# A Periodic West Nile Virus Transmission Model with Stage-Structured Host Population

Authors: Junli Liu; Tailei Zhang; Qiaoling Chen
Journal: Complexity
 (2020)

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## Abstract

In this paper, we study an avian (host) stage-structured West Nile virus model, which incorporates seasonality as well as stage-specific mosquito biting rates. We first introduce the basic reproduction numberR0 for this model and then show that the disease-free periodic solution is globally asymptotically stable when R0<1, while there exists at least one positive periodic solution and that the disease is uniformly persistent if R0>1. In the case where all coefficients are constants, for a special case, we obtain the global stability of the disease-free equilibrium, the uniqueness of the endemic equilibrium, and the permanence of the disease in terms of the basic reproduction number R0. Numerical simulations are carried out to verify the analytic result. Some sensitivity analysis of R0 is performed. Our finding shows that an increase in juvenile exposure will lead to more severe transmission. Moreover, we find that the ignorance of the seasonality may result in underestimation of the basic reproduction number R0.


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## Body

## 1. Introduction

West Nile virus (WNV) is a flavivirus commonly found in Africa, West Asia, and the Middle East. The virus can infect humans, birds, mosquitoes, horses, and some other mammals. It was believed that WNV is maintained in nature in a mosquito-bird-mosquito transmission cycle [1–4]. Since the first outbreak in New York in 1999, West Nile virus remains an annual public health concern. Therefore, it is important to understand the transmission dynamics of WNV in the mosquito-bird population.Mathematical models have played a significant role in understanding and analyzing the WNV transmission dynamics. Wonham et al. [5] presented a single-season susceptible-infectious-removed (SIR) model for WNV transmission in the mosquito-bird population. Lewis et al. [6] developed a reaction-diffusion model for the spatial spread of WNV, established the existence of traveling waves, and calculated the spatial spread rate of infection. Liu et al. [7] formulated a patchy model and studied the impact of directional dispersal of birds on the spatial spreading of WNV. Cruz-Pacheco et al. [8] studied the impact of seasonal variations on the dynamics of WNV infection. Seasonality of WNV has also been discussed in [9, 10]. In a more recent work by Fan et al. [11], they formulated and analyzed a single-season model for the transmission of WNV between vector mosquitoes and avian hosts, and their model incorporated maturation delay for mosquitoes. Further work has also been carried out to model the WNV transmission dynamics; see, e.g., Jiang et al. [12], Bergsman et al. [13], Zhang et al. [14], Xu et al. [15], Lin and Zhu [16], Chen et al. [17], and the references therein.Though mathematical modelling for the transmission of WNV has been studied extensively, the effect of within-species variability, such as differences among host age groups, along with species-specific host feeding preferences of vectors seem to have received little attention. It is obvious that since newly hatched, or nestlings, have little feather coverage, are largely immobile, and have more exposure to mosquito bites relative to older birds. The fledglings that have left the nest have nearly complete feather coverage and exhibit some antimosquito defensive behaviors. The fledglings are more exposed to vectors than mature adult birds, but less than nestlings [18]. Also, the climate effect on the dynamics of vector population and the biting rate from mosquitoes to birds cannot be neglected. Transmission of WNV in most regions of the United States is highly seasonal. Seasonality is strongly influenced by climatic factors. Due to these reason, in [18], Robertson and Cailloue¨t [18] developed a host-vector model for WNV transmission dynamics across a single season, which incorporated the host stage-structure as well as stage-specific biting rates of vectors on avian hosts. They mainly studied the model from the numerical simulation point of view.In this paper, by taking into account the seasonality, the avian stage-structure and within-species heterogeneity, we derive a periodic model to describe the dynamics of WNV transmission. The rest of the paper is organized as follows. In Section2, we present the model rigorously and study its well-posedness, and then we introduce the basic reproduction number R0. In Section 3, we study the threshold dynamics in terms of R0. In Section 4, for a special case, we study the autonomous case of the periodic model and prove the global stability of the disease-free equilibrium and the permanence of the system. In Section 5, we carry out numerical simulations to illustrate the obtained results. A brief discussion section completes the paper.


