The Diffusive Model for West Nile Virus on a Periodically Evolving Domain

In this paper, we investigate the impact of a periodically evolving domain on the dynamics of the diffusive West Nile virus. A reaction-diffusion model on a periodically and isotropically evolving domain which describes the transmission of the West Nile virus is proposed. In addition to the classical basic reproduction number, the spatial-temporal basic reproduction number depending on the periodic evolution rate is introduced and its properties are discussed. Under some conditions, we explore the long-time behavior of the virus. )e virus will go extinct if the spatial-temporal basic reproduction number is less than or equal to one. )e persistence of the virus happens if the spatial-temporal basic reproduction number is greater than one. We consider special case when the periodic evolution rate is equivalent to one to better understand the impact of the periodic evolution rate on the persistence or extinction of the virus. Some numerical simulations are performed in order to illustrate our analytical results. Our theoretical analysis and numerical simulations show that the periodic change of the habitat range plays an important role in theWest Nile virus transmission, in particular, the increase of periodic evolution rate has positive effect on the spread of the virus.


Introduction
West Nile virus (WNv) is an arbovirus with natural transmission cycle between mosquitoes and birds. When infected mosquitoes bite birds or other animals including humans, they transmit the virus [1]. WNv is different from other arbovirus since it involves a cross infection between birds (hosts) and mosquitoes (vector) and those birds might travel with spatial boundaries. Also, WNv can be passed via vertical transmission from mosquito to its offspring which increases the survival of the virus [2]. WNv was first isolated and identified in 1937 from the blood of a febrile woman in the West Nile province of Uganda during research on yellow fever virus [3]. It is worth mentioning that WNv is endemic in some temperate and tropical regions such as Africa and the Middle East; it has now spread to North America; the first epidemic case was introduced in New York City in 1999 and then propagated across the USA [4][5][6]. e USA had experienced one of its worst epidemics in 2012; there were 5387 cases of infections in humans [7]. As we know, there are no indications that the spread of the virus has stopped. Consequently, it is very necessary to acquire some insights into the propagation of WNv in the mosquito-bird population.
Mathematical nonspatial models have been proposed and analyzed in an attempt to study the transmission dynamics of WNv, in order to elucidate control strategies [2,6,8,9]. It is essential to study and understand its temporal and spatial spread, but most of the models are focused on the nonspatial transmission dynamics of the virus between birds and mosquitoes.
With respect to spatial models of WNv, Lewis et al. [4] studied the spatial spread of WNv to describe the movement of birds and mosquitoes, established the existence of travelling waves, and calculated the spatial spreading rate of the infection. e effects of vertical transmission in the spatial dynamics of the virus for different bird species were proposed by Maidana and Yang in [10], and they studied the travelling wave solutions of the model to determine the speed of virus dissemination. Liu et al. [11] presented the directional dispersal of birds and impact on spatial spreading of WNv. Likewise, Lin and Zhu studied spatial spreading model and dynamics of WNv in birds and mosquitoes with free boundary [12].
To investigate the existence of travelling wave and calculate the spatial spread rate of infection, Lewis et al. in [4] proposed the following simplified WNv model: where the positive constants N b and A m denote the total population of birds and adult mosquitoes; I b (x, t) and I m (x, t) represent the populations of infected birds and mosquitoes at the location x in the habitat Ω ⊂ R N and at time t ≥ 0, respectively, and I b (x, 0) + I m (x, 0) > 0. e parameters in the above system are defined as follows: (i) α m , α b : WNv transmission probability per bite to mosquitoes and birds, respectively (ii) β b : biting rate of mosquitoes on birds (iii) d m : adult mosquito death rate (iv) c b : bird recovery rate from WNv (v) D 1 , D 2 : diffusion coefficients for birds and mosquitoes, respectively As in [13], throughout this paper, we assume that the mosquitoes' population does not diffuse (D 2 � 0).
For the corresponding spatially independent model of (1), the basic reproduction number is such that for 0 < R 0 < 1, the virus always vanishes, while for R 0 > 1, a nontrivial epidemic level appears, which is globally asymptotically stable in the positive quadrant [4]. As pointed out in [14], the basic reproduction number R 0 is a very important concept in epidemiology and it defined as an expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population, and mathematically it is introduced as the dominant eigenvalue of a positive linear operator. It is important to mention that usually the basic reproduction numbers for the nonspatial models are calculated by the next generation matrix method [15], while for the spatially dependent systems, the numbers could expressed in terms of the principal eigenvalue of related eigenvalue problem [16] or the spectral radius of next infection operator [17]. e dynamics of the spatial dependence model (1) has been studied. e existence and nonexistence of the coexistence states in a heterogeneous environment have been investigated in [18]. e impact of the environmental heterogeneity and seasonal periodicity on the transmission of WNv was considered in [19].
In recent years, the impact of change of the habitat range on biological population has attracted much attention. We know there are two aspects: one is the domain changing with unknown boundary, which describes the domain change induced by the activity of population itself, and the other is the domain changing with known boundary, which characterizes the domain change induced by objective environments. For the domain changing with unknown boundary, many researchers have proposed and considered the free boundary problem, for example, [20][21][22][23] for the persistence of invasive species and [24,25] for the transmission of diseases. In addition, Tarboush et al. [13] discussed the corresponding free boundary problem to model (1). Wang et al. investigated the spreading speed for a WNv model with free boundary in a homogeneous environment [26]. eir results indicated that the asymptotic spreading speed of the WNv model with free boundary is strictly less than that of the corresponding model in Lewis et al. [4]. For the domain changing with known boundary, there are also some papers, for instance, [27][28][29][30] for a growing domain and [31][32][33][34] for a periodically evolving domain. In [31], the authors introduced the periodically evolving domain into a single-species diffusion logistic model and studied the influence of periodic evolution on the survival and extinction of species. Recently, Zhang and Lin considered the diffusive model for Aedes aegypti mosquito on a periodically evolving domain in order to explore the diffusive dynamics of Aedes aegypti mosquito [32]. eir results indicated that the periodic domain evolution has a significant impact on the dispersal of Aedes aegypti mosquito. To investigate the impact of periodically evolving domain on the mutualism interaction of two species, Adam et al. [33] studied a mutualistic model on the periodic evolving domain. ey suggested that the periodic evolution of domain places significant influence on the interaction of two species. Zhu et al. [34] used a periodic evolving domain to investigate the gradual transmission of a dengue fever model. ey found that the periodic domain evolution has a significant effect on the transmission of dengue.
In this paper, we will consider the impact of the periodic evolving domain on the dynamics of a diffusive WNv model corresponding to system (1). We followed the methods of Adam et al. [33], Zhang and Lin [32], and Zhu et al. [34]. e rest of this paper is organized as follows. We will present the formulation of our problem in Section 2. In Section 3, we introduce the spatial-temporal basic 2 Complexity reproduction number and present its properties. e existence and nonexistence of the periodic solutions on a periodically evolving domain Ω t are discussed in Section 4. Section 5 is devoted to the attractivity of periodic solutions on a periodically evolving domain Ω t . In Section 6, we deal with the existence, nonexistence, and attractivity of the periodic solutions on a fixed domain Ω 0 . Some numerical simulations are given in Section 7. Section 8 provides some conclusions.

