# On a Five-Dimensional Nonautonomous Schistosomiasis Model with Latent Period

Authors: Shujing Gao; Yanfei Dai; Dehui Xie
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

A five-dimensional nonautonomous schistosomiasis model which include latent period is proposed and studied. By constructing several auxiliary functions and using some skills, we obtain some sufficient conditions for the extinction and permanence (uniform persistence) of infectious population of the model. New threshold values of integral form are obtained. For the corresponding autonomous schistosomiasis model, our results are consistent with the past results. For the periodic and almost periodic cases, some corollaries for the extinction and permanence of the disease are established. In order to illustrate our theoretical analysis, some numerical simulations are presented.


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## Body

## 1. Introduction

Schistosomiasis is a chronic, parasitic disease caused by blood flukes (trematode worms) of the genusSchistosoma. More than 207 million people, 85% of them living in Africa, are infected worldwide, with an estimated 700 million people at risk in 74 endemic countries. Schistosomiasis is prevalent in tropical and subtropical areas, especially in poor communities without access to safe drinking water and adequate sanitation [1]. In the developing countries, schistosomiasis is frequently a serious health problem [2].The first mathematical models for schistosomiasis were those developed by Macdonald in [3] and Hairston in [4]. Since then, a number of mathematical models have been proposed. Barbour [5] studied the prevalence of infection in the definitive host population and the snails, Chiyaka and Garira [6] were concerned with the combined effect of the inclusion the dynamics of the miracidia and cercariae populations and a density dependent infection term, Bhunu et al. [7] developed a mathematical model of the transmission dynamics of HIV/AIDS in the presence of schistosomiasis in the human-snail hosts, Yang and Xiao [8] presented a dynamic model ofSchistosoma japonicum transmission that incorporated effects of the prepatent periods of the different stages ofSchistosoma into Barbour’s model, Qi and Cui [9] established a diffusive system of partial differential equations considering the diffusion of human and snail hosts, and Qi et al. [10] developed a schistosomiasis model of the prevalence of infection in theRattus norvegicus andOncomelania snail, in which the authors considered a schistosomiasis model on theRattus norvegicus in Qianzhou and Zimuzhou, two islets in the center of Yangtze River near Nanjing, China. There have been no human residents or other livestock on the islets; thus the mammalianRattus norvegicus is the only definitive host while snail serves as the unique intermediate host; the authors set up the compartmental model as follows:(1)dxsdt=Ax-μxxs-βxxsyi,dxidt=βxxsyi-μxxi-αxxi,dysdt=Ay-μyys-βyxiys,dyedt=βyxiys-μy+θye,dyidt=θye-μyyi-αyyi.Here, xs, xi represent the number of susceptible and infectedRattus norvegicus, respectively. ys, ye, yi are the number of susceptible, infected and preshedding, and infected and sheddingOncomelaniasnail, respectively. The other parameters interpretation of model (1) is shown in Parameters Section. All the parameters in the model are assumed to be nonnegative constants.According to the concept of the next generation of matrix [11], the authors obtain the basic reproduction number [12] of system (1) as follows: (2)R0=AxAyθβxβyμxμyμx+αxμy+αyμy+θ3.And they prove that the infection will go extinct if R0≤1 and schistosomiasis will be prevalent if R0>1.Nonautonomous phenomena, however, are dominating in real epidemic systems. Many diseases show seasonal behavior and taking into account the seasonal influence in epidemic models is very necessary. The nonautonomous phenomenon come from various sources, such as the variation of transmission rate and fluctuations in death and birth rates. Particularly, these coefficients of models usually vary with time when long-time dynamical behaviors are studied for an epidemic system. More and more realistic and interesting nonautonomous models can be found in papers [13–24] and the references cited therein.Motivated by the ideas of the above aspects, system (1) is modified to take into account factors mentioned above where all coefficients vary with time to give a more appropriate result and better understanding of the prevalence of schistosomiasis. Thus, we refer to the following nonautonomous system for the study of schistosomiasis:(3)dxstdt=Axt-μxtxst-βxtxstyit,dxitdt=βxtxstyit-μxtxit-αxtxit,dystdt=Ayt-μytyst-βytxityst,dyetdt=βytxityst-μyt+θtyet,dyitdt=θtyet-μytyit-αytyit with initial value (4)xs0>0,xi0>0,ys0>0,ye0≥0,yi0>0.In this paper, our purpose is to obtain the weaker conditions for the permanence and extinction of schistosomiasis. We will establish the sufficient conditions for the permanence and extinction of the disease and give some new threshold values of the integral form. When system (3) reduces to periodic or almost periodic case, the basic reproduction number is obtained (see Corollaries 18 and 19 given in Section 5). Our results show that the threshold value acts as a sharp threshold for the permanence and extinction of the disease.This paper is organized as follows. In Section2 we present preliminary setting and propositions which are used to analyze the long-time behavior of system (3) in the following sections. In Sections 3 and 4, we prove our main theorems on the extinction and permanence of infectious population of system (3). In Section 5, we derive some corollaries for the extinction and permanence of infectious population of system (3) for some special cases. In Section 6, we provide numerical examples to illustrate the validity of our analytical results.


