# Threshold Dynamics in a Periodic Three-Patch Rift Valley Fever Virus Transmission Model

Authors: Buyu Wen; Zhidong Teng; Wenlin Liu
Journal: Complexity
 (2019)

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## Abstract

This paper investigates a three-patch Rift Valley fever virus transmission model with periodic coefficients. The basic reproduction numberRi0i=1,2,3 is calculated for the model by using the next infection operator method. The threshold conditions on the extinction and permanence in the mean of the disease with probability one. The Rift Valley fever disease will be permanent in the j-th j=1,2,3 patch if j≥i,Ri0>1 and dies out in the j-th patch if j<i. The numerical simulations are given to confirm the theoretical results.


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## Body

## 1. Introduction

Rift Valley fever (RVF) is an acute viral zoonosis caused by the Rift Valley fever virus (RVF virus). This disease is primarily transmitted among domestic animals, such as sheep, cattle, buffalo, goat, and camels, with the main route of infection being mosquito bites (see [1–4]). RVF virus infection in livestock animals can produce high rate of abortion in pregnant animals and significant morbidity and mortality rate in young ruminants. During the disease epidemic among animals, humans may be infected with Rift Valley fever virus through bites from infected mosquitoes or contacts with the blood, body fluids, and organs of infected animals (see [5, 6]).In the 1930s, RVF virus was discovered for the first time near Naivasha Lake in the Rift Valley of Kenya. In 1977, a human RVF virus epidemic is reported in Egypt, which is the largest epidemic. Subsequently, RVF virus outbreaks have been reported in Saharan and North Africa. After that, this disease occurred in Saudi Arabia and Yemen outside the Africa in the 2000s. These disease outbreaks result in high mortality and abortion in young ruminants and case significant economic losses in Africa and Middle East. With the enlargement of scope of RVF virus infection, there is a growing concern that this disease will outbreak further in other parts of Asia and Europe. In recent years, numerous mathematical models have been developed to investigate RVF virus outbreaks (see [1, 3, 6–15]).Research results show that, in Northeastern Africa, human activities, including those associated with the Eid al-Adha feast, along with a combination of climatic factors such as rainfall level and hydrological variations, contribute to the transmission and dispersal of the disease pathogen. Moreover, sporadic outbreaks may occur when the two events occur together: (1) abundant livestock is recruited into areas at risk from RVF due to the demand for the religious festival and (2) abundant numbers of mosquitoes emerge. In [8, 14, 15], the authors consider the religious festival and discuss three-patch model for RVF virus transmission.On the one side, a religious festival, the Eid al-Adha feast, at which time large numbers of livestock are driven towards the site of the feast. In Africa and the Middle East, the invasion path follows the same route that people use to travel to the Nile Delta, where the important Islamic festival Greater Bairam is held. Geographically, the disease seems to be along a path from southwest to northeast of Africa. This is consistent with the way to Mecca, Saudi Arabia’s capital (see [15]).On the other side, the number of mosquitoes is affected by temperature and humidity, the greater the humidity, the more the number of mosquitoes, the humidity at about thirty degrees is the most suitable for mosquitoes to grow and propagate, and lower than ten degrees will enter hibernation. Due to rainfall and temperature, in fact, the mosquito population densities follow seasonal fluctuations, achieving maximum numbers during the rainy season and reaching lowest level during the winter season (see [4, 7, 8]). The abundant numbers of mosquitoes corresponds to an increased number of infected ruminants. Consequently, RVF virus infections fluctuate over time and highly link to seasonal variations.The autonomous models have been discussed in [3, 11–15]. At the same time, we can see that Rift Valley fever will change periodically with the religious festivals and mosquitoes’ periodicity. Recently, in [8], Xiao et al. proposed a three-patch periodic model for RVF virus transmission with the periodic recruitment rate of livestock in first patch, the periodic carrying capacities for mosquitoes in three patches, the periodic natural death rate for livestock in the last patch, and the periodic migration rate. In order to seek the seasonal and festival-driven impacts for RVF outbreaks, the authors in [8] investigated the dynamical behaviors of the model just by using the numerical simulation method.The main purpose of this paper is to discuss the threshold dynamics of a periodic three-patch Rift Valley fever virus transmission model, and thereout, we will make up the deficiency for the research given in [8]. By applying the methods in given [16–21], we calculate basic reproduction number Ri0i=1,2,3 in i-patch of model 1i By using the theory of persistence for dynamical systems, we will establish that Ri0 serves as a threshold value for Rift Valley fever virus dynamics in periodic case.The organization of this paper is as follows. In the next section, we develop a periodic three-patch model to describe the transmission of Rift Valley fever virus and present some useful lemmas. In Section3, we will propose the main results, namely, we will obtain the extinction or permanence of all positive solutions of model 1i. In Section 4, we will give the numerical simulations to verify our theoretical results.


## 2. Model

We investigated the festival-driven and seasonal impacts on the patterns of RVF outbreaks among livestock in Africa and Middle East.We assume that a disease invades and subdivides the target livestock into four classes: the susceptible classSi, the exposed class Ei, the infectious class Ii, and the recovered class Ri; the female mosquitoes are divided into three compartments: uninfected compartment Ui, exposed compartment Li, and infected compartment Vi in each patch.We assume thatr1t is the number of livestock imported daily at time t in 1-patch, αit is the transmission rate of disease from mosquitoes to livestock at time t in i-patch, βit is the transmission rate of disease from livestock to mosquitoes at time t in i-patch, μit is the time-dependent natural death rate for livestock in i-patch (average death rate for different livestock, i.e., cattle and sheep), νit is the time-dependent natural death rate for female mosquitoes in i-patch, ξit is the rate at which immunity is lost after recovery for livestock at time t in i-patch, εit is the rate of becoming infectious for livestock at time t in i-patch, ηit is the rate of becoming infectious for mosquitoes at time t in i-patch, γit is the recovery rate of leaving the infectious class for livestock at time t in i-patch, bit is the time-dependent birth rate for mosquitoes in i-patch, Mit is the time-dependent carrying capacity of mosquitoes in i-patch, δit is the disease-induced death rate for livestock at time t in i-patch, Cit is moment speed of animals from i-patch to i+1-patch, and di is the length of journey for animals within patch i. Motivated by the above assumption, we propose the following periodic three-patch RVF virus transmission model.
1idSitdt=rit+Ci−1tdi−1Si−1t−αitSitVit−μitSit+ξitRit−CitdiSit,dEitdt=Ci−1tdi−1Ei−1t+αitSitVit−μit+εitEit−CitdiEit,dIitdt=Ci−1tdi−1Ii−1t+εitEit−μit+γit+δitIit−CitdiIit,dRitdt=Ci−1tdi−1Ri−1t+γitIit−μit+ξitRit−CitdiRit,dUitdt=bit−νitUit1−Uit+Lit+VitMit−βitIitUit,dLitdt=βitIitUit−νit+ηitLit,dVitdt=ηitLit−νitVit,i=1,2,3,where r2t=r3t=C0t=C3t=0.For any continuous and bounded functionft defined for R+=0,∞, we denote fi=inft∈R+ft and fu=supt∈R+ft.Based on the biological background of model1i, the initial condition for model 1i (for any i=1,2,3) has in the following form:
(2)Si0≥0,Ei0≥0,Ii0≥0,Ri0≥0,Ui0>0,Li0≥0,Vi0≥0.For model1i, we give assumptions as follows:H1 functions r1t,αit,βit,μit,C1t,C2t,νit,ξit,εit,ηit,γit,bit,Mit and δiti=1,2,3 are continuous and ω-periodic defined on R+=0,∞. Furthermore, functions αit, βit μit, εit, νit, bit, γit, ηit, and Mit are positive, and functions C1t, C2t, ξit, and δit are nonnegative.H2∫0ωr1tdt>0,∫0ωbit−νitdt>0 and bit≥νiti=1,2,3 for all t≥0.Consider aω-periodic linear equation as follows:
(3)dxdt=at−ctx,where at and ct are ω-periodic continuous functions defined for t≥0. We have the following lemma.Lemma 1 (see [17]).
Assume that∫0ωatdt>0,∫0ωctdt>0 and ct≥0 for all t≥0. Then, equation (3) has a unique positive ω-periodic solution x∗t which is globally attractive.Furthermore, we consider aω-periodic logistic equation as follows:
(4)dxdt=xat−ctx,where at and ct are ω-periodic continuous functions defined for t≥0. We have the following lemma.Lemma 2 (see [18]).
Assume that∫0ωatdt>0,∫0ωctdt>0 and ct≥0 for all t≥0. Then, equation (4) has a unique positive ω-periodic solution x∗t which is globally attractive.WhenE1t=0,I1t=0,L1t=0,V1t=0 and R1t=0, we obtain the disease-free subsystem of model 11:
(5)dS1tdt=r1t−μ1t+C1td1S1t,dU1tdt=b1t−ν1tU1t1−U1tM1t.As consequence of Lemma1 and Lemma 2, we have the result on the existence of disease-free periodic solution of model 11.Lemma 3.
System (5) has a unique positive ω-periodic solution S1∗t,U1∗t which is globally attractive.From Lemma3, model 11 has a unique disease-free periodic solution E10=S1∗t,0,0,0,U1∗t,0,0.LetAt be a continuous, cooperative, irreducible, and ω-periodic n×n matrix function; we consider a linear system:
(6)dxtdt=Atxt.LetΦAt be the fundamental solution matrix of system (6) satisfying initial condition ΦA0=I, where I is an unit matrix. Let ρΦAω be the spectral radius of matrix ΦAω. Since At is continuous, cooperative, and irreducible, we obtain that ΦAt is nonnegative for all t≥0. By Perron-Frobenius theorem, ρΦAω is the principal eigenvalue of ΦAω in the sense that it is simple and admits a positive eigenvector υ∗. We present the following lemma.Lemma 4 (see [19]).
Letp=1/ωlnρΦAω. Then, there exists a positive ω-periodic function υt such that xt=eptυt is a solution of system (6).Now, we calculate the basic reproduction number of model11 by applying the next-generation matrix approach which is given in [20]. Firstly, we easily validate that model 11 satisfies the conditions A1−A7 given in [20]. Let
(7)F=F1t=000α1tS1∗t00β1tU1∗t000000000,V=B1t=μ1t+ε1t+C1td10000ν1t+η1t00−ε1t0μ1t+γ1t+δ1t+C1td100−η1t0ν1t.LetYt,s with t≥s be the evolution operator of linear ω-periodic system:
(8)dytdt=−B1tyt.Then for anys∈ℝ and t≥s,dYt,s/dt=−B1tYt,s and Ys,s=I, where I is the unit matrix. Let ℂω be the ordered Banach space of all ω periodic continuous functions ϕ:R→R4. Assume that ϕs∈ℂω is the initial distribution of infectious individuals. Then, F1sϕs is the rate at which new infections are produced by infected individuals who were introduced into the population at time s. When t≥s, then Yt,sF1sϕs represents the distribution of those infected individuals who were newly infected individuals at time s and remain in the infected compartments at time t. Hence, the cumulative distribution of new infections at time t produced by all those infected individuals ϕs introduced before t is given by a linear operator as follows:
(9)ℒ1ϕt=∫−∞tYt,sF1sϕsds=∫0∞Yt,t−aF1t−aϕt−ada.Applying the results obtained in [20], the basic reproduction number R10 for model 11 is defined as spectral radius ρℒ1 of operator ℒ1, that is, R10=ρℒ1.Furthermore, using Theorem 2.2 given in [20], we also can obtain the following results on basic reproduction number R10.Lemma 5 (see [20]).
The following conclusions hold.(a)
R10=1 if and only if ρΦF1−B1ω=1
(b)
R10>1 if and only if ρΦF1−B1ω>1
(c)
R10<1 if and only if ρΦF1−B1ω<1
(d)
Disease-free periodic solutionE10 for model 11 is locally asymptotically stable if R10<1 and unstable if R10>1Remark 1.
In particular, when periodic systems (1) degenerate into autonomous systems, by the next-generation matrix method (see [22]), we can obtain
(10)F=000α1S1∗00β1U1∗000000000,V=μ1+ε1+C1d10000ν1+η100−ε10μ1+γ1+δ1+C1d100−η10ν1,where S1∗=r1/μ1+C1/d1 and U1∗=M1. Then,
(11)R10=ρFV−1=α1r1ε1η1β1M1ν1μ1+C1/d1μ1+γ1+δ1+C1/d1μ1+ε1+C1/d1ν1+η1.


