A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment

In this paper, we study the effects of diffusion and advection for an SIS epidemic reaction-diffusion-advection model in a spatially and temporally heterogeneous environment. We introduce the basic reproduction number \begin{document} $\mathcal{R}_{0}$ \end{document} and establish the threshold-type results on the global dynamics in terms of \begin{document} $\mathcal{R}_{0}$ \end{document} . Some general qualitative properties of \begin{document} $\mathcal{R}_{0}$ \end{document} are presented, then the paper is devoted to studying how the advection and diffusion of the infected individuals affect the reproduction number \begin{document} $\mathcal{R}_{0}$ \end{document} for the special case that \begin{document} $γ(x,t)-β(x,t) = V(x,t)$ \end{document} is monotone with respect to spatial variable \begin{document} $x$ \end{document} . Our results suggest that if \begin{document} $V_{x}(x,t)≥0,\not\equiv0$ \end{document} and \begin{document} $V(x, t)$ \end{document} changes sign about \begin{document} $x$ \end{document} , the advection is beneficial to eliminate the disease, whereas if \begin{document} $V_{x}(x,t)≤0,\not\equiv0$ \end{document} and \begin{document} $V(x, t)$ \end{document} changes sign about \begin{document} $x$ \end{document} , the advection is bad for the elimination of disease.

1. Introduction. The SIS (susceptible-infected-susceptible) models provide essential frames in studying the dynamics of disease transmission in the field of theoretical epidemiology. Recently Allen et al. [3] proposed a frequency-dependent SIS model with a no-flux boundary condition S+I + γ(x)I, x ∈ Ω, t > 0, I t = d I ∆I + β(x)SI S+I − γ(x)I, x ∈ Ω, t > 0, ∂S ∂n = ∂I ∂n = 0, x ∈ ∂Ω, t > 0, S(x, 0) = S 0 (x) ≥ 0, I(x, 0) = I 0 (x) ≥ 0, x ∈ Ω, (1) to investigate the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, where S(x, t) and I(x, t) denote the density of susceptible and infected individuals in a given spatial region Ω, which is assumed to be a bounded domain in R m (m ≥ 1) with smooth boundary ∂Ω; the positive constants d S and d I are diffusion coefficients for the susceptible and infected populations; the positive functions β(x) and γ(x) are Hölder continuous onΩ and represent the rates of disease transmission and recovery at location x, respectively. The homogeneous Neumann boundary conditions mean that there is 4558 DANHUA JIANG, ZHI-CHENG WANG AND LIANG ZHANG no population flux across the boundary ∂Ω and both the susceptible and infected individuals live in a self-contained environment.
Regarding (1), the main results of [3] concern the existence, uniqueness and asymptotic behaviors of the endemic equilibrium as the diffusion rate of the susceptible individuals approaches to zero. Allen et al. [2] also investigated a discrete SIS model associated with (1). Peng and Liu [19] discussed the global stability of the endemic equilibrium in some special cases. Further results concerning the asymptotic behavior of the endemic equilibrium of (1) were derived by [18,20]. Peng and Zhao [21] treated the system (1) with β and γ being functions of spatiotemporal variables and temporally periodic, and found that the combination of spatial heterogeneity and temporal periodicity can enhance the persistence of the disease. Huang et al. [11] studied the global dynamics of system (1) subject to the Dirichlet boundary conditions. Ge et al. introduced a free boundary model for characterizing the spreading front of the disease in [7]. Li et al. [13] performed qualitative analysis on an SIS epidemic reaction-diffusion system with a linear source in spatially heterogeneous environment, which the main feature lies in that its total population number varies compared to model (1). In these works the populations are assumed to adopt random diffusion in the habitats.