Model Formulation
Motivated by [27], we let Ω t ⊂ R n (n ≥ 1) be a periodically evolving domain and zΩ t be the evolving boundary. For any point, x(t) ∈ Ω t satisfies x(t + T) � x(t) for some positive constant T. Also, we assume that the domain Ω t grows uniformly and isotropically, that is, where ρ(t) ∈ C 1 [[0, T]; (0, ∞)] and y represents the spatial coordinates of the initial domain Ω 0 . Moreover, According to the principle of mass conservation and Reynolds transport theorem [35], in this paper, we will focus on the following problem: where I b (x(t), t) and I m (x(t), t) represent the densities of infected birds and mosquitoes at position x(t) ∈ Ω t and time t, respectively, and I b,0 (x) and I m,0 (x) are positive smooth functions in Ω 0 . e functions α b (x(t), t), , and d m (x(t), t) are all sufficiently smooth, T-periodic, and strictly positive when t ≥ 0. According to [36,37], the evolution of domain Ω t generates a flow velocity a(x(t), t). In addition, the evolving domain Ω t represents two kinds of extra terms into the problem, one of which is the advection terms ∇I b · a and ∇I m · a representing the transport of material around Ω t at a rate determined by the flow a, and the other is the dilution terms (∇ · a)I b and (∇ · a) I m due to local volume expansion.
Since problem (4) is involving the terms of advection and dilution, it is not easy to study the long-time behavior of its solutions; therefore, we will transform the continuously deforming domain in problem (4) to a fixed domain by using Lagrangian transformations [29,36].
We suppose that a(x(t), t) � _ x(t) is the flow velocity, which is identical to the domain velocity.
is means that a � _ and assume erefore, problem (4) is transformed to Complexity 3 with the periodic condition and under the initial condition Moreover, we assume that the functions α b (y, t), α m (y, t), β b (y, t), c b (y, t), and d m (y, t) ∈ C α,α/2 (Ω 0 × [0, ∞)) for some α ∈ (0, 1); all are positive bounded in the sense that there exist constants α

