Abstract
This paper concerns with detailed analysis of a reaction-diffusion host-pathogen model with space-dependent parameters in a bounded domain. By considering the fact the mobility 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 the mobility 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.
1 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 u1(x, t), u2(x, t) and u3(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,
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 ∈ ℝn with smooth boundary ∂Ω,
where
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 d1 and d2,
Here, d1 and d2 are not necessarily equal. The simplest growth term for susceptible host, y(x) − μ(x)u1, is used. As pointed in [31] that distinct dispersal rates for hosts brings difficulty in proving the boundedness of the solution. The existence of global attractor needs appealing the general result in [8, Theorem 2.4.6] and Arezelà-Ascoli Theorem, so that the the asymptotic smoothness of semiflow is used instead of weak compactness by verifying the k-contraction condition. The authors studied the threshold dynamics of (1.3) and the effects of the spatial heterogeneity on disease dynamics. Wu and Zou in [31] also explored the asymptotic profiles of positive steady state for the case where d1 → 0 and d2 → 0.
Subsequently, inspired by the work [31], Shi et al. [23] revisited the model (1.2) by incorporating the horizontal disease transmission. The authors in [23] established the threshold dynamics, and performed the bifurcation analysis for steady state solution. Based on framework of [31], Wang and Wang [32] further extended the model (1.3) by adding the horizontal transmission term to (1.3). In [32], u1, u2 and u3 are used to stand for respectively the density of susceptible, infected individuals and the concentration of vibrios in the environment. The horizontal transmission is termed as direct person-to-person transmission route in cholera dynamics. The complete analysis of (1.3) with horizontal transmission term demonstrated the threshold dynamics of (1.3) and the effect of the spatial heterogeneity on disease dynamics, and also revealed that ignoring horizontal transmission will underestimate the risk of disease transmission.
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-infected-susceptible (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,
where S(x, t) and I(x, t) represent respectively the density of susceptible and infected individuals. y(x) and β(x) are respectively the space-dependent recovery rate and disease transmission rate. dS and dI 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 (dS → 0) can eliminate the disease, provided that the disease is of low risk (i.e., β(·) < y(·) 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 dS and dI 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 dS → 0. Additionally, disease can not be controlled when dS → 0 and the total population accounts large, and inversely, disease disappears in certain area when dI → 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 y(x) represent the death rates of susceptible and infected host; b(x) is the recruitment rate; βi(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
Note that when d1 = d2 = 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, u1, u2 can respectively stands for uninfected individuals, infected individuals, and u3 stands for the infected mosquitoes in the local landscape; 2) In H1N1 and seasonal influenzas, u1, u2 represent respectively the uninfected and infected individuals, and u3 represents the contaminated environment such as classrooms, or other public places; 3) In the transmission of avian influenza, u1, u2 can respectively stand for the uninfected migratory birds, infected migratory birds, and u3 stands for infected domestic poultry. On the other hand, (1.5) can be used to describe the transmission of cholera in the sense that u1, u2 and u3 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, ℜ0. We also establish that ℜ0 can be equivalently characterized as the principal spectral conditions. In section 4, with the ℜ0, detailed analysis are carried out on the threshold dynamics of (1.5), that is, ℜ0 predicts whether or not the disease persist. In a critical case that ℜ0 = 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 d1 → 0, d1 → ∞, d2 → 0 and d2 → ∞. Finally, detailed conclusions are drawn and some discussion is presented.
2 Well-posedness of the problem
The main aim of this section is to confirm that (1.5) has a global compact attractor. Throughout of the paper,
where G(·) = y(·), δ(·), ϱ(·), ζ (·), β1(·), β2(·), respectively.
