Dynamics of an epidemic model with advection and free boundaries

Abstract: This paper deals with the propagation dynamics of an epidemic model, which is modeled by a partially degenerate reaction-diffusion-advection system with free boundaries and sigmoidal function. We focus on the effect of small advection on the propagation dynamics of the epidemic disease. At first, the global existence and uniqueness of solution are obtained. And then, the spreading-vanishing dichotomy and the criteria for spreading and vanishing are given. Our results imply that the small advection make the disease spread more difficult.


Introduction
In order to describe the evolution of fecal-oral transmitted diseases in the Mediterranean regions, Capasso and Paveri-Fontana [1] proposed the following model where a, b, c are all positive constants, u(t) and v(t) denote the concentration of the infectious agent in the environment and the infective human population respectively. The coefficients a and b are the intrinsic decay rates of the infectious agent and the infective human population respectively, c represents the multiplication rate of the infectious agent due to the human infected population. The function G(u) stands for the force of infection of the human population due to the concentration of infectious agent. We assume that G(u) satisfies the two specific cases: (i) a monotone increasing function with constant concavity; (ii) a sigmoidal function of bacterial concentration tending to some finite limit, and with zero gradient at zero. These two cases contain most of the features of forces of infection in real epidemics. For some epidemic, if the density of infectious agent is small, the force of infection of the humans will be weak and may tend to zero, and the function G will satisfy case (ii). In this paper, we focus on such case, and assume that the function G : R + → R + satisfies: (G1) G ∈ C 2 (R + ), G(0) = 0, G (z) > 0 for any z > 0 and lim z→∞ G(z) = 1; (G2) there exists ξ > 0 such that G (z) > 0 for z ∈ (0, ξ) and G (z) < 0 for z ∈ (ξ, ∞).
Under two specific cases stated above, the global dynamics of the cooperative system (1.1) has been described in detail in [2]. It follows from [2,Theorem 4.3] that the global dynamics of (1.1) under conditions (G1) and (G2) can be described as follows: (i) If θ < 1 and G(z) z < ab c for any z > 0, then the trivial solution is the only equilibrium for problem (1.1) and it is globally asymptotically stable in R + × R + .
(iii) If θ < 1 and G(z 1 ) z 1 > ab c for some z 1 > 0, then problem (1.1) has three equilibrium points: where 0 < K 1 < K 2 are the positive roots of G(z) − ab c z = 0. In this case, E 1 is a saddle point, E 0 and E 2 are stable nodes.
In 1997, Capasso and Wilson [3] further considered spatial variation and studied the problem where Ω is bounded. By some numerical simulation, they speculated that the dynamical behavior of system (1.2) is similar to the ODE case. To understand the dispersal process of epidemic from outbreak to an endemic, Xu and Zhao [4] studied the bistable traveling waves of (1.2) in x ∈ R. The epidemic always spreads gradually, but the works mentioned above are hard to explain this gradual expanding process. To describe such a gradual spreading process, Du and Lin [5] introduced the free boundary condition to study the invasion of a single species. They considered the problem and showed that (1.3) admits a unique solution which is well-defined for all t ≥ 0 and spreading and vanishing dichotomy holds. Moreover, the criteria for spreading and vanishing are obtained: (i) for h 0 ≥ π 2 d a , the species will spread; (ii) for h 0 < π 2 d a and given u 0 (x), there exists µ * such that the species will spread for µ > µ * , and the species will vanish for 0 < µ ≤ µ * . Finally, they gave the spreading speed of the spreading front when spreading occurs. Since then, many problems with free boundaries and related problems have been investigated, see e.g. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and their references.
In 2016, Ahn et al. [23] considered (1.2) with monostable nonlinearity and free boundaries. They obtained the global existence and uniqueness of the solution and spreading and vanishing dichotomy. Furthermore, by introducing the so-called spatial-temporal risk index ab ≤ 1, the epidemic will vanish; (ii) if R F 0 (0) ≥ 1, the epidemic will spread; (iii) if R F 0 (0) < 1, epidemic will vanish for the small initial densities; (iv) if R F 0 (0) < 1 < R 0 , epidemic will spread for the large initial densities. Recently, Zhao et al. [24] determined the spreading speed of the spreading front of problem described in [23].
Inspired by the work [23], we want to study (1.2) with bistable nonlinearity and free boundaries. Meanwhile, we also want to consider the effect of the advection. In 2009, Maidana and Yang [25] studied the propagation of West Nile Virus from New York City to California. In the summer of 1999, West Nile Virus began to appear in New York City. But it was observed that the wave front traveled 187 km to the north and 1100 km to the south in the second year. Therefore, taking account of the advection movement has the greater realistic significance. Recently, there are some works considering the advection. In 2014, Gu et al. [26] was the first time to consider the long-time behavior of problem (1.3) with small advection. Then, the asymptotic spreading speeds of the free boundaries was given in [27]. For more general reaction term, Gu et al. [10] studied the long time behavior of solutions of Fisher-KPP equation with advection β > 0 and free boundaries. For single equation with advection, there are many other works. For example, [28][29][30][31][32][33][34] and their references. Besides, there are also several works devoted to the system with small advection, such as, [35][36][37][38][39][40] and their references.
Taking account of the effect of advection, we consider where we use the changing region (g(t), h(t)) to denote the infective environment of disease, where the free boundaries x = g(t) and x = h(t) represent the spreading fronts of epidemic. Since the diffusion coefficient of v is much smaller than that of u, we assume that the diffusion coefficient of v is zero. When u spreads into a new environment, some humans in the new environment may be infected. Hence, we can use (g(t), h(t)) to represent the habit of infective humans. We use I 0 (−h 0 , h 0 ) to denote the initial infective environment of epidemic. The initial functions u 0 (x) and v 0 (x) satisfy where p > 3. The derivation of the stefan conditions h (t) = −µu x (t, h(t)) and g (t) = −µu x (t, g(t)) can be found in [41,42]. In this paper, we always assume that G satisfies (G1)-(G2) and (G3) G(z) is locally Lipschitz in z ∈ R + , i.e., for any L > 0, there exists a constant ρ(L) > 0 such that Furthermore, we assume that 0 < β < β * with The rest of this paper is organized as follows. In Section 2, the global existence and uniqueness of solution, comparison principle and some results about the principal eigenvalue are given. Section 3 is devoted to the long time behavior of (u, v). We get a spreading and vanishing dichotomy and give the criteria for spreading and vanishing. Finally, we give some discussions in Section 4.