## 3. Main Results

For model1i, we investigate the persistence and extinction of positive solutions on the three patches.


### 3.1. The First Patch

In this section, we discuss model11. Firstly, on the nonnegativity, positivity, and boundedness of solutions for model 11, we have the following result.Theorem 1.
LetS1t,E1t,I1t,R1t,U1t,L1t,V1tbe the solution of model 11 with initial condition (2). Then, S1t,E1t,I1t,R1t,U1t,L1t,V1t is nonnegative for all t≥0 and ultimately bounded, and when S10>0,E10>0,I10>0,R10>0,U10>0,L10>0 and V10>0, then S1t,E1t,I1t,R1t,U1t,L1t,V1t is also positive for all t>0.Proof 1.
In fact, by the continuous dependence of solutions with respect to initial values, we only need to prove that whenS10>0,E10>0,I10>0,R10>0,U10>0,L10>0 and V10>0, then S1t,E1t,I1t,R1t,U1t,L1t,V1t is positive for all t>0. From the fifth equation of model 11, we have for any t≥0.
(12)dU1tdt=b1t−ν1t1−U1t+L1t+V1tM1t−β1tI1tU1t.
SinceU10>0, we get
(13)U1t=U10exp∫0tb1s−ν1s1−U1s+L1s+V1sM1s−β1sI1sds>0.Definem1t=mint≥0S1t,E1t,I1t,R1t,L1t,V1t. Obviously, m10=minS10,E10,I10,R10,L10,V10>0. We only need to prove m1t>0 for all t≥0. Suppose that there exists a t0>0 such that m1t0=0 and m1t>0 for all t∈0,t0. Then, we only need to discuss the following six cases: (1) m1t0=S1t0, (2) m1t0=E1t0, (3) m1t0=I1t0, (4) m1t0=R1t0, (5) m1t0=L1t0, and (6) m1t0=V1t0.Now, we only give the proof for case (2). The remaining cases can be proved in a similar manner. Letm1t0=E1t0. Since m1t>0 for all t∈0,t0, then V1t>0, S1t>0, and hence,
(14)dE1tdt≥−μ1t+ε1t+C1td1E1t,forallt∈0,t0.Integrating from0 to t0, we get
(15)0=E1t0≥E10exp−∫0t0μs+ε1s+C1sd1ds>0,which leads to a contradiction.Next, we show the boundedness of nonnegative solutions of model11. Let N1t=S1t+E1t+I1t+R1t, then we get
(16)dN1tdt=r1t−μ1t+C1td1N1t−δ1tI1t≤r1t−μ1t+C1td1N1t.Hence, we havelimsupt→∞N1t≤r1u/μ1i+C1/d1i, which implies that S1t,E1t,I1t and R1t are ultimately bounded. Since for any t≥0,
(17)dU1tdt≤b1t−ν1tU1t1−U1tM1t,using Lemma 3; we further have limsupt→∞U1t≤limsupt→∞U1∗t=max0≤t≤ωU1∗t. Hence, for any ϵ0>0, there is a T0>0 such that I1t≤r1u/μ1i+C1/d1i+ϵ0 and U1t≤P1+ϵ0 for all t≥T0, where P1=max0≤t≤ωU1∗t. From the sixth equation of model 11, we have
(18)dL1tdt≤β1tP2−ν1t+η1tL1t,for all t≥T0, where P2=r1u/μ1i+C1t/d1i+ϵ0P1+ϵ0. Hence, we can obtain limsupt→∞L1t≤β1uP2/ν1+η1i From this, for any ϵ1>0, there is a T1>0 such that L1t≤β1uP2/ν1+η1i+ϵ1 for all t≥T1. Hence, from the last equation of model 11, we further have
(19)dV1tdt≤−ν1tV1t+P3η1t,for all t≥T1, where P3=β1uP2/ν1+η1i+ϵ1. This shows that limsupt→∞V1t≤P3η1u/ν1i. From above discussions, we finally obtain that all solutions of model 11 with initial condition (2) are ultimately bounded. This completes the proof.Now, we show thatR10 serves as a threshold parameter. We prove that when R10<1, then disease-free periodic solution E10 is globally asymptotically stable, that is to say, the RVF virus is extinct in the first patch, and when R10>1, then model 11 is permanent, that is to say, the RVF virus is existent in the population in the first patch.Theorem 2.
IfR10<1, then disease-free periodic solution E10 of model 11 is globally asymptotically stable, and if R10>1, then E10 is unstable.Proof 2.
From conclusiond of Lemma 5, it follows that if R10>1, then E10 is unstable, and if R10<1, then E10 is locally asymptotically stable.We now prove that ifR10<1, then E10 is globally attractive. Let R10<1, from conclusion c of Lemma 5, then ρΦF1−B1ω<1. We can choose constant ς>0 small enough such that ρΦF1−B1+ςHω<1, where
(20)Ht=000α1t00β1t000000000.From model11, we have
(21)dN1tdt=r1t−μ1t+C1td1N1t−δ1tI1t≤r1t−μ1t+C1td1N1t,dU1tdt≤b1t−ν1tU1t1−U1tM1t.From Lemma3, for above given ς>0, there exists a T1>0 such that N1t=S1t+I1t+E1t+R1t≤S1∗t+ς and U1t≤U1∗t+ς for any t≥T1. Hence, from model 11, we obtain that for all t≥T1
(22)dE1tdt≤α1tS1∗t+ςV1t−μ1t+ε1t+C1td1E1t,dL1tdt≤β1tU1∗t+ςtI1t−ν1t+η1tL1t,dI1tdt=ε1tE1t−μ1t+γ1t+δ1t+C1td1I1t,dV1tdt=−ν1tV1t+η1tL1t.Consider the following auxiliary system:(23)dE^1tdt=α1tS1∗t+ςV^1t−μ1t+ε1t+C1td1E^1t,dL^1tdt=β1tU1∗t+ςtI^1t−ν1t+η1tL^1t,dI^1tdt=ε1tE^1t−μ1t+γ1t+δ1t+C1td1I^1t,dV^1tdt=−ν1tV^1t+η1tL^1t.It follows from Lemma4 that there is a positive ω-periodic function υt=υ1t,υ2t,υ3at,υ4tT such that E^1t,L^1t,I^1t,V^1tT=eptυt is a solution of system (23), where p=1/ωlnρΦF1−B1+ςHω.DenoteJt=E1t,L1t,I1t,V1tT. We can choose constant τ>0 such that JT1≤τepT1υT1. According to the comparison theorem of the vector form (see [23]), we obtain Jt≤τeptυt for any t≥T1. By ρΦF1−B1+ςHω<1, we know p=1/ωlnρΦF1−B1+ςHω<0. Then, we conclude that limt→∞Jt=0, that is to say, limt→∞E1t,L1t,I1t,V1t=0,0,0,0. By the equations of R1, S1, and U1 of model 11, we also obtain limt→∞S1t,R1t,U1t=S1∗t,0,U1∗t. Hence, disease-free periodic solution E10 is globally attractive. This shows that disease-free periodic solution E10 is globally asymptotically stable. This completes the proof.Theorem 3.
IfR10>1, then the disease in model 11 is permanent.Proof 3.
From conclusionb of Lemma 5, we know that R10>1 if and only if ρΦF1−B1ω>1. We can choose a small enough constant σ>0 such that ρΦF1−B1+ςHω>1, where Nt is given in (20). From H1 and H2, we can choose ϵ1>0 small enough such that b1t−ν1t−β1tϵ1>0 for all t≥0. Consider the following auxiliary equations:
(24)dS~1tdt=r1t−α1tS~1tϵ1−μ1t+C1td1S~1tand
(25)dU~1tdt=b1t−ν1tU~1t1−U~1tM1t−β1tU~1tϵ1,from Lemma 1 and Lemma 2 we can get equations (24) and (25) admit globally uniformly attractive positive ω-periodic solution S1ϵ1t and U1ϵ1t, respectively. By the continuity of solutions with respect to parameter ϵ1, for above σ>0, there exists a 0<ϵ0<ϵ1 such that for all t∈0,ω
(26)S1ϵ0t>S1∗t−σ2,U1ϵ0t>U1∗t−σ2.
Define the sets as follows:(27)X1=S1,E1,I1,R1,U1,L1,V1:S1>0,E1≥0,I1≥0,R1≥0,U1>0,L1≥0,V1≥0,X10=S1,E1,I1,R1,U1,L1,V1∈X1:E1>0,I1>0,L1>0,V1>0,∂X10=X1X10=S1,E1,I1,R1,U1,L1,V1∈X1:E1I1L1V1=0.Define further the Poincaré mapP:X→X1 as follows:
(28)Px0=uω,x0,x0=S10,E10,I10,R10,U10,L10,V10∈X1,where ut,x0=S1t,E1t,I1t,R1t,U1t,L1t,V1t is the solution of model 11 satisfying initial condition u0,x0=x0. It follows from Theorem 1 that all solutions of model 11 are ultimately bounded. Therefore, map P is point dissipative and compact on X1.We define(29)M∂=x0=S10,E10,I10,R10,U10,L10,V10∈∂X10:Pmx0∈∂X10,forallm≥0,where P0x0=x0,P1x0=Px0 and Pmx0=PPm−1x0 for m=1,2,…. Now, we claim M∂=S1,0,0,R1,U1,0,0:S1>0,R1≥0,U1>0. In fact, from model 11, we directly obtain that the solution of model 11 with initial value S10,0,0,R10,U10,0,0∈∂X10 has the form S1t,0,0,R1t,U1t,0,0. Hence,
(30)PmS10,0,0,R10,U10,0,0=S1mω,0,0,R1mω,U1mω,0,0∈∂X0,for all m≥0. This shows S10,0,0,R10,U10,0,0∈M∂. Therefore, S1,0,0,R1,U1,0,0:S1>0,R1≥0,U1>0⊆M∂.Next, we proveM∂⊆S1,0,0,R1,U1,0,0:S1>0,R1≥0,U1>0. Suppose that there is a S10,E10,I10,R10,U10,L10,V10∈M∂ such that S10,E10,I10,R10,U10,L10,V10∉S1,0,0,R1,U1,0,0:S1>0,R1≥0,U1>0.We firstly havemaxE10,I10,L10,V10>0. Let S1t,E1t,I1t,R1t,U1t,L1t,V1t be the solution of model 11 with initial value S10,E10,I10,R10,U10,L10,V10 at t=0. From Theorem 1, we firstly have that S1t,E1t,I1t,R1t,U1t,L1t,V1t is nonnegative for all t≥0. Furthermore, by S10>0 and U10>0, from the first and fifth equations of model 11, we also have that S1t>0 and U1t>0 for all t≥0, respectively.IfE10>0, then by the second equation of model 11, we obtain
(31)dE1tdt≥−μ1t+ε1t+C1td1E1t,forallt≥0.Integrating this inequality from0 to t, we have
(32)E1t≥E10exp−∫0tμ1s+ε1s+C1sd1ds>0,forallt≥0.Further, from the third equation of model11 we can obtain
(33)I1t>I10exp−∫0tμ1s+γ1s+δ1s+C1sd1ds≥0,forallt>0.From the sixth equation of model11, we get
(34)dL1tdt>−ν1t+η1tL1t,forallt≥0.Hence,(35)L1t>L10exp−∫0tν1s+η1sds≥0,forallt>0.Lastly, from the equation ofV1 for model 11, we also have V1t>0 for all t>0. Therefore, we finally obtain E1tI1tL1tV1t>0 for all t>0.Similarly, ifI10>0 or L10>0 or V10>0, we also can obtain E1tI1tL1tV1t>0 for all t>0. This shows that S10,E10,I10,R10,U10,L10,V10∉M∂, which leads to a contradiction. This implies that M∂⊆S1,0,0,R1,U1,0,0:S1>0,R1≥0,U1>0. Hence, M∂=S1,0,0,R1,U1,0,0:S1>0,R1≥0,U1>0.InM∂, model 11 degenerates to
(36)dS1tdt=r1t−μ1t+C1td1S1t+ξ1tR1t,dR1tdt=−μ1t+ξ1t+C1td1R1t,dU1tdt=b1t−ν1tU1t1−U1tM1t.We easily prove that system (36) has a globally asymptotically stable ω-periodic solution S1∗t,0,U1∗t, where S1∗t,U1∗t is given in Lemma 3. It is clear that map P has a unique globally attractive fixed point restricted in M∂, which is E1=S1∗0,0,0,0,U1∗0,0,0.Now, we defineWsE1=x0:Pmx0→E1,m→∞, which is said to be stable set of E1. We show that
(37)WsE1∩X0=0.According to the continuity of solutions with respect to the initial values, for above given constantϵ0, there exists δ>0 such that for any x0=S10,E10,I10,R10,U10,L10,V10∈X0 with x0−E1≤δ, it follows that ut,x0−ut,E1≤ϵ0 for all t∈0,ω, where ut,x0=S1t,E1t,I1t,R1t,U1t,L1t,V1t is the solution of model 11 with initial condition S10,E10,I10,R10,U10,L10,V10=x0 and ut,E1=S1∗t,0,0,0,U1∗t,0,0. Since ut,E1 is ω-periodic, we have ut1,E1=ut2,E1 for any t1,t2≥0 with t2=kω+t1, where k is an integer. We claim that
(38)limsupm→∞Pmx0−E1≥δ,forallx0=S10,E10,I10,R10,U10,L10,V10∈X0.Suppose that (38) does not hold. Then, we have
(39)limsupm→∞Pmx0−E1<δ,forsomex0=S10,E10,I10,R10,U10,L10,V10∈X0.Without loss of generality, we assume thatPmx0−E1<δ for all m≥0. Thus,
(40)ut,Pmx0−ut,E1≤ϵ0,forallt∈0,ω,m≥0.For anyt≥0, let t=t′+nω, where t′∈0,ω and n=t/ω are the greatest integers less than or equal to t/ω. Then, we have ut,x0−ut,E1=ut′,Pmx0−ut′,E1<ϵ0,foranyt≥0.It follows that0<E1t<ϵ0, 0<L1t<ϵ0, 0<I1t<ϵ0, and 0<V1t<ϵ0 for all t≥0. Thus, from model 11, we have the following:
(41)dS1tdt≥r1t−α1tS1tϵ0−μ1t+C1td1S1t,dU1tdt≥b1t−ν1tU1t1−U1tM1t−β1t+2b1t−ν1tM1tU1tϵ0,for any t≥0. Using the comparison principle, we obtain
(42)St≥S~1ϵ0t,Ut≥U~1ϵ0t,for any t≥0, where S~1ϵ0t and U~1ϵ0t are the solutions of equations (20) and (23) with respect to parameter ϵ0 satisfying initial conditions S~1ϵ00=S10 and U~1ϵ00=U10, respectively.From Lemma1 and Lemma 2, equations (24) and (25) with respect to the parameter ϵ0 have globally uniformly attractive solutions S1ϵ0t and U1ϵ0t, respectively. Hence, there exists t2>0 such that
(43)S~1ϵ0t≥S1ϵ0t−σ2,U~1ϵ0t≥U1ϵ0t−σ2,for all t≥t2. By (26), (42), and (43), we obtain S1t≥S~1ϵ0t≥S1ϵ0t−σ/2>S1∗t−σ and U1t≥U~1ϵ0t≥U1ϵ0t−σ/2>U1∗t−σ for all t≥t2. Thus, for all t≥t2, it holds that
(44)dE1tdt≥α1tS1∗t−σV1t−μ1t+ε1t+C1td1E1t,dL1tdt≥β1tU1∗t−σI1t−ν1t+η1tL1t,dI1tdt=ε1tE1t−μ1t+γ1t+δ1t+C1td1I1t,dV1tdt=−ν1tV1t+η1tL1t.Consider the following auxiliary system:(45)dE~1tdt=α1tS1∗t−σV~1t−μ1t+ε1t+C1td1E~1t,dL~1tdt=β1tU1∗t−σI~1t−ν1t+η1tL~1t,dI~1tdt=ε1tE~1t−μ1t+γ1t+δ1t+C1td1I~1t,dV~1tdt=−ν1tV~1t+η1tL~1t.System (45) can be rewritten in terms of the matrices F1t and B1t as follows:
(46)ddtE~1L~1I~1V~1=F1t−B1t+ςHtE~1L~1I~1V~1.It follows from Lemma4 that there is a positive ω-periodic function wt=w1t,w2t,w3at,w4tT such that E~1t,L~1t,I~1t,V~1tT=eθtwt is a solution of system (46), where θ=1/ωlnρΦF1−B1+ςHω.Now, we denoteJt=E1t,L1t,I1t,V1tT. We can choose small enough constant τ>0 such that Jt2≥τeθt2wt2. Thus, by comparison theorem of the vector form, it follows that Jt≥τeθtwt for all t≥t2. By ρΦF1−B1+ςHω>1, we obtain θ=1/ωlnρΦF1−B1+ςHω>0. Hence, limt→∞Jt=∞, which implies that
(47)limt→∞E1t=∞,limt→∞L1t=∞,limt→∞I1t=∞,limt→∞V1t=∞,which leads to a contradiction. Hence, claim (25) holds. This shows WsE1∩X0=0.It is clear that fixed pointE1 of P is globally attractive restricting on M∂. Hence, E1 is acyclic in M∂. From the theorems established in [19] on the uniform persistence of dynamical systems, we obtain that P is uniformly persistent with respect to X0,∂X0, which implies that model 11 is permanent. This completes the proof.