Γ is the Green function of
and
We further, denote
According to [22, Section 7.1],
forms a C0 semigroup on 𝕏 preserving 𝕏+, namely, 𝒜(t)𝕏+ ⊂ 𝕏+, t ≥ 0 (see, for example, [19]). Further, let
By these settings, we can rewrite (1.5) as
We can easily check that
In fact, it is easy to verify that for any φ ∈ 𝕏+ and small enough h ≥ 0,
As just a consequence of [22, Corollary 4], we have
Lemma 2.1
For any ϕ ∈ 𝕏+, (1.5) with (1.6) admits a unique nonnegative solution u(·, t) := (u1(·, t), u2(·, t), u3(·, t)) on
In what follows, we shall confirm tmax = ∞by verifying the boundedness of u(·, t) in Ω × (0, tmax).
2.1 Existence of the global solution
Theorem 2.1
For any ϕ ∈ 𝕏+, (1.5) with (1.6) admits a unique nonnegative solution defined on
is ultimately bounded.
Proof. We first prove that the boundedness of u1. From (1.5), u1(x, t) is governed by
By the comparison principle, we directly have
where
where
Next, we verify that the solution u2 and u3 of (1.5) are ultimately bounded. For this purpose, we first establish the uniform estimate in L1(Ω) × L1(Ω) and then pass it to the uniform estimate in C(Ω) × C(Ω).
u(·, t) satisfies the L1 bounded estimate, i.e.,
where 𝕄1 is a positive constant.
By (1.5), we have
where
Further, we have
It follows that
Hence, by taking 𝕄1 = max{M1, M 2}, the assertion directly follows.
Fork ≥ 0, u2 and u3 satisfies the
where M 2k 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 u2 by
Recall that
Hence, (2.9) becomes
where
By lim supt→∞ ∥u1(·, t)∥L1 ≤ M0, there is t0 > 0 such that for t ≥ t0,
and
Applying Young’s inequality:
Thus, (2.10) can be estimated by
where
We multiply
Again applying Young’s inequality (by setting
Hence (2.15) becomes
Combined with (2.14) and (2.17), we obtain
where
Applying interpolation inequality:
Let
Thus, (2.18) becomes
where r* = min
It then follows from (2.8) that lim
Thus, according to continuous embedding Lq(Ω) ⊂ Lp(Ω), q ≥ p ≥ 1, we have
where M p > 0, independent of initial conditions. Denote by Ya, 0 ≤ a ≤ 1 the fractional power space. By [31, Lemma 2.4], we obtain that
2.2 Asymptotic smoothness of Y(t)
We refer the readers to consult [8, 36] for the definition of κ(·), the Kuratowski measure of noncompactness.
Lemma 2.2
The semigroup Y(t) is a κ-contraction on 𝕏+, that is, for any bounded set 𝔹 ⊆ 𝕏+,
where
Proof. For t ≥ 0, let Y(t) = Y1(t) + Y2(t), where
and
By [31, Lemma 2.5], Y1(t)𝔹 is precompact. Thus, we have κ(Y1(t)𝔹) = 0. We estimate Y2(t) as
Hence,
that is, Y(t) is κ-contraction on 𝕏+. This completes the proof. □
Theorem 2.2
(1.5) possesses a global compact attractor in 𝕏+, denoted by 𝒜0.
Proof. By Theorem 2.1, the solution of (1.5) exists globally and Y(t) is point dissipative (see Lemma 2.2). Asymptotic smoothness condition of Y(t) is implied by κ-contraction condition. Hence, by [8, Theorem 2.4.6], system (1.5) possesses a global compact attractor, which attracts every bounded set in 𝕏+. □
3 Basic reproduction number
Obviously, (1.5) has a disease-free steady state
where
Substituting (u2(·, t), u3(·, t)) := eλt(ϕ2(·), ϕ3(·)) into (3.1) gets
We rewrite
It is easy to see that both 𝓑 and B are resolvent-positive operators [27]. Denote by T(t)(resp.
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 ℜ0 is defined as
The following result is a consequence of [27, 30].
Lemma 3.1
Let 𝓑 and ℜ0 be defined in (3.4) and (3.2), respectively. s(𝓑) has the same sign as ℜ0 − 1.
