Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates

Abstract: This paper concerns with detailed analysis of a reaction-diffusion host-pathogenmodel with spacedependent parameters in a bounded domain. By considering the fact themobility of host individuals playing a crucial role in disease transmission, we formulate the model by a system of degenerate reaction-diffusion equations, where host individuals disperse at distinct rates and themobility of pathogen is ignored in the environment. We first establish the well-posedness of the model, including the global existence of solution and the existence of the global compact attractor. The basic reproduction number is identified, and also characterized by some equivalent principal spectral conditions, which establishes the threshold dynamical result for pathogen extinction and persistence. When the positive steady state is confirmed, we investigate the asymptotic profiles of positive steady state as host individuals disperse at small and large rates. Our result suggests that small and large diffusion rate of hosts have a great impacts in formulating the spatial distribution of the pathogen.


Introduction
In recent years, the studies of some reaction-diffusion host-pathogen models have received much attentions, as the investigation of these systems allow us to get better understanding the interactions between host and pathogens and the mechanisms of the disease spread. Let u 1 (x, t), u 2 (x, t) and u 3 (x, t) be the densities of susceptible hosts, infected hosts and pathogen particles at the spatial location x and time t. Laplacian operator dΔ accounts for the host movement (d > 0 is the diffusion coefficient). In [6], under a one-dimensional and unbounded domain, the authors considered the following model, x ∈ R, t > 0. (1.1) where r and K represent respectively the reproductive rate and the carrying capacity of the susceptible and infected hosts; β and α are the transmission coefficient and disease-induced mortality rate; λ and δ are respectively the pathogens production rate from infected hosts and the pathogens' decay rate. All above mentioned coefficients in (1.1) are assumed to be positive constants. With the consideration of spatial spread of pathogens, the authors in [6] investigated the existence problem of traveling wave solution.
In reality, the habitats where hosts live should be a spatially bounded domain. Wang et al. [29] took the model (1.1) as a basis and extended it to a more general model in bounded spatial domain Ω ∈ R n with smooth boundary ∂Ω, where ∂u1 ∂n stands for the differentiation along the unit outward normal n to ∂Ω; β(x)(u 1 + u 2 )u 3 represents the consumption of the pathogen due to the interaction with the hosts. Unlike in model (1.1), where parameters β, K, λ are constants, model (1.2) allows the space-dependent parameter functions, β(·), K(·) and λ(·), which are used to obey the spatial heterogeneity arising from the variance in environmental conditions (for example, temperature and humidity etc). Compared to model (1.1), the space-dependent functions β(x), K(x) and λ(x) are assumed to be continuous and positive in Ω. Since the mobility of pathogen is ignored in the domain, the semiflow induced by solution lacks of compactness. The authors in [29] overcame this difficulty by verifying the k-contracting condition. The basic reproduction number (BRN) is proved to be a threshold index for the dynamics of (1.2), and defined by adopting the concept of next generation operator (NGO). Bifurcation analysis for steady state solution are also carried out when space-dependent parameters are used.
Note that the susceptible and infected hosts in (1.1) and (1.2) share the same diffusion rate. Wu and Zou [31] further generalized the model (1.2) with distinct diffusion rates d 1  Our current work is also inspired by a series of works on diffusive models for disease dynamics in the spatial heterogeneous environment. These models are in the form of reaction-diffusion susceptible-infectedsusceptible (SIS) equations in spatially bounded domain and the main concern are how the spatial heterogeneity and the diffusion affect the disease spread and control [1,9,10,21,[31][32][33]. In an earlier article [1], the authors studied a SIS model with frequency-dependent interaction, , Ω (S(x, 0) + I(x, 0))dx ≡ N > 0, (1.4) where S(x, t) and I(x, t) represent respectively the density of susceptible and infected individuals. γ(x) and β(x) are respectively the space-dependent recovery rate and disease transmission rate. d S and d I represent respectively the diffusion rates of susceptible and infected individuals. The total number of human individuals remains constant N. The main concern in the aspect of biological implication is: limiting the flow of susceptible individuals (d S → 0) can eliminate the disease, provided that the disease is of low risk (i.e., β(·) < γ(·) for x ∈ Ω). This pioneering work start up the investigation that how the spatial heterogeneity and the diffusion affect the disease spread and control. Subsequently, the result that limiting the flow of the infected individuals can not eliminate the disease (see in [21]) revealed that d S and d I play different role in disease control. Motivated by meaningful and important aspect of spatial heterogeneity of environment and distinct dispersal rates, Wu and Zou [33] further modified the model (1.4) by replacing the frequency-dependent interaction with mass action mechanisms. They showed that an additional condition on the total population is needed for disease control if d S → 0. Additionally, disease can not be controlled when d S → 0 and the total population accounts large, and