### 3.2. The Second Patch

In this section, we discuss model12. First of all, we have the following result on the positivity and ultimate boundedness of solutions in model 12.Theorem 4.
LetS2t,E2t,I2t,R2t,R2t,L2t,V2t be the solution of model 12 with initial condition (2). Then, S2t,E2t,I2t,R2t,R2t,L2t,V2t is nonnegative for all t≥0 and ultimately bounded, and when S20>0,E20>0,I20>0,R20>0,R20>0,L20>0 and V20>0, then S2t,E2t,I2t,R2t,R2t,L2t,V2t is also positive for all t>0.Proof 4.
Using the same method given in Theorem1, we can prove the positivity of solutions for model 12. Next, we show the boundedness of solutions for model 12. Let N2t=S2t+E2t+I2t+R2t, then we get
(48)dN2tdt=C1td1N1t−μ2t+C2td2N2t−δ2tI2t≤C1td1N1t−μ2t+C2td2N2t.By the comparison theorem and Lemma1, we get that there exists constant M~1>0 such that limsupt→∞N2t≤M~1, which implies that S2t,E2t,I2t and R2t are ultimately bounded. Since for any t≥0,
(49)dU2tdt≤b2t−ν2tU2t1−U2tM2t,then there exists constant M~2>0 such that limsupt→∞U2t≤M~2. Hence, for any ϵ~>0, there is a T~0>0 such that I2t≤M~1+ϵ~0≔Q1 and U2t≤M~2+ϵ~0≔Q2 for all t≥T~0. From the sixth equation of model 12, we have
(50)dL2tdt≤β2tQ1Q2−ν2t+η2tL2t,for all t≥T~0. Hence, we can obtain that there exists M~3>0 such that limsupt→∞L2t≤M~3. From this, for any ϵ~1>0, there is a T~1≥T~0 such that L2t≤M~3+ϵ~1≔Q3 for all t≥T~1. Hence, from the last equation of model 12, we further have
(51)dV2tdt≤−ν2tV2t+Q3η2t,for all t≥T~1. This shows that there exists M~4>0 such that limsupt→∞V2t≤M~4. Hence, for any ϵ~2>0, there is a T~2≥T~1 such that V2t≤M~4+ϵ~2≔Q4 for all t≥T~2.From above discussions, we finally obtain that all solutions of model12 with initial condition (2) are ultimately bounded. This completes the proof.Assume firstlyR10>1. From above result, without loss of generality, we can assume that there are positive constants m1 and K2 such that minS1t,E1t,I1t,R1t,U1t,L1t,V1t≥m1 for any solution S1t,E1t,I1t,R1t,U1t,L1t,V1t of model 11 and maxS1t,E1t,I1t,R1t,U1t,L1t,V1t≤K2 for any solution S2t,E2t,I2t,R2t,U2t,L2t,V2t of model 12. Thus, from model 12, we further have
(52)dS2tdt≥C1td1m1−α2tS2tK2−μ2tS2t−C2td2S2t,dE2tdt≥C1td1m1−μ2t+ε2tE2t−C2td2E2t,dI2tdt≥C1td1m1−μ2t+γ2t+δ2tI2t−C2td2I2t,dR2tdt≥C1td1m1−μ2t+ξ2tR2t−C2td2R2t,dU2tdt≥b2t−ν2tU2t1−U2t+2K2M2t−β2tK2U2t,dL2tdt=β2tI2tU2t−ν2t+η2tL2t,dV2tdt=η2tL2t−ν2tV2t.Consider auxiliary equations as follows:(53)dS¯2tdt=C1td1m1−α2tK2+μ2t+C2td2S¯2t,dE¯2tdt=C1td1m1−μ2t+ε2t+C2td2E¯2t,dI¯2tdt=C1td1m1−μ2t+γ2t+δ2t+C2td2I¯2t,dR¯2tdt=C1td1m1−μ2t+ξ2t+C2td2R¯2t,dU¯2tdt=b2t−ν2t1−2K2M2t−β2tK2U¯2t−b2t−ν2tM2tU¯22t,dL¯2tdt=β2tI¯2tU¯2t−ν2t+η2tL¯2t,dV¯2tdt=η2tL¯2t−ν2tV¯2t.Firstly, we consider the livestock equations of model (53) by Lemma 1, and we have
(54)liminft→∞S¯2t=C1/d1m1iα2K2+μ2+C2/d2u,liminft→∞E¯2t=C1/d1m1iμ2+ε2+C2/d2u,liminft→∞I¯2t=C1/d1m1iμ2+γ2+δ2+C2/d2u,liminft→∞R¯2t=C1/d1m1iμ2+ξ2+C2/d2u.Next, we consider the mosquito system. From the fifth equation of model (53), we have
(55)liminft→∞U¯2t=b2−ν21−2K2/M2−β2K2ib2−ν2/M2u.Therefore, for anyϵ~3>0, there is a T~3>0 such that for all t≥T~3,
(56)I¯2t≥C1/d1m1iμ2+γ2+δ2+C/d2u−ϵ~3≔Q5,U¯2t≥b2−ν21−2K2/M2−β2K2ib2−ν2/M2u−ϵ~3≔Q6.Considering the sixth equation of model (53), we have
(57)dL¯2tdt≥β2tQ5Q6−ν2t+η2tL¯2t,then we can obtain liminft→∞L¯2t=β2Q5Q6i/ν2+η2u. Thus, for any ϵ~4>0, there is a T~4>0 such that L¯2t≥β2Q5Q6i/ν2+η2u−ϵ~4≔Q7. Considering the last equation of model (53), we have
(58)dV¯2tdt≥η2tQ7−ν2tV¯2t,and then, we can obtain liminft→∞V¯2t=η2Q7i/ν2u. Using comparison theorem, we can obtain the theorem as follows.Theorem 5.
IfR10>1, the disease in model 12 is permanent.Next, we assume ifR10<1, then each nonnegative solution of model 11 converges to disease-free periodic solution E10=S1∗t,0,0,0,U1∗t,0,0, and we further obtain the limiting system of model 12 as follows:
(59)dS2tdt=C1td1S1∗t−α2tS2tV2t−μ2tS2t+ξ2tR2t−C2td2S2t,dE2tdt=α2tS2tV2t−μ2t+ε2tE2t−C2td2E2t,dI2tdt=ε2tE2t−μ2t+γ2t+δ2tI2t−C2td2I2t,dR2tdt=γ2tI2t−μ2t+ξ2tR2t−C2td2R2t,dU2tdt=b2t−ν2tU2t1−U2t+L2t+V2tM2t−β2tI2tU2t,dL2tdt=β2tI2tU2t−ν2t+η2tL2t,dV2tdt=η2tL2t−ν2tV2t.From a similar argument as in the above to model11, we can obtain that model (59) has a unique disease-free periodic solution E20=S2∗t,0,0,0,U2∗t,0,0, where S2∗t,U2∗t is the unique positive ω-periodic solution.
(60)dS2tdt=C1td1S1∗t−μ2t+C2td2S2t,dU2tdt=b2t−ν2tU2t1−U2tM2t.Furthermore, by the same way as in the above for model11, we can obtain the basic reproduction number R20=ρℒ2 for model (59), where the operator ℒ2 can be defined similarly to ℒ1.It is clear that model (59) is very similar with model 11. Therefore, using the theory of limiting systems and the nearly same arguments as in the above, we can prove the following theorems.Theorem 6.
LetR10<1 if R20<1, then disease-free periodic solution E20 of model (59) is globally asymptotically stable for model 12, and if R20>1, then E20 is unstable for model 12.Theorem 7.
IfR10<1 and R20>1, then the disease in model 12 is permanent.