Spatial-Temporal Basic Reproduction Number
In this section, we will introduce the spatial-temporal basic reproduction number R 0 (ρ) and exhibit its properties. To address this, we consider the following linearized periodic reaction-diffusion problem: Employing the ideas go back to [17,32,38], and we let C T be the ordered Banach space consisting of all T-periodic and continuous functions from R to C(Ω 0 , R)‖·‖ with the maximum norm C + T � η ∈ C T : η(t)y ≥ 0, for all t ∈ R, y ∈ Ω 0 } and the positive cone η ∈ C T . For any given η(y, t) � η(t)y, we have m � (m 1 , m 2 ) ∈ C T × C T . Next, we suppose that is the density distribution at the spatial location y ∈ Ω 0 and time s and let Φ(t, s), t ≥ s { } be the evolution family determined by Define the operator G(t) by where Now under the same boundary conditions in problem (11), we let Φ 1 (t, s), t ≥ s and Φ 2 (t, s), t ≥ s be the evolution families determined by the first equation and second equation in problem (11), respectively. Moreover, let A and B be two bounded linear operator defined by for Consequently, we define the spatial-temporal basic reproduction number of system (10), that is, where r(L) is spectral radius of the operator L.
With the above discussion, we have the following result (see [19,32] for more details). Complexity and λ 0 is the principal eigenvalue of the eigenvalue problem In what follows, we will present the properties of the spatial-temporal basic reproduction number R 0 (ρ), that is, μ 0 . Note that problem (17) is degenerate, so we will not able to derive the existence of the principal eigenvalue by using Krein-Rutman theorem [39] because of the lack of compactness for the solution semigroup. erefore, we first transform the equations of (17) to one equation. To achieve this, let and then the second equation of (17) can be written as which gives rise to Together with the periodic condition ψ(y, 0) � ψ(y, T), direct computations yield Here, one can easily see that the function G(ϕ(y, t)) is monotonically nondecreasing with respect to β b (y, t) and decreasing with respect to d m (y, t). Hence, problem (17) reduces to the following form: Complexity By employing the integration by parts to problem (23) and denoting the principal eigenvalue of (17) as μ 0 , we obtain where For formula (24), in some special cases, we can replace the sign of inequality ( ≤ ) by equality sign by using the variational methods [40][41][42].
In order to give the relationship between the spatialtemporal basic reproduction number R 0 (ρ) and the periodic evolution rate ρ(t), we adopt the notation Consequently, we have the following result.
Theorem 1. e following assertions are valid: (17) is expressed by Furthermore, in the sense that ρ(t) � 1.
where p(t) and q(t) are functions to be determined later and Together with (17), we obtain where and (R 0 (ρ); p(t)φ(y), q(t)φ(y)) is the unique principal eigenpair of problem (17). Rewriting (32) as 6 Complexity and integrating from 0 to T yield By using Hölder inequality, one can easily obtain that e proof of assertion (a) is completed. For assertion (b), since we assumed that all coefficients are constants, we can get directly from (27).
To prove the rest of assertion (b), let ρ(t) ≡ 1, that is, Ω t � Ω 0 is a fixed domain and rewrite (32) as which are explicitly given by According to (38), direct computations yield which is consistent with the result given from the variational method.

Periodic Solutions on Evolving Domain
In this section, we discuss the existence and nonexistence of T-periodic solutions. To begin, we first consider the T-periodic boundary problem corresponding to (7) and (8): where

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For later analysis, we give the following definition of upper and lower solutions.
is called ordered upper and lower solutions of problem (42) Now we are in a position to state the existence and nonexistence of T-periodic solutions to problem (42) as well as problems (7) and (8). To begin with, in the following result we give the existence of T-periodic solution.
To establish the nonexistence of a T-periodic solution to problem (42), we have the following result. □ Theorem 3. If R 0 (ρ) ≤ 1, then problem (42) has no positive T-periodic solution.

