Global stability of a SIR epidemic model with nonlinear incidence rate and time delay

Abstract

In this paper, a SIR epidemic model with nonlinear incidence rate and time delay is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease free equilibrium is discussed. It is proved that if the basic reproductive number $R_0>1$, the system is permanent. By comparison arguments, it is shown that if $R_0<1$, the disease free equilibrium is globally asymptotically stable. If $R_0>1$, by means of an iteration technique and Lyapunov functional technique, respectively, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium.

Introduction

Let $S\left(t\right)$ denote the number of members of a population susceptible to the disease, $I\left(t\right)$ the number of infective members and $R\left(t\right)$ the number of members who have been removed from the possibility of infection through full immunity. In [7], Cooke formulated a SIR model with time delay effect by assuming that the force of infection at time $t$ is given by $\beta S\left(t\right)I(t-\tau )$, where $\beta$ is the average number of contacts per infective per day and $\tau >0$ is a fixed time during which the infectious agents develop in the vector, and it is only after that time that the infected vector can infect a susceptible human. Cooke considered the following model

$$\dot{S}\left(t\right)=B-{\mu }_{1}S\left(t\right)-\beta S\left(t\right)I(t-\tau )\text{,}$$$$\dot{I}\left(t\right)=\beta S\left(t\right)I(t-\tau )-({\mu }_2+\gamma )I\left(t\right)\text{,}$$$$\dot{R}\left(t\right)=\gamma I\left(t\right)-{\mu }_{3}R\left(t\right)\text{,}$$

where parameters ${\mu }_1,{\mu }_2,{\mu }_3$ are positive constants representing the death rates of susceptibles, infectives, and recovered, respectively. It is natural biologically to assume that ${\mu }_1\le min\{{\mu }_2,{\mu }_3\}$. The parameters $B$ and $\gamma$ are positive constants representing the birth rate of the population and the recovery rate of infectives, respectively. Much attention has been paid to the analysis of the stability of the disease free equilibrium and the endemic equilibrium of system (1.1) (see, for example, [1], [2], [3], [4], [5], [7], [10], [12], [13], [14], [15]). In [3], Beretta et al. considered the global stability of the disease free equilibrium and the endemic equilibrium of system (1.1). They showed that the disease free equilibrium is globally stable for any delay $\tau$ while the endemic equilibrium is not feasible. By constructing a suitable Lyapunov functional, sufficient conditions were derived to guarantee that if the endemic equilibrium is feasible, it is also globally stable for the delay being sufficiently small. In [12], Ma et al. derived an explicit expression of the lower bound of the component $I\left(t\right)$ of solution of system (1.1) which was proposed as an open problem. They therefore gave an estimation of the length of the time delay ensuring the global asymptotic stability of the endemic equilibrium.

Incidence rate plays an important role in the modelling of epidemic dynamics. It has been suggested by several authors that the disease transmission process may have a nonlinear incidence rate. In many epidemic models, the bilinear incidence rate $\beta SI$ and the standard incidence rate $\beta SI/N$ are frequently used. The bilinear incidence rate is based on the law of mass action. This contact law is more appropriate for communicable diseases such as influenza etc., but not for sexually transmitted diseases. It has been pointed out that for standard incidence rate, it may be a good approximation if the number of available partners is large enough and everybody could not make more contacts than is practically feasible. After studying the cholera epidemic spread in Bari in 1973, Capasso and Serio [6] introduced a saturated incidence rate $g\left(I\right)S$ into epidemic models, where $g\left(I\right)$ tends to a saturation level when $I$ gets large, i.e., 

$$g\left(I\right)=\frac{\beta I}{1+\alpha I}\text{,}$$

where $\beta I$ measures the infection force of the disease and $1/(1+\alpha I)$ 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 $\beta IS$, 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.

Motivated by the work of Beretta et al. [3], Capasso and Serio [6] and Ma et al. [12], in this paper, we are concerned with the effect of time delay and nonlinear incidence rate on the dynamics of a SIR epidemic model. To this end, we consider the following delay differential equations

$$\dot{S}\left(t\right)=B-{\mu }_{1}S\left(t\right)-\frac{\beta S\left(t\right)I(t-\tau )}{1+\alpha I(t-\tau )}\text{,}$$$$\dot{I}\left(t\right)=\frac{\beta S\left(t\right)I(t-\tau )}{1+\alpha I(t-\tau )}-({\mu }_2+\gamma )I\left(t\right)\text{,}$$$$\dot{R}\left(t\right)=\gamma I\left(t\right)-{\mu }_{3}R\left(t\right)\text{,}$$

where ${\mu }_1\le min\{{\mu }_2,{\mu }_3\}$. The initial conditions for system (1.2) take the form

$$S\left(\theta \right)={\varphi }_1\left(\theta \right)\text{,}I\left(\theta \right)={\varphi }_2\left(\theta \right)\text{,}R\left(\theta \right)={\varphi }_3\left(\theta \right)\text{,}$$$${\varphi }_i\left(\theta \right)\ge 0\text{,}\theta \in [-\tau ,0],{\varphi }_i\left(0\right)>0\text{,}i=1,2,3\text{,}$$

where $({\varphi }_1\left(\theta \right),{\varphi }_2\left(\theta \right),{\varphi }_3\left(\theta \right))\in C([-\tau ,0],{\mathbb{R}}_{+0}^3)$, the Banach space of continuous functions mapping the interval $[-\tau ,0]$ into ${\mathbb{R}}_{+0}^3$, where ${\mathbb{R}}_{+0}^3=\{(x_1,x_2,x_3):x_i\ge 0,i=1,2,3\}$.

It is well known by the fundamental theory of functional differential equations [9], system (1.2) has a unique solution $(S\left(t\right),I\left(t\right),R\left(t\right))$ satisfying the initial conditions (1.3). It is easy to show that all solutions of system (1.2) with initial conditions (1.3) are defined on $[0,+\infty )$ and remain positive for all $t\ge 0$.

The organization of this paper is as follows. In the next section, by analyzing the corresponding characteristic equations, the local stability of a disease free equilibrium and an endemic equilibrium of system (1.2) is discussed. In Section 3, it is proved that system (1.2) is permanent if the basic reproductive number $R_0>1$. In Section 4, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Alternatively, by constructing a suitable Lyapunov functional, it is shown that if $R_0>1$, the endemic equilibrium of system Eq. (1.2) is globally stable provided that the parameter $\alpha$ and the time delay $\tau$ are sufficiently small. By comparison arguments, it is proved that if $R_0<1$, the disease free equilibrium is globally stable. A brief discussion is given in Section 5 to conclude this work.