## 2. Notations and Preliminaries

At first, we put the following assumptions for system (3).(H1)
FunctionsAxt, μxt, βxt, αxt, Ayt, μyt, βyt, αyt, and θt are positive, bounded, and continuous on [0,+∞).(H2)
There exist constantsωi>0i=1,2,3,4,5,6 such that (5)liminft→+∞⁡∫tt+ω1Axsds>0,liminft→+∞⁡∫tt+ω2Aysds>0,liminft→+∞⁡∫tt+ω3μxsds>0,liminft→+∞⁡∫tt+ω4μysds>0,liminft→+∞⁡∫tt+ω5βxsds>0,liminft→+∞⁡∫tt+ω6βysds>0.For any continuous periodic functionf with period ω>0, we denote by f¯ the average value of ft where f¯=1/ω∫0ωftdt.For any continuous almost periodic functionf, we denote by mf the average value of ft where mf=limt→+∞⁡1/t∫0tftdt.Remark 1.
When system (3) degenerates into ω-periodic system, that is to say, all the coefficients of system (3) are all nonnegative and nonzero, continuous periodic functions with period ω>0, then (H2) is equivalent to the following cases: (6)A¯x>0,A¯y>0,μ¯x>0,μ¯y>0,β¯x>0,β¯y>0.Remark 2.
When system (3) degenerates into almost periodic system, that is to say, all the coefficients in it are all nonnegative and nonzero, continuous almost periodic functions, then (H2) is equivalent to the following cases: (7)mAx>0,mAy>0,mμx>0,mμy>0,mβx>0,mβy>0. In what follows, we denote Nxt≔xst+xit,Nyt≔yst+yet+yit. We also define that N_xt,N¯xt,N_yt,N¯yt,x_st,y_st are the solutions of the following equations with initial value x_s0=xs0>0,  y_s0=ys0>0, N_x0=N¯x0=Nx0=xs0+xi0,  N_y0=N¯y0=Ny0=ys0+ye0+yi0, respectively: (8)dNxtdt=Axt-μxt+αxtNxt,(9)dNxtdt=Axt-μxtNxt,(10)dNytdt=Ayt-μyt+αytNyt,(11)dNytdt=Ayt-μytNyt,(12)dxstdt=Axt-μxt+βxtN¯ytxst,(13)dystdt=Ayt-μyt+βytN¯xtyst.Obviously, from system (3) and the comparison theorem, we have (14)N_xt≤Nxt≤N¯xt,N_yt≤Nyt≤N¯yt,x_st≤xst≤N¯xt,y_st≤yst≤N¯yt.Furthermore, we have the following result.Lemma 3.
Suppose that assumptions (H1) and (H2) hold. Then the following results hold.(a)
There exist constantsm1>0 and M1>0, which are independent of the choice of initial value Nx0>0, such that (15)0<m1<liminft→+∞⁡N_xt≤liminft→+∞⁡Nxt≤limsupt→+∞⁡Nxt≤limsupt→+∞⁡N¯xt<M1<+∞.(b)
There exist constantsm2>0 and M2>0, which are independent of the choice of initial value Ny0>0, such that (16)0<m2<liminft→+∞⁡N_yt≤liminft→+∞⁡Nyt≤limsupt→+∞⁡Nyt≤limsupt→+∞⁡N¯yt<M2<+∞.(c)
Each fixed solutionN_xt of (8), N¯xt of (9), N_yt of (10), and N¯yt of (11) is bounded and globally uniformly attractive on R+.(d)
When (8) is ω-periodic, then (8) has a unique nonnegative ω-periodic solution N_xt which is globally uniformly attractive. And (9)–(11) have similar results.(e)
When (8) is almost periodic, then (8) has a unique nonnegative almost periodic solution N_xt which is globally uniformly attractive. And (9)–(11) have similar results.(f)
Ifμxt>0 for all t≥0 and 0<liminft→∞⁡Axt/μxt+αxt≤limsupt→∞⁡Axt/μxt+αxt<∞, then for any solution N_x(t) of (8), we have (17)Axμx+αxm≤liminft→+∞⁡N_xt≤limsupt→+∞⁡N_xt≤Axμx+αxM,where(18)Axμx+αxm=liminft→+∞⁡Axtμxt+αxt,Axμx+αxM=limsupt→+∞⁡Axtμxt+αxt.(g)
Ifμxt>0 for all t≥0 and 0<liminft→∞⁡Axt/μxt≤limsupt→∞⁡Axt/μxt<∞, then, for any solution N¯xt of (9), we have (19)Axμxm≤liminft→+∞⁡N¯xt≤limsupt→+∞⁡N¯xt≤AxμxM,where(20)Axμxm=liminft→+∞⁡Axtμxt,AxμxM=limsupt→+∞⁡Axtμxt.(h)
Ifμyt>0 for all t≥0 and 0<liminft→∞⁡Ayt/μyt+αyt≤limsupt→∞⁡Ayt/μyt+αyt<∞, then, for any solution N_yt of (10), we have (21)Ayμy+αym≤liminft→+∞⁡N_yt≤limsupt→+∞⁡N_yt≤Ayμy+αyM,where(22)Ayμy+αym=liminft→+∞⁡Aytμyt+αyt,Ayμy+αyM=limsupt→+∞⁡Aytμyt+αyt.(i)