## 2. Model Formulation

Our model was built on the framework of [18]. The host population is divided into three stages, nestlings, fledglings, and adults. Each stage the birds are classified into three categories: the susceptible class NS,FS,AS, infectious class NI,FI,AI, and recovered class NR,FR,AR. The total number of nestlings, fledglings, and adults are denoted by NT=NS+NI+NR, FT=FS+FI+FR, and AT=AS+AI+AR, respectively. The mosquito population is divided into three epidemiological classes: the susceptible class MS, exposed class ML, and infectious class MI. The total number of mosquito population is denoted by M, and then M=MS+ML+MI.Susceptible nestlings are produced at ratebt, mosquito bites are distributed among the avian stage classes. The fraction of bites going to nestlings, fledglings, and adults is given by αNt, αFt, and αAt, respectively:(1)αNt=aNtNTtaNtNTt+aFtFTt+aAtATt,αFt=aFtFTtaNtNTt+aFtFTt+aAtATt,αAt=aAtATtaNtNTt+aFtFTt+aAtATt,where aNt, aFt, and aAt are the exposure coefficients for each stage.Susceptible nestlings, fledglings, and adults can move into the infectious nestling, fledgling, and adult class, respectively, upon being bitten by an infectious mosquito. The probability of mosquito to bird transmission per bite are assumed to be stage-dependent and is given byβit, i=N,F,A. The stage-dependent disease-induced death rate is given by νit, i=N,F,A, and each stage recovers at rate γit, i=N,F,A. Nestlings mature into fledglings at rate mNt and fledglings mature into adults at rate mFt. Natural mortality rates are given by μit, i=N,F,A. The probability of bird-mosquito transmission per bite is also assumed to be stage-dependent and is denoted by δit, i=N,F,A. Latent mosquitoes transition to infectious mosquitoes at rate kt. Due to their relatively short life-cycle, we assume that mosquitoes never recover from infection after they are infected, and their infective period ends with their death. We assume all mosquitoes are born into the susceptible class. A density-dependent logistic-type birth rate (due to competition for environmentally limited breeding sites) with intrinsic per capita birth rate rt and carrying capacity Kt is assumed, and the death rate for the mosquito population is denoted by μMt. We assume the biting rate on competent avian hosts is at, then the force of infection at time t for the susceptible nestlings, fledglings, adults, and susceptible mosquitoes are, respectively, given by(2)λ1t≔atαNtβNtMItNStNTt=ataNtβNtMItNStaNtNTt+aFtFTt+aAtATt,λ2t≔atαFtβFtMItFStFTt=ataFtβFtMItFStaNtNTt+aFtFTt+aAtATt,λ3t≔atαAtβAtMItAStATt=ataAtβAtMItAStaNtNTt+aFtFTt+aAtATt,λ4t≔atαNtδNtNItNTt+αFtδFtFItFTt+αAtδAtAItATtMSt=ataNtδNtNIt+aFtδFtFIt+aAtδAtAItaNtNTt+aFtFTt+aAtATtMSt.Consequently, we propose the following model of West Nile virus:(3)dNStdt=bt−λ1t−σNtNSt,dNItdt=λ1t−σNt+ηNtNIt,dNRtdt=γNtNIt−σNtNRt,dFStdt=mNtNSt−λ2t−σFtFSt,dFItdt=λ2t+mNtNIt−σFt+ηFtFIt,dFRtdt=γFtFIt+mNtNRt−σFtFRt,dAStdt=mFtFSt−λ3t−μAtASt,dAItdt=λ3t+mFtFIt−σAtAIt,dARtdt=γAtAIt+mFtFRt−μAtARt,dMStdt=rtMt1−MtKt−λ4t−μMtMSt,dMLtdt=λ4t−ktMLt−μMtMLt,dMItdt=ktMLt−μMtMIt,where(4)σNt=mNt+μNt,σFt=mFt+μFt,σAt=γAt+μAt+νAt,ηNt=γNt+νNt,ηFt=γFt+νFt.Note that if all the parameters in (3) are positive constants except for that bt and μMt are time-dependent, then this system has the same form as equation (3) in [18]. We assume that νit≥0, i=N,F,A, are continuous ω-periodic functions, and all the other parameters are continuous and positive ω-periodic functions for some ω>0. All parameters and their biological interpretations are given in Table 1.Table 1
Parameters of model (3) and their interpretations.
Parameter
Description
b
t
Nestling recruitment rate
a
t
Biting rate on competent avian hosts
a
N
t, aFt, aAt
Exposure coefficients of nestling, fledgling and adult
r
t
Mosquito per capita birth rate
K
t
Mosquito carrying capacity
μ
M
t
Vector mortality rate
1
/
k
t
Virus extrinsic incubation period
m
N
t, mFt
Maturation rates of nestling and fledgling
γ
N
t, γFt, γAt
Recovery rates of nestling, fledgling, and adult
μ
N
t, μFt, μAt
Natural mortality rates of nestling, fledgling, and adult
β
N
t, βFt, βAt
Susceptibility of nestling, fledgling and adult
δ
N
t, δFt, δAt
Infectivity of nestling, fledgling, and adult
ν
N
t, νFt, νAt
Disease-induced death rates of nestling, fledgling, and adultIn order to avoid the extinction of the mosquito population, we make the following assumption throughout this paper:(5)A1rt>μMt,∀t≥0.Since the carrying capacity of the mosquito populationKt is time-dependent, to prove the well-posedness of system (3), we need to impose a condition on this function. Throughout this paper, we require the following condition:(6)dKtdt≥−μMtKt.For the whole mosquito population, we have(7)dMtdt=rtMt1−MtKt−μMtMt.Then,Kt is an upper solution of (7), which means that if M0≤K0, then Mt≤Kt for all t≥0.For continuous positiveω-periodic function gt, let(8)gu=maxt∈0,ωgt,gl=mint∈0,ωgt.Model (3) is mathematically and epidemiologically well posed on the region Dt given by(9)Dt=NS,NI,NR,FS,FI,FR,AS,AI,AR,MS,MI,MR∈ℝ+120<NT+FT+AT≤buμ¯,MS+ML+MI≤Kt,where μ¯=minμNl,μFl,μAl.Theorem 1.
