Global Behaviors of a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate

Abstract

We study a class of discrete SIRS epidemic models with nonlinear incidence rate and disease-induced mortality. By using analytic techniques and constructing discrete Lyapunov functions, the global stability of disease-free equilibrium and endemic equilibrium is obtained. That is, if basic reproduction number , then the disease-free equilibrium is globally asymptotically stable, and if , then the model has a unique endemic equilibrium and when some additional conditions hold the endemic equilibrium also is globally asymptotically stable. By using the theory of persistence in dynamical systems, we further obtain that only when , the disease in the model is permanent. Some special cases of are discussed. Particularly, when , it is obtained that the endemic equilibrium is globally asymptotically stable if and only if . Furthermore, the numerical simulations show that for general incidence rate the endemic equilibrium may be globally asymptotically stable only as .

1. Introduction

During the past decades, no matter discrete epidemic models or continuous epidemic models, have been widely studied. Many important and interesting results can be found in [1–28] and the references cited therein. The main research subjects are the computation of the threshold value or basic reproduction number which distinguishes whether the infectious disease will persist or die out, the local and global stability of the disease-free equilibrium and endemic equilibrium, the extinction, persistence, and permanence of the disease, and the bifurcations, chaos, and more complex dynamical behaviors of the models.

Among these questions, global stability of equilibria has always been one of the research focuses and difficult problems. Many authors have investigated this question using the second Lyapunov method (see [29]). The most popular types of Lyapunov functions candidate for population biology models are the Volterra-type functions and the quadratic function . The former has been successfully applied for various disease propagation models by Korobeinikov and his coworkers (see [7–10] and the references cited therein). In [11], Li et al. presented an algebraic approach to prove the global stability, which can provide the method of constructing a Lyapunov function and prove the negative definiteness of the derivative. Recently, by combining Volterra functions and quadratic functions, Vargas-De-León has studied global stability of classic continuous SIS, SIR, and SIRS epidemic models with constant recruitment, disease-induced death, and standard incidence rate and bilinear incidence rate in [12, 13], respectively. McCluskey in [14–16] introduced the Lyapunov functional formed as to investigate global stability of endemic equilibrium of SEIR epidemic model with distributed delay or discrete delay.

It is well known that a crucial role in mathematical models of infectious disease is played by the so-called incidence rate, namely, a function describing the mechanism of transmission of the disease. In most epidemiological models, bilinear incidence rate and standard incidence rate are frequently used, where is the total number of the population and is the per capita contact rate. These incidences imply that the contact number between and is proportional to or . But the infection probability per contact is likely influenced by the number of infective and susceptible individuals, because more infective individuals can increase the infection risk and susceptible individuals would avoid the contact with infective individuals. Therefore, a number of nonlinear incidence rates are suggested by researchers. After studying the cholera epidemic spread in Bari in 1973, Capasso and Serio [17] introduced the saturated incidence rate into epidemic models. To incorporate the effect of the behavioral changes of the susceptible individuals, Liu et al. proposed the general incidence rate in [18], where , and . The special cases when and are given different values have been used by many authors (see, e.g., Korobeinikov and Maini [6], Ruan and Wang [19], and Xiao and Ruan [20]).

However, until now, to the best of our knowledge, there are few search results about global stability of equilibria for discrete SIRS model with nonlinear incidence rate. Hu et al. in [28] discussed local stability and complex dynamical behaviors for a class of discrete SIRS epidemic models with general nonlinear incidence rate discretized by the forward Euler scheme. Enatsu et al. in [22] proposed a class of discrete SIR epidemic models with bilinear incidence rate, which are derived from continuous SIR epidemic models with distributed delays by using a variation of the backward Euler method, and obtained that global stability of disease-free equilibrium and endemic equilibrium. Muroya et al. in [23] discussed global stability and permanence of a discrete epidemic model with bilinear incidence rate and for disease with immunity and latency spreading in a heterogeneous host population, which is also discretized from the continuous case by using the backward Euler method. In [24], Enatsu et al. studied a class of discrete SIR epidemic models with nonlinear incidence rates and distributed delays, which are derived from the corresponding continuous SIR epidemic models by applying a variation of the backward Euler discretization. Using discrete-time analogue of Lyapunov functionals, the global asymptotic stability of the disease-free equilibrium and endemic equilibrium is fully determined by the basic reproduction number , when the infection incidence rate has a suitable monotone property.

Motivated by the fact that discrete epidemic models are more appropriate approach to understand disease transmission dynamics and to evaluate eradication policies because they permit arbitrary time step units, preserving the basic features of corresponding continuous models, in this paper, we will extend a discrete-time analogue of Lyapunov techniques proposed in [25–27] to the following discrete SIRS epidemic models with nonlinear incidence rate , which is established by using the backward Euler scheme (see [30, 31]) to discretize the corresponding continuous SIRS epidemic model: We will investigate the global behaviors of solutions of model (1). By constructing new discrete Lyapunov functions, we will establish some new criteria on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium for model (1). By using the theory of persistence in dynamical systems, we will further obtain the sufficient and necessary conditions for the permanence of the disease for model (1).