Ifμyt>0 for all t≥0 and 0<liminft→∞Ayt/μyt≤limsupt→∞Ayt/μyt<∞, then, for any solution N¯y(t) of (11), we have (23)Ayμym≤liminft→+∞⁡N¯yt≤limsupt→+∞⁡N¯yt≤AyμyM,where(24)Ayμym=liminft→+∞⁡Aytμyt,AyμyM=limsupt→+∞⁡Aytμyt.Lemma 4.
Suppose that assumptions (H1) and (H2) hold. Then the solution(xst,xit,yst,yet,yit) of system (3) with initial value (3) exists and is uniformly bounded. Furthermore, for any t>0, we have(25)xst>0,xit>0,yst>0,yet≥0,yit>0.
For anyp>0,q>0, and t>0 we define (26)G1p,t=1+1pθt-αyt,G2p,q,t=1qβxtN¯xt+μyt+αyt-1pθt-μxt-αxt,G2′p,q,t=1qβxtx_st+μyt+αyt-1pθt-μxt-αxt,W1p,t=pyet-yit,W2q,t=qyit-xit,where xs,xi,ys,ye,yi is any solution of system (3).We use the following two results in order to investigate the long-time behavior of system (3).Lemma 5.
If there exist positive constantsp>0 and T1>0 such that G1p,t<0 for all t≥T1, then there exists T2≥T1 such that either W1p,t>0 for all t≥T2 or W1p,t≤0 for all t≥T2.Proof.
Suppose that there does not existT2≥T1 such that W1p,t>0 for all t≥T2 or W1p,t≤0 for all t≥T2 hold. Then there necessarily exists s1≥T1 such that W1p,s1=0 and dW1p,t/dt∣t=s1<0. Hence we have (27)pyes1=yis1,(28)dW1p,tdtt=s1=pβys1xis1yss1-pμys1+θs1yes1-θs1yes1+μys1+αys1yis1>μys1+αys1yis1-μys1+1+1pθs1pyes1.
Substituting (27) into (28), we have (29)0>-1+1pθs1-αys1pyes1=-pyes1G1p,s1. From Lemma 4, we have G1p,s1>0, which is a contradiction.Lemma 6.
If there exist positive constantsp>0,q1>0,q2>0, and T1>0 such that G1p,t<0,G2(p,q1,t)<0,G2′(p,q2,t)>0 for all t≥T1, then there exists T2≥T1 such that for every t≥T2, we have either W2(qi,t)>0 or W2(qi,t)≤0, i=1,2.Proof.
From Lemma5, we know that there exists T2′≥T1 such that either W1p,t>0 for all t≥T2′ or W1p,t≤0 for all t≥T2′.
For the first case, ifW1p,t>0 for all t≥T2′, we will show that there exists T3≥T1 such that either W2(q1,t)>0 for all t≥T3 or W2(q1,t)≤0 for all t≥T3. Suppose that there does not exist T3≥T1 such that W2(q1,t)>0 for all t≥T3 or W2(q1,t)≤0 for all t≥T3 hold. Then there necessarily exists s2≥T1 such that W2q1,s2=0 and dW2q1,t/dt∣t=s2<0. Hence we have (30)q1yis2=xis2,(31)dW2q1,tdtt=s2=q1θs2yes2-μys2+αys2q1yis2-βxs2xss2yis2+μxs2+αxs2xis2≥q1pθs2yis2-μys2+αys2q1yis2-βxs2xss2yis2+μxs2+αxs2xis2=q1yis21pθs2-μys2-αys2-1q1βxs2xss2+μxs2+αxs2xis2. Substituting (30) into (31), we have (32)0>q1yis21pθs2+μxs2+αxs2-μys2-αys2-1q1βxs2xss2≥-q1yis2G2p,q1,s2.From Lemma 4, we have G2p,q1,s2>0, which is a contradiction.
For the second case, ifW1p,t≤0 for all t≥T2′, we will prove that there exists T4≥T1 such that either W2q2,t>0 for all t≥T4 or W2q2,t≤0 for all t≥T4. Suppose that there does not exist T4≥T1 such that W2q2,t>0 for all t≥T4 or W2q2,t≤0 for all t≥T4 hold. Then there necessarily exists s3≥T1 such that W2q2,s3=0 and dW2q2,t/dt∣t=s3>0. Hence we have(33)q2yis3=xis3,(34)dW2q2,tdtt=s3=q2θs3yes3-μys3+αys3q2yis3-βxs3xss3yis3+μxs3+αxs3xis3≤q2pθs3yis3-μys3+αys3q2yis3-βxs3xss3yis3+μxs3+αxs3xis3=-q2yis31q2βxs3xss3+μys3+αys3-1pθs3+μxs3+αxs3xis3. Substituting (33) into (34), we have (35)0<-q2yis31q2βxs3xss3+μys3+αys3-1pθs3-μxs3-αxs3≤-q2yis3G2′p,q2,s3. From Lemma 4, we have G2′(p,q2,s3)<0, which is a contradiction.
If we letT2=max⁡T3,T4, then the results of the lemma hold.