inversely, disease disappears in certain area when d I → 0. In contrast with [1,21,33], Li et al. [9,10] analyzed an spatial SIS model with linear source and logistic source (which allows varying total population). With small and large diffusion rates, both of the works revealed that disease can not be controlled, which is not a good situation in disease control. Based on the above mentioned works, we continue to explore how diffusion rates and the spatial heterogeneity affect the dynamics of (1.3) by incorporating the frequency-dependent interaction used in (1.4), and thus, can be considered as a continuation of the work [1,9,21,[31][32][33]. We shall explore the following system, with the initial condition where ζ (x) and γ(x) represent the death rates of susceptible and infected host; b(x) is the recruitment rate; β i (x)(i = 1, 2) represent the disease transmission rate; δ(x) is the pathogens' decay rate; ϱ(x) is the pathogens' production rate. By replacing the frequency-dependent interaction with mass action mechanisms (direct person-to-person transmission route), model (1.5) reduced to the model in [32]. Even though (1.5) bears a resemblance to that in [32], there is one major difference: frequency-dependent interaction β1(x)u1u2 u1+u2 assumes bounded infection force, while unbounded infection mechanism for mass action term.
Note that when d 1 = d 2 = 0 and all parameters are space-independent, the reduced system of (1.5) is the same as the model [24] with standard incidence function for the direct disease transmission and bilinear incidence function for indirect disease transmission. It is mentioned in [24] that disease transmission within and without groups may be different. In this sense, it is natural to assume that the contact probability between a susceptible and an infected host is decreasing function of total host population (standard incidence function) and the contact probability between a susceptible host and pathogen is a constant (bilinear incidence function). Meanwhile, the model studied in [24] can also provide us a biological interpretation for our model (1.5) that: 1) In Zika virus transmission, u 1 , u 2 can respectively stands for uninfected individuals, infected individuals, and u 3 stands for the infected mosquitoes in the local landscape; 2) In H1N1 and seasonal influenzas, u 1 , u 2 represent respectively the uninfected and infected individuals, and u 3 represents the contaminated environment such as classrooms, or other public places; 3) In the transmission of avian influenza, u 1 , u 2 can respectively stand for the uninfected migratory birds, infected migratory birds, and u 3 stands for infected domestic poultry. On the other hand, (1.5) can be used to describe the transmission of cholera in the sense that u 1 , u 2 and u 3 stand for respectively the density of susceptible, infected individuals and the concentration of vibrios in the environment. However, it is well-known that those who recovered from cholera do lose immunity. A realistic excuse to justify the hypothesis that infected individuals do not lose immunity may be that if we care about this model during one outbreak, then those who recovered are very likely to remain immune throughout the outbreak.
We plan to proceed this paper as follows. Section 2 shall pay attention to the well-posedness of (1.5) with (1.6) such as, the global existence and uniqueness and ultimate boundedness of solution of (1.5), the asymptotic smoothness condition of semiflow, the existence of global compact attractor. In Section 3, we identify the basic reproduction number, 0 . We also establish that 0 can be equivalently characterized as the principal spectral conditions. In section 4, with the 0 , detailed analysis are carried out on the threshold dynamics of (1.5), that is, 0 predicts whether or not the disease persist. In a critical case that 0 = 1, the global asymptotic stability of disease free steady state is also addressed. Section 5 is spent on the dynamics of (1.5) in homogeneous case. We addressed the existence of unique positive equilibrium and local stability. The global attractivity of positive equilibrium with additional condition is achieved by the technique of Lyapunov function. Section 6 is devoted to exploring the asymptotic profiles of the positive steady state for the case that Finally, detailed conclusions are drawn and some discussion is presented.
In what follows, we shall confirm t max = ∞ by verifying the boundedness of u(·, t) in Ω × (0, tmax).

Existence of the global solution
is ultimately bounded.
• u(·, t) satisfies the L 1 bounded estimate, i.e., lim sup where M 1 is a positive constant. By (1.5), we have It follows that lim sup Hence, by taking M 1 = max{M 1 , M 2 }, the assertion directly follows. • For k ≥ 0, u 2 and u 3 satisfies the L 2 k bounded estimate, that is, where M 2 k is a positive constant. We shall prove (2.7) by induction. Obviously, k = 0 holds. Suppose that (2.7) holds for k −1. Then for We multiply u 2 by u 2 k −1 2 , and then integrate it over Ω, Recall that Hence, (2.9) becomes Applying Young's inequality: Thus, (2.10) can be estimated by , and then integrate it over Ω, Again applying Young's inequality (by setting ϵ 2 = δ * 4ϱ * , p = 2 k /(2 k − 1) and q = 2 k ), we have Hence (2.15) becomes Combined with (2.14) and (2.17), we obtain Applying interpolation inequality: Let Thus, (2.18) becomes It then follows from (2.8) that lim sup t→∞ Thus, according to continuous embedding where M p > 0, independent of initial conditions. Denote by Ya , 0 ≤ a ≤ 1 the fractional power space. By [31, This proves the ultimate boundedness of the solution of (1.5). The proof is cpmplete.

