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

In this paper, we study an avian (host) stage-structured West Nile virus model, which incorporates seasonality as well as stagespecific mosquito biting rates. We first introduce the basic reproduction number R0 for this model and then show that the diseasefree 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.


Introduction
West Nile virus (WNV) is a flavivirus commonly found in Africa, West Asia, and the Middle East. e 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][2][3][4]. Since the first outbreak in New York in 1999, West Nile virus remains an annual public health concern. erefore, 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.
ough mathematical modelling for the transmission of WNV has been studied extensively, the effect of withinspecies 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. e fledglings that have left the nest have nearly complete feather coverage and exhibit some antimosquito defensive behaviors. e 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 Caillou€ et [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. ey 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. e rest of the paper is organized as follows. In Section 2, we present the model rigorously and study its wellposedness, and then we introduce the basic reproduction number R 0 . In Section 3, we study the threshold dynamics in terms of R 0 . 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.

Model Formulation
Our model was built on the framework of [18]. e host population is divided into three stages, nestlings, fledglings, and adults. Each stage the birds are classified into three categories: the susceptible class (N S , F S , A S ), infectious class (N I , F I , A I ), and recovered class (N R , F R , A R ). e total number of nestlings, fledglings, and adults are denoted by N T � N S + N I + N R , F T � F S + F I + F R , and A T � A S + A I + A R , respectively. e mosquito population is divided into three epidemiological classes: the susceptible class M S , exposed class M L , and infectious class M I . e total number of mosquito population is denoted by M, and then M � M S + M L + M I .
Susceptible nestlings are produced at rate b(t), mosquito bites are distributed among the avian stage classes. e fraction of bites going to nestlings, fledglings, and adults is given by α N (t), α F (t), and α A (t), respectively: , where a N (t), a F (t), and a A (t) 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. e probability of mosquito to bird transmission per bite are assumed to be stage-dependent and is given by β i (t), i � N, F, A. e stage-dependent disease-induced death rate is given by ] i (t), i � N, F, A, and each stage recovers at rate c i (t), i � N, F, A. Nestlings mature into fledglings at rate m N (t) and fledglings mature into adults at rate m F (t). Natural mortality rates are given by μ i (t), i � N, F, A. e probability of bird-mosquito transmission per bite is also assumed to be stage-dependent and is denoted by δ i (t), i � N, F, A. Latent mosquitoes transition to infectious mosquitoes at rate k(t). Due to their relatively short lifecycle, 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 r(t) and carrying capacity K(t) is assumed, and the death rate for the mosquito population is denoted by μ M (t). We assume the biting rate on competent avian hosts is a(t), then the force of infection at time t for the susceptible nestlings, fledglings, adults, and susceptible mosquitoes are, respectively, given by , , , 2 Complexity Consequently, we propose the following model of West Nile virus: Note that if all the parameters in (3) are positive constants except for that b(t) and μ M (t) are time-dependent, then this system has the same form as equation (3) in [18].
We assume that ] i (t) ≥ 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.
In order to avoid the extinction of the mosquito population, we make the following assumption throughout this paper: Since the carrying capacity of the mosquito population K(t) is time-dependent, to prove the well-posedness of system (3), we need to impose a condition on this function. roughout this paper, we require the following condition: For the whole mosquito population, we have en, K(t) is an upper solution of (7), which means that if For continuous positive ω-periodic function g(t), let Model (3) is mathematically and epidemiologically well posed on the region D(t) given by where μ � min μ l N , μ l F , μ l A .

Theorem 1.
For any initial value z in D(t), system (3) has a unique nonnegative solution through z for all t ≥ 0, and all solutions are ultimately bounded and uniformly bounded.
Proof. Let G(t, z) be the vector field described by (3) with z ∈ D(t). en, G(t, z) is continuous and Lipschitizian in z on each compact subset of R 1 × D(t). Clearly, G i (t, z) ≥ 0 whenever z ≥ 0 and z i � 0, i � 1, . . . , 12. It follows from eorem 5.2.1 in Smith [19] that there exists a unique nonnegative solution for system (3) through z ∈ D(t) in its maximal interval of existence. e total numbers for birds satisfies and it follows that for each z ∈ D(t), t ≥ 0. By Lemma 1 in Teng et al. [20], (7) has a unique positive ω-periodic solution M * (t) which is globally asymptotically stable with respect to M(0) > 0, that is, It follows that the solutions are ultimately bounded. Hence, every solution of (3) exists for all t ≥ 0. Since M(t) ≤ K(t), for all t ≥ 0, and K(t) is a positive ω-periodic function, this implies that all solutions are uniformly bounded.
From the proof of eorem 1, we know that any solution of (3) with initial condition in D(t) will remain in D(t) for all t ≥ 0, then D(t) is positively invariant.
We define the diseased classes as the birds or mosquito populations that are either exposed or infectious, i.e., N I , F I , A I , M L , and M I . To find the disease-free state, letting It is easy to see that the ordinary differential system en, by the third and fifth equations in (14), there exist positive ω-periodic solutions F * S (t) and is the positive periodic solution of (7). ere exists another trivial periodic solution, . Let A(t) be a continuous, cooperative, irreducible, and periodic n × n matrix function with period ω > 0 and Φ A (t) be the fundamental solution matrix of the linear ordinary differential equation: Let r(Φ A (ω)) be the spectral radius of Φ A (ω). By Perron-Frobenius theorem, r(Φ A (ω)) is the principle eigenvalue of Φ A (ω), in the sense that it is simple and admits an eigenvector v * ≫ 0. e following lemma is useful for our discussion in the Section 3.
en, there exists a positive ω-periodic function v(t) such that e pt v(t) is a solution of (15).
Nestling recruitment rate a(t) Biting rate on competent avian hosts Exposure coefficients of nestling, fledgling and adult r(t) Mosquito per capita birth rate Maturation rates of nestling and fledgling Recovery rates of nestling, fledgling, and adult Natural mortality rates of nestling, fledgling, and adult Susceptibility of nestling, fledgling and adult Infectivity of nestling, fledgling, and adult Disease-induced death rates of nestling, fledgling, and adult 4 Complexity Now we introduce the basic reproduction number of (3), by applying the theory in Wang and Zhao [22] with where where I is a 5×5 identity matrix. Let C ω be the ordered Banach space of all ω-periodic function from R ⟶ R 5 , which is equipped with maximum norm ‖·‖ ∞ and the positive cone C + ω � ϕ ∈ C ω : ϕ(t) ≥ 0, for any t ∈ R}. Consider the following linear operator L: Finally, we can define the spectral radius of L as the basic reproduction number R 0 of (3) as follows: From the abovementioned discussion, we obtain the following results for the local asymptotic stability of the disease-free periodic solution Lemma 2 (see [22], eorem 2.2). e following statements are valid: Let W(t, λ) be the monodromy matrix of the following linear ω-periodic system: with parameter λ ∈ (0, ∞). Since F(t) is nonnegative and − V(t) is cooperative, it follows that r(W(ω, λ)) is continuous and nonincreasing for λ ∈ (0, ∞) and lim λ⟶∞ r(W(ω, λ)) < 1. en, we have the following result, which will be used in our numerical calculation of the basic reproduction number in Section 5.