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
That is, 𝓛 = −FB−1. Further, 𝓑 = B + F can be viewed as the perturbation of B. According to [27, Theorem 3.5], s(𝓑) has the same sign as r(−FB−1) − 1 = ℜ0 − 1. □
For the convenience of forthcoming discussions, we next claim that ℜ0 has relationship with other important indicators:
Lemma 3.2
Let ℜ0 be defined by (3.4). Then we have
(i)
(ii) ℜ0 − 1 and s(𝓑) have the same sign as η0, where η0 is the principal eigenvalue of
Proof. We first prove (i). We rewrite
where
As F21 = 0, F22 = 0, by [30, Theorem 3.3], we know that
Hence,
Therefore, ℜ0 satisfies
that is,
This proves (i).
We next prove (ii). In fact, eigenvalue problem (3.8) has a principle eigenvalue η0, associated with a positive eigenfunction ϕ* on Ω, i.e.,
We then multiply the first equation of (3.10) and (3.9) by φ and ϕ*, respectively, then integrate and subtract the equation, obtaining
Since both
Due to the assertion in (i) of Lemma 3.2 and variational formula,
which indicate how ℜ0 depends on the diffusion coefficient d2 (see also in [1, Theorem 2] and [31]).
Theorem 3.1
Let ℜ0 be defined by (3.11). Then we have
(i) Fix d1 > 0, then ℜ0 is decreasing with respect to d2, and satisfies
(ii) Fix d2 > 0, then
and
(iii) If
(iv) If
The following statements come from [13, Theorem 1.1].
Remark 1
It is clear that the asymptotic behavior of the unique positive solution
Remark 2
Fix d2 > 0, η0 of the eigenvalue problem (3.8) satisfies
where
Remark 3
ℜ0,0 − 1 has the same sign as
Theorem 3.2
If ℜ0 ≥ 1 (or s(𝓑) ≥ 0), s(𝓑) is the principal eigenvalue of (3.3).
Proof. In fact, we see from (3.1) that
where
By [31, Lemma 2.5], the compactness of T3(t) directly follows. Further,
that is, ∥T2(t)∥ ≤ e−δ* t.
Hence, for any bounded set 𝕊 in 𝕐,
Thus, T(t) is a κ-contraction on 𝕐, that is, the essential growth bound, ωess(Π(t)) ≤ −δ and the essential spectral radius
It is well-known that ω(T(t)), the exponential growth bound (defined by
With the assumption that s(𝓑) ≥ 0, the spectral radius of T(t),
Consequently, re(T(t)) < r(T(t)). With the help of the generalized Krein-Rutman Theorem (see, for example, [18]), we complete the proof. □
The following result indicates that for a special case, s(𝓑) is the principal eigenvalue of (3.3) without any limitations.
Lemma 3.3
Suppose that δ(x) ≡ δ. s(𝓑) is the principal eigenvalue of (3.3).
Proof. Let
Denote C1 :=
admits one principal eigenvalue,
Then
According to [30, Theorem 2.3], the assertion directly follows. We complete the proof. □
4 Threshold dynamics
We now study the threshold dynamics of model (1.5) as ℜ0 ≤ 1 and ℜ0 > 1.
Theorem 4.1
If ℜ0 ≤ 1 (or s(𝓑) ≤ 0), then E0 is globally asymptotically stable.