Preliminaries
Firstly, we prove the existence and uniqueness of the solution.
Proof. This proof can be done by the similar arguments in [43]. But there are some differences. Hence, we give the details. Let and z(t, y) = v t, (h(t) − g(t))y + h(t) + g(t) 2 .
It is easy to see that D T D 1T × D 2T × D 3T is a complete metric space with the metric For any given (w, g, h) ∈ D T , there exist some ξ 1 , ξ 2 ∈ (0, t) such that Thus, A(g(t), h(t)) and B(g(t), h(t), y) are well-defined. By the definition of w, we have Since |w(t, y)| ≤ w 0 L ∞ + 1 for (t, y) ∈ T , we have For u defined as (2.6) and any given x ∈ [g(T ), h(T )], we consider the following ODE problem By the similar arguments as the step 1 in the proof of [44,Lemma 2.3], it is easy to show that (2.7) Then For this v, we can get For (w, g, h) and z obtained above, we consider the following problem Applying standard L p theory and the Sobolev imbedding theorem, we can have there exists T 2 ∈ (0, T 1 ] such that (2.8) admits a unique solution w(t, y) and where C 2 is a constant depending only on h 0 , α and , h(τ))w y (τ, 1)dτ, , h(t))w y (t, 1), Now, we can define the mapping F : Obviously, D T is a bounded and closed convex set of Define as before, .
Then it is easy to see that (