## 3. Extinction of Infectious Population

In this section, we obtain sufficient conditions for the extinction of the infectious population of system (3). The definition of the extinction is as follows.Definition 7.
We say that the infectivexi(t) of system (3) is extinct if (36)limt→+∞⁡xit=0.Similarly, we can define the extinction ofyit. Next, one of the main results of this paper is investigated.Theorem 8.
If there exist positive constantsλ>0,p>0,q>0, and T1>0 such that (37)R1λ,p≔limsupt→+∞⁡∫tt+λ1pθs-μys-αysds<0,(38)R1′λ,q≔limsupt→+∞⁡∫tt+λ1qβxsN¯xs-μxs-αxsds<0,(39)R1∗λ,p,q≔limsupt→+∞⁡∫tt+λpqβysN¯ys-μys-θsds<0 and G1p,t<0,G2p,q,t<0 for all t≥T1, then limt→+∞xit=0 and limt→+∞yit=0.Proof.
Since there existp>0,T1>0 such that G1p,t<0 holds for all t≥T1, from Lemma 5, we only have to consider the following two cases:(i)
pyet≤yit for all t≥T2.(ii)
pyet>yit for all t≥T2.
First we consider case (i). Since we havepye(t)≤yi(t) for all t≥T2, it follows from the fifth equation of system (3) that (40)dyitdt=θtyet-μytyit-αytyit≤1pθt-μyt-αytyit
for allt≥T2. Hence we have (41)yit≤yiT2exp⁡∫T2t1pθs-μys-αys
for allt≥T2. Now it follows from (37) that there exist constants δ1>0 and T3>T2 such that (42)∫tt+λ1pθs-μys-αysds<-δ1
for allt≥T3. From (41) and (42) we have limt→+∞yi(t)=0. Then the second equation of the corresponding limit system (3) is dxi(t)/dt=-(μx(t)+αx(t))xi(t). From (i) and (H2) we can easily see that limt→+∞xi(t)=0.
Next we consider case (ii). Because there existp>0,q>0 such that G2(p,q,t)<0 for all t≥T1, then from the proof of Lemma 6, we know that there exists T2′≥max{T1,T2} such that either W2(q,t)>0 for all t≥T2′ or W2(q,t)≤0 for all t≥T2′. Thus we also have two cases to discuss.
On the one hand, ifW2(q,t)≤0 holds for all t≥T2′, that is, yi(t)≤(1/q)xi(t) holds for all t≥T2′, then from the second equation of system (3) we obtain (43)dxitdt=βxtxstyit-μxt+αxsxit≤1qβxtN¯xt-μxt-αxtxit for all t≥T2′. Hence, we get (44)xit≤xiT2′exp⁡∫T2′t1qβxsN¯xs-μxs-αxsds for all t≥T2′. From (38) we see that there exist constants δ2>0 and T4>T2′ such that (45)∫tt+λ1qβxsN¯xs-μxs-αxsds<-δ2 for all t≥T4. From (44) and (45), we have limt→+∞xi(t)=0.
On the other hand, ifW2(q,t)>0 for all t≥T2′ hold, then we have xi(t)<qyi(t)<pqye(t) for all t≥T2′. From the fourth equation of system (3), we have (46)dyetdt=βytxityst-μyt+θtyet<pqβytN¯yt-μyt-θtyetfor all t≥T2′. Hence we have (47)yet<yeT2′exp⁡∫T2′tpqβysN¯ys-μys-θs. From (39), we see that there exist constants δ3>0 and T5>T2′ such that (48)∫tt+λpqβysN¯ys-μys-θsds<-δ3 for all t≥T5. From (47) and (48), we have limt→+∞ye(t)=0. Thus the fifth equation of the corresponding limit system (3) is dyi(t)/dt=-(μy(t)+αy(t))yi(t). From (i) and (H2), we can easily get that limt→+∞yi(t)=0. In a similar way, we can also prove limt→+∞xi(t)=0.Remark 9.
We assume that all coefficients of system (3) are given by identically constant functions; then system (3) becomes an autonomous system. Conditions (37)–(39) are as follows: (49)1pθ-μy-αy<0,1qβxAxμx-μx-αx<0,pqβyAyμy-μy-θ<0.These inequalities show that θβxAx/μx(μy+αy)(μx+αx)<pq<μy(μy+θ)/βyAy, which implies R0<1; here R0 is given in (2). This shows that the result is consistent with the literature [10].Remark 10.
It is obvious that if there exist positive constantsp>0,q>0, and T1>0 such that (50)limsupt→+∞⁡1pθt-μyt-αyt<0,limsupt→+∞⁡1qβxtN¯xt-μxt-αxt<0,limsupt→+∞⁡pqβytN¯yt-μyt-θt<0and G1p,t<0,G2p,q,t<0 for all t≥T1, then limt→+∞xit=0 and limt→+∞yit=0.