For any initial valuez in Dt, system (3) has a unique nonnegative solution through z for all t≥0, and all solutions are ultimately bounded and uniformly bounded.Proof.
LetGt,z be the vector field described by (3) with z∈Dt. Then, Gt,z is continuous and Lipschitizian in z on each compact subset of ℝ1×Dt. Clearly, Git,z≥0 whenever z≥0 and zi=0,i=1,…,12. It follows from Theorem 5.2.1 in Smith [19] that there exists a unique nonnegative solution for system (3) through z∈Dt in its maximal interval of existence. The total numbers for birds satisfies(10)dNT+FT+ATdt=bt−μNtNT−μFtFT−μAtAT−νNtNI−νFtFI−νAtAI,and it follows that(11)bl−μ^+ν^NT+FT+AT≤dNT+FT+ATdt≤bu−μ¯NT+FT+AT,where μ^=maxμNu,μFu,μAu and ν^=maxνNu,νFu,νAu. Thus, we have(12)0<NTt+FTt+ATt≤buμ¯,for each z∈Dt, t≥0.
By Lemma1 in Teng et al. [20], (7) has a unique positive ω-periodic solution M∗t which is globally asymptotically stable with respect to M0>0, that is,(13)limt⟶∞Mt−M∗t=0,forM0>0.
It follows that the solutions are ultimately bounded. Hence, every solution of (3) exists for all t≥0. Since Mt≤Kt, for all t≥0, and Kt is a positive ω-periodic function, this implies that all solutions are uniformly bounded.
From the proof of Theorem1, we know that any solution of (3) with initial condition in Dt will remain in Dt for all t≥0, then Dt is positively invariant.
We define the diseased classes as the birds or mosquito populations that are either exposed or infectious, i.e.,NI,FI,AI,ML, and MI. To find the disease-free state, letting NI=FI=AI=ML=MI=0, we then obtain(14)dNStdt=bt−σNtNSt,dNRtdt=−σNtNRt,dFStdt=mNtNSt−σFtFSt,dFRtdt=mNtNRt−σFtFRt,dAStdt=mFtFSt−μAtASt,dARtdt=mFtFRt−μAtARt,dMStdt=rtMSt1−MStKt−μMtMSt.
It is easy to see that the ordinary differential systemdNSt/dt=bt−σNtNSt has a globally attractive positive ω-periodic solution NS∗t, that is, limt⟶∞NSt−NS∗t=0. Then, by the third and fifth equations in (14), there exist positive ω-periodic solutions FS∗t and AS∗t such that limt⟶∞FSt−FS∗t=0 and limt⟶∞ASt−AS∗t=0. Hence, there is only one disease-free state, E0t=NS∗t,0,0,FS∗t,0,0,AS∗t,0,0,M∗t,0,0, where M∗t is the positive periodic solution of (7). There exists another trivial periodic solution, E00t=NS∗t,0,0,FS∗t,0,0,AS∗t,0,0,0,0,0.
Letℝn,ℝ+n be the standard ordered n-dimensional Euclidean space with a norm ⋅. For u,v∈ℝn, we denote u≥v, if u−v∈ℝ+n; u>v, if u−v∈ℝ+n\0; u≫v, if u−v∈Intℝ+n.
LetAt be a continuous, cooperative, irreducible, and periodic n×n matrix function with period ω>0 and ΦAt be the fundamental solution matrix of the linear ordinary differential equation:(15)x˙=Atx.
LetrΦAω be the spectral radius of ΦAω. By Perron–Frobenius theorem, rΦAω is the principle eigenvalue of ΦAω, in the sense that it is simple and admits an eigenvector v∗≫0. The following lemma is useful for our discussion in the Section 3.Lemma 1 (see [21], Lemma 1).
Letp=1/ωlnrΦAω. Then, there exists a positive ω-periodic function vt such that eptvt is a solution of (15).Now we introduce the basic reproduction number of (3), by applying the theory in Wang and Zhao [22] with(16)Ft=0000bNtNS∗tΘ0000bFtFS∗tΘ0000bAtAS∗tΘcNtM∗tΘcFtM∗tΘcAtM∗tΘ0000000,Vt=σNt+ηNt0000−μNtσFt+ηFt0000−mFtσAt00000kt+μMt0000−ktμMt,where bxt=ataxtβxt, cxt=ataxtδxt(x=N,F or A), and Θ=aNtNS∗t+aFtFS∗t+aAtAS∗t.LetYt,s be a 5×5 matrix solution of the system:(17)∂∂tYt,s=−VtYt,s,for anyt≥s,Ys,s=I,where I is a 5×5 identity matrix.LetCω be the ordered Banach space of all ω-periodic function from ℝ⟶ℝ5, which is equipped with maximum norm ⋅∞ and the positive cone Cω+=ϕ∈Cω:ϕt≥0,for anyt∈ℝ. Consider the following linear operator L:Cω⟶Cω by(18)Lϕt=∫0+∞Yt,t−aFt−aϕt−ada,for anyt∈ℝ,ϕ∈Cω.Finally, we can define the spectral radius ofL as the basic reproduction number R0 of (3) as follows:(19)R0=rL.From the abovementioned discussion, we obtain the following results for the local asymptotic stability of the disease-free periodic solutionE0t=NS∗t,0,0,FS∗t,0,0,AS∗t,0,0,M∗t,0,0.Lemma 2 (see [22], Theorem 2.2).
The following statements are valid:(i)
R0=1 if and only if rΦF−Vω=1(ii)
R0>1 if and only if rΦF−Vω>1(iii)
R0<1 if and only if rΦF−Vω<1Thus, the disease-free periodic solutionE0t=NS∗t,0,0,FS∗t,0,0,AS∗t,0,0,M∗t,0,0 of (3) is asymptotically stable if R0<1 and unstable if R0>1.LetWt,λ be the monodromy matrix of the following linear ω-periodic system:(20)dWtdt=−Vt+FtλWt,t∈ℝ,with parameter λ∈0,∞. Since Ft is nonnegative and −Vt is cooperative, it follows that rWω,λ is continuous and nonincreasing for λ∈0,∞ and limλ⟶∞rWω,λ<1. Then, we have the following result, which will be used in our numerical calculation of the basic reproduction number in Section 5.Lemma 3 (see [22], Theorem 1).
The following statements are valid:(i)
IfrWω,λ=1 has a positive solution λ0, then λ0 is an eigenvalue of L, and hence R0>0(ii)
IfR0>0, then λ=R0 is the unique solution of rWω,λ=1(iii)
R0=0 if and only if rWω,λ<1 for all λ>0.