The organization of this paper is as follows. In Section 2, the existence of equilibria and positivity of solutions for model (1) are given. In Section 3, the results on the global asymptotic stability of the disease-free equilibrium and endemic equilibrium for model (1) are stated and proved. In Section 4, the results on the permanence of the disease in model (1) are established. In Section 5, the global asymptotic stability of the endemic equilibrium of model (1) for the special case is discussed. Finally, some examples are given to illustrate the main theoretical results in Section 6.

2. Equilibria and Positivity

For model (1), , , and represent the numbers of susceptible, infectious, and recovered individuals at th generation, respectively. The parameters , , , and are positive constants and is nonnegative constant in which is the recruitment rate into the population, is the natural death rate, is the disease-induced death rate, is the recovery rate of the infectious individuals, is the rate of losing immunity, implies that the recovered individuals would lose the immunity, and implies that the recovered individuals acquire permanent immunity. The spread of disease can be described by general form with incidence rate ; that is, the incidence rate depends on the number of the susceptible individuals and the number of the infectious individuals. This generalizes the bilinear incidence rate (i.e., ), saturated incidence rate with respect to (i.e., ), and saturated incidence rate with respect to (i.e., ), where , , and are constants, which denotes the contact coefficient and the saturated coefficient, respectively.

The initial condition for model (1) is given by

In this paper, for functions and , we firstly introduce the following assumption. and are positive, monotonically increasing, and continuous differentiable functions defined for all and , the derivative exists, and . Furthermore, is nonincreasing for all .

Remark 1. Assumption is basic for model (1). In fact, for many special cases of , for example, , , and , is always satisfied.

In order to obtain the existence of disease-free equilibrium and endemic equilibrium of model (1), we introduce a constant We have the following result.

Theorem 2. Assume that holds.(1)When , then model (1) has only a unique disease-free equilibrium .(2)When , then model (1) shows a unique endemic equilibrium , except for , where , , and satisfy

Proof. Obviously, model (1) always has a disease-free equilibrium . From (4), we have Hence, and from the second equation of (4) we further have When , let Then by we obtain Let ; then we obviously have . From , is monotonically decreasing for , and hence is monotonically increasing for . Thus, from (9), we obtain that when equation has not any solution in and when equation has a unique positive solution in . This shows that when model (1) does not have any endemic equilibrium. When , let and then is a unique endemic equilibrium of model (1). This completes the proof.

From Theorem 2, we can claim that the basic reproduction number of model (1) is . On the positivity and ultimate boundedness of solutions of model (1), we obtain the following theorem.

Theorem 3. Assume that holds. Let be the solution of model (1) with initial conditions (2); then is positive for any and ultimately bounded.

Proof. Let be any solution of model (1) with initial conditions (2). Further, let ; then model (1) is equivalent to the following form: In the following, we will use the induction to prove the positivity of . When , we have From (13)–(15) we see that as long as is confirmed, then , , and will be whereafter confirmed.
Firstly, we prove that if , then and . From (14), we directly obtain when . Let , and from (15) we obtain It is obvious that, when , is monotonically increasing for . Obviously, is a continuous function for . Since and , we obtain that has a unique positive solution . Therefore, we further have . Furthermore, we also have .
Let ; then from (13) we see that must satisfy the following equation: where Denote Obviously, . Let ; then when we have . We also have that is monotonically decreasing with respect to . Hence, by , is also monotonically decreasing with respect to . From the expression of and , we obtain that is monotonically increasing for . Obviously, is a continuous function for . Since there exists a unique such that .
Now, we show that is a unique solution of on . Otherwise, there is a such that . Since , we have when . From , we have for any ; hence from we further have . On the other hand, since and , we obtain , which leads to a contradiction.
Therefore, we certainly have . From the above discussions, we finally have , , and .
When , we obtain Obviously, using a similar argument in the above process, we also can obtain , , and . Lastly, by using the induction, we can finally obtain , , and for all .
From the third equation of model (11), we have Since comparison equation, has a globally asymptotically stable equilibrium , from the comparison principle of difference equations (see [32]), we finally obtain Therefore, is also ultimately bound. This completes the proof.

3. Global Stability

Now, we are concerned with the global asymptotic stability of disease-free equilibrium and endemic equilibrium of model (1), respectively.

Theorem 4. Assume that holds. Then disease-free equilibrium of model (1) is globally asymptotically stable if and is globally attractive if .