In some circumstance, populations may take biased or passive movement in certain direction, e.g., due to external environmental forces such as water flow [14,15,16], wind [6] and so on, which usually can be described by adding an advection term to the existing reaction-diffusion models. Recently, Cui et al. [4,5] considered an SIS epidemic with advection, the results in [4] are in strong contrast with the case of no advection, and the results in [5] suggest that advection can help speed up the elimination of disease. For the SIS epidemic reaction-diffusionadvection model in [4,5], it was assumed that the rates of disease transmission and recovery depend only on the spatial variable. However, the rates of disease transmission and disease recovery may be spatially and temporally heterogeneous. Typically, they vary periodically in time, for example, due to the seasonal fluctuation and periodic availability of vaccination strategies.
Here, we consider the following SIS epidemic reaction-diffusion-advection model in a spatially heterogeneous and temporally periodic environment: where the functions β(x, t) and γ(x, t) represent the rates of disease transmission and recovery at location x and time t, respectively; L is the size of the habitat, and we call x = 0 the upstream end and x = L the downstream end; q is the effective speed of the current (sometimes we call q the advection speed/rate, and we remark here that q should be non-negative since x = L is defined to be the downstream end). Here we impose no-flux boundary conditions at the upstream and downstream ends, respectively. It means that there is no population net flux across the boundary x = 0 and x = L.
As the term SI/(S + I) is a Lipschitz continuous function of S and I in the open first quadrant, we can extend its definition to the entire first quadrant by defining it to be zero when either S = 0 or I = 0. Following [3], we further assume that at the initial time, there is a positive number of infected individuals, that is, (A1) S(x, 0), I(x, 0) are continuous on [0, L], S(x, 0) ≥ 0, I(x, 0) ≥ 0 for x ∈ (0, L), and L 0 I(x, 0)dx > 0. Concerning the functions β(x, t) and γ(x, t), throughout this paper we impose the following hypothesis: (A2) β(x, t) and γ(x, t) are Hölder continuous and positive on [0, L] × R, and periodic in time with the same period ω for some constant ω > 0. As discussed in [4,5], we know that (2) has a unique classical solution (S, I) ∈ C 2,1 ([0, L] × (0, ∞)) which exists globally in time. Let be the total number of individuals in (0, L) at t = 0. Summing two equations of (2) and integrating over (0, L) gives Thus the total population size is constant in time, i.e., which also shows that both S(·, t) L 1 ([0,L]) and I(·, t) L 1 ([0,L]) are uniformly bounded for t ∈ [0, ∞). From now on, we always assume that (A1) and (A2) hold and N is a given positive constant throughout this paper. We say (S,Ĩ) is an ω-periodic solution of (2), if it is a nonnegative classical solution of the associated periodic-parabolic problem: By a disease-free ω-periodic solution of (2), we mean that (S,Ĩ) is a nonnegative solution to (5) in whichĨ ≡ 0 on [0, L] × R; and (S,Ĩ) is said to be an endemic ω-periodic solution ifĨ ≥ 0, ≡ 0 on [0, L] × R. It is easy to observe from (5) that the unique disease-free ω-periodic solution is (S * , 0) = ( qN e (q/d S )x d S (e qL/d S −1) , 0) (see [4]). It follows from the maximum principle for parabolic equations that the endemic To give biological interpretations of our analytical results more clearly, we adopt the terminology analogous to that in [3,21]. We say that x is a low-risk site if the local disease transmission rate ω 0 β(x, t)dt is lower than the local disease recovery rate ω 0 γ(x, t)dt. A high-risk site is defined in a reversed manner. We also call that The main goal of our current work is to investigate the effect of advection and diffusion in a spatially heterogeneous and temporally periodic environment on the persistence and extinction of the infectious disease for (2). Of particular interest is the basic reproduction number which will serve as the threshold value for persistence and extinction of the disease.