## 3. Threshold Dynamics

In this section, we study the global dynamics of system (3) in terms of the basic reproduction number. The following theorem shows that when the disease-induced death rates are equal to zero, and then the disease will be cleared from the populations if R0<1.Theorem 2.
LetA1 hold. If R0<1 and νNt=νFt=νAt=0, then the disease-free periodic solution E0t=NS∗t,0,0,FS∗t,0,0,AS∗t,0,0,M∗t,0,0 is globally asymptotically stable in the sense that for any u0=x0∈Dt with u100+u110+u120>0, and the solution ut,x0 of system (3) through x0 at t=0 satisfies(21)limt⟶∞ut,x0−NS∗t,0,0,FS∗t,0,0,AS∗t,0,0,M∗t,0,0=0.IfR0>1, E0t=NS∗t,0,0,FS∗t,0,0,AS∗t,0,0,M∗t,0,0 is unstable.Proof.
By Lemma2, we know that E0t is unstable if R0>1, and if R0<1, then E0t is locally stable. Hence, it is sufficient to show the global attractivity of E0t for R0<1.
LetMεt=Fεt−Vt, with(22)Fɛt=0000bNtNS∗t+εΘε0000bFtFS∗t+εΘε0000bAtAS∗t+εΘεcNtM∗t+εΘεcFtM∗t+εΘεcAtM∗t+εΘε0000000,where Θε=aNtNS∗t−ε+aFtFS∗t−ε+aAtAS∗t−ε. By Lemma 2, if R0<1, we have rΦF−Vω<1, and we can restrict ϵ>0 small enough such that rΦMεω<1.
SinceνNt=νFt=νAt=0, we then get the following equations:(23)dNTtdt=bt−σNtNTt,dFTtdt=mNtNTt−σFtFTt,dATtdt=mFtFTt−μAtATt.
It is easy to see that system (23) has a globally attractive ω-periodic solution NS∗t,FS∗t,AS∗t, that is, limt⟶∞NTt−NS∗t=0, limt⟶∞FTt−FS∗t=0, and limt⟶∞ATt−AS∗t=0. Therefore, there exists an ε>0 and a T1>0 such that, for any t≥T1, we have(24)NS∗t−ε<NTt<NS∗t+ε,FS∗t−ε<FTt<FS∗t+ε,AS∗t−ε<ATt<AS∗t+ε.
Thus, whent≥T1, we have(25)dNItdt≤bNtNS∗t+εΘεMIt−σNt+ηNtNIt,dFItdt≤bFtFS∗t+εΘεMIt+mNtNIt−σFt+ηFtFIt,dAItdt≤bAtAS∗t+εΘεMIt+mFtFIt−σAtAIt,dMLtdt≤cNtNIt+cFtFIt+cAtAItM∗t+εΘε,dMItdt=ktMLt−μMtMIt.
By Lemma1, there exists a positive ω-periodic function vt such that ut=eptvt is a solution of u′t=Mεtut, where p=1/ωlnrΦMεω<0. We see that ut⟶0 as t⟶∞. Using the standard comparison principle, we have(26)limt⟶∞NIt,FIt,AIt,MLt,MIt=0,0,0,0,0.
It then follows from the theory of asymptotically periodic semiflow (see [23], Theorem 3.2.1]) that(27)limt⟶∞NRt,FRt,ARt=0,0,0,limt⟶∞NSt−NS∗t,FSt−FS∗t,ASt−AS∗t,MSt−M∗t=0,0,0,0.
This completes the proof.
The following result shows the uniform persistence of the disease ifR0>1.Theorem 3.
LetA1 hold. If R0>1, then system (3) admits at least one positive periodic solution, and there exists a positive constant η>0 such that any solution NSt,NIt,NRt,FSt,FIt,FRt,ASt,AIt,ARt,MSt,MLt,MIt of (3) in Dt with NI0>0, FI0>0, AI0>0, ML0>0, and MI0>0 satisfies(28)liminft⟶∞NIt,FIt,AIt,MLt,MIt≥η,η,η,η,η.Proof.
Define(29)X≔ℝ+12,X0≔NS,NI,NR,FS,FI,FR,AS,AI,AR,MS,ML,MI∈X:NI>0,FI>0,AI>0,ML>0,MI>0,∂X0≔X\X0.
LetP be the Poincare′ map associated with (3), i.e.,(30)Px0=uω,x0,forx0∈ℝ+12,where ut,x0 is the unique solution of (3) with u0,x0=x0.
It then suffices to show that (3) is uniformly persistent with respect to X0,∂X0. Note that both X and X0 are positively invariant. Further, Theorem 1 means that (3) is point dissipative. It is apparent that ∂X0 is relatively closed in X.
Setx0=NS0,NI0,NR0,FS0,FI0,FR0,AS0,AI0,AR0,MS0,ML0,MI0∈X0. Let ut,x0=NSt,x0,NIt,x0,NRt,x0,FSt,x0,FIt,x0,FRt,x0,ASt,x0,AIt,x0,ARt,x0,MSt,x0,MLt,x0,MIt,x0 be the solution of (3) through x0 at t=0. Let M1=NS0∗,0,0,FS0∗,0,0,AS0∗,0,0,MS0∗,0,0, and M2=NS0∗,0,0,FS0∗,0,0,AS0∗,0,0,0,0,0, where NS0∗0=NS∗0, FS0∗0=FS∗0, AS0∗0=AS∗0, and MS0∗0=MS∗0.
Sincelimx0⟶M1ut,x0−ut,M1=0 uniformly for t∈0,ω, then for any ε>0, there exists δ1>0 such that.(31)ut,x0−ut,M1≤ε,∀t∈0,ω,x0−M1≤δ1.
Then, we have the following claim.Claim 1.
lim
sup
m
⟶
∞
d
P
m
x
0
,
M
1
≥
δ
1
,
for all
x
0
∈
X
0
.
If not, then(32)limsupm⟶∞dPmx0,M1<δ1,for some x0∈X0. Without loss of generality, we can assume that(33)dPmx0,M1<δ1,for all m>0. Then, we know that(34)ut,Pmx0−ut,M1≤ε,∀t∈0,ω,for anym>0.
For anyt≥0, let t=mω+t1, where t1∈0,ω and m=t/ω, which is the greatest integer less than or equal to t/ω. Then, we have(35)ut,x0−ut,M1=ut1,Pmx0−ut1,M1≤ε,∀t≥0.
