omega ,$$

where $S(t,x)$ and $I(t,x)$ represent the density of susceptible and infected individuals at time $t$ and location $x$, respectively, while $P(t,x)$ denotes the concentration of cholera bacteria in the water source at time $t$ and location $x$; $D_1\left(x\right),D_2\left(x\right)$ and $D_3\left(x\right)$ are the diffusion coefficients measuring the mobility of susceptible and infected individuals and cholera at location $x$, respectively; $b\left(x\right)$ is the recruitment rate of susceptible individuals; $a\left(x\right)$ means the natural death rate of susceptible individuals and infected individuals; $m\left(x\right)$ stands for the natural death rate of cholera bacteria; $c\left(x\right)$ denotes the recovery rate from infection and shedding rate of cholera bacteria from infected individuals; $f(x,S,I)$ and $g(x,S,P)$ are general nonlinear incidence functions for direct transmission between susceptible and infected individuals, and indirect transmission between susceptible and cholera bacteria, respectively. The habitat $\Omega \subset {\mathbb{R}}^n$ is a bounded domain with smooth boundary $\partial \Omega$, the Neumann boundary condition assumes zero flux on the boundary which means that no population flux crosses the boundary $\partial \Omega$. The initial functions, $(S_0\left(x\right),I_0\left(x\right),P_0\left(x\right)),x\in \bar{\Omega }$ are nonnegative continuous functions. In [8], the well-posedness, global asymptotical stability (${\mathcal{R}}_0<1$) and the uniform persistence( ${\mathcal{R}}_0>1$) of CFSS of system (1.1) have been well studied. Now a natural question arises: How about the dynamics of the CFSS for system (1.1) when ${\mathcal{R}}_0=1$? It is known that such a critical case cannot be directly obtained by the method for the case of ${\mathcal{R}}_0>(<)1$, and the main purpose of this paper is devoted to this question. Motivated by the argument developed recently in [9], [10], [11], we can indeed show that the CFSS is globally asymptotically stable if ${\mathcal{R}}_0=1$. The obtained theoretical result is nontrivial and can be considered as a continuation work of [8].

The rest of this paper is organized as follows. In Section 2, we obtain the basic reproduction number. In Section 3, we analyze the globally asymptotically stable of CFSS of system (1.1) when ${\mathcal{R}}_0=1$. Finally, we demonstrate the effectiveness of these theoretical results with numerical simulations in Section 4.