## 4. Permanence of Infectious Population

In this section, we obtain sufficient conditions for the permanence of infectious population of system (3).The definition of the permanence is as follows.Definition 11.
We say that the infectivexi(t) of system (3) is permanent if there exist positive constants xi1>0 and xi2>0, which are independent of the choice of initial value satisfying (3), such that (51)0<xi1≤liminft→+∞⁡xit≤limsupt→+∞⁡xit≤xi2.
Similarly, we can give the definition of the permanence ofyi(t).
Let the function(52)bt,u1,u2≔3βxtβytθtu1tu2t3-μxt+αxt+μyt+αyt+μyt+θtand let N_x(t) be some fixed solution of (8) with initial value N_x(0)>0andN_y(t) be some fixed solution of (10) with initial value N_y(0)>0.We have the following theorem.Theorem 12.
If there is a constantλ1>0 such that (53)R0∗=liminft→+∞⁡∫tt+λ1bs,N_xs,N_ysds>0, then the infective xi(t) and yi(t) of system (3) are permanent.Proof.
SinceN_x(t)≤Nx(t), N_y(t)≤Ny(t) for all t≥0, then, from system (3), we have (54)dxitdt≥βxtN_xt-xityit-μxtxit-αxtxit,dyetdt≥βytN_yt-yet-yitxit-μyt+θtyet,dyitdt=θtyet-μytyit-αytyit.Let us consider the following system: (55)dx¯itdt=βxtN_xt-x¯ity¯it-μxtx¯it-αxtx¯it,dy¯etdt=βytN_yt-y¯et-y¯itx¯it-μyt+θty¯et,dy¯itdt=θty¯et-μyty¯it-αyty¯it.
Firstly, we prove that the numberR0∗ is independent of the choice of N_x(t) and N_y(t). In fact, (c) of Lemma 3 implies that, for any sufficiently small ϵ>0 (<min⁡(m1,m2)), any solution N_x∗(t) of (8) with initial value N_x∗(0)>0, and any solution N_y∗(t) of (10) with initial value N_y∗(0)>0, there exists T>0 such that as t≥T(56)N_xt-ϵ≤N_x∗t≤N_xt+ϵ,N_yt-ϵ≤N_y∗t≤N_yt+ϵ,N_xt≥m1,N_yt≥m2.
Hence,(57)bt,N_xt-ϵ,N_yt-ϵ≤bt,N_x∗t,N_y∗t≤bt,N_xt+ϵ,N_yt+ϵ.
Fort≥T, since the inequality a1+a2+a33≤a13+a23+a33 holds for all ai≥0(i=1,2,3) and from Taylor’s theorem of binary function, we obtain (58)liminft→+∞⁡∫tt+λ1bs,N_xs+ϵ,N_ys+ϵds≤R0∗+M1+M2ϵ3+ϵ233λ1supt≥0βxtβytθt3,liminft→+∞⁡∫tt+λ1bs,N_xs-ϵ,N_ys-ϵds≥R0∗-ϵλ1M2m123+M1m223supt≥0βxtβytθt3. By the arbitrariness of ϵ, we finally obtain(59)liminft→+∞⁡∫tt+λ1bs,N_x∗s,N_y∗sds=R0∗.This shows that R0∗ is independent of the choice of N_x(t) and N_y(t). From inequalities (15) and (16), we can set T largely enough such that (60)m1<N_xt≤Nxt≤N¯xt<M1,(61)m2<N_yt≤Nyt≤N¯yt<M2 for all t≥T. From (53), for sufficiently small η>0, there exists T1≥T such that (62)∫tt+λ1bs,N_xs,N_ysds>η for all t≥T1. We define (63)βx+≔supt≥0⁡βxt,μx+≔supt≥0⁡μxt,βy+≔supt≥0⁡βyt,μy+≔supt≥0⁡μyt,αx+≔supt≥0⁡αxt,θ+≔supt≥0⁡θt,αy+≔supt≥0⁡αyt.From (60)–(63), we see that, for positive constants η1<η, there exist small ϵi>0,(i=1,2,3) and T2≥T1 such that(64)∫tt+λ1bs,N_xs-ϵ1-κ1ϵ2,N_ys-κ2ϵ1-κ3ϵ2-ϵ3ds>η1,(65)N_xt-ϵ1-κ1ϵ2>m1,(66)N_yts-κ2ϵ1-κ3ϵ2-ϵ3>m2hold for all t≥T2, where κ1≔βx+M1ω3,κ2≔βy+M2ω4,κ3≔1+κ1κ2. From (H2), we can choose ϵ1,ϵ2 such that both the following two equations hold for all t≥T2. (67)∫tt+ω3βxsM1ϵ2-μxs+αxsϵ1ds<-η1,(68)∫tt+ω4βysM2ϵ1+κ1ϵ2-μys+θsϵ3ds<-η1.