It follows thatNS∗t−ε≤NSt≤NS∗t+ε,0<NIt≤ε,0≤NRt≤ε, FS∗t−ε≤FSt≤FS∗t+ε,0<FIt≤ε,0≤FRt≤ε, AS∗t−ε≤ASt≤AS∗t+ε,0<AIt≤ε,0≤ARt≤ε, and M∗t−ε≤MSt≤M∗t+ε,0<MLt≤ε,0<MIt≤ε, and this, together with (3), implies(36)J′t≥Aεt−VtJt,where Jt=NIt,FIt,AIt,MLt,MItT and(37)Aεt=0000bNtNS∗t−εΔε0000bFtFS∗t−εΔε0000bAtAS∗t−εΔεcNtM∗t−εΔεcFtM∗t−εΔεcAtM∗t−εΔε0000000,where Δε=aNtNS∗t+3ε+aFtFS∗t+3ε+aAtAS∗t+3ε. By Lemma 2, we know that rΦF−Vω>1, then we can choose ε>0 small enough, such that rΦAε−Vω>1. Again by Lemma 1 and the standard comparison principle, there exists a positive ω-periodic function v˜t such that Jt≥eμtv˜t, where μ=1/ωlnrΦAε−Vω>0, which implies that NIt⟶∞,FIt⟶∞,AIt⟶∞,MLt⟶∞,MIt⟶∞, which leads to a contradiction.
SinceM∗t is a positive periodic solution, we can choose a small positive number δ2 such that(38)3δ2<inft≥0M∗t.
Sincelimx0⟶M2ut,x0−M2=0 uniformly for t∈0,ω, there exists η1>0 such that(39)ut,x0−M2≤δ2,for allt∈0,ω,x0−M2≤η1.Claim 2.
lim
sup
n
⟶
∞
d
P
n
x
0
,
M
2
≥
η
1
,
for all
x
0
∈
X
0
.
Suppose, by contradiction, thatlimsupn⟶∞dPnx0,M2<η1,for somex0∈X0, without loss of generality, we can assume that(40)dPnx0,M2<η1,for alln>0.
For anyt≥0, let t=nω+t2, where t2∈0,ω and n=t/ω. Then, we have(41)ut,x0−M2=ut2,Pnx0−M2≤δ2,∀t≥0.
Hence,MSt≤δ2, MLt≤δ2, MIt≤δ2, and Mt≤3δ2 when t≥0. Since M0=MS0+ML0+MI0>0, we have limt⟶∞Mt−M∗t=0, a contradiction.
Then, we define(42)M∂≔N˜S,N˜I,N˜R,F˜S,F˜I,F˜R,A˜S,A˜I,A˜R,M˜S,M˜L,M˜I∈∂X0:PmN˜S,N˜I,N˜R,F˜S,F˜I,F˜R,A˜S,A˜I,A˜R,M˜S,M˜L,M˜I∈∂X0,∀m≥0,D1≔u∈X:ui0,x0=0,∀i∈2,5,8,11,12,D2≔u∈X:ui0,x0=0,∀i=10,11,12.
We claim thatM∂=D1∪D2. We first prove that M∂D1∪D2. For any ut,x0∈D2, it is easy to see that uit,x0=0, i=10,11,12. Hence, D2⊂M∂. For any ut,x0∈D1, we can define V˜t≥0 such that V˜it≡0 for all t≥0 and i=2,5,8,11,12. Let V˜it i=1,3,4,6,7,9,10 satisfy the following equations:(43)dV˜1tdt=bt−σNtV˜1t,dV˜3tdt=−σNtV˜3t,dV˜4tdt=mNtV˜1t−σFtV˜4t,dV˜6tdt=mNtV˜3t−σFtV˜6t,dV˜7tdt=mFtV˜4t−μAtV˜7t,dV˜9tdt=mFtV˜6t−μAtV˜9t,dV˜10tdt=rtV˜10t1−V˜10tKt−μMtV˜10t,with V˜i0=ui0,x0, i=1,3,4,6,7,9,10. Then, V˜t is a solution of (3) through x0. By the uniqueness of the solution, we have ut,x0=V˜t for all t≥0, and hence D1⊂M∂. To prove the claim, it then suffices to show that M∂⊂D1∪D2. For any ut,x0∈∂X0\D1∪D2, we have ∑i=1012ui0,x0>0, and hence limt⟶∞∑i=1012uit,x0−M∗t=0. From the first, fourth, seventh, and tenth of (3), we have u1t,x0>0, u4t,x0>0, u7t,x0>0, and u10t,x0>0 for all t>0. If u20,x0>0, by (3), we get ujt,x0>0 for all t>0 and j∈2,3,5,6,8,9,11,12. If u50,x0>0, by the fifth and sixth equations of (3), we have u5t,x0>0 and u6t,x0>0 for all t>0. Then, by the eighth and ninth equation of (3), we have u8t,x0>0 and u9t,x0>0 for all t>0. Then, we have u11t,x0>0 and u12t,x0>0 and u2t,x0>0 and u3t,x0>0 for all t>0. For the case u80,x0>0 or u110,x0>0 or u120,x0>0, we can similarly prove that uit,x0>0 for i∈1,2,…,12. Therefore, we have ut,x0∈X0 for all t>0. This implies that for any x0∈∂X0\D1∪D2, we have Put,x0∉∂X0 for all t>0, and hence M∂⊂D1∪D2. It then follows that M1 and M2 are disjoint, compact, and isolated invariant sets for P in M∂, and ∪u∈M∂ωu=M1,M2. Furthermore, no subset of M1andM2 forms a cycle in M∂ (and hence in ∂X0). In view of the two claims above, we see that M1 and M2 are isolated invariant sets for P in X, and WsMi∩X0=∅ for all i=1,2, where WsMi is the stable set of Mi for P.
Applying the acyclicity theory on uniform persistence for maps (see [23], Theorem 1.3.1 and Remark 1.3.1), we see that P:X⟶X is uniformly persistent with respect to X0,∂X0. It follows from [23], Theorem 3.1.1, that the solution of (3) is uniformly persistent.
Furthermore, by [23], Theorem 1.3.6, P has a fixed point N¯S0,N¯I0,N¯R0,F¯S0,F¯I0,F¯R0,A¯S0,A¯I0,A¯R0,M¯S0,M¯L0,M¯I0∈X0. Then, we see that N¯S0,N¯R0,F¯S0,F¯R0,A¯S0,A¯R0,M¯S0 are all nonnegative, and N¯I0,F¯I0,A¯I0,M¯L0,andM¯I0 are all positive. We further prove that N¯S0,N¯R0,F¯S0,F¯R0,A¯S0,A¯R0,andM¯S0 are all positive. Suppose not, if N¯S0=0, then from the first equation of (3), we obtain(44)dN¯tdt=bt−a1tN¯t,with N¯S0=N¯Snω=0, n=1,2,3,…, where a1t=bNtM¯It/aNtN¯Tt+aFtF¯Tt+aAtA¯Tt+σNt. Then, we obtain(45)N¯St=N¯S0+∫0tbρe∫0ρa1sdsdρe∫0ta1sds,a contradiction. Thus N¯S0>0. Similarly, we can prove that N¯R0>0, F¯S0>0, F¯R0>0, A¯S0>0, A¯R0>0, and M¯S0>0. Thus, ut,N¯S0,N¯I0,N¯R0,F¯S0,F¯I0,F¯R0,A¯S0,A¯I0,A¯R0,M¯S0,M¯L0,M¯I0∈Intℝ+12 and N¯St,N¯It,N¯Rt,F¯St,F¯It,F¯Rt,A¯St,A¯It,A¯Rt,M¯St,M¯Lt,M¯It is a positive ω-periodic solution of (3) due to the definition of semiflow P. Hence, the proof is complete.


