# Positive Periodic Solutions for an Generalized SIS Epidemic Model with Time-Varying Coefficients and Delays

Authors: Zhiwen Long
Journal: Discrete Dynamics in Nature and Society
 (2018)

---

## 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.


---

## Body

## 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: (1)i′t=β1-itit-β1-it-ωit-ωe-dω-dit,where i(t)=I(t)/N(t), N(t)=S(t)+I(t), S(t), I(t), and N(t) denote the susceptible numbers, infective numbers, and total numbers at time t, respectively. Here β is the transmission rate, ω>0 designates the average latent period of the disease, and d 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: (2)i′t=βt1-itit-βt1-it-ωtit-ωte-dtωt-dtit,where β(t), ω(t), d(t) are nonnegative bounded and continuous T-periodic functions.For convenience, we introduce the following notations.(3)g+=supt∈R⁡gt,g-=inft∈R⁡gt,C=C-ω+,0,R,C+=C-ω+,0,R+,itθ=it+θ.By virtue of the biological interpretation of model (2), one can find that all solutions of (2) should remain in the interval [0,1]. Then, we introduce the initial value conditions of (2) as follows: (4)it0=φ,φ∈C-ω+,0,0,1⊂C+andi0>0.In addition, define a continuous map f:R×C+→R by(5)ft,φ=βtφ01-φ0-βtφ-ωt1-φ-ωte-dtωt-dtφ0.Clearly, the existence and uniqueness of the solution of (2) with the initial condition (4) are guaranteed by which f is a locally Lipschitz map with respect to φ∈C+. Let i(t;t0,φ) be a solution of the initial value problem (2) and (4) and [t0,η(φ)) be the maximal right-interval of the existence of i(t;t0,φ).Letθ(t)=βt1-e-dtωt/dt, gt,x=βt1-x-1/4xe-dtωt/d(t)-1,t∈R. In the following, we always assume that θ(t)>2, for all t¨¨¨¨¨∈R. Fix t∈R; it is not difficult to verify that g(t,x) is increasing on (0,1/2e-d(t)ω(t)/2) and decreasing on (1/2e-d(t)ω(t)/2,+∞) about the variable x. Note that (6)gt,12e-dtωt/2≥gt,12=βt1-e-dtωt2dt-1=θt2-1>0,∀t∈R.And(7)gt,x→-∞asx→0+;gt,x→-∞asx→+∞.For fixed t∈R, we denote the unique zero solution of g(t,x) on (0,1/2e-d(t)ω(t)/2) by κ(t). For any κ∈(κ+,1/2), denote M=1-κ, Cκ=φ∈C∣φθ∈κ,Mfor allθ∈-ω+,0. Furthermore, it is not hard to see that (8)gt,κ>0,∀t∈R.


## 2. Preliminary Results

The following lemmas are helpful to prove our main results in Section3.Lemma 1.
Assume thatd->0, ω->0, and (9)2dt<βt1-e-dtωt<dtκfor allt∈R.Then, for φ∈Cκ,(10)ηφ=∞andit;t0,φ∈Cκfort≥t0.Proof.
For the sake of simplicity of notations, leti(t)=i(t;t0,φ) for all t∈[t0-ω+,η(φ)). We first show that (11)it>κforallt∈t0,ηφ.Arguing by contradiction, if this is not true, then there must exist T1∈(t0,η(φ)) such that (12)iT1=κ,it>κforallt∈t0-ω+,T1,which, with the help of (8), (9), and (12), entails that (13)i′T1=βT11-iT1iT1-βT11-iT1-ωT1iT1-ωT1e-dT1ωT1-dT1iT1≥βT11-κκ-βT14e-dT1ωT1-dT1κ=dT1κβT11-κ-1/4κe-dT1ωT1dT1-1>0,which is contrary to the fact that i′(T1)≤0. Hence (11) holds.
Next, we demonstrate that(14)it<Mfort∈t0,ηφ.If not, there must exist T2∈(t0,η(φ)) such that (15)iT2=M,it<Mforallt∈t0-ω+,T2,which, together with (9) and (15), suggests that (16)i′T2=βT21-iT2iT2-βT21-iT2-ωT2iT2-ωT2e-dT2ωT2-dT2iT2≤βT21-MM-βT21-MMe-dT2ωT2-dT2M=MβT21-M1-e-dT2ωT2-dT2<0,which is a contradiction with i′(T2)≥0 and hence (14) is true.
In view of (9), (11), and (14), we can show that (17)κ<it<Mforallt∈t0-ω+,ηφ.From Theorem 2.3.1 in [8], we can easily obtain η(φ)=+∞. This completes the proof.Lemma 2.
Assuming that the conditions of Lemma1 are established, we further assume that (18)dt>βt1-2κ1+e-dtωtfort∈R.Then there exists a positive λ such that (19)it;t0,φ-it;t0,φ∗≤Kφ,φ∗e-λtforφ,φ∗∈Cκandt≥t0,where Kφ,φ∗=eλt0maxθ∈-ω+,0⁡φθ-φ∗θ+1.Proof.
ConsiderΓ:[0,1]→R defined by(20)Γu=supt∈R⁡-dt-1-2κβt1+e-dtωteuω+-u,foru∈0,1.Clearly, Γ is continuous. Note that Γ(0)=supt∈R⁡-dt-1-2κβt1+e-dtωt<0, which follows from the continuity and periodicity of d(t), ω(t), and β(t), then there are η>0 and 1>λ>0 such that Γ(λ)<-η or(21)-dt-1-2κβt1+e-dtωteλω+-λ<-η<0fort∈R.
Letφ, φ∗∈Cκ. For simplicity, denote i(t;t0,φ) and i(t;t0,φ∗) by i(t) and i∗(t), respectively. From Lemma 1, we have(22)κ<it,i∗t<Mforallt∈t0-ω+,∞.Let y(t)=i(t)-i∗(t) for t∈t0-ω+,∞. Then, for t≥t0, (23)y′t=-βtit+i∗t-1+dtyt+βte-dtωtit-ωt+i∗t-ωt-1yt-ωt.
Consider the Lyapunov functional(24)Vt=yteλtfort≥t0.We claim that V(t)≤Kφ,φ∗ for t≥t0. Otherwise, there exists t∗>t0 such that(25)Vt∗=Kφ,φ∗andVt<Kφ,φ∗forallt∈t0-ω+,t∗.Then (26)0≤D-Vt∗≤-βt∗it∗+i∗t∗-1+dt∗yt∗eλt∗+βt∗e-dt∗ωt∗it∗-ωt∗+i∗t∗-ωt∗-1yt∗-ωt∗eλt∗+λyt∗eλt∗=-βt∗it∗+i∗t∗-1+dt∗-λyt∗eλt∗+βt∗e-dt∗ωt∗it∗-ωt∗+i∗t∗-ωt∗-1yt∗-ωt∗eλt∗-ωt∗eλωt∗≤-βt∗2κ-1+dt∗-λyt∗eλt∗+2M-1βt∗e-dt∗ωt∗yt∗-ωt∗eλt∗-ωt∗eλωt∗≤-dt∗+βt∗2κ-1-λ+1-2κβt∗e-dt∗ωt∗eλω+Kφ,φ∗≤-dt∗-βt∗1-2κ1+e-dt∗ωt∗eλω+-λKφ,φ∗<-ηKφ,φ∗<0,a contradiction. Consequently, we infer that (19) holds. The proof is completed.