### 3.3. The Third Patch

In this section, we discuss model13. For model 13, the discussions are nearly similar to model 12. We here directly state the following main results.Theorem 8.
LetS3t,E3t,I3t,R3t,U3t,L3t,V3t be the solution of model 13 with initial condition (2). Then, S3t,E3t,I3t,R3t,U3t,L3t,V3t is nonnegative for all t≥0 and ultimately bounded, and when S30>0,E30>0,I30>0,R30>0,U30>0,L30>0 and V30>0, then S3t,E3t,I3t,R3t,U3t,L3t,V3t is also positive for all t>0.Theorem 9.
IfR10>1 or R10<1,R20>1, the disease in model 13 is permanent.IfR10<1 and R20<1, then any nonnegative solution S2t,E2t,I2t,R2t,U2t,L2t,V2t of model 12 converges to disease-free periodic solution S2∗t,0,0,0,U2∗t,0,0 of model (59). Thus, we obtain the limiting system of model 13 as follows:
(61)dS3tdt=C2td2S2∗−α3tS3tV3t−μ3tS3t+ξ3tR3t,dE3tdt=α3tS3tV3t−μ3t+ε3tE3t,dI3tdt=ε3tE3t−μ3t+γ3t+δ3tI3t,dR3tdt=γ3tI3t−μ3t+ξ3tR3t,dU3tdt=b3t−ν3tU3t1−U3t+L3t+V3tM3t−β3tI3tU3t,dL3tdt=β3tI3tU3t−ν3t+η3tL3t,dV3tdt=η3tL3t−ν3tV3t.From a similar argument as in the above to model12, we can obtain that model (61) has a unique disease-free periodic solution E30=S3∗t,0,0,0,U3∗t,0,0, where S3∗t,U3∗t is the unique positive ω-periodic solution.
(62)dS3tdt=C2td2S2∗t−μ3tS3t,dU3tdt=b3t−ν3tU3t1−U3tM3t.By the same way as in the above for model11, we can obtain the basic reproduction number R30=ρℒ3 for model (61), where the operator ℒ3 can be defined similarly to ℒ1. Thus, we further have the following results.Theorem 10.
LetR10<1 and R20<1, if R30<1, then disease-free periodic solution E30 of model (61) is globally asymptotically stable for model 13, and if R30>1, then E30 is unstable for model 13.Theorem 11.
IfR10<1,R20<1 and R30>1, then the disease in model 13 is permanent.