Proof. We divide the proof into two parts. One is ℜ0 < 1, the other is ℜ0 = 1. When ℜ0 < 1, we know from [30, Theorem 3.1] that E0 is locally asymptotically stable. We are mow in a position to consider the global attractivity of E0. Fix ϵ0 > 0. By (2.6), there is t1 > 0 such that for all (x, t) ∈ Ω‾× [t1,∞),
Let (û2(·, t), û3(·, t)) is the solution of the following problem
where
By the comparison principle (see, e.g. [16]),
Let Tϵ0 (t) be the positive semigroup induced by 𝓑ϵ0 . We next aim to prove that the exponential growth bound of Tε0 (t), ωϵ0 is negative. Let
It follows from (ii) of Lemma 3.2,ℜ0 < 1 and the continuous dependence of
When ℜ0 = 1, the global asymptotic stability result of E0 needs to combine with the local stability and global attractivity of E0 (see also in [3, 4, 31, 32]). We first confirm the local asymptotic stability of E0. Let ϵ > 0 be given. Choosing ρ > 0 such that for any u0 = (u10, u20, u30), we have ∥ u0 − E0 ∥≤ ρ.
Let us introduce
Hence, the first equation of (1.5) becomes
Solving (4.3) gives
where
where
By a zero trick and the definition of T(t) (positive semigroup generated by 𝓑), we have
Under the condition that ℜ0 = 1 (or s(𝓑) = 0), we know that ∥ T(t) ∥≤ Mτ for t ≥ 0 and Mτ > 0. Hence, by (4.6), we have
where
Further from (4.7) and (4.8), we know that
Thus, we consider the system
Obviously, u1(·, t) ≥ ϑ(·, t), for all
Solving (4.10) yields
where A1(t) is defined in (2.1) and satisfies ∥A1(t)∥ ≤ K0eζ*t for some constant K0. Hence,
Let
Choose ρ > 0 small enough that
By a zero trick,
From
Combined with (4.11) and (4.12), one can get
By choosing ρ sufficiently small,
where ε can be chosen as a small number. This proves the local stability of E0.
Secondly, we confirm the global stability of E0. For any u0 ∈ 𝕏+, we set
and
where (ϕ2, ϕ3) is the positive eigenfunction of s(𝓑) = 0. Recall that (1.5) has a connected global compact attractor in 𝕏+, denoted by 𝒜0 (see Theorem 2.2).
We claim that for any u0 ∈ 𝒜0, ω(u0) ∈ ∂𝕏1. It is clear that for all t > 0, σ(t; u0) > 0 is strictly decreasing. In fact, for
such that
where
Furthermore, by (4.14) and comparison principle, one can see that
Since t̃ 0 is arbitrary, σ(t; u0) is strictly decreasing.
Next, we pay attention to the the semiflow Υ(t), generated by (1.5) (see Theorem 2.1). Let
This means that σ(t; v) = σ*. Repeating the previous arguments, σ(t; v) is strictly decreasing when v2 ≠0 or v3 ≠0. This gives a contradiction. So, we have v2 = v3 = 0. When t → ∞, we get (u2(·, t), u3(·, t)) → (0, 0). It follows that
The following result indicates that (1.5) is uniformly persistent when ℜ0 > 1. Before going into details, we set
and
where Y(t) is the semiflow generated by (1.5).
Theorem 4.2
If ℜ0 > 1, then (1.5) is uniformly persistent, i.e., for any ϕ ∈ 𝕏+ with ϕ2(·)/ ≡ 0, or ϕ3(·)/ ≡ 0, there exists σ > 0,
Moreover, (1.5) possesses at least one positive steady state.
Proof. Theorem 4.2 is achieved by the following claims. The first two claims are the direct consequences as those in [31, Theorem 3.10] with slight modifications. We present them here without proof.
Claim 1 Υ(t)𝕏0 ⊆ 𝕏0 for all t ≥ 0.
Claim 2 ω(u0) = {E0}, for any u0 ∈ M∂, where ω(u0) is the ω limit set of u0.
Claim 3 lim supt→∞ ∥Υ(t)ϕ − E0∥ ≥ σ, ∀ϕ ∈ 𝕏0, where σ is a positive constant.