In this paper, we first introduce the basic reproduction number R 0 for (2), which contains the results in [21] for the case q = 0 and in [4,5] for the case β(x, t) ≡ β(x), γ(x, t) ≡ γ(x) as special cases. The threshold-type dynamics for the system (2) is established by monotone dynamics, which says that if R 0 ≤ 1, the disease-free solution is globally stable, while if R 0 > 1, (2) admits at least one endemic ωperiodic solution and the disease is uniformly persistent. Compared to the previous references mentioned above [3,4,13,18,19,20,21], where the global asymptotic stability of the disease-free equilibrium was proved under the stronger assumption R 0 < 1, our general results that disease-free solution is globally stable provided R 0 ≤ 1 for nonautonomous system, seems to be new and applicable for other SIS tye PDE models. Let In particular, we consider the special case: V (x, t) is monotone with respect to spatial variable x. Biologically, if V (x, t) is monotone increasing with respect to x, i.e., V x (x, t) ≥ 0, ≡ 0, it means that for any given time, the spatial change rate of disease recovery γ x (x, t) is greater than the spatial change rate of disease transmission β x (x, t), while if V (x, t) is monotone decreasing with respect to x, it has the reversed biological meaning. Then we study the effect of the advection rate q and mobility of the infected individuals d I on R 0 , and the results obtained are explicit.
The rest of our paper is arranged as follows. In Section 2, we first introduce the basic reproduction number R 0 and then establish the threshold dynamics in the terms of R 0 . Section 3 is concerned with some general qualitative properties of R 0 . Section 4 is devoted to the effect of the advection rate q and mobility of the infected individuals d I on R 0 when V (x, t) is monotone with respect to spatial variable x.

2.
Threshold dynamics in terms of R 0 . In this section, we first introduce the basic reproduction number R 0 for the periodic reaction-diffusion-advection system (2) and then we establish its threshold-type dynamics in terms of R 0 . As a first step, we need to define the next infection operator for (2), which arises from the combination of the idea in [25] for periodic ordinary differential models and that in [26] for autonomous reaction-diffusion systems, see also [21].
Since the environment of our model (2) is periodic in time, we let X = C([0, L], R) denote the Banach space of continuous functions on the interval [0, L] with the supremum norm u ∞ = max x∈[0,L] |u(x)| and C ω be the ordered Banach space consisting of all ω-periodic and continuous functions from R to X, which is equipped with the maximum norm · and the positive cone For any given φ ∈ C ω , we also use the notation φ(x, t) := φ(t)(x). Let Γ(t, s) be the evolution operator of the reaction-diffusion-advection equation It follows from the standard semigroup theory that there exist positive constants K and c 0 such that Let φ ∈ C ω and suppose that φ(x, s) is the density distribution of the infected individuals at the spatial location x ∈ (0, L) and time s. Then the term β(x, s)φ(x, s) means the density distribution of the new infections produced by the infected individuals who were introduced at time s. Thus, for given t ≥ s, Γ(t, s)β(x, s)φ(x, s) is the density distribution at location x of those infected individuals who were newly infected at time s and remain infected at time t. Therefore, represents the density distribution of the accumulative new infections at location x and time t produced by all those infected individuals φ(x, s) introduced at all the previous time to t.
As in [25] and [21], we introduce the linear operator L : C ω → C ω : which we call as the next infection operator. Under our assumption on β and γ, it is easy to see that L is continuous, compact on C ω and positive (namely, L(C + ω ) ⊂ C + ω ). We define the spectral radius of L as the basic reproduction number R 0 = ρ(L) for system (2).
In what follows, we first want to obtain an equivalent characterization of the basic reproduction number R 0 . This thus leads us to consider the following linear periodic-parabolic eigenvalue problem By [10, Theorem 16.1], problem (6) has a unique principal eigenvalue µ 0 , which is positive and corresponds to an eigenvector ψ ∈ C ω and ψ > 0 on [0, L] × R. Furthermore, we have the following observation, and the proof is similar to the argument of Lemma 2.1 in [21].