Threshold Dynamics
In this section, we study the global dynamics of system (3) in terms of the basic reproduction number. e 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 R 0 < 1.
Since ] N (t) � ] F (t) � ] A (t) � 0, we then get the following equations: It is easy to see that system (23) has a globally attractive ) � 0, and lim t⟶∞ (A T (t) − A * S (t)) � 0. erefore, there exists an ε > 0 and a T 1 > 0 such that, for any t ≥ T 1 , we have us, when t ≥ T 1 , we have 6 Complexity

(26)
It then follows from the theory of asymptotically periodic semiflow (see [23], eorem 3.2.1]) that is completes the proof. e following result shows the uniform persistence of the disease if R 0 > 1.
Let P be the Poincare ′ map associated with (3), i.e., where u(t, x 0 ) is the unique solution of (3) with u(0, x 0 ) � x 0 . It then suffices to show that (3) is uniformly persistent with respect to (X 0 , zX 0 ). Note that both X and X 0 are positively invariant. Further, eorem 1 means that (3) is point dissipative. It is apparent that zX 0 is relatively closed in X.
en, we have the following claim.

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: [4,24,25] with λ 1 (t) � (aβM I (t)N S (t))/(N T (t) + F T (t) + A T (t)), .
It is easy to show that P 0 is a saddle and is unstable. By eorem 2, we know that P 1 is globally asymptotically stable in Ω\ P 0 if R 0 < 1 and is unstable for R 0 > 1. Also, by eorem 3, if R 0 > 1, there exists a positive constant η 1 > 0 such that any solution (N S (t), (52)

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. e time unit is taken as month. Baseline parameters are listed in Table 2.

Long Term Behavior.
We take a(t) � 30.4(0.133 + 0.1 sin πt/6), r(t) � 30.4(0.537 + 0.5 sin πt/6), μ M (t) � 30.4(0.096 + 0.09 sin πt/6), b � 3040, K � 5 × 10 5 , a N � 5, and a F � a A � 1, which are chosen or adapted from Robertson and Caillou€ et [18]. Using Lemma 3, we can numerically compute the basic reproduction number R 0 . With this set of parameters, we have R 0 � 0.8644 < 1, and the infection is extinct in host and mosquito populations. If the nestling exposure coefficient increases to a N � 10, then R 0 � 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 R 0 < 1 and R 0 > 1. ese simulations are consistent with our theoretical results in Section 3.

Sensitivity
Analysis of R 0 . In order to provide some effective control measures about mosquitoes, in this section, we will perform some sensitivity analyses of the basic reproduction number R 0 in terms of some coefficients in our model (3).
We first discuss four important factors to control the disease: the mosquito carrying capacity K; the biting rate of mosquitoes on birds a(t); the death rate of mosquitoes μ M (t); and the recruitment rate of mosquitoes r(t). Obviously the biting rate of mosquitoes on birds can be reduced if the population size of mosquitoes decreases. Let a N � a F � a A � 1, by replacing a(t), μ M (t), and r(t) with a respectively, and keeping the other parameter values the same as those in Figure 1. Figure 2 reflects the relationship between R 0 and these parameters. Our numerical analysis shows that R 0 is a decreasing function of c, l, and f 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. Secondly, we explore the effect of host stage exposure heterogeneity. We take parameter values as those in Figure 1. Figure 3 shows that the larger the exposure coefficients, the larger the basic reproduction number R 0 becomes. Moreover, this figure shows that juvenile (nestling or fledgling) exposure has a larger impact on the WNV transmission than that of the adult.
irdly, Figure 4 compares R 0 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. is implies that, without seasonality, the risk of infection will be underestimated. erefore, considering the influence of temperature, we should make more efforts to control the spread of WNV.

Discussion
In this paper, we have analyzed a mathematical WNV model, which was presented in Robertson and Caillou€ et [18]. e 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 R 0 < 1, and the disease will persist if R 0 > 1. For the corresponding autonomous system, for a special case, we establish a threshold result in terms of the basic reproduction number R 0 . e 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 Figure 3 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.
us, it is necessary to include the seasonality and stage-dependent host differences in mosquito exposure in the models.

Data Availability
All data generated or analyzed during this study are included in this article.