## 3.1. The First Patch

In this section, we discuss model11. Firstly, on the nonnegativity, positivity, and boundedness of solutions for model 11, we have the following result.Theorem 1.
LetS1t,E1t,I1t,R1t,U1t,L1t,V1tbe the solution of model 11 with initial condition (2). Then, S1t,E1t,I1t,R1t,U1t,L1t,V1t is nonnegative for all t≥0 and ultimately bounded, and when S10>0,E10>0,I10>0,R10>0,U10>0,L10>0 and V10>0, then S1t,E1t,I1t,R1t,U1t,L1t,V1t is also positive for all t>0.Proof 1.
In fact, by the continuous dependence of solutions with respect to initial values, we only need to prove that whenS10>0,E10>0,I10>0,R10>0,U10>0,L10>0 and V10>0, then S1t,E1t,I1t,R1t,U1t,L1t,V1t is positive for all t>0. From the fifth equation of model 11, we have for any t≥0.
(12)dU1tdt=b1t−ν1t1−U1t+L1t+V1tM1t−β1tI1tU1t.
SinceU10>0, we get
(13)U1t=U10exp∫0tb1s−ν1s1−U1s+L1s+V1sM1s−β1sI1sds>0.Definem1t=mint≥0S1t,E1t,I1t,R1t,L1t,V1t. Obviously, m10=minS10,E10,I10,R10,L10,V10>0. We only need to prove m1t>0 for all t≥0. Suppose that there exists a t0>0 such that m1t0=0 and m1t>0 for all t∈0,t0. Then, we only need to discuss the following six cases: (1) m1t0=S1t0, (2) m1t0=E1t0, (3) m1t0=I1t0, (4) m1t0=R1t0, (5) m1t0=L1t0, and (6) m1t0=V1t0.Now, we only give the proof for case (2). The remaining cases can be proved in a similar manner. Letm1t0=E1t0. Since m1t>0 for all t∈0,t0, then V1t>0, S1t>0, and hence,
(14)dE1tdt≥−μ1t+ε1t+C1td1E1t,forallt∈0,t0.Integrating from0 to t0, we get
(15)0=E1t0≥E10exp−∫0t0μs+ε1s+C1sd1ds>0,which leads to a contradiction.Next, we show the boundedness of nonnegative solutions of model11. Let N1t=S1t+E1t+I1t+R1t, then we get
(16)dN1tdt=r1t−μ1t+C1td1N1t−δ1tI1t≤r1t−μ1t+C1td1N1t.Hence, we havelimsupt→∞N1t≤r1u/μ1i+C1/d1i, which implies that S1t,E1t,I1t and R1t are ultimately bounded. Since for any t≥0,
(17)dU1tdt≤b1t−ν1tU1t1−U1tM1t,using Lemma 3; we further have limsupt→∞U1t≤limsupt→∞U1∗t=max0≤t≤ωU1∗t. Hence, for any ϵ0>0, there is a T0>0 such that I1t≤r1u/μ1i+C1/d1i+ϵ0 and U1t≤P1+ϵ0 for all t≥T0, where P1=max0≤t≤ωU1∗t. From the sixth equation of model 11, we have
(18)dL1tdt≤β1tP2−ν1t+η1tL1t,for all t≥T0, where P2=r1u/μ1i+C1t/d1i+ϵ0P1+ϵ0. Hence, we can obtain limsupt→∞L1t≤β1uP2/ν1+η1i From this, for any ϵ1>0, there is a T1>0 such that L1t≤β1uP2/ν1+η1i+ϵ1 for all t≥T1. Hence, from the last equation of model 11, we further have
(19)dV1tdt≤−ν1tV1t+P3η1t,for all t≥T1, where P3=β1uP2/ν1+η1i+ϵ1. This shows that limsupt→∞V1t≤P3η1u/ν1i. From above discussions, we finally obtain that all solutions of model 11 with initial condition (2) are ultimately bounded. This completes the proof.Now, we show thatR10 serves as a threshold parameter. We prove that when R10<1, then disease-free periodic solution E10 is globally asymptotically stable, that is to say, the RVF virus is extinct in the first patch, and when R10>1, then model 11 is permanent, that is to say, the RVF virus is existent in the population in the first patch.Theorem 2.
IfR10<1, then disease-free periodic solution E10 of model 11 is globally asymptotically stable, and if R10>1, then E10 is unstable.Proof 2.
From conclusiond of Lemma 5, it follows that if R10>1, then E10 is unstable, and if R10<1, then E10 is locally asymptotically stable.We now prove that ifR10<1, then E10 is globally attractive. Let R10<1, from conclusion c of Lemma 5, then ρΦF1−B1ω<1. We can choose constant ς>0 small enough such that ρΦF1−B1+ςHω<1, where
(20)Ht=000α1t00β1t000000000.From model11, we have
(21)dN1tdt=r1t−μ1t+C1td1N1t−δ1tI1t≤r1t−μ1t+C1td1N1t,dU1tdt≤b1t−ν1tU1t1−U1tM1t.From Lemma3, for above given ς>0, there exists a T1>0 such that N1t=S1t+I1t+E1t+R1t≤S1∗t+ς and U1t≤U1∗t+ς for any t≥T1. Hence, from model 11, we obtain that for all t≥T1
(22)dE1tdt≤α1tS1∗t+ςV1t−μ1t+ε1t+C1td1E1t,dL1tdt≤β1tU1∗t+ςtI1t−ν1t+η1tL1t,dI1tdt=ε1tE1t−μ1t+γ1t+δ1t+C1td1I1t,dV1tdt=−ν1tV1t+η1tL1t.Consider the following auxiliary system:(23)dE^1tdt=α1tS1∗t+ςV^1t−μ1t+ε1t+C1td1E^1t,dL^1tdt=β1tU1∗t+ςtI^1t−ν1t+η1tL^1t,dI^1tdt=ε1tE^1t−μ1t+γ1t+δ1t+C1td1I^1t,dV^1tdt=−ν1tV^1t+η1tL^1t.It follows from Lemma4 that there is a positive ω-periodic function υt=υ1t,υ2t,υ3at,υ4tT such that E^1t,L^1t,I^1t,V^1tT=eptυt is a solution of system (23), where p=1/ωlnρΦF1−B1+ςHω.DenoteJt=E1t,L1t,I1t,V1tT. We can choose constant τ>0 such that JT1≤τepT1υT1. According to the comparison theorem of the vector form (see [23]), we obtain Jt≤τeptυt for any t≥T1. By ρΦF1−B1+ςHω<1, we know p=1/ωlnρΦF1−B1+ςHω<0. Then, we conclude that limt→∞Jt=0, that is to say, limt→∞E1t,L1t,I1t,V1t=0,0,0,0. By the equations of R1, S1, and U1 of model 11, we also obtain limt→∞S1t,R1t,U1t=S1∗t,0,U1∗t. Hence, disease-free periodic solution E10 is globally attractive. This shows that disease-free periodic solution E10 is globally asymptotically stable. This completes the proof.Theorem 3.
IfR10>1, then the disease in model 11 is permanent.Proof 3.
From conclusionb of Lemma 5, we know that R10>1 if and only if ρΦF1−B1ω>1. We can choose a small enough constant σ>0 such that ρΦF1−B1+ςHω>1, where Nt is given in (20). From H1 and H2, we can choose ϵ1>0 small enough such that b1t−ν1t−β1tϵ1>0 for all t≥0. Consider the following auxiliary equations:
(24)dS~1tdt=r1t−α1tS~1tϵ1−μ1t+C1td1S~1tand
(25)dU~1tdt=b1t−ν1tU~1t1−U~1tM1t−β1tU~1tϵ1,from Lemma 1 and Lemma 2 we can get equations (24) and (25) admit globally uniformly attractive positive ω-periodic solution S1ϵ1t and U1ϵ1t, respectively. By the continuity of solutions with respect to parameter ϵ1, for above σ>0, there exists a 0<ϵ0<ϵ1 such that for all t∈0,ω
(26)S1ϵ0t>S1∗t−σ2,U1ϵ0t>U1∗t−σ2.
Define the sets as follows:(27)X1=S1,E1,I1,R1,U1,L1,V1:S1>0,E1≥0,I1≥0,R1≥0,U1>0,L1≥0,V1≥0,X10=S1,E1,I1,R1,U1,L1,V1∈X1:E1>0,I1>0,L1>0,V1>0,∂X10=X1X10=S1,E1,I1,R1,U1,L1,V1∈X1:E1I1L1V1=0.Define further the Poincaré mapP:X→X1 as follows:
(28)Px0=uω,x0,x0=S10,E10,I10,R10,U10,L10,V10∈X1,where ut,x0=S1t,E1t,I1t,R1t,U1t,L1t,V1t is the solution of model 11 satisfying initial condition u0,x0=x0. It follows from Theorem 1 that all solutions of model 11 are ultimately bounded. Therefore, map P is point dissipative and compact on X1.We define(29)M∂=x0=S10,E10,I10,R10,U10,L10,V10∈∂X10:Pmx0∈∂X10,forallm≥0,where P0x0=x0,P1x0=Px0 and Pmx0=PPm−1x0 for m=1,2,…. Now, we claim M∂=S1,0,0,R1,U1,0,0:S1>0,R1≥0,U1>0. In fact, from model 11, we directly obtain that the solution of model 11 with initial value S10,0,0,R10,U10,0,0∈∂X10 has the form S1t,0,0,R1t,U1t,0,0. Hence,
(30)PmS10,0,0,R10,U10,0,0=S1mω,0,0,R1mω,U1mω,0,0∈∂X0,for all m≥0. This shows S10,0,0,R10,U10,0,0∈M∂. Therefore, S1,0,0,R1,U1,0,0:S1>0,R1≥0,U1>0⊆M∂.Next, we proveM∂⊆S1,0,0,R1,U1,0,0:S1>0,R1≥0,U1>0. Suppose that there is a S10,E10,I10,R10,U10,L10,V10∈M∂ such that S10,E10,I10,R10,U10,L10,V10∉S1,0,0,R1,U1,0,0:S1>0,R1≥0,U1>0.We firstly havemaxE10,I10,L10,V10>0. Let S1t,E1t,I1t,R1t,U1t,L1t,V1t be the solution of model 11 with initial value S10,E10,I10,R10,U10,L10,V10 at t=0. From Theorem 1, we firstly have that S1t,E1t,I1t,R1t,U1t,L1t,V1t is nonnegative for all t≥0. Furthermore, by S10>0 and U10>0, from the first and fifth equations of model 11, we also have that S1t>0 and U1t>0 for all t≥0, respectively.IfE10>0, then by the second equation of model 11, we obtain
(31)dE1tdt≥−μ1t+ε1t+C1td1E1t,forallt≥0.Integrating this inequality from0 to t, we have
(32)E1t≥E10exp−∫0tμ1s+ε1s+C1sd1ds>0,forallt≥0.Further, from the third equation of model11 we can obtain
(33)I1t>I10exp−∫0tμ1s+γ1s+δ1s+C1sd1ds≥0,forallt>0.From the sixth equation of model11, we get
(34)dL1tdt>−ν1t+η1tL1t,forallt≥0.Hence,(35)L1t>L10exp−∫0tν1s+η1sds≥0,forallt>0.Lastly, from the equation ofV1 for model 11, we also have V1t>0 for all t>0. Therefore, we finally obtain E1tI1tL1tV1t>0 for all t>0.Similarly, ifI10>0 or L10>0 or V10>0, we also can obtain E1tI1tL1tV1t>0 for all t>0. This shows that S10,E10,I10,R10,U10,L10,V10∉M∂, which leads to a contradiction. This implies that M∂⊆S1,0,0,R1,U1,0,0:S1>0,R1≥0,U1>0. Hence, M∂=S1,0,0,R1,U1,0,0:S1>0,R1≥0,U1>0.InM∂, model 11 degenerates to
(36)dS1tdt=r1t−μ1t+C1td1S1t+ξ1tR1t,dR1tdt=−μ1t+ξ1t+C1td1R1t,dU1tdt=b1t−ν1tU1t1−U1tM1t.We easily prove that system (36) has a globally asymptotically stable ω-periodic solution S1∗t,0,U1∗t, where S1∗t,U1∗t is given in Lemma 3. It is clear that map P has a unique globally attractive fixed point restricted in M∂, which is E1=S1∗0,0,0,0,U1∗0,0,0.Now, we defineWsE1=x0:Pmx0→E1,m→∞, which is said to be stable set of E1. We show that
(37)WsE1∩X0=0.According to the continuity of solutions with respect to the initial values, for above given constantϵ0, there exists δ>0 such that for any x0=S10,E10,I10,R10,U10,L10,V10∈X0 with x0−E1≤δ, it follows that ut,x0−ut,E1≤ϵ0 for all t∈0,ω, where ut,x0=S1t,E1t,I1t,R1t,U1t,L1t,V1t is the solution of model 11 with initial condition S10,E10,I10,R10,U10,L10,V10=x0 and ut,E1=S1∗t,0,0,0,U1∗t,0,0. Since ut,E1 is ω-periodic, we have ut1,E1=ut2,E1 for any t1,t2≥0 with t2=kω+t1, where k is an integer. We claim that
(38)limsupm→∞Pmx0−E1≥δ,forallx0=S10,E10,I10,R10,U10,L10,V10∈X0.Suppose that (38) does not hold. Then, we have
(39)limsupm→∞Pmx0−E1<δ,forsomex0=S10,E10,I10,R10,U10,L10,V10∈X0.Without loss of generality, we assume thatPmx0−E1<δ for all m≥0. Thus,
(40)ut,Pmx0−ut,E1≤ϵ0,forallt∈0,ω,m≥0.For anyt≥0, let t=t′+nω, where t′∈0,ω and n=t/ω are the greatest integers less than or equal to t/ω. Then, we have ut,x0−ut,E1=ut′,Pmx0−ut′,E1<ϵ0,foranyt≥0.It follows that0<E1t<ϵ0, 0<L1t<ϵ0, 0<I1t<ϵ0, and 0<V1t<ϵ0 for all t≥0. Thus, from model 11, we have the following:
(41)dS1tdt≥r1t−α1tS1tϵ0−μ1t+C1td1S1t,dU1tdt≥b1t−ν1tU1t1−U1tM1t−β1t+2b1t−ν1tM1tU1tϵ0,for any t≥0. Using the comparison principle, we obtain
(42)St≥S~1ϵ0t,Ut≥U~1ϵ0t,for any t≥0, where S~1ϵ0t and U~1ϵ0t are the solutions of equations (20) and (23) with respect to parameter ϵ0 satisfying initial conditions S~1ϵ00=S10 and U~1ϵ00=U10, respectively.From Lemma1 and Lemma 2, equations (24) and (25) with respect to the parameter ϵ0 have globally uniformly attractive solutions S1ϵ0t and U1ϵ0t, respectively. Hence, there exists t2>0 such that
(43)S~1ϵ0t≥S1ϵ0t−σ2,U~1ϵ0t≥U1ϵ0t−σ2,for all t≥t2. By (26), (42), and (43), we obtain S1t≥S~1ϵ0t≥S1ϵ0t−σ/2>S1∗t−σ and U1t≥U~1ϵ0t≥U1ϵ0t−σ/2>U1∗t−σ for all t≥t2. Thus, for all t≥t2, it holds that
(44)dE1tdt≥α1tS1∗t−σV1t−μ1t+ε1t+C1td1E1t,dL1tdt≥β1tU1∗t−σI1t−ν1t+η1tL1t,dI1tdt=ε1tE1t−μ1t+γ1t+δ1t+C1td1I1t,dV1tdt=−ν1tV1t+η1tL1t.Consider the following auxiliary system:(45)dE~1tdt=α1tS1∗t−σV~1t−μ1t+ε1t+C1td1E~1t,dL~1tdt=β1tU1∗t−σI~1t−ν1t+η1tL~1t,dI~1tdt=ε1tE~1t−μ1t+γ1t+δ1t+C1td1I~1t,dV~1tdt=−ν1tV~1t+η1tL~1t.System (45) can be rewritten in terms of the matrices F1t and B1t as follows:
(46)ddtE~1L~1I~1V~1=F1t−B1t+ςHtE~1L~1I~1V~1.It follows from Lemma4 that there is a positive ω-periodic function wt=w1t,w2t,w3at,w4tT such that E~1t,L~1t,I~1t,V~1tT=eθtwt is a solution of system (46), where θ=1/ωlnρΦF1−B1+ςHω.Now, we denoteJt=E1t,L1t,I1t,V1tT. We can choose small enough constant τ>0 such that Jt2≥τeθt2wt2. Thus, by comparison theorem of the vector form, it follows that Jt≥τeθtwt for all t≥t2. By ρΦF1−B1+ςHω>1, we obtain θ=1/ωlnρΦF1−B1+ςHω>0. Hence, limt→∞Jt=∞, which implies that
(47)limt→∞E1t=∞,limt→∞L1t=∞,limt→∞I1t=∞,limt→∞V1t=∞,which leads to a contradiction. Hence, claim (25) holds. This shows WsE1∩X0=0.It is clear that fixed pointE1 of P is globally attractive restricting on M∂. Hence, E1 is acyclic in M∂. From the theorems established in [19] on the uniform persistence of dynamical systems, we obtain that P is uniformly persistent with respect to X0,∂X0, which implies that model 11 is permanent. This completes the proof.