Asymptotic smoothness of Υ(t)
We refer the readers to consult [8,36] for the definition of κ(·), the Kuratowski measure of noncompactness.

Lemma 2.2. The semigroup Υ(t) is a κ-contraction on X + , that is, for any bounded set
and that is, Υ(t) is κ-contraction on X + . This completes the proof.

Basic reproduction number
Obviously, (1.5) has a disease-free steady state E 0 = (u 0 The linearization of (1.5) at E 0 leads to the following cooperative system, We rewrite It is easy to see that both B and B are resolvent-positive operators [27]. Denote by T(t)(resp.T(t)) : Y → Y the positive semigroup generated by B (resp. B). Since both B and B are cooperative for any x ∈ Ω, we get that T(t)Y + ⊆ Y+ (resp.T(t)Y+ ⊆ Y+). Throughout of the paper, we denote by s(Q) = sup{Reλ, λ ∈ σ(Q)} the spectral bound of Q and r(Q) = sup{|λ|, λ ∈ σ(Q)}, the the spectral radius of Q.
Following the standard procedures in [27,30], we define the NGO L as that is, within the infection period, L[ϕ](·) represents the total new infections distribution from initial distribution ϕ. Then, the BRN 0 is defined as The following result is a consequence of [27,30]. Proof. Due to the fact that B is resolvent-positive operator, we then have Due to s(B) < 0, we can let λ = 0 in (3.5), leading to For the convenience of forthcoming discussions, we next claim that 0 has relationship with other important indicators:λ 0 and η 0 .
Due to the assertion in (i) of Lemma 3.2 and variational formula, which indicate how 0 depends on the diffusion coefficient d 2 (see also in [1, Theorem 2] and [31]).
Theorem 3.1. Let 0 be defined by (3.11). Then we have (i) Fix d 1 > 0, then 0 is decreasing with respect to d 2 , and satisfies The following statements come from [13, Theorem 1.1].
The following result indicates that for a special case, s(B) is the principal eigenvalue of (3.3) without any limitations.
Proof. Define after elementary calculations, we can obtain By using u 1 ≤ u 1 + u 2 , we have By assumption (H1), we have ⎛ Hence, According to the LaSalle's invariance principle, the global stability of E * (5.1) is confirmed. This completes the proof.

Remark 5. In Theorem 5.3, (H1) is a technical condition such that the derivation of Lyapunov function is less than zero. It is significant to establish the global asymptotic stability of endemic equilibrium by constructing a suitable Lyapunov function in epidemiology. Our model (1.5) includes two types of infection: frequency dependent and mass action, which leads to challenging issue in proving global asymptotic stability of endemic equilibrium by Lyapunov function. Similar arguments can be found in (C1)-(C3) in the proof of Theorem 2.3 of [24].
6 Asymptotic profiles of the positive steady state Theorem 4.2 indicates that when 0 > 1, (6.1) admits at least one positive steady state E * , which is the positive solution of This section is devoted to the investigation of the asymptotic profiles of E * for the cases that d 1 → 0, d 1 → ∞, d 2 → 0 and d 2 → ∞. As exploration of such problem can achieve better understanding the spatial distribution of disease. We begin with the preliminary estimates of solutions of (6.1).
Lemma 6.1. Let (u 1 , u 2 , u 3 ) be any positive solution of (6.1). We then directly have (i) For any d 1 , d 2 > 0, we have the upper bound of u 1 : (ii) Fix d 2 > 0, for any d 1 > 0, we have the upper bound of u 2 and lower bound of u 1 : where the positive constants C i (i = 1, 2) do not depend on d 1 > 0.
(ii) Making the sum of u 1 , u 2 of (6.1) and integrating over Ω lead to (6.5) Fix d 2 > 0, it follows from statement (i) that By using [20, Lemma 2.2] (see also [12]), we get a Harnack-type inequality of u 2 as follows where C > 0 does not depend on d 1 . In what follows, we permit it changing from place to place. Based on (6.4) and (6.6), we have ( 6 . 7 ) Set u 1 (x 1 ) = min x∈Ω u 1 (x). By applying the maximum principle, we have Hence, we get the lower bound of u 1 .