## 4. Autonomous Case of System (3)

In this section, we study the corresponding autonomous system of system (3), that is, all the coefficients of system (3) are constants, and also we assume that the disease-induced death rates of infected birds are zero. Based on the references and also mathematical tractability, we also make the following assumptions: aN=aF=aA≔a˜, βN=βF=βA≔β [4, 24, 25] δN=δF=δA≔δ, mN=mF≔m, γN=γF=γA≔γ, and μN=μF=μA≔μ. In this case, system (3) becomes(46)dNStdt=b−λ1t−m+μNSt,dNItdt=λ1t−m+μ+γNIt,dNRtdt=γNIt−m+μNRt,dFStdt=mNSt−λ2t−m+μFSt,dFItdt=λ2t+mNIt−m+μ+γFIt,dFRtdt=γFIt+mNRt−m+μFRt,dAStdt=mFSt−λ3t−μASt,dAItdt=λ3t+mFIt−μ+γAIt,dARtdt=γAIt+mFR−μARt,dMStdt=rMt1−MtK−λ4t−μMMSt,dMLtdt=λ4t−kMLt−μMMLt,dMItdt=kMLt−μMMIt,with λ1t=aβMItNSt/NTt+FTt+ATt,λ2t=aβMItFSt/NTt+FTt+ATt,λ3t=aβMItASt/NTt+FTt+ATt,λ4t=aδNIt+FIt+AItMSt/NTt+FTt+ATt.It is easy to show that(47)Ω=NS,NI,NR,FS,FI,FR,AS,AI,AR,MS,MI,MR∈ℝ+120<NT+FT+AT≤bμ,MS+ML+MI≤K,is the positive invariant set for system (46).Corresponding toA1, we also need the following assumption:(48)A2r>μM.It is clear that system (46) always has two equilibria P0=NT∗,0,0,FT∗,0,0,AT∗,0,0,0,0,0 and P1=NT∗,0,0,FT∗,0,0,AT∗,0,0,M∗,0,0, where NT∗=b/m+μ, FT∗=bm/m+μ2, AT∗=bm2/μm+μ2, and M∗=Kr−μM/r.In order to find any endemic equilibrium of system (46), we introduce the basic reproduction number R0. According to the concepts of the next generation matrix and reproduction number presented in [26, 27], R0 is given by the following expression:(49)R0=ka2βμδKr−μMrbμMk+μMμ+γ.Let the right-hand sides be zero, then system (46) admits another equilibrium: P∗=NS∗,NI∗,NR∗,FS∗,FI∗,FR∗,AS∗,AI∗,AR∗,MS∗,MI∗,MR∗, where(50)NS∗=b2aβμMI∗+bm+μ,NI∗=abβμMI∗aβμMI∗+bm+μm+μ+γ,NR∗=γNI∗m+μ,FS∗=mbNS∗aβμMI∗+bm+μ,FI∗=aβμMI∗FS∗+bmNI∗bm+μ+γ,FR∗=γFI∗+mNR∗m+μ,AS∗=mbFS∗μaβMI∗+b,AI∗=aβμMI∗AS∗+bmFE∗bμ+γ,AR∗=γAI∗+mFR∗μ,MS∗=μMKr−μMraδ+μM,MR∗=μMMI∗k,and MI∗ satisfies equation f1MI∗=f2MI∗, where(51)f1MI∗=bmaβμMI∗+bm+μ+b2m2aβμMI∗+bm+μaβμMI∗+bμ+1,f2MI∗=μMk+μMμ+γaβμMI∗+bm+μβδa2μ2kM∗−k+μMMI∗.Note thatf10=m+μ/μ, f1x is decreasing with respect to x>0 and limx⟶+∞f1x=1. Also, f20=μMk+μMμ+γbm+μ/kβδa2μ2M∗ and f2′x=μMk+μMμ+γaβμkM∗+bm+μk+μM/βδa2μ2kM∗−k+μMx2>0 for MI∗∈0,kM∗/k+μM, and limx⟶kM∗/k+μM=f2x=+∞. Then, f1x=f2x has a unique positive root in 0,kM∗/k+μM, if and only if f10>f20, which is equivalent to R0>1. Therefore, if R0>1, system (46) has a unique endemic equilibrium P∗.It is easy to show thatP0 is a saddle and is unstable. By Theorem 2, we know that P1 is globally asymptotically stable in Ω\P0 if R0<1 and is unstable for R0>1. Also, by Theorem 3, if R0>1, there exists a positive constant η1>0 such that any solution NSt,NIt,NRt,FSt,FIt,FRt,MSt,MLt,MIt of (46) with NI0>0, FI0>0, AI0>0, ML0>0, and MI0>0 satisfies(52)liminft⟶∞NIt,FIt,AIt,MLt,MIt≥η1,η1,η1,η1,η1.


