Abstract
This paper mainly explores a generalized SIS epidemic model with time-varying coefficients and delays. By employing Lyapunov function method and differential inequality approach, a sufficient criterion to guarantee the existence and exponential stability of positive periodic solutions for the addressed model is obtained, which complements the earlier publications. Particularly, an example and its numerical simulations are given to demonstrate our theoretical results.
1. Introduction
In the study of infectious dynamics, under the assumption that there is no mortality due to illness, A. Iggidr, K. Niri b, and E. Ould Moulay Ely [1] proposed the following delayed SIS epidemic model: where , , , , and denote the susceptible numbers, infective numbers, and total numbers at time , respectively. Here is the transmission rate, designates the average latent period of the disease, and represents the natural death rate. A more detailed description of the model can be found in [2]. Recently, some criteria ensuring the global attractivity of the endemic equilibrium of (1) have been established in [3]. On the one hand, any biological or environmental parameters are naturally subject to fluctuation in time and it is more realistic to consider the model with time-varying coefficients and delays. One can easily find that (1) can be extended to the case with time-varying delay and time-varying coefficients when d(t) has a special form. On the other hand, in the real world, the model coefficients are usually assumed to be periodic because the correlation coefficients are susceptible to the change of climate and other factors. In fact, periodic phenomena are found in the spread of many infectious diseases such as influenza and chickenpox. Therefore it is worthwhile to investigate how periodic solutions arise and the stability of periodic solutions in an epidemiological model (see, for instance, [4–7]). However, to the best of our knowledge, there are no existing papers on positive periodic solutions of (1). According to the previous analysis, in this paper, our goal is to study the existence and the exponential stability of positive periodic solutions of the following generalized SIS model with time-varying delays and coefficients: where , , are nonnegative bounded and continuous -periodic functions.
For convenience, we introduce the following notations.
By virtue of the biological interpretation of model (2), one can find that all solutions of (2) should remain in the interval . Then, we introduce the initial value conditions of (2) as follows: In addition, define a continuous map byClearly, the existence and uniqueness of the solution of (2) with the initial condition (4) are guaranteed by which is a locally Lipschitz map with respect to . Let be a solution of the initial value problem (2) and (4) and be the maximal right-interval of the existence of .
Let , . In the following, we always assume that , for all . Fix ; it is not difficult to verify that is increasing on and decreasing on about the variable . Note that AndFor fixed , we denote the unique zero solution of on by . For any , denote , . Furthermore, it is not hard to see that
2. Preliminary Results
The following lemmas are helpful to prove our main results in Section 3.
Lemma 1. Assume that , , and Then, for ,
Proof. For the sake of simplicity of notations, let for all . We first show that Arguing by contradiction, if this is not true, then there must exist such that which, with the help of (8), (9), and (12), entails that which is contrary to the fact that . Hence (11) holds.
Next, we demonstrate that If not, there must exist such that which, together with (9) and (15), suggests that which is a contradiction with and hence (14) is true.
In view of (9), (11), and (14), we can show that From Theorem 2.3.1 in [8], we can easily obtain . This completes the proof.
Lemma 2. Assuming that the conditions of Lemma 1 are established, we further assume that Then there exists a positive such that where .
Proof. Consider defined byClearly, is continuous. Note that , which follows from the continuity and periodicity of , , and , then there are and such that orLet , . For simplicity, denote and by and , respectively. From Lemma 1, we haveLet for . Then, for , Consider the Lyapunov functionalWe claim that for . Otherwise, there exists such thatThen a contradiction. Consequently, we infer that (19) holds. The proof is completed.
3. Main Results
Combined with Lemmas 1 and 2, we have the following theorem.
Theorem 3. Under the assumptions of Lemma 2, then system (2) has exactly one positive -periodic solution which is exponentially stable.
Proof. Fix a ; let be a solution of system (2) and (4). According to Lemma 1, we haveFrom the periodicity of coefficients and delays for system (2), for any nonnegative integer , we get which implies that is also a solution to system (2) on . Denote . In view of Lemma 2, for any nonnegative integer and , we obtain where .
Now, we claim that is convergent on any compact interval as . For an arbitrary subset , we can pick a nonnegative integer satisfying for . Then for and , one can see thatwhich implies that converges uniformly to a continuous function . Due to the arbitrariness of , one can easily find that as for . Moreover, we getNext we show that is a -periodic solution of (2). The periodicity can be obtained immediately from the factfor all . Noting that is a solution to (2), for . Letting gives us for ; namely, is a solution to (2) on . Finally, by the same method as that in the proof of Lemma 2, we can show that is exponentially stable. This finishes the proof.
4. An Example
Example 1. Regard the following generalized SIS epidemic model with time-varying delays and coefficients:Then and Choose , then . By a simple computation, we can easily check that all the conditions of Theorem 3 are satisfied. Therefore, system (35) with initial values in has a unique positive -periodic solution which is exponentially stable. The numerical simulations in Figure 1 strongly support the conclusion.
Remark 5. To the author’s knowledge, it is the first time to focus attention on the exponential stability of positive periodic solutions for (2), which is the generalization of (1). By constructing invariant sets ingeniously, under appropriate conditions, we represent that all solutions of the addressed model converge exponentially to the positive periodic solution. In addition, one can find that all the results in [1, 3–6] cannot be applicable to show the global exponential stability on the positive periodic solution of (35), which implies that the results of this paper are new and complement of previously known results. In future, we will consider the properties of almost periodic solutions and pseudo almost periodic solutions of the model.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
This work was supported by the Natural Scientific Research Fund of Hunan Province of China (grant no. 2018JJ2194).