## 3.2. The Second Patch

In this section, we discuss model12. First of all, we have the following result on the positivity and ultimate boundedness of solutions in model 12.Theorem 4.
LetS2t,E2t,I2t,R2t,R2t,L2t,V2t be the solution of model 12 with initial condition (2). Then, S2t,E2t,I2t,R2t,R2t,L2t,V2t is nonnegative for all t≥0 and ultimately bounded, and when S20>0,E20>0,I20>0,R20>0,R20>0,L20>0 and V20>0, then S2t,E2t,I2t,R2t,R2t,L2t,V2t is also positive for all t>0.Proof 4.
Using the same method given in Theorem1, we can prove the positivity of solutions for model 12. Next, we show the boundedness of solutions for model 12. Let N2t=S2t+E2t+I2t+R2t, then we get
(48)dN2tdt=C1td1N1t−μ2t+C2td2N2t−δ2tI2t≤C1td1N1t−μ2t+C2td2N2t.By the comparison theorem and Lemma1, we get that there exists constant M~1>0 such that limsupt→∞N2t≤M~1, which implies that S2t,E2t,I2t and R2t are ultimately bounded. Since for any t≥0,
(49)dU2tdt≤b2t−ν2tU2t1−U2tM2t,then there exists constant M~2>0 such that limsupt→∞U2t≤M~2. Hence, for any ϵ~>0, there is a T~0>0 such that I2t≤M~1+ϵ~0≔Q1 and U2t≤M~2+ϵ~0≔Q2 for all t≥T~0. From the sixth equation of model 12, we have
(50)dL2tdt≤β2tQ1Q2−ν2t+η2tL2t,for all t≥T~0. Hence, we can obtain that there exists M~3>0 such that limsupt→∞L2t≤M~3. From this, for any ϵ~1>0, there is a T~1≥T~0 such that L2t≤M~3+ϵ~1≔Q3 for all t≥T~1. Hence, from the last equation of model 12, we further have
(51)dV2tdt≤−ν2tV2t+Q3η2t,for all t≥T~1. This shows that there exists M~4>0 such that limsupt→∞V2t≤M~4. Hence, for any ϵ~2>0, there is a T~2≥T~1 such that V2t≤M~4+ϵ~2≔Q4 for all t≥T~2.From above discussions, we finally obtain that all solutions of model12 with initial condition (2) are ultimately bounded. This completes the proof.Assume firstlyR10>1. From above result, without loss of generality, we can assume that there are positive constants m1 and K2 such that minS1t,E1t,I1t,R1t,U1t,L1t,V1t≥m1 for any solution S1t,E1t,I1t,R1t,U1t,L1t,V1t of model 11 and maxS1t,E1t,I1t,R1t,U1t,L1t,V1t≤K2 for any solution S2t,E2t,I2t,R2t,U2t,L2t,V2t of model 12. Thus, from model 12, we further have
(52)dS2tdt≥C1td1m1−α2tS2tK2−μ2tS2t−C2td2S2t,dE2tdt≥C1td1m1−μ2t+ε2tE2t−C2td2E2t,dI2tdt≥C1td1m1−μ2t+γ2t+δ2tI2t−C2td2I2t,dR2tdt≥C1td1m1−μ2t+ξ2tR2t−C2td2R2t,dU2tdt≥b2t−ν2tU2t1−U2t+2K2M2t−β2tK2U2t,dL2tdt=β2tI2tU2t−ν2t+η2tL2t,dV2tdt=η2tL2t−ν2tV2t.Consider auxiliary equations as follows:(53)dS¯2tdt=C1td1m1−α2tK2+μ2t+C2td2S¯2t,dE¯2tdt=C1td1m1−μ2t+ε2t+C2td2E¯2t,dI¯2tdt=C1td1m1−μ2t+γ2t+δ2t+C2td2I¯2t,dR¯2tdt=C1td1m1−μ2t+ξ2t+C2td2R¯2t,dU¯2tdt=b2t−ν2t1−2K2M2t−β2tK2U¯2t−b2t−ν2tM2tU¯22t,dL¯2tdt=β2tI¯2tU¯2t−ν2t+η2tL¯2t,dV¯2tdt=η2tL¯2t−ν2tV¯2t.Firstly, we consider the livestock equations of model (53) by Lemma 1, and we have
(54)liminft→∞S¯2t=C1/d1m1iα2K2+μ2+C2/d2u,liminft→∞E¯2t=C1/d1m1iμ2+ε2+C2/d2u,liminft→∞I¯2t=C1/d1m1iμ2+γ2+δ2+C2/d2u,liminft→∞R¯2t=C1/d1m1iμ2+ξ2+C2/d2u.Next, we consider the mosquito system. From the fifth equation of model (53), we have
(55)liminft→∞U¯2t=b2−ν21−2K2/M2−β2K2ib2−ν2/M2u.Therefore, for anyϵ~3>0, there is a T~3>0 such that for all t≥T~3,
(56)I¯2t≥C1/d1m1iμ2+γ2+δ2+C/d2u−ϵ~3≔Q5,U¯2t≥b2−ν21−2K2/M2−β2K2ib2−ν2/M2u−ϵ~3≔Q6.Considering the sixth equation of model (53), we have
(57)dL¯2tdt≥β2tQ5Q6−ν2t+η2tL¯2t,then we can obtain liminft→∞L¯2t=β2Q5Q6i/ν2+η2u. Thus, for any ϵ~4>0, there is a T~4>0 such that L¯2t≥β2Q5Q6i/ν2+η2u−ϵ~4≔Q7. Considering the last equation of model (53), we have
(58)dV¯2tdt≥η2tQ7−ν2tV¯2t,and then, we can obtain liminft→∞V¯2t=η2Q7i/ν2u. Using comparison theorem, we can obtain the theorem as follows.Theorem 5.
IfR10>1, the disease in model 12 is permanent.Next, we assume ifR10<1, then each nonnegative solution of model 11 converges to disease-free periodic solution E10=S1∗t,0,0,0,U1∗t,0,0, and we further obtain the limiting system of model 12 as follows:
(59)dS2tdt=C1td1S1∗t−α2tS2tV2t−μ2tS2t+ξ2tR2t−C2td2S2t,dE2tdt=α2tS2tV2t−μ2t+ε2tE2t−C2td2E2t,dI2tdt=ε2tE2t−μ2t+γ2t+δ2tI2t−C2td2I2t,dR2tdt=γ2tI2t−μ2t+ξ2tR2t−C2td2R2t,dU2tdt=b2t−ν2tU2t1−U2t+L2t+V2tM2t−β2tI2tU2t,dL2tdt=β2tI2tU2t−ν2t+η2tL2t,dV2tdt=η2tL2t−ν2tV2t.From a similar argument as in the above to model11, we can obtain that model (59) has a unique disease-free periodic solution E20=S2∗t,0,0,0,U2∗t,0,0, where S2∗t,U2∗t is the unique positive ω-periodic solution.
(60)dS2tdt=C1td1S1∗t−μ2t+C2td2S2t,dU2tdt=b2t−ν2tU2t1−U2tM2t.Furthermore, by the same way as in the above for model11, we can obtain the basic reproduction number R20=ρℒ2 for model (59), where the operator ℒ2 can be defined similarly to ℒ1.It is clear that model (59) is very similar with model 11. Therefore, using the theory of limiting systems and the nearly same arguments as in the above, we can prove the following theorems.Theorem 6.
LetR10<1 if R20<1, then disease-free periodic solution E20 of model (59) is globally asymptotically stable for model 12, and if R20>1, then E20 is unstable for model 12.Theorem 7.
IfR10<1 and R20>1, then the disease in model 12 is permanent.


## 3.3. The Third Patch

In this section, we discuss model13. For model 13, the discussions are nearly similar to model 12. We here directly state the following main results.Theorem 8.
LetS3t,E3t,I3t,R3t,U3t,L3t,V3t be the solution of model 13 with initial condition (2). Then, S3t,E3t,I3t,R3t,U3t,L3t,V3t is nonnegative for all t≥0 and ultimately bounded, and when S30>0,E30>0,I30>0,R30>0,U30>0,L30>0 and V30>0, then S3t,E3t,I3t,R3t,U3t,L3t,V3t is also positive for all t>0.Theorem 9.
IfR10>1 or R10<1,R20>1, the disease in model 13 is permanent.IfR10<1 and R20<1, then any nonnegative solution S2t,E2t,I2t,R2t,U2t,L2t,V2t of model 12 converges to disease-free periodic solution S2∗t,0,0,0,U2∗t,0,0 of model (59). Thus, we obtain the limiting system of model 13 as follows:
(61)dS3tdt=C2td2S2∗−α3tS3tV3t−μ3tS3t+ξ3tR3t,dE3tdt=α3tS3tV3t−μ3t+ε3tE3t,dI3tdt=ε3tE3t−μ3t+γ3t+δ3tI3t,dR3tdt=γ3tI3t−μ3t+ξ3tR3t,dU3tdt=b3t−ν3tU3t1−U3t+L3t+V3tM3t−β3tI3tU3t,dL3tdt=β3tI3tU3t−ν3t+η3tL3t,dV3tdt=η3tL3t−ν3tV3t.From a similar argument as in the above to model12, we can obtain that model (61) has a unique disease-free periodic solution E30=S3∗t,0,0,0,U3∗t,0,0, where S3∗t,U3∗t is the unique positive ω-periodic solution.
(62)dS3tdt=C2td2S2∗t−μ3tS3t,dU3tdt=b3t−ν3tU3t1−U3tM3t.By the same way as in the above for model11, we can obtain the basic reproduction number R30=ρℒ3 for model (61), where the operator ℒ3 can be defined similarly to ℒ1. Thus, we further have the following results.Theorem 10.
LetR10<1 and R20<1, if R30<1, then disease-free periodic solution E30 of model (61) is globally asymptotically stable for model 13, and if R30>1, then E30 is unstable for model 13.Theorem 11.
IfR10<1,R20<1 and R30>1, then the disease in model 13 is permanent.