We prove Claim 3 by the way of contradiction. Assume to the contrary that for σ > 0, there exists t̄1 > 0 such that
Hence (u2(·, t; ϕ), u3(·, t; ϕ)) is the upper solution of
where
Combined with Lemma 3.2 and the continuous dependence of s(𝓑) ≥ 0 on σ, choosing σ > 0 small enough that s(𝓑σ) > 0, associated with positive eigenvalue function
It is easy to see that for
Since s(𝓑σ) > 0, u2(·, t; ϕ), u3(·, t; ϕ) → ∞as t → ∞, a contradiction.
Similar to [25, Theorem 3] (see also in [31, Theorem 3.10]), let us define a distance function ρ(·) : 𝕏+ → [0,∞) by
we conclude that
Recall from Theorem 2.1 that there exist M∞ > 0 and t2 > 0 such that u2(·, t, ϕ), u3(·, t, ϕ) ≤ M∞, (x, t) ∈ Ω × [t2,∞). It then follows that
Thus, there exists a constant σ2, lim inft→∞ u1(·, t, ϕ) ≥ σ2. Let σ0 = min{σ1, σ2}. The uniform persistence is obtained. Further from [15, Theorem 4.7], (1.5) possesses at least one positive steady state. This completes the proof. □
5 Spatially homogeneous case
When
system (1.5) becomes
subject to the same initial and boundary condition as in (1.5). Clearly, system (5.1) possesses a disease-free equilibrium
In this circumstance, Theorems 4.1 and 4.2 still hold for (5.1).
Theorem 5.1
If [ℜ0] < 1, then E0 is globally asymptotically stable; while if [ℜ0] > 1, then (5.1) is uniformly persistent.
The positive equilibrium of (5.1) (whenever it exists) should satisfy
To make
where 𝔵 = ϱβ2(y − ζ) > 0, 𝔹 = bϱβ2 + β1yδ + yδζ − y2δ and ℂ = byδ > 0. Isolating the square root and squaring both sides gives 𝔹 + 2𝔵b/ζ > 0 and
Simplifying the last inequality yields ℂζ 2 < 𝔹bζ +𝔵b2. We make use of the expressions of 𝔵, 𝔹, ℂ and rewrite the inequality as
By expanding and canceling, we obtain yδζ < bϱβ2 + β1δζ, which is the same as [ℜ0] > 1. Furthermore, coupling [ℜ0] > 1 and y > ζ implies 𝔹 + 2𝔵b/ζ > 0. Thus, if [ℜ0] > 1 and y ≥ ζ, (5.1) possesses a unique positive equilibrium.
Remark 4
Note that y and ζ are the death rates of infected and susceptible host, respectively. Due to the disease burden, it is realistic to assume that y ≥ ζ. Combined with the analysis above, (5.1) possesses a unique positive equilibrium.
Theorem 5.2
Suppose that y ≥ ζ hold. If [ℜ0] > 1, then unique positive equilibrium
Proof. Linearizing system (5.1) at
where χ = (u1, u2, u3), D = diag(d1, d2, 0) and
Hence, the characteristic equation of
where l is the wavenumber,
and
It then follows that a1(l2) > 0, a2(l2) > 0 and a3(l2) > 0. Further, after elementary calculations, a1(l2)a2(l2)− a3(l2) > 0. Hence, the local stability of
Theorem 5.3
Assume that
Proof. Define
Since
after elementary calculations, we can obtain
By using u1 ≤ u1 + u2, we have
By assumption (H1), we have
Hence,
Hence, the largest invariant subset
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 ℜ0 > 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 d1 → 0, d1 → ∞, d2 → 0 and d2 → ∞. 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 (u1, u2, u3) be any positive solution of (6.1). We then directly have
(i) For any d1, d2 > 0, we have the upper bound of u1:
(ii) Fix d2 > 0, for any d1 > 0, we have the upper bound of u2 and lower bound of u1:
where the positive constants Ci (i = 1,2) do not depend on d1 > 0.
Proof. (i) Set
Thus, we know that
this gives the upper bound of u1.