It then easily follows that (R 0 , ϕ) satisfies which can be rewritten as It is also well known (see [10,Theorem 7.2]) that R 0 is the principal eigenvalue of the adjoint problem of (8): To obtain a better understanding of R 0 , sometimes we need to resort to the following periodic-parabolic eigenvalue problem Let λ 0 be the unique principal eigenvalue of (9). Then we have the following result.
Proof. We rewrite (9) as the following (10) Due to [10, Theorem 7.2], we know that λ 0 is also the principal eigenvalue of the adjoint problem of (10), that is, there exists We multiply the equation (8) by θ * and (11) by ϕ, respectively, integrate over (0, L) × (0, ω) by parts, and then subtract the resulting equation to obtain x ϕθ * dxdt are both positive, we conclude that 1 − 1 R0 and λ 0 have the opposite signs, which thereby deduces our result. Next, we establish the threshold dynamics behavior of (2) in terms of R 0 . We start with the uniform bound of the solution when the initial data (S 0 , I 0 ) satisfy (A1), which is fundamental in determining the long-time behavior of solutions of (2). In fact, under hypothesis (3)(and so (4) holds), it can be concluded that for any fixed q ≥ 0, S(·, t) L ∞ ([0,L]) and I(·, t) L ∞ ([0,L]) are also uniformly bounded in [0, ∞), by following the argument in [1](see also Exercise 4 of Section 3.5 in [9]). Lemma 2.3. There exists a positive constant C independent of the initial data (S 0 , I 0 ) satisfying (A1) such that for the corresponding unique solution (S, I) of (2), we have Now we are in a position to prove a threshold type result on the global dynamics of (2), which implies that the basic reproduction number R 0 can be used to predict the extinction and persistence of the disease.
Theorem 2.4. The following statements are valid.
has at least one endemic ω-periodic solution, and for any solution (S, I) of (2) with the initial data (S 0 , I 0 ) satisfying (A1), there exists a constant η > 0 such that Proof. (i) By the strong maximum principle [23], both S(x, t) and S(x, t).
, we obtain thatĪ(x, t) = 0 is globally attractive for T in X + , and the claim follows. Since We obtain that S j (x, t) = qN e (q/d S )x d S (e qL/d S −1) = S * . Then we have lim t→+∞ S j (x, t) = S * for each j ∈ R. Hence, S(x, t) → S * as t → ∞. This completes the proof of assertion (i).
(ii) We appeal to the theory of uniform persistence and coexistence states developed in [17,27] for periodic semiflow to prove (ii). We denote By the standard regularity theory for parabolic equations and the property (4), for every (S 0 , I 0 ) ∈ U , (2) admits a unique solution ψ(t, (S 0 , I 0 )) = (S, I) ∈ U , which exists for any t ≥ 0. We now define an ω-periodic semiflow Φ(t) : U → U by We first prove the uniform persistence of the Poincaré map T : With the above argument, it is easy to see that T : U → U is a continuous, point dissipative and compact map with T (U 0 ) ⊂ U 0 and T (∂U 0 ) ⊂ ∂U 0 . By [8,Theorem 2.4.7], T : U → U has a global attractor. Denote by ω(S 0 , I 0 ) the associated omega-limit set of (S 0 , I 0 ) ∈ U . When (S 0 , I 0 ) ∈ ∂U 0 , we know that I(x, t) ≡ 0, and so S(x, t) satisfies the following: Claim. There exists a real number δ > 0 such that It suffices to prove that there exist δ 0 > 0 such that for any (S 0 , I 0 ) ∈ B(Q, δ 0 ) ∩ U 0 , where B(Q, δ 0 ) is the δ 0 -neighborhood of Q, there exists n = n 0 ≥ 1 such that T n ((S 0 , I 0 )) ∈ B(Q, δ 0 ).
3. Qualitative properties of R 0 . In this section, we will present some quantitative properties for the basic reproduction number R 0 . First of all, when β(x, t) − γ(x, t) or both β(x, t) and γ(x, t) are spatially homogeneous, we have: Lemma 3.1. The following assertions hold.