## 4. Numerical Simulation

In this section, we will present some examples and the numerical simulations to confirm our theoretical results. The numerical values of most parameters in the following examples are adopted from Xiao et al. [8], Gao et al. [13], and Gaff et al. [14]. And the numerical simulation method of basic reproduction number is presented of Posny and Wang [24]. We also give our numerical simulation with reference to [25].In the following numerical examples, we choose the initial valueS10=1800, E10=0, I10=50, R10=100, U10=1000, L10=0, V10=100, S20=0, E20=0, I20=0, R20=0, U20=8000, L20=0, V20=0, S30=0, E30=0, I30=0, R30=0, U30=1500, L30=0 and V30=0 in model 11–13 which is given in [8].Example 1.
We assume that in model11–13, only the capacity of mosquitoes is periodic. Thus, we take r1t=300, C1t=C2t=200, αit=0.003, β1t=0.008, β2t=β3t=0.0008, εit=0.6, μ1t=μ2t=0.0012, μ3t=0.4, ξit=0.005, d1=100, d2=800, δit=0.1, ηit=0.5, γit=0.4, bit=0.45, νit=0.06, where i=1,2,3,M1t=10001+a1sin2πt/T2+ϕ, M2t=80001+a1sin2πt/T2+ϕ and M3t=15001+a1sin2πt/T2+ϕ, where a1=0.3, T2=365, and ϕ=π.It is clear thatH1 and H2 are satisfied. By the numerical calculations, we get the basic reproduction number R10=ρℒ1=2.1374>1. Therefore, from Theorems 3, 5, and 9, we obtain that the disease in model 11–13 in three patches is permanent. The numerical simulations are given in Figure 1.Example 2.
We only consider the capacity of mosquitoes, and the parameters incorporating both seasonal and festival impacts are periodic. In model11–13, we take r1t in the following form:
(63)r1j=r1j−1+dr1,k−n<j≤k,r1j−1−dr¯1,k<j≤k+m,r10,otherwise,with k=315,n=m=10,dr1=dr~1=20,r10=300 and j∈1,354. We also take C1t, C2t, and μ3t using the same definition as in (63) with dCi=dC¯i=98i=1,2, dμ3=dμ¯3=0.02, C10=C20=200 and μ30=0.4. The other parameters in model 11–13 are chosen as in Example 1.Figure 1
The graphs of solutionSit,Eit,Iit,Rit,Uit,Lit,Vit,i=1,2,3 of model 11–13. From (c) and (g), we obtain that the disease is permanent in model 11–13 in three patches.
(a)
(b)
(c)
(d)
(e)
(f)
(g)It is clear thatH1 and H2 are satisfied. By the numerical calculations, we get the basic reproduction number R10=ρℒ1=2.0891>1. Therefore, from Theorems 3, 5, and 9, we obtain that the disease in model 11–13 in three patches is permanent. The numerical simulations are given in Figure 2.Example 3.
In model11–13, we take β1t=0.0081+a1sin2πt/T2+ϕ, β2t=β3t=0.00081+a1sin2πt/T2+ϕ, bit=0.451+a1sin2πt/T2+ϕ, ηit=0.51+a1sin2πt/T2+ϕ and νit=0.061+a1sin2πt/T2+ϕ, where a1,T, and ϕ are given as in Example 1, where i=1,2,3. We also take αit, εit, ξit, δit, γit, μ1t and μ2t using the same definition as in (63) with dαi=dα¯i=0.0001, dεi=dε¯i=0.01, dξi=dξ¯i=0.0001, dδi=dδ¯i=0.01, dγi=dγ¯i=0.04,dμ1=dμ¯1=dμ2=dμ¯2=0.00005, αi0=0.003, εi0=0.6, ξi0=0.005, δi0=0.1, γi0=0.4 and μ10=μ20=0.0012, where i=1,2,3. We further choose r1t,C1t,C2t and μ3t as in Example 2 and Miti=1,2,3, d1, and d2 as in Example 1.Figure 2
The graphs of solutionSit,Eit,Iit,Rit,Uit,Lit,Vit,i=1,2,3 of model 11–13. From (c) and (g), we obtain that the disease is permanent in model 11–13 in three patches.
(a)
(b)
(c)
(d)
(e)
(f)
(g)It is clear thatH1 and H2 are satisfied. By the numerical calculations, we get the basic reproduction number R10=ρℒ1=2.1334>1. Therefore, from Theorems 3, 5, and 9, we obtain that the disease in model 11–13 in three patches is permanent. The numerical simulations are given in Figure 3.Example 4.
In model11–13, take βit=0.000081+a1sin2πt/T2+ϕ and bit=0.251+a1sin2πt/T2+ϕ for i=1,2,3. We also take C1t, C2t, and αit using the same definition as in (63) with dC1=dC¯1=dC2=dC¯2=98, dαi=dα¯i=0.00001, C10=C20=40 and αi0=0.0003, where i=1,2,3. The other parameters in model 11–13 are chosen as in Example 3.Figure 3
The graphs of solutionSit,Eit,Iit,Rit,Uit,Lit,Vit,i=1,2,3 of model 11–13. From (c) and (g), we obtain that the disease is permanent in model 11–13 in three patches.
(a)
(b)
(c)
(d)
(e)
(f)
(g)It is clear thatH1 and H2 are satisfied. By the numerical calculations, we get the basic reproduction numbers R10=ρℒ1=0.4158<1 and R20=ρℒ2=5.2205>1. Therefore, from Theorem 9, we obtain that the disease in model 11–13 dies out in the first patch and is permanent in the second and third patches. The numerical simulations are given in Figure 4.Example 5.
In model11–13 take β3t=0.00081+a1sin2πt/T2+ϕ,η2t=0.11+a1sin2πt/T2+ϕ. We also take αit and εiti=1,2,3 using the same definition as in (63) with dα1=dα¯1=dα2=dα¯2=0.00001, dα3=dα¯3=0.0001, dε2=dε¯2=0.001, dε3=dε¯3=0.01, α10=α20=0.00003, α30=0.003 and ε20=0.02ε30=0.9. The other parameters in model 11–13 are chosen as in Example 4.Figure 4
The graphs of componentsIit and Viti=1,2,3 of the solution for model 11–13. We see that the disease dies out in the first patch and is permanent in the second and third patches.
(a)
(b)It is clear thatH1 and H2 are satisfied. By the numerical calculations, basic reproduction numbers are R10=ρℒ1=0.1315<1,R20=ρℒ2=0.7547<1, and R30=ρℒ3=5.5289>1. Therefore, from Theorem 11, we obtain that the disease in model 11–13 dies out in first and second patches and is permanent in third patch. The numerical simulations are given in Figure 5.Example 6.
In model11–13, take η2t=0.11+a1sin2πt/T2+ϕ for i=1,2,3. We also take αit and ε2t using the same definition as in (63) with dαi=dα¯i=0.00001i=1,2,3,dε2=dε¯2=0.001,αi0=0.00003i=1,2,3 and ε20=0.02. The other parameters in model 11–13 are chosen as in Example 4.Figure 5
The graphs of componentsIit and Viti=1,2,3 of the solution for model 11–13. We see that the disease dies out in first and second patches and is permanent in third patch.
(a)
(b)It is clear thatH1 and H2 are satisfied. By the numerical calculations, basic reproduction numbers are R10=ρℒ1=0.1315<1, R20=ρℒ2=0.7546<1, and R30=ρℒ3=0.1664<1. Therefore, from Theorem 10, we obtain that the disease in model 11–13 in three patches dies out. The numerical simulations are given in Figure 6.Figure 6
The graphs of componentsIit and Viti=1,2,3 of the solution of model 11–13. We see that the disease dies out in three patches.
(a)
(b)


## 5. Discussion

From the analysis and simulations of the model, we found that the basic reproduction numberRi0i=1,2,3 which was calculating by using the next infection operator approach determined whether the disease was persistent or extinct.For the model1i, if Ri0>1, the disease is persistent in the patch; if the time-dependent birth rate or the transmission rate of disease of the disease or the rate of becoming infectious for livestock and female mosquitoes decreases, the basic reproductive number Ri0<1, then the disease dies out. This means that importation of livestock (especially the major festivals such as Eid al-AdhA) results in more livestock and increases the chance of disease outbreaks; the transportation of livestock from one patch to the other patch increases the chance of disease outbreaks in the other path but reduces the chance of disease outbreaks in the one path. Also, as summer temperatures rise, rain is abundant, mosquitoes are increasing, and disease carriers are increasing which increases the chance of disease outbreaks. That is abundances of hosts and vectors increase the probability of large disease outbreaks.According to our work, diseases will occur during religious festivals which hold in summer. But this is not entirely consistent with the actual situation. In the model, human activity is reflected by periodic parameters, and human compartments are not directly considered, but real-time changes in human activity have different effects on the pattern of disease. Since we assumed well-mixed populations of livestock in each patch and identical movement rates regardless the clinical stage of livestock during transportation, the model may overestimate the real situation. Implementations of various kinds of vaccination strategies and unpredicted religious festival preparations or other stochastic events will also change the pattern of disease outbreaks. These factors will also be our future research work.We see that some basic theoretical results are established for the Rift Valley fever virus transmission model. The deficiency presented in [8] is made up. There is a shortage that in this paper we do not establish a theoretical result on the local and global stability of the model when R0>1. It will be studied in our future works.



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*Source: 7896946-2019-01-09.xml*