First we claim thatlimsupt→+∞y¯i(t)>ϵ2. In fact, if it is not true, then there exists T3≥T2 such that (69)y¯it≤ϵ2for all t≥T3. Suppose that x¯i(t)≥ϵ1 for all t≥T3. Then, from (60) and (69) we have (70)x¯it=x¯iT3+∫T3tβxsN_xs-x¯isy¯is-μxs+αxsx¯isds≤x¯iT3+∫T3tβxsM1ϵ2-μxs+αxsϵ1dsfor all t≥T3. Thus, from (67), we have limt→+∞x¯i(t)=-∞, which contradicts with (b) of Lemma 3. Therefore we see that there exist s1≥T3 such that x¯i(s1)<ϵ1. Suppose that there exists s2≥s1 such that x¯i(s2)>ϵ1+κ1ϵ2. Then, we see that there necessarily exists s3∈(s1,s2) such that x¯i(s3)=ϵ1 and x¯i(t)>ϵ1 for all t∈(s3,s2]. It is easy to see that there exists an integer n that satisfies s2∈[s3+nω3,s3+(n+1)ω3). Then, from (67), we have(71)ϵ1+κ1ϵ2<x¯is2=x¯is3+∫s3s2βxsN_xs-x¯isy¯is-μxs+αxsx¯isds<ϵ1+∫s3s3+nω3+∫s3+nω3s2βxsM1ϵ2-μxs+αxsϵ1ds≤ϵ1+∫s3+nω3s2βxsM1ϵ2ds<ϵ1+κ1ϵ2,which is a contradiction. Therefore, we see that (72)x¯it≤ϵ1+κ1ϵ2for all t≥s1. In a similar way, from (68), we can show that there exists s1~≥s1 such that (73)y¯et≤κ2ϵ1+κ1κ2ϵ2+ϵ3for all t≥s1~.
Fort≥0, we define a differentiable function V(t)=x¯i(t)y¯e(t)y¯i(t). When t≥s1~, from (69)–(73), we have (74)dVtdt55=βxtN_xt-x¯ity¯ety¯i2t+βytN_yt-y¯et-y¯itx¯i2ty¯it+θtx¯ity¯e2t-μxt+2μyt+θt+αxt+αytVt≥βxtN_xt-ϵ1-κ1ϵ2y¯ety¯i2t+βytN_ys-κ2ϵ1-κ3ϵ2-ϵ3x¯i2ty¯it+θtx¯ity¯e2t-μxt+2μyt+θt+αxt+αytVt.Since the inequality a+b+c≥3abc3 holds for all a,b,c≥0 and from (65) and (66), we have (75)dVtdt55≥bt,N_xt-ϵ1-κ1ϵ2,N_yt-κ2ϵ1-κ3ϵ2-ϵ3Vt.Integrating the above inequality from s1~ to t, we have (76)Vt≥Vs1~exp⁡∫s1~tbs,N_xs-ϵ1-κ1ϵ2,N_ys-κ2ϵ1-κ3ϵ2-ϵ3dsfor all t≥s1~. By (62), we obtain limt→+∞V(t)=∞. This contradicts with the boundedness of x¯i(t), y¯e(t), and y¯i(t). From this contradiction, we finally conclude limsupt→+∞y¯i(t)>ϵ2.
Next, we prove(77)liminft→+∞⁡y¯it≥y¯i1, where y¯i1>0 is a constant given in the following lines. For convenience sake, we let ω∗ be the least common multiple of ω3, ω4, and λ1. If we define (78)liminft→+∞⁡bt,N_xt-ϵ1-κ1ϵ2,N_yt-κ2ϵ1-κ3ϵ2-ϵ3≔m_ for ϵ>0, there exists T3~(≥T2) such that (79)bt,N_xt-ϵ1-κ1ϵ2,N_yt-κ2ϵ1-κ3ϵ2-ϵ3>m_-ϵ for all t≥T3~. Then, from inequalities (64) and (67)-(68) and (H2), we see that there exist constants T4~(≥T3~),λ2>max⁡ω5,ω6, here we choose λ2 is an integral multiple of ω∗, and η2>0 such that (80)∫tt+λ3βxsM1ϵ2-μxs+αxsϵ1ds<-M1,(81)∫tt+λ3βysM2ϵ1+κ1ϵ2-μys+θsϵ3ds<-M2,(82)∫tt+λ3bs,N_xs-ϵ1-κ1ϵ2,N_ys-κ2ϵ1-κ3ϵ2-ϵ3ds>η2,(83)∫tt+λ3βxsds>η2,(84)∫tt+λ3βysds>η2 for all t≥T4~, λ3≥λ2, and λ3 is an integral multiple of ω. Let K0>0 be a positive integer satisfying (85)e-μx++αx++μy++θ+λ2η22m1m2ν2ν3eK0η2+m_-ϵω∗>ϵ1+κ1ϵ2ϵ3+ϵ1+κ1ϵ2κ2,where ν2≔ϵ2e-3μy++αy+λ2,ν3≔e-2μx++αx+λ2η2m1ν2. Since we have proved limsupt→+∞yi(t)>ϵ2, there are only two possibilities as follows:(i)
There existsT5~≥T4~ such that y¯i(t)≥ϵ2 for all t≥T5~.(ii)
y¯i(t) oscillates about ϵ2 for large t≥T4~.
In case (i), we haveliminft→+∞y¯i(t)≥ϵ2≔y¯i1. In case (ii), then there necessarily exist two constants t1,t2≥T4~(t2≥t1) such that (86)y¯it1=y¯it2=ϵ2,y¯it<ϵ2,∀t∈t1,t2.Suppose that t2-t1≤(K0+3)λ2. Then, from the third equation of system (55), we have (87)dy¯itdt≥-μy++αy+y¯it.Hence, we obtain (88)y¯it≥y¯it1exp⁡∫t1t-μy++αy+ds≥ϵ2e-μy++αy+K0+3λ2≔y¯i1for all t∈t1,t2. Suppose that t2-t1>K0+3λ2. Then, from (87), we have (89)y¯it≥ϵ2e-μy++αy+K0+3λ2=y¯i1 for all t∈(t1,t1+K0+3λ2). Now, we are in a position to show that y¯i(t)≥y¯i1 for all