The proof of Lemma 3.1 is similar to that of [21,Lemma 2.3]. Secondly, if β(x, t) − γ(x, t) or both β(x, t) and γ(x, t) depend on the spatial variable alone, we have the following results.
and the following statements about R 0 hold: that is, ϕ x ≡ 0 and equivalently, ϕ(x, t) ≡ ϕ(t). Hence, (7) becomes equivalent to Then it is easy to see that  From (7), it can be seen that R 0 is a smooth function of d I and q. If the advection rate q = 0, we denote the basic reproduction number by R 0 , so R 0 is a smooth function of d I only. The basic reproduction number R 0 was introduced in [21], where R 0 is a threshold value for the global dynamics. Next, by using Propositions 1, 2 and 3 in Appendix, we immediately establish the limiting behaviors of R 0 as follows.
(ii) For any given d I > 0, we have where R 0 = µ 0 is the principal eigenvalue of (6) corresponding to q = 0.
Let γ(x, t) − β(x, t) = V (x, t). We see that V (x, t) may change sign with respect to x or t. In particular, if V (x, t) stays the same sign, we have the following results. Proof. (i) We subtract both sides of (7) by β(x, t)ϕ, multiply by e (q/d I )x ϕ, and integrate by parts over (0, L) × (0, ω) to obtain which implies that R 0 < 1.
(iii) By virtue of statements (i) and (ii), we can easily see that (iii) holds.

4.
Special case: γ(x, t)−β(x, t) is monotone with respect to spatial variable x. In general, the principal eigenvalue λ 0 of problem (9) does not enjoy any monotonicity property on either d I or q. In this section, we investigate the monotonicity results of λ 0 on d I and q when γ(·, t), β(·, t) ∈ C 1 ([0, L]) and γ(x, t)−β(x, t) is monotone with respect to spatial variable x. Precisely, let γ(x, t) − β(x, t) = V (x, t), we consider V (x, t) satisfying one of the following assumptions: We remark that it is biologically reasonable for the assumptions of γ(x, t)−β(x, t). Due to the spatial heterogeneity of environment and movements of individuals, the rate of disease transmission β(x, t) and the rate of disease recovery γ(x, t) will change with spatial locations, and the spatial change rate of them maybe different. Assumption (C1) implies that for any time, the spatial change rate of disease recovery γ x (x, t) is greater than the spatial change rate of disease transmission β x (x, t), while assumption (C2) means that the spatial change rate of disease recovery γ x (x, t) is less than the spatial change rate of disease transmission β x (x, t).

4.1.
Monotonicity of λ 0 in d I and q. In this subsection, we explore the monotonicity of the principal eigenvalue λ 0 with respect to d I and q. Note that γ(x, t) − β(x, t) = V (x, t), then the eigenvalue problem (9) reads as  
Lemma 4.1. For any given d I , q > 0, the following assertions hold.
Proof. It is well known (see, e.g., [12]) that λ 0 and the associated principal eigenfunction θ are C 1 -function of q. For notational simplicity, we denote ∂θ ∂q by θ and ∂λ0 ∂q by λ 0 . Then, we differentiate (19) with respect to q to obtain We rewrite (22) as Let ψ be the principal eigenfunction corresponding to λ 0 in (21). We now multiply the equation of (23) by ψ and integrate the resulting equation over (0, L)×(0, ω) In view of the equation of ψ in (21), we find From (24) and (25), it immediately follows that Thanks to Lemma 4.1 and the positivity of θ and ψ, if (C1) holds, we have λ 0 > 0, while if (C2) holds, we have λ 0 < 0. So our conclusions are established. Proof. Clearly, λ 0 and the associated principal eigenfunction θ are C 1 -function of d I (see, e.g., [12]). For simplicity, we denote ∂θ ∂d I by θ and ∂λ0 ∂d I by λ 0 . Then, differentiating (19) with respect to d I , we have which can be further rewritten as Let ψ be the principal eigenfunction corresponding to λ 0 which satisfies (21). Multiplying the equation of (26) by ψ and integrating the resulting equation over (0, L) × (0, ω), we obtain Substituting −ψ t = d I ψ xx + qψ x − V (x, t)ψ + λ 0 ψ to (27), we get By virtue of (21), it is easy to see that ξ(x, t) = ψ(x, −t) solves By the similar arguments as [22, Lemma 4.1], we have that if assumption (C1) holds, then ξ x (x, t) = ψ x (x, −t) < 0, while if assumption (C2) holds, then ξ x (x, t) = ψ x (x, −t) > 0. In light of this fact and Lemma 4.1, if (C2) holds, it follows from (28) that λ 0 > 0 for all q > 0. The proof is complete. (28), we see that if assumption (C1) holds, we are not sure whether λ 0 is monotone with respect to d I for any given q > 0.