Attractivity of Periodic Solutions
In this section, we first construct the true solutions of problem (42) and then present the attractivity of T-periodic solutions to problems (8) and (9) in relation to the minimal and maximal T-periodic solution of problems (8) and (9). In what follows, we construct the true solutions of problem (42) by using the monotone iterative scheme. Let where f m � min (

−∞,∞)×[0,T] f(t) and f M � max (−∞,∞)×[0,T] f(t) for any given continuous T-periodic function f.
It is easy to verify that both F 1 and F 2 are nondecreasing with respect to u and v. en, problem (42) is equivalent to Using (u (0) , v (0) ) � (N b , A m ) and (u (0) , v (0) ) � (δϕ, δψ) as an initial iteration, one can construct a sequence (u (i) , v (i) ) from the iteration process , v with the periodic condition where i � 1, 2, . . .. Under condition R 0 (ρ) > 1, we know that (N b , A m ) and (δϕ, δψ) are ordered upper and lower solution of problem (42) . Taking (N b , A m ) and (δϕ, δψ) as initial iteration and employing ideas of [43] with the monotonicity of f 1 and f 2 , it follows that the well-defined sequences governed by (49) and (50) possess the monotone property (51) erefore, the pointwise limits exist and their limits possess the relation  (42) in Ω 0 .
From the above conclusions, we have the following result. (42), respectively, and then the sequences (u (i) , v (i) ) and (u (i) , v (i) ) provided from (49), (50) converge monotonically from above to a maximal solution (u, v) and from below to a minimal solution (u, v) in Ω 0 , respectively, and satisfy the relation

Theorem 4. Let (u, v) and (u, υ) be a pair of ordered upper and lower solutions of
For problems (8) and (9), (u, v) and (u, v) defined in (44) are also the ordered upper and lower solutions provided the initial condition is replaced by (55) Applying (u (0) , v (0) ) � (u, v) and (u (0) , v (0) ) � (u, v) as an initial iteration again, we denote the sequences generated by (49) as (u (i) (56) e following three lemmas follow from [43], so we omit their proofs here.
In the next theorem, we present the attractivity of Tperiodic solutions to problems (7) and (9) in relation to the maximal and minimal T-periodic solution of problems (7) and (8).
When it comes to assertion (b), in fact, it is easy to see that (N b , A m ) and (0, 0) are a pair of ordered upper and lower solutions of problems (7) and (8). Using the same argument as in assertion (a), as well as the fact that (0, 0) is the unique solution to problems (7) and (8), we can conclude that the solution (u, v)(y, t; η 1 , η 2 ) of problem (7), associated with any nonnegative initial function pair (η 1 (y), η 2 (y)), possesses the convergence property which is equivalent to

The Impact of Evolving Domain
To better understand the impact of periodic evolving domain, in this section, we assume that ρ(t) ≡ 1, that is, Ω t � Ω 0 is a fixed domain, and then problem (7) becomes with the periodic condition U(y, 0) � U(y, T), and under the initial condition U(y, 0) � η 1 (y) � I b,0 (y), V(y, 0) � η 2 (y) � I m,0 (y), By the similar arguments as in Section 2, we have the following eigenvalue problem corresponding to problems (71) and (72): where (Φ, Ψ) is the eigenfunction corresponding to the principal eigenvalue and R * 0 � R 0 (1) is the principal eigenvalue of the eigenvalue problem Moreover, for any (η 1 , η 2 ) ∈ S * 0 , on Ω 0 ×[0, ∞). (78) uniformly for y ∈ Ω 0 , where anks to the above analysis, here we adopt the integral average value ρ − 2 � 1/T T 0 (1/ρ 2 (t))dt generated by the evolution rate ρ(t). It is easy to see that the spreading or vanishing of the virus on periodically evolving domain depends on the spatial-temporal basic reproduction number R 0 (ρ), while on the fixed domain, it depends on R * 0 . When ρ − 2 < 1, we have R 0 (ρ) > 1, which means that the spreading of the virus has increased. Meanwhile, if ρ − 2 > 1, then R 0 (ρ) ≤ 1, which implies that the spreading of the virus has decreased. When the domain is fixed, the parallel results hold with R * 0 � R 0 (1), then the virus in the case of vanishing.