t∈[t1+(K0+3)λ2,t2). Suppose that x¯i(t)≥ϵ1 for all t∈[t1,t1+λ2]. Then, from (80), we have (90)x¯it1+λ2≤x¯it1+∫t1t1+λ2βxsM1ϵ2-μxs+αxsϵ1ds<M1-M1=0, which is a contradiction. Therefore, there exists s4∈[t1,t1+λ2] such that xi(s4)<ϵ1. Then, similar to the proof that limsupt→+∞y¯i(t)>ϵ2, we can show that x¯i(t)≤ϵ1+κ1ϵ2 for all t≥s4. Similarly, from (75), we can show that there exists s4~∈[t1,t1+λ2] such that y¯e(t)≤ϵ3+κ2ϵ1+κ1κ2ϵ2 for all t≥s4~. Thus we have (91)x¯it≤ϵ1+κ1ϵ2,y¯et≤ϵ3+κ2ϵ1+κ1κ2ϵ2for all t≥t1+λ2≥max(s4,s4~). From (87), we have (92)y¯it≥ν2=ϵ2e-3μy++αy+λ2>y¯i1for all t∈[t1,t1+3λ2]. Thus, from (65), (91), and (92), we have (93)dx¯itdt=βxtN_xt-x¯ity¯it-μxt+αxtx¯it≥βxtm1ν2-μx++αx+x¯it for all t∈[t1+λ2,t1+3λ2]. Hence, from (83), we obtain (94)x¯it1+3λ2≥e-μx++αx+t1+3λ2x¯it1+2λ2eμx++αx+t1+2λ2+∫t1+2λ2t1+3λ2eμx++αx+sβxsm1ν2ds≥e-μx++αx+t1+3λ2∫t1+2λ2t1+3λ2eμx++αx+sβxsm1ν2ds≥e-μx++αx+λ2η2m1ν2,(95)x¯it≥e-μx++αx+tx¯it1+λ2eμx++αx+t1+λ2+∫t1+λ2teμx++αx+sβxsm1ν2ds≥e-μx++αx+t1+3λ2∫t1+λ2t1+2λ2eμx++αx+sβxsm1ν2ds≥e-2μx++αx+λ2η2m1ν2=ν3for all t∈[t1+2λ2,t1+3λ2]. Thus, from (66), (91), (92), and (95), we have (96)dy¯etdt=βytN_yt-y¯et-y¯itx¯it-μyt+θty¯et≥βytm2ν3-μyt+θty¯etfor all t∈[t1+2λ2,t1+3λ2]. Hence, from (84), we obtain (97)y¯et1+3λ2≥e-μy++θ+t1+3λ2y¯et1+2λ2eμy++θ+t1+2λ2+∫t1+2λ2t1+3λ2βysm2ν3eμy++θ+sds≥e-μy++θ+t1+3λ2∫t1+2λ2t1+3λ2βysm2ν3eμy++θ+sds≥e-μy++θ+λ2η2m2ν3.
We claim thaty¯i(t)>y¯i1 for all t∈(t1+(K0+3)λ2,t2). If it is not true, there exists a t0>0 such that t0∈(t1+(K0+3)λ2,t2),y¯i(t0)=y¯i1 and y¯i(t)≥y¯i1 for all t∈(t1,t0). Then there exists m∈N such that t0∈[t1+(K0+3)λ2+mω,t1+(K0+3)λ2+(m+1)ω). Let V(t)=x¯i(t)y¯e(t)y¯i(t). From (74), (75), (91), and (92), the derivative of V(t) along solutions of (55) satisfies(98)dVtdt55≥bt,N_xt-ϵ1-κ1ϵ2,N_yt-κ2ϵ1-κ3ϵ2-ϵ3Vt for all t∈[t1+λ2,t2]. Integrating the above inequality from t1+3λ2 to t0, we have (99)x¯it0y¯et0y¯it0≥x¯it1+3λ2y¯et1+3λ2y¯it1+3λ2exp⁡∫t1+3λ2t0bt,N_xt-ϵ1-κ1ϵ2,N_yt-κ2ϵ1-κ3ϵ2-ϵ3dt≥e-μx++αx++μy++θ+λ2η22m1m2ν2ν3y¯i1exp⁡∫t1+3λ2t1+K0+3λ2+∫t1+K0+3λ2t1+K0+3λ2+mω+∫t1+K0+3λ2+mωt0bt,N_xt-ϵ1-κ1ϵ2,N_yt-κ2ϵ1-κ3ϵ2-ϵ3dt≥e-μx++αx++μy++θ+λ2η22m1m2ν2ν3y¯i1eK0η2+m_-ϵω∗>ϵ1+κ1ϵ2ϵ3+ϵ1+κ1ϵ2κ2y¯i1.Thus, x¯i(t0)y¯e(t0)>(ϵ1+κ1ϵ2)[ϵ3+(ϵ1+κ1ϵ2)κ2], which contradicts with (91).
Therefore,y¯i(t)≥y¯i1 is valid for all t∈[t1,t2], which implies liminft→+∞y¯i(t)≥y¯i1>0. According to the comparison theorem, we have liminft→+∞yi(t)≥y¯i1>0. Since limsupt→+∞yi(t)≤limsupt→+∞Ny(t)≤M2<+∞, the infective yi(t) of system (3) is permanent.
Furthermore, from system (3), there exists xi1>0 such that liminft→+∞xi(t)≥xi1>0. Since limsupt→+∞xi(t)≤limsupt→+∞Nx(t)≤M1<+∞, the infective xi(t) of system (3) is also permanent. This completes the proof.Remark 13.
For the corresponding autonomous system of system (3), condition (53) can become as follows: (100)3βxβyθAxμx+αxAyμy+αy3>μx+αx+μy+αy+μy+θ≥3μx+αxμy+αyμy+θ3. This shows that βxβyθ(Ax/μx)(Ay/μy)3>(μx+αx)(μy+αy)(μy+θ)3, which implies R0>1; here R0 is given in (2). This means that the result is consistent with the literature [10].Remark 14.
It is easy to prove that ifliminft→+∞b(t,N_x(t),N_y(t))>0, then the infective yi(t) of system (3) is permanent.


