Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment

Abstract

The dynamical behaviors of an SIR epidemic model with nonlinear incidence and treatment is investigated. It is assumed that treatment rate is proportional to the number of infectives below the capacity and is a constant when the number of infectives is greater than the capacity. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low. Theoretical and numerical results suggest that decreasing the basic reproduction number below one is insufficient for disease eradication.

Introduction

It has been suggested by several authors that the disease transmission process may have a nonlinear incidence rate. This allows one to include behaviour changes and prevent unbounded contact rate [1], [2]. A particular example of such an incidence rate is given by $\frac{\lambda I^p}{1+\alpha I^q}$, with $p\text{,}q\text{,}\alpha \text{,}\lambda >0$. After studying the cholera epidemic spread in Bari in 1973, Capasso and Serio [3] introduced a saturated incidence rate $g(I)S$ into epidemic models, where $g(I)$ tends to a saturation level when I gets large, i.e., $g(I)=\frac{\beta I}{1+\alpha I}$, where $\beta I$ measures the infection force of the disease and $\frac{1}{1+\alpha I}$ measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals. This incidence rate seems more reasonable than the bilinear incidence rate $g(I)S=\beta \mathit{IS}$, because it includes the behavioral change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters.

In this paper, we shall investigate a model which includes a nonlinear incidence rate $\frac{\lambda I}{1+\alpha I}$ and treatment. The treatment is an important method to decrease the spread of diseases such as measles, tuberculosis and flu [4], [5], [6]. In classical epidemic models, the treatment rate of infectives is assumed to be proportional to the number of the infectives. This is unsatisfactory because the resources for treatment should be quite large. In fact, every community should have a suitable capacity for treatment. If it is too large, the community pays for unnecessary cost. If it is small, the community has the risk of the outbreak of a disease. Thus, it is important to determine a suitable capacity for the treatment of a disease. In paper [7], Wang and Ruan adopt a constant treatment, which simulates a limited capacity for treatment. Note that a constant treatment is suitable when the number of infectives is large. Wang modified (see [8]) it into

$$T(I)=\left\{\begin{array}{cc}\hfill \mathit{rI}\hfill & \hfill \text{if}0⩽I⩽I_0\text{,}\hfill \\ \hfill k\hfill & \hfill \text{if}I>I_0\text{,}\hfill \end{array}\right.$$

where $k={\mathit{rI}}_0$. This means that the treatment rate is proportional to the number of the infectives when the capacity of treatment is not reached, and otherwise takes the maximal capacity. This renders for example the situation where patients have to be hospitalized: the number of hospitalized beds is limited. This is true also for the case where medicines are not sufficient. Evidently, this improves the classical proportional treatment and the constant treatment in [7].

We will consider a population that is divided into three types: susceptible, infective and removed. Let $S\text{,}I\text{,}R$ denote the numbers of susceptible, infective, and removed, respectively. Then the model to be studied takes the following form:

$$\left\{\begin{array}{c}\frac{\mathit{dS}}{\mathit{dt}}=A-\mathit{dS}-\frac{\lambda \mathit{SI}}{1+\alpha I}\text{,}\hfill \\ \frac{\mathit{dI}}{\mathit{dt}}=\frac{\lambda \mathit{SI}}{1+\alpha I}-(d+\gamma +ϵ)I-T(I)\text{,}\hfill \\ \frac{\mathit{dR}}{\mathit{dt}}=\gamma I+T(I)-\mathit{dR}\text{,}\hfill \end{array}\right.$$

where A is the recruitment rate of the population. d the nature death rate of the population. $\gamma$ the nature recovery rate of infective individuals, $ϵ>0$ the disease-related death rate. $\lambda$ the infection coefficient. It is assumed that all the parameters are positive constants. Clearly, $R_{+}^3$ is positively invariant for system (1.2). Since the first two equations in (1.2) are independent of the variable R, it suffices to consider the following reduced model:

$$\left\{\begin{array}{c}\frac{\mathit{dS}}{\mathit{dt}}=A-\mathit{dS}-\frac{\lambda \mathit{SI}}{1+\alpha I}\text{,}\hfill \\ \frac{\mathit{dI}}{\mathit{dt}}=\frac{\lambda \mathit{SI}}{1+\alpha I}-(d+\gamma +ϵ)I-T(I).\hfill \end{array}\right.$$

The purpose of this paper is to show that (1.3) has a backward bifurcation if the capacity for treatment is small. We will prove that (1.3) has bistable endemic equilibria if the capacity is low.

This organization of this paper is as follows. In the following section, we study the bifurcation of (1.3). In Section 3 we present a global analysis. Some numerical simulations and a brief discussion are given in Sections 4 Simulation, 5 Concluding remarks, respectively.