By Propositions 1, 2, 3 in the Appendix and the similar arguments to Lemma 3.3(i), we have the following results. In this subsection, we study the effects of d I and q on R 0 when V (x, t) satisfies assumption (C1) or (C2), and V (x, t) solely changes sign about spatial factor x, which means the sign of V (x, t) completely depends on x. Specifically, we investigate the impact of d I and q on R 0 in different habitats. Here, according to the definitions of high-risk and low-risk habitat presented in Introduction, we see that (0, L) is a high-risk habitat if ω 0 L 0 V (x, t)dxdt < 0, and a low-risk habitat if ω 0 L 0 V (x, t)dxdt > 0. Theorem 4.5. Assume that V (x, t) solely changes sign about x. Then for given d I > 0, the following statements on R 0 hold. (c) in case of R 0 < 1, there exists a unique q + > 0 such that R 0 ≤ 1 for 0 < q ≤ q + , and R 0 > 1 for q > q + .
From the biological point of view, Theorem 4.5 reveals that for given diffusion rate of the infected individuals, the advection which causes the individuals to concentrate at the downstream end, has various influences on the disease in different environments.
Statement (i) illustrates that if the spatial change rate of disease recovery is greater than the spatial change rate of disease transmission and the downstream end is always a low-risk site for any time, we see that in the high-risk habitat, there exists a unique critical advection speed such that the disease persists if the advection speed is less than the critical value, and the disease will be eliminated if the advection rate is greater than the critical value, while in the low-risk habitat, if the disease can not persist without advection, then it can not persist with any advection, but if the disease persists without advection, there exists a unique critical advection speed such that the disease can be eliminated if and only if the advection speed is greater than the critical value. That is, all in all, the advection is beneficial to eliminate the disease in such a condition. Statement (ii) displays that if the spatial change rate of disease recovery is less than the spatial change rate of disease transmission and the downstream end is always a high-risk site for any time, we see that in the high-risk habitat, the disease persists for any advection speed, while in the low-risk habitat, if the disease persists without advection, then it persists with any advection, but if the disease can be eliminated without advection, then either the disease persists for any advection speed, or there exists a unique critical advection speed such that the disease will be eliminated if and only if the advection speed is less than the critical value. Therefore, in this case, the advection is bad for the elimination of the disease.
Theorem 4.6. Assume that V (x, t) solely changes sign about x. Then for given q > 0, the following statements on R 0 hold.
Then there exist 0 < d I ≤ d I , such that λ 0 > 0 provided 0 < d I < d I , and λ 0 < 0 Then either λ 0 > 0 for any d I > 0, or there exists two numbers 0 < d − Furthermore, it follows from Lemma 4.3 that λ 0 is monotone increasing in d I if (C2) holds. Then λ 0 < 0 for all d I > 0 for this case. On the other hand, if By virtue of the monotonicity of λ 0 in d I , there exists a unique critical number d * I > 0 such that λ 0 = 0 for d I = d * I , λ 0 < 0 for 0 < d I < d * I , and λ 0 > 0 for d I > d * I . Hence, it follows from Lemma 2.2 that assertion (ii) holds. Part (i) of Theorem 4.5 shows that for a given advection speed, if the spatial change rate of disease recovery is greater than the spatial change rate of disease transmission and the downstream end is always a low-risk site for any time, we see that in the high-risk habitat, the disease will be eliminated provided the diffusion rate of the infected individuals is small enough, while in the low-risk habitat, the disease will be eliminated provided the diffusion rate of the infected individuals is small or large enough.