## 3. Main Results

Combined with Lemmas1 and 2, we have the following theorem.Theorem 3.
Under the assumptions of Lemma2, then system (2) has exactly one positive T-periodic solution which is exponentially stable.Proof.
Fix aφ∈Cκ; let i(t)=i(t;t0,φ) be a solution of system (2) and (4). According to Lemma 1, we have(27)κ<it;t0,φ<Mforallt≥t0-ω+.From the periodicity of coefficients and delays for system (2), for any nonnegative integer h, we get (28)i′t+hT=βt+hT1-it+hTit+hT-βt+hT1-it+hT-ωt+hTit+hT-ωt+hTe-dt+hTωt+hT-dt+hTit+hT=βt1-it+hTit+hT-βt1-it+hT-ωtit+hT-ωte-dtωt-dtit+hTwhich implies that i(t+hT;t0,φ) is also a solution to system (2) on t0-ω+-hT,∞. Denote ψ=i(t+T;t0,φ). In view of Lemma 2, for any nonnegative integer h and t+hT≥t0, we obtain (29)it+h+1T;t0,φ-it+hT;t0,φ=it+hT;t0,ψ-it+hT;t0,φ≤Kφ,ψe-λt+hT,where Kφ,ψ=eλt0maxθ∈-ω+,0⁡φθ-ψθ+1.
Now, we claim thati(t+qT;t0,φ) is convergent on any compact interval as q→∞. For an arbitrary subset [a,b]⊂R, we can pick a nonnegative integer q0 satisfying t+q0T≥t0 for t∈[a,b]. Then for t∈[a,b] and q>q0, one can see that(30)it+qT=it+q0T+∑h=q0q-1it+h+1T-it+hTwhich implies that {i(t+qT)}q converges uniformly to a continuous function i∗(t). Due to the arbitrariness of [a,b], one can easily find that i(t+qT)→i∗(t) as q→∞ for t∈R. Moreover, we get(31)κ≤i∗t≤Mforallt∈R.
Next we show thati∗ is a T-periodic solution of (2). The periodicity can be obtained immediately from the fact(32)i∗t+T=limq→∞⁡it+T+qT=limq+1→∞⁡it+q+1T=i∗tfor all t∈R. Noting that i(t+qT) is a solution to (2), (33)it+qT-it0+qT=∫t0tβs1-is+qTis+qT-βs1-is+qT-ωsis+qT-ωse-dsωs-dsis+qTdsfor t≥t0. Letting q→∞ gives us (34)i∗t-i∗t0=∫t0tβs1-i∗si∗s-βs1-i∗s-ωsi∗s-ωse-dsωs-dsi∗sdsfor t≥t0; namely, i∗ is a solution to (2) on [t0-ω+,∞). Finally, by the same method as that in the proof of Lemma 2, we can show that i∗(t) 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:(35)i′t=0.32+0.08sin⁡t1-itit-0.32+0.08sin⁡t1-it-40+10sin⁡tit-40+10sin⁡te-0.08+0.02sin⁡t40+10sin⁡t-0.08+0.02sin⁡tit.Then (36)e-4≤e-dtωt≤e-1.8≈0.1653,forallt∈R,and κ+≈0.23025. Choose κ=1/3, then 1/4<κ. By a simple computation, we can easily check that all the conditions of Theorem 3 are satisfied. Therefore, system (35) with initial values in φ∈Cκ has a unique positive 2π-periodic solution i∗(t) which is exponentially stable. The numerical simulations in Figure 1 strongly support the conclusion.Figure 1
Numerical solutionsi(t) of model (35) for initial value φ(t)=0.4,0.5,0.6,t∈[-50,0].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.



---

*Source: 7296320-2018-10-11.xml*