## 5. Numerical Simulations

In this section, we will perform some numerical simulations to illustrate the results of previous sections and explore the effects of some parameters on the WNV transmission. First from [18], we can determine some realistically feasible constant parameters. We assume that the disease-induced death rates of nestling, fledgling, and adult are zero. The time unit is taken as month. Baseline parameters are listed in Table 2.Table 2
Parameters of model (3) and their interpretations.
Parameter
Value
Reference
b
Varies
a
N, aF, aA
Varies
K
Varies
1
/
k
0.106
×
30.4
[5]
m
N, mF
30.4
/
14
[24]
γ
N, γF, γA
30.4
/
3
[4]
μ
N, μF, μA
0.0014
×
30.4
[25]
β
N, βF, βA
1
[4]
δ
N, δF, δA
0.36
[4]


### 5.1. Long Term Behavior

We takeat=30.40.133+0.1sinπt/6, rt=30.40.537+0.5sinπt/6, μMt=30.40.096+0.09sinπt/6, b=3040, K=5×105, aN=5, and aF=aA=1, which are chosen or adapted from Robertson and Cailloue¨t [18]. Using Lemma 3, we can numerically compute the basic reproduction number R0. With this set of parameters, we have R0=0.8644<1, and the infection is extinct in host and mosquito populations. If the nestling exposure coefficient increases to aN=10, then R0=1.0832>1, and then the infection is persistent in host and mosquito populations. Figure 1 shows the long-term behavior of the infectious birds and mosquitoes when R0<1 and R0>1. These simulations are consistent with our theoretical results in Section 3.Figure 1
Long-term behavior of the infectious host and vector populations. (a)R0<1 and (b) R0>1.
(a)
(b)


### 5.2. Sensitivity Analysis ofR0

In order to provide some effective control measures about mosquitoes, in this section, we will perform some sensitivity analyses of the basic reproduction numberR0 in terms of some coefficients in our model (3).We first discuss four important factors to control the disease: the mosquito carrying capacityK; the biting rate of mosquitoes on birds at; the death rate of mosquitoes μMt; and the recruitment rate of mosquitoes rt. Obviously the biting rate of mosquitoes on birds can be reduced if the population size of mosquitoes decreases. Let aN=aF=aA=1, by replacing at, μMt, and rt with a⌣t=1−cat, μ⌣Mt=lμMt, and r⌣t=1−frt, respectively, and keeping the other parameter values the same as those in Figure 1. Figure 2 reflects the relationship between R0 and these parameters. Our numerical analysis shows that R0 is a decreasing function of c,l,andf and an increasing function of K; to reduce the basic reproduction number below 1, we should at least decrease the recruitment rate of mosquitoes by 32% or increase the death rate of mosquitoes by 50% from the current level. Figure 2(b) shows that we should keep c>0.273 to control WNV transmission. Hence, we may take some vector control measures to control WNV, such as using products to kill mosquito larvae and adult mosquitoes, and we can also reduce mosquito breeding sites by eliminating standing pools of water, such as in old tires, buckets, gutters, and swimming pools.Figure 2
R
0 vs K, c, l, and f. (a) Relationship between R0 and K. (b) Relationship between R0 and c. (c) Relationship between R0 and l. (d) Relationship between R0 and f. In (b, c, and d), we fix K=1.5×106.
(a)
(b)
(c)
(d)Secondly, we explore the effect of host stage exposure heterogeneity. We take parameter values as those in Figure1. Figure 3 shows that the larger the exposure coefficients, the larger the basic reproduction number R0 becomes. Moreover, this figure shows that juvenile (nestling or fledgling) exposure has a larger impact on the WNV transmission than that of the adult.Figure 3
Effect of host stage exposure on the basic reproduction numberR0.
(a)
(b)
(c)Thirdly, Figure4 compares R0 of the periodic system and the autonomous system. For the periodic system, the parameters are the same as those in Figure 2(b); for the autonomous system, the parameters are the average values of those in the periodic system. We can see that the basic reproduction number of the autonomous one is always less than that of the periodic one when c varies ranging from 0 to 1. This implies that, without seasonality, the risk of infection will be underestimated. Therefore, considering the influence of temperature, we should make more efforts to control the spread of WNV.Figure 4
The curves of the basic reproduction numbers versusc.