(ii) Making the sum of u1, u2 of (6.1) and integrating over Ω lead to
Notice that u2 solves
Fix d2 > 0, it follows from statement (i) that
By using [20, Lemma 2.2] (see also [12]), we get a Harnack-type inequality of u2 as follows
where C > 0 does not depend on d1. In what follows, we permit it changing from place to place.
Based on (6.4) and (6.6), we have
Set
From (6.7), we have
Hence, we get the lower bound of u1. □
Remark 6
Note that the statement (ii) hold for any d1 > 0and d2 ≥ 1. Actually, we can get (6.6) for any d2 ≥ 1.
6.1 Profile as d1 → 0
In view of Theorems 3.1 and 4.2, (6.1) possesses at least a positive solution for sufficiently small d1 when ℜ0,0 > 1. This subsection is spent on exploring the asymptotic profile of the positive solution of (6.1) with d1 → 0.
Theorem 6.1
Fix d2 > 0, and let d1 → 0. If ℜ0,0 > 1, then up to a subsequence of d1, every positive solution (u1(·; d1), u2(·; d1), u3(·; d1)) of (6.1) satisfies
where
and Φu2 is a positive solution to
Proof. We deal with the proof by two steps.
Step 1. Convergence of u2. Observe that u2 satisfies
From the estimates in Lemmma 6.1, we know that ∀ p > 1,
Combined with the standard Lp-estimate for elliptic equations and Sobolev embedding theorem (see, e.g. [7]),
From the third equation of (6.1), it is clear that
Denoted by d1n the subsequence of d1 → 0, satisfying d1n → 0 as n → ∞. With this subsequence, (u1n , u2n , u3n) := (u1(x; d1n), u2(x; d1n), u3(x; d1n)) of (6.1) with d1 = d1n satisfies
where Φu2 ∈ C 1
We next prove that Φu2 > 0 on
Choose
Hence, u1n satisfies
and
respectively, with
and
It follows from the super-subsolution argument that
By applying the singular perturbation theory technique (see, e.g. [5, Lemma 2.4] or [13, Lemma 2.1]), we know that as n → ∞,
Hence,
Since ϵ is arbitrary, (6.16) further implies that
We now pay attention to u2 equation of (6.1) with u2n:
For all n ≥ 1, define
As above, with the aid of standard compactness argument, as n → ∞,
where
From [20, Lemma 2.2] (see also [12]), it follows that the Harnack-type inequality of
Step 2. Convergence of u1. From (6.1), we know that u1n solves:
By (6.20), choose ϵ > 0 small enough that 0 < Φu2 − ϵ ≤ u2n ≤ Φu2 + ϵ on
where
Consider the following problem
Then, for sufficiently large C > 0, (u1n , C) is a pair of sub-supersolution of (6.21) on
Hence,
On the other hand,
with
where
Since
then as n → ∞,
As a result, by (6.18), Φu2 satisfies (6.9). This proves Theorem 6.1. □
6.2 Profile as d2 → 0
This subsection is spent on exploring the asymptotic profile of the positive solution of (6.1) with d2 → 0. In this case, because of mathematical difficulty, we only consider the limiting profile of positive solutions in 1-D space. Without loss of generality, we take Ω = (0, 1).
Theorem 6.2
Assume that
where
and Ψu1 solves
Proof. From the estimats (6.2) and (6.4), together with the elliptic L1-estimate theory (see, e.g. [2] or [12, Lemma 2.2]),
Choose p large enough, combined with the Sobolev embedding theorem, that
Denoted by d2k the subsequence of d2 → 0, satisfying d2k → 0 as k → ∞. With this subsequence, (u1k , u2k , u3k) := (u1(·; d2k), u2(·; d2k), u3(·; d2k)) of (6.1) with d2 = d2k satisfies u1k → Ψu1 in C([0, 1]) as k → ∞.
We note that u2k satisfies
By a similar super-subsolution argument (see Theorem 6.1), we get
where Ψu2 is given by (6.24). Moreover, from the expression of Ψu2, it is clear that Ψu1 solves (6.2). The proof is complete.