Proof. Calculating the linearization system of model (1) at equilibrium , we have From the second equation of system (25), we have When , we obtain Therefore, . By we further obtain and . This shows that is locally stable when . Since the case is a critical one for model (1), in the following, we discuss global attractivity of disease-free equilibrium when .
Let be any positive solution of model (1) with initial conditions (2). We need to consider the following two cases.
Case  1. for all .
Case  2. There exists an integer such that .
For Case  1, from (24), we directly have From third equation of (11), we further obtain
For Case  2, by using the iterative computations to inequality (22), we can obtain for all . Hence, for all . From , we further obtain Since from the second equation of model (1), it follows that, for all , If , then Hence, is nonincreasing for . Consequently, exists and .
Suppose ; then from the second and third equations of model (11), we can obtain that and exist, and From , it follows that exists. Obviously, we have , , and . Taking from the both sides of model (1), we can obtain the following equations: Hence, is an equilibrium of model (1). However, from Theorem 2, we see that when , (36) only has a unique solution , , and . This leads to a contradiction. Therefore, we have .
Therefore, we always have . By (35), it follows that and . Consequently, . This shows that disease-free equilibrium is globally attractive when . This completes the proof.

In order to obtain the global asymptotic stability of endemic equilibrium of model (1), we need the following assumptions.()For any , ()For any ,

Theorem 5. Assume that hold. If , then endemic equilibrium of model (1) is globally asymptotically stable.

Proof. We firstly define the auxiliary functions as follows: where and . From , we easily obtain that when and when
Let be any positive solution of model (1) with initial condition (2). By computing , we have From , it follows that Hence,
By computing , we also have
Further, by computing , we have
Finally, by computing , we further have
Now, we define a Lyapunov function as follows: where and are positive constants which will be chosen in the following. It is obvious that from (40) and (41) for all and if and only if . By computing we have Choose constants and as follows: Then we further have Noting that , for all , then we have From , it follows that when and when . Hence, we have the following inequality: Furthermore, from , we also have the following inequalities: which implies that
From (53), (54), and (56), we further obtain From and , we finally have for all . Obviously, if and only if , , and for all . Therefore, using the theorems of stability of the difference equations (see Theorem  6.3 in [33]), we obtain that is globally asymptotically stable. This completes the proof.

As a special case of model (1), we consider the rate of losing immunity in model (1); that is, model (1) degenerates into a SIR epidemic model. Then, in the above calculation of , we can directly obtain the following inequality without and : We have for all and if and only if and for all . Therefore, as a consequence of Theorem 5, we have the following result.

Corollary 6. Assume that holds and in model (1). Then endemic equilibrium is globally asymptotically stable if and only if .

Remark 7. By comparing the results obtained in [24], then, from Corollary 6, we see that Theorem 5 is a direct extension of the corresponding result given in [24] on the global stability of the endemic equilibrium in the nondelayed case and the recovered individuals are in a position to lose the immunity.

Remark 8. For general model (1), we spontaneously expect that as long as basic reproduction number , then model (1) shows a unique endemic equilibrium which is globally asymptotically stable. However, it is a pity that, in Theorem 5, in order to obtain the global asymptotic stability of endemic equilibrium , we need to introduce some additional conditions, that is, and . Furthermore, from the proof of Theorem 5, we easily see that assumptions and only are used to ensure for all . Therefore, an important open problem is whether we can directly prove for all without assumptions and and further obtain the global asymptotic stability of endemic equilibrium of model (1) only when basic reproduction number .

4. Permanence of Disease

In this section, we will use the theory of persistence in general discrete dynamical systems to study the permanence of model (1). We will obtain that the disease in model (1) is permanent only when basic reproduction number and assumption holds.

Let be a metric space with metric and let be a continuous map. For any , the sequence defined by for any integer is said to be a solution sequence through , and the omega limit set of is defined by there is a sequence such that . For a nonempty set , we further define the stable set of by .

Let be a nonempty open set of . We denote

Lemma 9. Let be a continuous map. Assume that the following conditions hold.() is compact and point dissipative, and .()There exists a finite sequence of compact and isolated invariant sets such that(a) for any and ;(b);(c)no subset of forms a cycle in ;(d) for each .Then is uniformly persistent with respect to ; that is, there exists a constant such that for all .

Here, the definitions on the compactness and point dissipativity of map and the definitions on the compactness, isolated invariance, and the cycle in for sequence can be found in [34]. Furthermore, Lemma 9 can be obtained from Theorem , Theorem , Remark , and Theorem given by Zhao in [34].

On the permanence of the disease for model (1), we have the following result.

Theorem 10. Assume that holds. Then, the disease in model (1) is permanent; that is, there are two constants such that for any positive solution of model (1) if and only if .

Proof. From Theorem 4 we see that the necessity is obvious. Now, we only need to prove the sufficiency. Define two sets as follows: We have For any initial point , let be the solution of model (1) through . We define map by .
From the positivity and ultimate boundedness of solutions of model (1), we obtain and is also point dissipative.
By observing the proof of Theorem 3, we see that, since and are continuous with respect to and , respectively, is also continuous with respect to . Hence, , as the solution of , is also continuous for . Similarly, from the expression of and the continuity of with respect to , we obtain that is continuous with respect to . Hence, , as the solution of , is also continuous for . Therefore, we finally obtain that map is continuous on . From this, we obtain that also is compact.
In , we have , and hence satisfies Obviously, we can obtain as . This shows that