Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment.

This paper is concerned with a strongly-coupled elliptic system, which describes a West Nile virus (WNv) model with cross-diffusion in a heterogeneous environment. The basic reproduction number is introduced through the next generation infection operator and some related eigenvalue problems. The existence of coexistence states is presented by using a method of upper and lower solutions. The true positive solutions are obtained by monotone iterative schemes. Our results show that a cross-diffusive WNv model possesses at least one coexistence solution if the basic reproduction number is greater than one and the cross-diffusion rates are small enough, while if the basic reproduction number is less than or equal to one, the model has no positive solution. To illustrate the impact of cross-diffusion and environmental heterogeneity on the transmission of WNv, some numerical simulations are given.


1.
Introduction. Infectious diseases have been attracting considerable attention in recent years, and various epidemic models have been proposed and analyzed for prevention and control strategies, especially for vector borne diseases [15,39]. For example, a West Nile virus (WNv) is an arbovirus of the Flavivirus kind in the family Flaviviridae that causes the epidemics of febrile illness and sporadic encephalitis [7]. WNv is found in temperate and tropical regions of the world, it was first isolated and identified from the blood of a febrile Ugandan woman during research on yellow fever virus in 1937 [3].
Although WNv is widely distributed in Africa, the Middle East, Asia and southern Europe, in North America, the first infected case was detected in 1999 during an outbreak of encephalitis in New York city [3,21,26,39]. Since 1999 this virus has spread spatially and prevail in much of North America [8,21], it is evident that the spread of WNv comes from the interplay of disease dynamics and bird and mosquito movement.
To the best of our knowledge, currently there are no effective vaccine or medicine for WNv. To reduce the rates of WNv infection, anti-WNv efforts are primarily based on personal protective measures like insect repellent and protective clothing, and public heath measures [4].
Many mathematical models for WNv have been proposed and analyzed, however most of the models are focused on the non-spatial transmission dynamics [4,36,39]. In fact, the spatial spreading is an important factor to affect the persistence and eradication of WNv. In 2006, Lewis et al. [21] investigated the spatial spread of WNv to describe the movement of birds and mosquitoes. The reaction-diffusion model was extended from the non-spatial model for cross infection between birds and mosquitoes that was proposed and developed by Wonham et al. in [39].
To utilize the cooperative characteristic of cross-infection dynamics and estimate the spatial spread rate of infection, Lewis et al. in [21] proposed the following simplified WNv model where the positive constants N b and A m denote the total population of birds and adult mosquitos; I b (x, t) and I m (x, t) represent the populations of infected birds and mosquitos 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. The parameters in the above system are defined as follows: • α m , α b : WNv transmission probability per bite to mosquitoes and birds, respectively; • β b : biting rate of mosquitoes on birds; • d m : adult mosquitos death rate; • γ b : bird recovery rate from WNv. Here, the positive constants D 1 and D 2 are diffusion coefficients for birds and mosquitoes, respectively.
Considering the spatially-independent model one can see that if R 0 (:= ) < 1, the virus will vanish eventually, while for R 0 > 1, a nontrivial epidemic level appears and is globally asymptotically stable in the positive quadrant [21]. For the diffusive model (1), Lewis et al. proved the existence of traveling wave and calculated the spatial spread rate of infection [21]. The corresponding free boundary problem describing the expanding process has been discussed in [34]. It is worth mentioning that WNv usually spreads from one area to another because of the diffusions of birds and mosquitoes, so that its transmission is affected not only by the characteristics of pathogens, but also by the spatial difference of environment in which birds or mosquitoes reside. Considering the complexity of diffusions and the heterogeneity of environment, the system (1) can be extended to the following strongly-coupled parabolic system and the corresponding elliptic problem with Dirichlet boundary conditions becomes where , α m (x) and d m (x) are all sufficiently smooth and strictly positive functions defined on Ω. d i (i = 1, 2) is positive constants represent the free-diffusion coefficients of population I b (x) and I m (x), respectively. α i , β i and γ i (i = 1, 2) are nonnegative constants; α i , β i and γ i are the self-diffusion coefficients, the cross-diffusion rates and cross-diffusion pressures, respectively. The homogeneous Dirichlet boundary condition in (4) means that there is no infection on the boundary and outside of the domain Ω. More specifically, the diffusion terms can be written as The terms represent the self-diffusions and the terms represent the cross-diffusions. Here the term  Problem (1) is weakly-coupled parabolic system which only consider the random diffusion in a homogeneous