.4) and (2.5). Denoting
(2.11) Using the L p estimates for parabolic equations and Sobolev imbedding theorem, we obtain where C 4 depends on C 2 , C 3 and the functions A and B. Next we should estimate z 1 − z 2 C( T ) . For convenience, we define By direct calculations, we have It will be divided into the following three cases.
where C 6 only depends on h 0 and w 2x C( T 3 ) . By W(0, y) = 0 and Sobolev imbedding theorem, we have where C 9 depends on C 4 , C 5 and C 8 ; C 10 depends on C 1 , C 5 and C 6 . If T ∈ min T 3 , (2C 9 ) − 2 α T 4 , (2.20) In the following, we estimate G C 1 ([0,T ]) and H C 1 ([0,T ]) . Since G(0) = G (0) = 0, we have By (2.11), we have where C 11 depends on µ, A and h 0 . It follows from the proof of [45, Theorem 1.1] that we have where C 12 and C 13 do not depend on T . Therefore, we have where C 14 depends on C 2 , C 10 , C 11 and C 13 . Similarly, there exists C 15 such that Then it follows from the arguments in [23] that we can get the following estimates.
Just like the proof of [37, Theorem 3.2], we can obtain the global existence and uniqueness. Theorem 2.4. Assume that g, h ∈ C 1 ([0, +∞)), u(t, x), v(t, x) ∈ C(D) ∩ C 1,2 (D), and (u, v, g, h) satisfies Then the solution (u, v, g, h) of the free boundary problem (1.4) satisfies In the following part, we consider the following eigenvalue problem (2.23) Denote by λ 0 (l) the principal eigenvalue of problem (2.23) with some fixed l.
Lemma 2.6. λ 0 (l) has the following form: Proof. We choose β to be small and determine it later. By a simple calculation, we can achieve the characteristic equation and let µ i (i = 1, 2) be the roots of (2.24). Then the solution of (2.23) is where c 1 and c 2 will be determined later. Since φ(−l) = φ(l) = 0, we can derive that ≥ 0, we have φ ≡ 0, which is a contradiction. Hence, (2.24) has two complex roots: Then When k = 0, λ attain its minimum, we have and the corresponding eigenfunction φ(x) = e β 2d x cos π 2l x . Then we have the following properties about λ 0 (l).
Proof. By the expression of λ 0 (l) in Lemma 2.6, the proof of lemma is obvious. We omit it here.

Spreading and vanishing
Firstly, we give the definitions of spreading and vanishing of the disease: and spreading happens if Then, we give the following lemmas. , h(t)]) and g, h ∈ C 1+ α 2 ([0, ∞)) for some α > 0. We further assume that w 0 (x) ∈ X 1 (h 0 ). If (w, g, h) satisfies

3)
and Proof. It can be proved by the similar arguments in [16, Theorem 4.2].
By above Lemmas 3.2 and 3.3, we can derive the following result.
Proof. Firstly, we can use the method in the proof of [46, Theorem 2.1] to get Proof. We prove this result by constructing the appropriate upper solution. Let φ be the corresponding eigenfunction of λ 0 (h 0 ). Since λ 0 (h 0 ) > 0, we can choose some small δ such that Direct computations yield for all t > 0 and −σ(t) < x < σ(t), where ξ ∈ (0, u). Let Since u ≤ εe β 2d h 0 δ , we can choose δ to be sufficiently small such that B > 0. Noting that then we have If u 0 and v 0 are sufficiently small such that Applying Theorem 2.4 gives that h(t) ≤ σ(t) and g(t) ≥ −σ(t). Hence, h ∞ − g ∞ ≤ 2h 0 (1 + δ) < ∞. By Theorem 3.4, we have lim By Lemma 3.6, we can derive the following corollary directly.
Theorem 3.11. Assume that cG (0) , v * (x)) will be given in the proof.

Discussion
In this paper, we have dealt with a partially degenerate epidemic model with free boundaries and small advection. At first, we obtain the global existence and uniqueness of the solution. Then the effect of small advection is considered. We have proved that the results is similar to that in [20,23] under the condition 0 < β < β * . But we should explain that, for the case that cG (0) ab > 1 and β ≥ 2 d cG (0) b − a , the criteria for spreading and vanishing is hard to get by using the results of eigenvalue problem to construct the suitable upper and lower solution. We will study it in the future. When spreading occurs, the precise long-time behavior also needs a further consideration.
In order to study the spreading of disease, the asymptotic spreading speed of the spreading fronts is one of the most important subjects. To estimate the precise asymptotic spreading speed, we need to study the corresponding semi-wave problem or some other new technique. This may be not an easy task and deserves further study. We will consider it in another paper.
Due to the advection term, we find that the spreading barrier l * becomes larger if we increase the size of β for β ∈ (0, β * ). This means that if β ∈ (0, β * ), the more lager the size of advection is, the more difficult the disease will spread. This result may provide us a suggestion in controlling and preventing the disease. It may be an effective measure to make the infectious agents move along a certain direction by artificial means.