Numerical Simulation and Discussion
In this section, we first carry out numerical simulations to illustrate the theoretical results obtained in previous sections. Our focus is the impact of periodic evolving domain on the transmission of the West Nile virus (WNv).
For simplicity, first we fix Ω 0 � (0, 1), and then change the value of the evolution rate ρ(t) to observe the long time behavior of problems (7) and (9).

Example 1.
In systems (7) and (9), we fix β b � 0.3 and d m � 0.029 with ρ(t) � 1. Direct calculations show that Hence, the solution of problems (7) and (9) tends to positive steady states (see Figures 1 and 2), which implies that the virus will persist in a fixed domain.

Example 2.
In systems (7) and (9), we choose β b � 0.09 and d m � 0.29 with ρ(t) � 1. Direct calculations show that It is easy to see that the solution of problems (7) and (9) decays to zero quickly (see Figures 3 and 4), which implies that the virus will be extinct in a fixed domain.
erefore, it is easy to see that the solution of problems (7) and (9) converges to a positive periodic steady state (see Figures 5 and 6), which means that the virus with periodically evolving domain will persist. Consequently, we can see that ρ −2 < 1 has positive effect on the persistence of WNv.
erefore, one can easily see that the solution of problems (7) and (9) tends to zero quickly (see Figures 7 and 8), which   (7) and (9), which implies that the domain is fixed when the evolution rate ρ(t) � 1.

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means that the virus with periodically evolving domain will be extinct. Consequently, we can say that ρ −2 > 1 has negative effect on the persistence of WNv.

Conclusions
Recently, the impact of periodic evolution domain has been attracting considerable attention. In [31], Jiang and Wang studied the impact of periodic evolution on the singlespecies diffusion logistic model. Asymptotic profile of a mutualistic model on a periodically evolving domain has been investigated by Adam et al. in [33]. e diffusive model for Aedes aegypti mosquito on a periodically evolving domain has been considered by Zhang and Lin in [32]. Zhu et al. [34] constructed a dengue fever model and studied its asymptotic profile on a periodically evolving domain. ese studies indicated that the periodic domain evolution has a significant impact on the dispersal of species and transmission of infectious diseases.
In this paper, we study a diffusive West Nile virus model with periodical and isotropic domain evolution. To circumvent the difficulty induced by the advection and dilution terms, we transform the model to a reaction-diffusion model in a fixed domain. We introduce the spatial-temporal basic reproduction number R 0 (ρ) depending on the periodic evolution rate ρ(t). In the case that all parameters are constants and ρ(t) ≡ 1, the explicit formula for the spatialtemporal basic reproduction number is presented ( eorem 1). Moreover, to better understand the impact of periodic evolution value on the persistence or extinction of the virus, we assume ρ(t) ≡ 1, that is, the periodic domain Ω t becomes a fixed domain Ω 0 . Furthermore, the notation ρ −2 � (1/T) T 0 1/ρ 2 (t)dt is utilized as an average value. Our results show that if R 0 (ρ) > 1 depending on the evolution rate ρ(t), then the virus will persist and all solutions possess the attractor 〈(u, v), (u, v)〉, which is the sector between the maximal and minimal T-periodic solutions (u, v) and (u, v) of problems (7) and (8) ( eorems 2-4) (a), whereas, if R 0 (ρ) ≤ 1, then any solution of problems (7) and (8) decays to (0, 0), that is, the virus is in the case of extinction ( eorem 3 and 5) (b). In the case that ρ(t) ≡ 1, we introduce R * 0 . For this case, if R * 0 > 1, the model admits a maximal and minimal T-periodic solutions, while if R * 0 ≤ 1, the model has no positive solution ( eorem 6 and 7). It is important to mention that numerical simulation in this paper is presented by using some parameters given in Lewis et al. [4], namely, A m /N b � 20, α b � 0.88, α m � 0.16, c b � 0.01, β b � 0.3, and d m � 0.029.
From our theoretical and numerical results, we believe that the periodic domain evolution has a significant impact on the transmission of WNv.

Data Availability
All data are provided in full in the numerical simulation and discussion section of this article.  (7) and (9), which implies that the domain is not periodically evolving when the evolution rate ρ(t) � e 0.2(cos(4t)−1) . 16 Complexity