environment. However, problem (4) implies that, in addition to the dispersive force, the diffusion also depends on population pressure from other population. This means that the population in (4) are not homogeneously distributed due to the consideration of self and cross diffusion terms. Moreover, the diffusive behavior in different populations also affect the distribution of resources. Thus, the consideration of diffusion and cross-diffusion effect is very reasonable and more close to reality, see for example, [29] for mixed-culture biofilm model, [17] for the tumor-growth model and [14,16,31,40] for the competition model. There are some valuable results about the roles of diffusion and cross-diffusion in the modeling of the dynamics of strongly coupled reaction-diffusion systems [5,11,13,16,19,22,25,31,30,38,40]. For instance, Shigesada et al. [31] proposed the strongly coupled elliptic system describing two species Lotka-Volterra competition model. Ko and Ryu studied a predator-prey system with cross-diffusion, representing the tendency of prey to keep away from its predators, under the homogeneous Dirichlet boundary conditions in [19]. Fu et al. investigated the global behavior of solutions for a Lotka-Volterra predator-prey system with prey-stage structure, under the homogeneous Neumann boundary conditions [11]. In 2014, Jia et al. [16] discussed a Lotka-Volterra competition reaction-diffusion system with nonlinear diffusion effects. In 2016, Braverman and Kamrujjaman [5] introduced a competitive-cooperative models with various diffusion strategies. More recently, Li et al. studied an effect of cross-diffusion on the stationary problem of a Leslie prey-predator model with a protection zone [22].
In recent years, researches on the existence and non-existence of the positive solutions for the dynamics of strongly-coupled elliptic systems have received comprehensive attention [16,19,20,28]. There are many standard approaches to derive the coexistence for the standard semi-linear parabolic system in mathematical models, such as construction of upper and lower solutions [12,16,18,28], bifurcation theory [6], fixed point theorem [19,42], ect. The upper and lower solutions method developed by Pao [27] is concise and effective to derive the coexistence. Based on the method, the coexistence for a general strongly-coupled system has been given in [28]. In [18], Kim and Lin studied the coexistence of three species in strongly coupled elliptic system. Gan and Lin in [12] considered the competitor-competitormutualist three species Lotka-Volterra model. Recently, Jia et al. [16] investigated the existence of the positive steady state solution of a Lotka-Volterra competition model with cross-diffusion.
Motivated by above problems, in this paper we are more interested in the nonnegative steady state solutions, that is, the coexistence of problem (4) describing a cross-diffusive WNv model in a heterogenous environment.
The plan of this paper is as follows: Section 2 is devoted to the basic reproduction number of problem (4) and its properties. The existence and non-existence of coexistence to (4) are discussed in Section 3. Finally, some numerical simulations and a brief discussion are given in Section 4.

2.
Basic reproduction numbers. In this section, we first present the basic reproduction number for problem (4) and its properties for the corresponding system in Ω. According to [10], the basic reproduction number is 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 was defined as the dominant eigenvalue of a positive linear operator. Usually the basic reproduction numbers for the spatially homogenous models were calculated by the next generation matrix method [35], while for the spatially-dependent models, the numbers could be presented in the term of the principal eigenvalue of related eigenvalue problem [1] or the spectral radius of next infection operator [37,41].
Considering the linearized problem of (3), we have We now consider the following linear reaction-diffusion system where In addition, the interval evolution of individuals in the infectious compartments is governed by the following linear system Let X 1 := C(Ω, R 2 ) and X + 1 := C(Ω, R 2 + ). Set T (t) be the solution semigroup on X 1 associated with system (7). We let Ψ = (φ, ψ) is the density distribution of u at the spatial location x ∈ Ω, we then see that T (t)Ψ := (T (t)φ, T (t)ψ) represents the remaining distribution of infective birds and mosquitoes at time t. Therefore, the distribution of total new infective members is Following the idea of [37,41], we define the linear operator It follows from the definition, we know that L is a continuous and positive operator which maps the initial infection distribution Ψ to the distribution of the total members produced during the infection period. Consequently, we define the spectral radius of L as the basic reproduction number of system (5), that is, . As in [23], to ensure the existence of the basic reproduction numbers we consider the following linear eigenvalue problem: For any R > 0, the system is strongly cooperative, that is, According to [2,9,33], for any R > 0, there exists a unique value µ := µ 1 (R), and called the principal eigenvalue, such that problem (8) admits a unique solution pair (φ R , ψ R ) (subject to constant multiples) with φ R > 0 and ψ R > 0 in Ω. Moreover, µ 1 (R) is algebraically simple and dominant, and the following properties hold.