## 5. Some Corollaries

As consequences of Theorems8 and 12, we have a series of corollaries as follows.Corollary 15.
Suppose thatμx(t)>0, μy(t)>0 for all t≥0 and 0<liminft→∞⁡Axt/μxt+αxt≤limsupt→∞⁡Ax(t)/μx(t)<∞, 0<liminft→∞⁡Ayt/μyt+αyt≤limsupt→∞⁡Ay(t)/μy(t)<∞. If there exist positive constants λ>0,  p>0,  q>0, and T1>0 such that (101)limsupt→+∞⁡∫tt+λ1pθs-μys-αysds<0,limsupt→+∞⁡∫tt+λ1qβxsAxμxM-μxs-αxsds<0,limsupt→+∞⁡∫tt+λpqβysAyμyM-μys-θsds<0and G1p,t<0,G2(p,q,t)<0 for all t≥T1, then the two infectious populations xi(t) and yi(t) of system (3) are extinct.Corollary 16.
If there is a constantλ1>0 such that (102)liminft→+∞⁡∫tt+λ1bs,Axμx+αxm,Ayμy+αymds>0, then the infective yi(t) of system (3) is permanent.Corollary 17.
If there exists a constantλ1>0 such that R¯01>1, where (103)R¯01=3βxβyθN_xN_y30μx+αx+μy+αy+μy+θ0. Here, (104)3βxβyθN_xN_y30=liminft→+∞⁡∫tt+λ13βxtβytθtN_xtN_yt3dt,μx+αx+μy+αy+μy+θ0=limsupt→+∞⁡∫tt+λ1μxt+αxt+μyt+αyt+μyt+θtdt,N_x(t) is some fixed solution of (8) with initial value N_x0>0andN_y(t) is some fixed solution of (10) with initial value N_y(0)>0. Then the infective yi(t) of system (3) is permanent.Corollary 18.
Suppose system (3) is ω-periodic; then the infective yi(t) of system (3) is permanent provided that (105)R¯02=3βxβyθN_xN_y3¯μx+αx+μy+αy+μy+θ¯>1. Here, N_x(t) is the globally uniformly attractive nonnegative ω-periodic solution of (8) and N_y(t) is the globally uniformly attractive nonnegative ω-periodic solution of (10).Corollary 19.
Suppose system (3) is almost periodic; then the infective yi(t) of system (3) is permanent provided that (106)R¯03=m3βxβyθN_xN_y3mμx+αx+μy+αy+μy+θ>1. Here, N_x(t) is the globally uniformly attractive nonnegative almost periodic solution of (8) and N_y(t) is the globally uniformly attractive nonnegative almost periodic solution of (10).Corollary 20.
Suppose thatμx(t)>0, μy(t)>0 for all t≥0 and 0<liminft→∞⁡Ax(t)/μxt+αxt≤limsupt→∞⁡Ax(t)/μx(t)<∞,  0<liminft→∞Ayt/μyt+αyt≤limsupt→∞⁡Ay(t)/μy(t)<∞. If there exist positive constants λ>0,p>0,q>0, and T1>0 such that (107)p>θ0μy+αy0,q>βxN¯x0μx+αx0,pq>μy+θ0βyN¯y0 and G1p,t<0,G2(p,q,t)<0 for all t≥T1, then the two infectious populations xi(t) and yi(t) of system (3) are extinct.
Here,(108)θ0=limsupt→+∞⁡∫tt+λθsds,μy+θ0=limsupt→+∞⁡∫tt+λμys+θsds,βxN¯x0=limsupt→+∞⁡∫tt+λβxsN¯xsds,βyN¯y0=liminft→+∞⁡∫tt+λβysN¯ysds,μy+αy0=liminft→+∞⁡∫tt+λμys+αysds,μx+αx0=liminft→+∞⁡∫tt+λμxs+αxsds.N¯x(t) is some fixed solution of (9) with initial value N¯x(0)>0andN¯y(t) is some fixed solution of (11) with initial value N¯y(0)>0.Corollary 21.
Suppose system (3) is ω-periodic; we suppose that μx(t)>0, μy(t)>0 for all t≥0 and 0<liminft→∞⁡Ax(t)/μx(t)+αx(t)≤limsupt→∞Ax(t)/μx(t)<∞, 0<liminft→∞⁡Ay(t)/μy(t)+αy(t)≤limsupt→∞Ay(t)/μy(t)<∞. Then if there exist positive constants λ>0,p>0,q>0, and T1>0 such that (109)p>θ¯μy+αy¯,q>βxN¯x¯μx+αx¯,pq<μy+θ¯βyN¯y¯ and G1p,t<0,G2(p,q,t)<0 for all t≥T1, then the two infectious populations xi(t) and yi(t) of system (3) are extinct. Here, N¯x(t) is the globally uniformly attractive nonnegative ω-periodic solution of (9) and N¯y(t) is the globally uniformly attractive nonnegative ω-periodic solution of (11).Corollary 22.
Suppose system (3) is almost periodic; also suppose that μx(t)>0,  μy(t)>0 for all t≥0 and 0<liminft→∞⁡Ax(t)/μxt+αxt≤limsupt→∞Axt/μxt<∞,  0<liminft→∞Ayt/μyt+αyt≤limsupt→∞⁡Ay(t)/μy(t)<∞. Then if there exist positive constants λ>0,p>0,q>0, and T1>0 such that (110)p>mθmμy+αy,q>mβxN¯xmμx+αx,pq<mμy+θmβyN¯y and G1p,t<0,G2(p,q,t)<0 for all t≥T1, then the two infectious populations xi(t) and yi(t) of system (3) are extinct. Here N¯x(t) is the globally uniformly attractive nonnegative almost periodic solution of (9) and N¯y(t) is the globally uniformly attractive nonnegative almost periodic solution of (11).


## 6. Numerical Simulation

Numerical verification of the results is necessary for completeness of the analytical study. In this section, we will present some numerical simulations to substantiate and discuss our analytical findings of system (3) by the means of the software Matlab.In order to testify the validity of our results, in system (3), fix Axt=0.5+0.1sin⁡t, Ay(t)=0.6+0.1sin⁡(t),  μx(t)=0.2+0.18sin⁡(t),  μy(t)=0.3+0.28sin⁡(t),  αx(t)=0.4+0.3sin⁡(t),  αy(t)=0.5+0.2sin⁡(t),  θ(t)=0.28+0.2sin⁡(t),  βx(t)=0.2+0.2sin⁡(t),  βy(t)=0.1+0.1sin⁡(t). It is easy to verify that (i) and (H2) hold. From (8)–(11), we have (111)limt→∞N_xt=0.5+0.1sin⁡t0.6+0.48sin⁡t,limt→∞N¯xt=0.5+0.1sin⁡t0.2+0.18sin⁡t,limt→∞N_yt=0.6+0.1sin⁡t0.8+0.48sin⁡t,limt→∞N¯yt=0.6+0.1sin⁡t0.3+0.28sin⁡t.Then system (3) becomes periodic with period 2π. We choose λ=2π,p=3, and q=1; then we have (112)G1p,t=-0.1267+0.0667sin⁡t<0,G2p,q,t=0.2+0.2sin⁡t0.5+0.1sin⁡t0.2+0.18sin⁡t-6775-715sin⁡t. Obviously, it is easy to see that G2p,q,t<0 for all t>0 from Figure 1. Furthermore, we can get that (113)3=p>θ¯μy+αy¯≃0.35,1=q>βxN¯x¯μx+αx¯≃0.8279,3=pq<μy+θ¯βyN¯y¯≃3.2402. Hence, from Corollary 21, we get that the disease will be extinct (see Figure 2). Increasing the infective rate to βxt=0.8+0.1sin⁡t,βyt=0.5+0.1sin⁡t, from Corollary 18, we can get R¯02=3βxβyθN_xN_y3¯/μx+αx+μy+αy+μy+θ¯=4.1087>1, which means the condition of Corollary 18 is satisfied. The conclusion of Corollary 18 is verified (see Figure 3).Figure 1
The figure shows that movement paths ofG2(p,q,t) as functions of time t.Figure 2
(a) shows movement paths ofxi, ye, and yi as functions of time t. The graph of the trajectory in (xi,ye,yi)-space is shown in (b). The disease is extinct.
(a) (b)Figure 3
(a) shows movement paths ofxi, ye, and yi as functions of time t. The graph of the trajectory in (xi,ye,yi)-space is shown in (b). System (3) is permanent.
(a) (b)



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*Source: 8707258-2016-09-22.xml*