Part (ii) shows that for a given advection speed, if the spatial change rate of disease recovery is less than the spatial change rate of disease transmission and the downstream end is always a high-risk site for any time, we see that in the high-risk habitat, the disease persists for any diffusion rate of the infected individuals, but in the low-risk habitat, there exists a unique critical diffusion rate of the infected individuals, such that the disease will be eliminated if and only if the diffusion rate of the infected individuals is greater than the critical value.
5. Discussion. In this paper, we have investigated an SIS epidemic reactiondiffusion-advection model where spatial heterogeneity and temporal periodicity are incorporated, which makes it more reasonable for describing the transmission of infectious disease. We have introduced the basic reproduction number R 0 and shown that R 0 predicts the threshold dynamics of (2) by persistence theory. Then we have discussed in detail the analytical properties of R 0 . The effects of advection and diffusion on the persistence of the disease are further studied for the special case that γ(x, t) − β(x, t) = V (x, t) is monotone with respect to spatial variable x and solely changes sign about x. The results show that if V x (x, t) ≥ 0, ≡ 0 and V (x, t) changes sign about x, the advection is beneficial to eliminate the disease, if V x (x, t) ≤ 0, ≡ 0 and V (x, t) changes sign about x, the advection is bad for the elimination of disease. In addition, the effects of diffusion on the persistence of the disease depends on the habitat is high-risk or low-risk.
In the future work, we would like to study the effects of advection and diffusion on the persistence of the disease provided that V (x, t) changes sign about spatial variable x and temporal variable t and V (x, t) has no monotonicity in x or t. In particular, we assume that V (x, t) = f (x)g(t), where f (x) is a Hölder continuous function on [0, L], and g(t) is a ω-periodic Hölder continuous function on R. Then we distinguish three different cases: (i) Spatial changing sign: f (x) changes sign in (0, L) and g(t) > 0 in R; (ii) Temporal changing sign: f (x) > 0 in (0, L) and g(t) changes sign in R; (iii) Full changing sign: both f (x) changes sign in (0, L) and g(t) changes sign in R. By Lemma 4.4, it is clear that when d I → 0 or q → ∞, the sign of the principal eigenvalue λ 0 depends on V (L, t), but how λ 0 varies with d I and q is unknown. We would like to discuss f (x) and g(t) change sign once firstly, then twice, and so forth.
The key problem is how to present the qualitative properties of λ 0 with respect to d I and q in this case to see the effects of advection and diffusion on the persistence of the disease. Therefore, it is a challenging and interesting work for us. 11731005, 11701242) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-ot09, lzujbky-2017-27) and NSF of Gansu Province, China (1606RJZA069).
Appendix. We consider the linear periodic-parabolic eigenvalue problem in one space dimension: where D, α, L are constants with D, L > 0, the functions a, h and V are Hölder continuous and periodic in t with the same period ω. Moreover, it is assumed that a(x, t), h(x, t) > 0, ∀(x, t) ∈ [0, L] × [0, ω]. The constants D and α stand for the diffusion and advection (or drift) coefficients, respectively.
Let us denote by λ N 1 (α, D) the principal eigenvalue of (29). In [22], for the weight function m(x, t) ≡ 1, the asymptotic behaviors of the principal eigenvalue λ = λ 1 as D goes to zero or infinity and α goes to infinity were investigated. By modifying the arguments in [22] slightly, we see that the following three results hold true.