6.3 Profile as d1 → ∞
In view of Theorems 3.1 and 4.2, (6.1) possesses at least a positive solution for sufficiently small d1 when ℜ0,∞ > 1. This subsection is spent on exploring the asymptotic profile of the positive solution of (6.1) with d1 → ∞.
Theorem 6.3
Assume that ℜ0,∞ > 1. Fix d2 > 0, and let d1 → ∞, then every positive solution (u1(·; d1), u2(·; d1), u3(·; d1)) of (6.1), up to a subsequence of d1, satisfies
where
and
Proof. Rewriting u1 equation of (6.1) as
From the estimates of u1 and u2 in Lemma 6.1, we know that
where C > 0 does not dependent on d1 ≥ 1. Combined with the standard Lp-estimate for elliptic equations and Sobolev embedding theorem,
Denoted by d1i the subsequence of d1 → ∞, satisfying d1i → 0 as i → ∞. With this subsequence, (u1i, u2i, u3i) := (u1(·; d1i), u2(·; d1i), u3(·; d1i)) of (6.1) with d1 = d1i satisfies u1i →
Similar arguments as before, as i → ∞,
where
As a result, ũ2i →
It can be seen that the Harnack-type inequality of
6.4 Profile as d2 → ∞
This subsection is spent on exploring the asymptotic profile of the positive solution of (6.1) with d2 → ∞.
Theorem 6.4
Assume
every positive solution (u1(·; d2), u2(·; d2), u3(·; d2)) of (6.1), up to a subsequence of d2, satisfies
where (u1∞, u2∞) solves
and u2∞ is a positive constant and u1∞ > 0 on
Proof. From the discussion in Lemma 6.1 and Remark 6, it is clear that the estimats (6.6) and (6.7) hold for all large d2, and C > 0 does not depend on d1 and d2 ≥ 1.
Denoted by d2j the subsequence of d2 → ∞, satisfying d2j → 0 as j → ∞. With this subsequence, combined with the standard Lp-estimate for elliptic equations and Sobolev embedding theorem, (u1j , u2j, u3j) := (u1(·; d2j), u2(·; d2j), u3(·; d2j)) of (6.1) with d2 = d2j satisfies
where u1∞ > 0 on
We next show that u2∞ is a positive constant. By a similar argument as before, suppose that u2∞ ≡ 0. For all j ≥ 1, define ũ2j := u2j/∥u2j∥L∞(Ω) that ∥ũ2j∥L∞(Ω) = 1, and ũ2j satisfies
Thus, if necessary, by passing to a further subsequence of d2j, one can get
On the other hand, since u2j → 0 as j → ∞, it follows from a super-subsolution argument that u1j →
a contradiction. Therefore, u2∞ is a positive constant, and (u1∞, u2∞) satisfies (6.28). The proof is complete. □
7 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 y(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 𝒜0 (see Theorem 2.2).
The BRN, ℜ0, 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 howℜ0 depends on the diffusion coefficient d1 for d1 → 0 and d1 → ∞ (see Theorem 3.1). We have also confirmed the global stability of E0 in a critical case that ℜ0 = 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 ℜ0 = 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 ℜ0 > 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 d1 → 0, d1 → ∞, d2 → 0 and d2 → ∞. When d1 → 0, our result (Theorem 6.1) demonstrate that host individuals distribute on
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],
Acknowledgments
J. Wang was supported by National Natural Science Foundation of China (nos. 12071115, 11871179), Natural Science Foundation of Heilongjiang Province (nos. LC2018002, LH2019A021) and Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems. R. Cui was supported by Natural Science Foundation of Heilongjiang Province (no. LH2020A012), Heilongjiang Postdoctoral Science Foundation (no. LBH-Q17086) and the Fundamental Research Funds for Heilongjiang Provincial Universities (no. 2018-KYYWF-0996).
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