## 5.1. Long Term Behavior

We takeat=30.40.133+0.1sinπt/6, rt=30.40.537+0.5sinπt/6, μMt=30.40.096+0.09sinπt/6, b=3040, K=5×105, aN=5, and aF=aA=1, which are chosen or adapted from Robertson and Cailloue¨t [18]. Using Lemma 3, we can numerically compute the basic reproduction number R0. With this set of parameters, we have R0=0.8644<1, and the infection is extinct in host and mosquito populations. If the nestling exposure coefficient increases to aN=10, then R0=1.0832>1, and then the infection is persistent in host and mosquito populations. Figure 1 shows the long-term behavior of the infectious birds and mosquitoes when R0<1 and R0>1. These simulations are consistent with our theoretical results in Section 3.Figure 1
Long-term behavior of the infectious host and vector populations. (a)R0<1 and (b) R0>1.
(a)
(b)


## 5.2. Sensitivity Analysis ofR0

In order to provide some effective control measures about mosquitoes, in this section, we will perform some sensitivity analyses of the basic reproduction numberR0 in terms of some coefficients in our model (3).We first discuss four important factors to control the disease: the mosquito carrying capacityK; the biting rate of mosquitoes on birds at; the death rate of mosquitoes μMt; and the recruitment rate of mosquitoes rt. Obviously the biting rate of mosquitoes on birds can be reduced if the population size of mosquitoes decreases. Let aN=aF=aA=1, by replacing at, μMt, and rt with a⌣t=1−cat, μ⌣Mt=lμMt, and r⌣t=1−frt, respectively, and keeping the other parameter values the same as those in Figure 1. Figure 2 reflects the relationship between R0 and these parameters. Our numerical analysis shows that R0 is a decreasing function of c,l,andf and an increasing function of K; to reduce the basic reproduction number below 1, we should at least decrease the recruitment rate of mosquitoes by 32% or increase the death rate of mosquitoes by 50% from the current level. Figure 2(b) shows that we should keep c>0.273 to control WNV transmission. Hence, we may take some vector control measures to control WNV, such as using products to kill mosquito larvae and adult mosquitoes, and we can also reduce mosquito breeding sites by eliminating standing pools of water, such as in old tires, buckets, gutters, and swimming pools.Figure 2
R
0 vs K, c, l, and f. (a) Relationship between R0 and K. (b) Relationship between R0 and c. (c) Relationship between R0 and l. (d) Relationship between R0 and f. In (b, c, and d), we fix K=1.5×106.
(a)
(b)
(c)
(d)Secondly, we explore the effect of host stage exposure heterogeneity. We take parameter values as those in Figure1. Figure 3 shows that the larger the exposure coefficients, the larger the basic reproduction number R0 becomes. Moreover, this figure shows that juvenile (nestling or fledgling) exposure has a larger impact on the WNV transmission than that of the adult.Figure 3
Effect of host stage exposure on the basic reproduction numberR0.
(a)
(b)
(c)Thirdly, Figure4 compares R0 of the periodic system and the autonomous system. For the periodic system, the parameters are the same as those in Figure 2(b); for the autonomous system, the parameters are the average values of those in the periodic system. We can see that the basic reproduction number of the autonomous one is always less than that of the periodic one when c varies ranging from 0 to 1. This implies that, without seasonality, the risk of infection will be underestimated. Therefore, considering the influence of temperature, we should make more efforts to control the spread of WNV.Figure 4
The curves of the basic reproduction numbers versusc.


## 6. Discussion

In this paper, we have analyzed a mathematical WNV model, which was presented in Robertson and Cailloue¨t [18]. The model incorporates the stage structure of avian, host stage exposure heterogeneity, and the seasonal climate effects on WNV transmission. We have obtained the basic reproduction number and showed that the disease-free periodic state is globally asymptotically stable if the disease-induced death rates are zero and R0<1, and the disease will persist if R0>1. For the corresponding autonomous system, for a special case, we establish a threshold result in terms of the basic reproduction number R0.The numerical simulations about the long-term behavior of the model solutions are consistent with the obtained analytic result. In order to seek for effective control measures to prevent outbreaks of WNV, we performed sensitive analysis of the basic reproduction number in terms of the mosquito carrying capacity, the recruitment rate of mosquitoes, the death rate of mosquitoes, and the biting rate of mosquitoes on birds. It is clear that if the mosquito population size decreases, then the biting rate of mosquitoes on birds can be reduced. Hence, the key to control the spread of WNV is to increase the death rate and decrease the recruitment rate of mosquitoes. To do this, we can reduce mosquito breeding sites, kill adult mosquitoes by adulticides, and kill mosquito larvae by larvicides. Furthermore, we compare the effect of host stage exposure heterogeneity, and Figure3 shows that increased exposure of all stages increases transmission. Moreover, this figure shows that the juvenile stage exposure has a larger impact on WNV transmission than that of the adult stage. Our numerical simulations also show that the risk of the disease will be underestimated if we do not consider the seasonality. Thus, it is necessary to include the seasonality and stage-dependent host differences in mosquito exposure in the models.



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