Remark 6.
Note that the statement (ii) hold for any d 1 > 0 and d 2 ≥ 1. Actually, we can get (6.6) for any d 2 ≥ 1.
respectively, with ∂u1n ∂ν = 0, x ∈ ∂Ω. Fix sufficiently large n, denoted by Un and Vn respectively the unique positive solution of the following two auxiliary systems: and uniformly on Ω.

Conclusion and Discussion
In current work, we explore the global dynamics and asymptotic profiles of a host-pathogen model in a spatially bounded domain. We formulate the model by a reaction-diffusion system with space dependent parameters, where host individuals disperse at distinct rates and the mobility of pathogen is ignored. Complete analysis of the system allow us to investigate how large or small flows of hosts affect the spatial spread of disease, and what is the role of spatial heterogeneity on disease transmission. We firstly establish that the existence of global solution, which is achieved by extending the local solution to a global one (see Lemma 2.1 and Theorem 2.1). To cope with the non-compactness of solution semiflow Υ(t), we utilize the Kuratowski measure of noncompactness to verify the asymptotic smoothness condition. Hence, by [8,Theorem 2.4.6], (1.5) possesses a global compact attractor in X + , denoted by A 0 (see Theorem 2.2).
The BRN, 0 , is identified as the spectral radius of NGO, and also characterized by some equivalent principal spectral conditions, which establishes the threshold dynamical result for pathogen extinction and persistence (see Theorem 4.1 and 4.2). Specifically, we demonstrate that how 0 depends on the diffusion coefficient d 1 for d 1 → 0 and d 1 → ∞ (see Theorem 3.1). We have also confirmed the global stability of E 0 in a critical case that 0 = 1. It should be pointed here that the method used in Theorem 4.1 can also be applied in Avian influenza dynamical model [28] and Ebola transmission model [35], where the dynamic behaviors for the case that 0 = 1 are still open. We left it as future investigation. In a homogeneous case and additional condition, we explore the global attractivity of PSS (positive equilibrium) by the technique of Lyapunov function.
When 0 > 1, (1.5) possesses at lease one positive steady state. To achieve better understanding the effects of the host's movements on the spatial distribution of pathogen, we explore the asymptotic profiles of positive steady state for the cases that d 1 → 0, d 1 → ∞, d 2 → 0 and d 2 → ∞. When d 1 → 0, our result (Theorem 6.1) demonstrate that host individuals distribute on Ω in a non-homogeneous way. Under the assumption that 1-D space, Ω = (0, 1), and the condition that x ∈ [0, 1] : δ(·)β 1 (·) + β 2 (·)ϱ(·)u 0 1 (·) > δ(·)γ(·) is non-empty (of which locations can be termed as the favorite or not favorite sites for pathogens), we see from Theorem 6.2 that the infected hosts will vanish in some place and distribute in the remaining place (due to 6.24) and the susceptible hosts stay inhomogeneously on the whole habitat. When d 1 → ∞ (d 2 → ∞), susceptible (resp. infected) hosts distribute eventually over Ω, and infective (resp. susceptible) hosts distribute on Ω in a non-homogeneous way (see Theorems 6.3 and Theorem 6.4). The condition in Theorem 6.4 that Ω [β 1 (·) + β 2 (·)ϱ(·)u 0 1 (·)/δ(·)]dx > Ω γ(·)dx is usually termed as the favorable domain for pathogens and also ensure the existence of the positive solution in Theorem 6.4). In summary, our result suggests that slow or fast movement of host individuals have a great impacts on the spatial distribution of the pathogens, which may help to design strategies for disease control and prevention.
On the other hand, our results on asymptotic profiles are established when positive steady state exists. With the same arguments as those in [31,32], u 0 1 (x) → b(x)/ζ (x) in C 1 (Ω) as d 1 → 0 and η 0 defined in 3.8 ensures that η 0 < 0. It follows that 0 < 1 if d 1 <d 1 , for somed 1 > 0. This together with Theorem 4.1 indicate that pathogens still can be eliminated with d 1 → 0. Compared to the results in [31,32], our results enrich the dynamical results on the asymptotic profiles. In fact, our results also reveal that disease can not be eliminated by fast movement of host individuals. It also remains an challenging problem to revisit Theorem 6.2 as d 2 → 0 without spatial dimension limitations. In a homogeneous case, it also remains an interesting problem to perform the bifurcation analysis on steady state solutions with specific bifurcation parameter.