With the above definition, we have the following relation between the two principal eigenvalues.
and λ 0 is the principal eigenvalue of the eigenvalue problem Proof. In fact, λ 0 = µ 1 (1). On the other hand, one can easily deduce from the monotonicity with respect to the coefficients in (8) that lim R→0 + µ 1 (R) < 0 and lim R→+∞ µ 1 (R) > 0, therefore R D 0 is the unique positive root of the equation µ 1 (R) = 0. The result follows from the monotonicity of µ 1 (R) with respect to R.

Remark 1.
Recalling that µ 1 is monotonically increasing with respect to β b (x), in the sense that in Ω, we deduce from Lemma 2.1 that R D 0 is monotonically increasing with respect to β b (x), and R D 0 > 1 if β b (x) is sufficiently large. If all coefficients are constant, we can provide an explicit formula for R D 0 , which is known as the basic reproduction number for the corresponding diffusive WNv model. (9), or the basic reproduction number for model (4), is expressed by where λ * is the principal eigenvalue of −∆ in Ω with null Dirichlet boundary condition.
Proof. Let ψ * be the eigenfunction corresponding to the principal eigenvalue (λ * ) of −∆ in Ω with null Dirichlet boundary condition and Then we know that (φ * , ψ * ) is a positive solution of problem (9) with R D 0 = √ P * , and (11) follows directly from the uniqueness of the principal eigenvalue of (9).
3. Coexistence. In this section, inspired by [16,18,20,28], we first study the existence of a coexistence solution to problem (4) by constructing upper and lower solutions and then we establish the non-existence of the coexistence solution to problem (4). For the convenience, we let where (Î b ,Î m ) and (Ĩ b ,Ĩ m ) are given in the following definition.
Next we are going to give a sufficient condition for problem (4) to possess a positive solution by constructing upper and lower solutions as in [28]. To achieve this, we first give an equivalent form of problem (4): where Taking Therefore, the inverse I b = g 1 (u, v), I m = g 2 (u, v) exist whenever (I b , I m ) ≥ (0, 0). Hence, problem (4) reduces to the following equivalent form where In addition, from an elementary computations one can check that which shows that I b = g 1 (u, v) is nondecreasing in both u and v, while I m = g 2 (u, v) is also nondecreasing in both u and v for all (I b , I m ) ≥ (0, 0). For the later analysis, we present the definition of upper and lower solutions to problem (13) as follows.
x ∈ Ω, For definiteness, we select I b = g 1 (ũ,ṽ),Ĩ m = g 2 (ũ,ṽ); Then the requirements of (Ĩ b ,Ĩ m ) and (Î b ,Î m ) in (14) are satisfied and those of (ũ,ṽ), (û,v) are reduced to Now we consider the monotonicity of F i (i = 1, 2). From direct computations it is easy to see that If we choose then F 1 and F 2 are increasing with respect to I b and I m , respectively, as long as On the other hand, the direct calculations show that ≤ 0 for any β1 γ1 and β2 γ2 , respectively. To ensure that ∂F1 ∂Im ≥ 0, ∂F2 ∂I b ≥ 0, we have to modify the upper solution and we seek from the first and second inequalities of (15). Let then for β 1 ≤ β * 1 and β 2 ≤ β * 2 , F i (i = 1, 2) is monotone nondecreasing with respect to I b and I m . Consequently, (Ĩ b ,Ĩ m ,ũ,ṽ), (Î b ,Î m ,û,v) are a pair of ordered upper and lower solutions of problem (13).
Using Theorem 2.1 of [28] leads to the following existence result : Theorem 3.2. If R D 0 > 1, problem (4) admits at least one coexistence solution (I b (x), I m (x)) provided that β 1 and β 2 are sufficiently small.
To establish the non-existence of the coexistence solution to problem (4), we have the following result.
) has no positive solution.
Proof. Suppose (I * b (x), I * m (x)) is a coexistence solution of problem (4), that is, First by the upper and lower solution method we know that ( which means that On the other hand, the principal eigenvalue λ 0 in problem (10) meets Comparing (20) with (21), we can easily deduce from the monotonicity with respect to the coefficients in (21) that λ 0 is monotone decreasing with respect to β b (x), which implies that λ 0 < 0. Recalling Theorem 2.2 we can get that R D 0 > 1, which is contrary to R D 0 ≤ 1.

Remark 2.
Assume that all coefficients of (4) are spatially-independent. R D 0 is represented by (11). If α * b , α * m or β * b is big, then R D 0 > 1 and problem (4) admits at least one coexistence solution provided that β 1 and β 2 are sufficiently small. On the other hand, if α * b , α * m or β * b is small enough, then R D 0 ≤ 1 and problem (4) has no positive solution.
Next we apply the monotone iterative schemes to construct the true solutions of (4). It follows from R D 0 > 1, we know that (M 1 , M 2 ) and (g 1 (δd 1 φ, δd 2 ψ), g 2 (δd 1 φ, δd 2 ψ)) are ordered upper and lower solution of problem (4), respectively. Using (Ī b , I (0) m ) = (g 1 (δd 1 φ, δd 2 ψ), g 2 (δd 1 φ, δd 2 ψ)) as two initial iterations, we can construct two sequences {(ū (n) ,v (n) )} and {(u (n) , v (n) )} from the iteration process where n = 1, 2, · · · . As in Lemma 3.1 of [28], the sequences {(ū (n) ,v (n) )} and {(u (n) , v (n) )} governed by (22) are well-defined and possess the monotone property Hence, the pointwise limits exist and their limits possess the relation for every n = 1, 2, · · · . The last three equations of (22) givē , v (n) ), , v (n) ), which is equivalent tō Now, by the above relation, letting n → ∞ and applying the standard regularity argument for elliptic boundary problems, we derive that (Ī b ,Ī m ) and (I b , I m ) satisfy which is equivalent to x ∈ Ω, Therefore (Ī b ,Ī m ) and (I b , I m ) are true solutions of (4). Moreover, (Ī b ,Ī m ) and (I b , I m ) are maximal and minimal solutions in the sense that (I b , I m ) is any other solution of (4) in the sector is the unique solution of (4) in Ω. To achieve this, in fact, a subtraction of the third equation from the first equation in (26) yields that The above conclusions lead to the following theorem. mosquitoes are also interacted by each other and reaction depends on spatial heterogeneity of the environment. Therefore, we introduce the cross-diffusion terms (1), which can better describe the interplay between birds and mosquitoes in diffusion. The main result of this paper is twofold. Firstly, we introduce a definition of R D 0 , which is known as the basic reproduction number of problem (4) (Theorem 2.2). In the case that all coefficients are constants, we provide an explicit formula for R D 0 (Theorem 2.3). Secondly, the coexistence of problem (4) is investigated by using method of upper and lower solutions and its associated monotone iterative schemes (Theorem 3.2 and Fig. 1) under condition R D 0 > 1 provided that β 1 and β 2 are sufficiently small, whereas if R D 0 ≤ 1, problem (4) has no coexistence solution (Theorem 3.3). Our results show that no existence exists for small WNv transmission probabilities (α m and α b ), and small biting rate of mosquitoes on birds (β b )  (Remark 2). Moreover, the coexistence solution of problem (4) is between the maximal and minimal solution (Ī b ,Ī m ) and (I b , I m ), respectively, and the true solution can be obtained by constructing the monotone iterative sequences (Theorem 3.4). However, the uniqueness of coexistence solution is still unclear.
We believe that the strongly-coupled problem (4) can produce much more complex dynamics of WNv than the weakly-coupled system (1). Such problems need further investigations. In fact, even for the corresponding parabolic problems with cross-diffusion, the existence of the solution is known only for some special cases, see [24,32] and references therein. To further investigate the effect of cross-diffusion in comparison to no cross-diffusion or small cross-diffusion, we come back to problem (3), Fig. 2 shows that the global solution of problem (3) exists and stabilizes to a positive steady-state for small cross-diffusion (β 1 = 0.132 and β 2 = 0.11), we can also see that the global solution of (3) exists for β 1 ≤ 0.132 and β 2 ≤ 0.11 by simulations. However, if we choose a little big cross-diffusion, for example, β 1 = 0.133 and β 2 = 0.11, we can see from Fig. 3 that the global solution of problem (3) does not exist. We leave it for future work.