The Traveling Wave Solutions in a Mixed-Diffusion Epidemic Model

: In this paper, we study the traveling wave solution of an epidemic model with mixed diffusion. First, we give two deﬁnitions of the minimum wave speeds and prove that they are equivalent. Second, the existence, decaying behavior, and uniqueness of traveling wave fronts are obtained. Third, the signs of minimum wave speeds are studied, and further, in two speciﬁc cases of the dispersal kernel, we show how to identify the signs of minimum wave speeds.


Introduction
This paper is devoted to studying the following epidemic model: where u(t, x) and v(t, x) in biology stand for the spatial concentration of an infectious agent and the spatial density of the infectious human population, respectively; α > 0 and β > 0 denote the natural death rates of the infectious agent and infectious humans; h(v) means the growth of the infectious agent caused by infectious humans; and g(u) is the infection rate of the human population under the assumption that the total susceptible human population is a constant during the evolution of the epidemic. The model (1) describes a positive feedback interaction between the concentration of infectious agent and the infectious human population; that is, a high concentration of infectious agent leads to a large infection rate in the human population, and as more people are infected, the growth rate of the infectious agent increases. This model is an extension to the classical SEIR (susceptible-exposed-infectious-recovered) model. There is a widely adopted numerical approach to the solution of epidemic phenomena based on the modification of SEIR model and similar ones, and the very recent contributions include [1][2][3]. The model (1) is a mixed-diffusion variant of the following classical epidemic model: where K(x − y) can be viewed as the probability of individuals moving from location y to location x (see [6]). Compared to the classical diffusion operator, the nonlocal dispersal operator describes the movements between not only adjacent but also nonadjacent spatial locations. Here the nonlocal dispersal of v can be thought as the long-distance movements of infectious humans across cites or countries by air traffic and other long-distance transportation. If the diffusion of infectious agent is also nonlocal, then (1) reduces to the following nonlocal dispersal model: In (3), the nonlocal dispersal operator K 1 * u − u means that long-distance movements of infectious agent happen; for example, the infectious agent can move among countries through the transportation of imported food or the flow of international rivers. The wave propagation phenomena, which are associated with the studies of traveling wave solutions and spreading speeds of systems (2) and (3), have been widely studied in the literature. For example, Hsu and Yang [7] considered the existence, uniqueness, and decaying behavior of traveling wave fronts of (2), and Wu and Hsu [8] studied the entire solutions of (2). We also refer to [9,10] for the traveling wave solution of (2) in the case d 2 = 0, and [11,12] for traveling wave solutions of a more general system that includes (2) as a special case. For the nonlocal dispersal model (3), we assume that K i satisfies R K i (x)e λx dx < +∞ for λ ∈ R. Li, Xu, and Zhang [13]; and Meng, Yu, and Hsu [14] studied the traveling wave solutions and entire solutions. We also refer to [15,16] for the traveling wave solutions and spreading speed of (3) in the case d 2 = 0. The spreading speed of (3) was studied by Bao et al. [17], Hu et al. [18], and Xu et al. [19].
The study of the following scalar dispersal equation with reaction: is also closely related to (1) and (3), where A is a dispersal operator. There are various forms of A, such as classical diffusion Au = ∆u, nonlocal dispersal Au = K * u − u, fractional Laplacian Au = −(−∆) α u (see [20] for a recent review), and variable-order Riemann-Liouville fractional derivatives defined by [21,22] x L D where Γ(·) is the gamma function. The variability and transition of fractional orders contribute to the detailed description of highly heterogeneous systems and complex phenomena. Such scenarios have motivated the formulation of variable-order fractional operators and related algorithms-for example, [23]. Consider the monostable case with f satisfying the following Fisher-KPP condition The different forms of A usually cause distinct wave propagation phenomena of (4). (i) When Au = ∆u, (4) is a classical reaction-diffusion equation and there is a unique traveling wave front for any speed c 2 f (0), but no traveling wave solution for the speed c ∈ (0, 2 f (0)). (ii) When Au = −(−∆) α u, (4) is a fractional diffusion equation with reaction, and there is no traveling wave solution for any speed c ∈ R. Moreover, it was shown that the front position propagates exponentially; see, e.g., [24][25][26]. To the best of our knowledge, there is no result about the propagation dynamics of variable-order fractional diffusion equations, and our work could possibly provide some basis for this topic.
For the case Au = K * u − u, the properties of K determine whether (4) admits a traveling wave front spreading at a finite speed or has exponentially propagating front position. More precisely, when the symmetric kernel K satisfies R K(x)e λx dx < +∞ for λ ∈ R, there exists c * > 0 such that (4) has a unique traveling wave front for any speed c c * , and no traveling wave solution with the speed c ∈ (0, c * ); see [27][28][29][30][31][32][33]. However, when K is "heavy-tailed", in the sense that |K (x)| = o(K(x)) as |x| → +∞, there is no traveling wave solution for any speed c ∈ R and the spatial propagation of front position is accelerated; see, e.g., [34][35][36][37]. In particular, when K(x) ∼ |x| −β as |x| → +∞, the front position propagates exponentially, which means the nonlocal dispersal case with an algebraic-tailed kernel has similar wave propagation properties to the case Au = −(−∆) α u.
To the best of our knowledge, there is no result about the traveling wave solutions of the mixed-diffusion model (1), although its background in biology is clear; that is, the movements of an infectious agent are local, but the long-distance movements of infectious human happen. Herein, we consider the traveling wave solutions and the minimum wave speeds in monostable system (1). A traveling wave solution of (1) is a solution of the special form (u(x, t), v(x, t)) = (φ(x − ct), ψ(x − ct)), which can be regarded as the dispersal process of epidemic from outbreak to an endemic. Usually, a non-decreasing or non-increasing traveling wave solution is called a traveling wave front. Note that we use the form (u(x, t), v(x, t)) = (φ(x − ct), ψ(x − ct)) to represent not only non-increasing but also non-decreasing traveling wave front. Therefore, no matter whether a traveling wave solution is non-increasing or non-decreasing, when its speed is positive, it propagates from left to right along the x-axis, and when its speed is negative, it propagates from right to left on the x-axis. In this paper, we study the "light-tailed" dispersal kernel, namely, R K(x)e λx dx < +∞, for λ ∈ R. Our results can be summarized from three angles.
First, we give two definitions of the minimum wave speeds. The first definition is related to the principal eigenvalue of a linear operator derived from (1), and this definition is common in the study of traveling wave solutions and spreading speeds in (2) and (3), and other related systems (see, e.g., [17][18][19]38,39]). The second definition is related to the root number of an eigenvalue equation, and this definition is used to study the traveling wave solutions in [7,13]. Moreover, we prove that these two definitions are equivalent.
Second, we consider the traveling wave solutions of (1). Motivated by the works of [7,13,40,41], we change the traveling wave solution problem into investigating the fixed point of a nonlinear operator, and the existence of traveling wave front is obtained by constructing a pair of upper and lower solutions and applying the Schauder's fixed point theorem. The decaying behavior and uniqueness of traveling wave fronts are also obtained.
Third, we study the signs of minimum wave speeds. In (1), the kernel function K(·) is assumed to be asymmetric. As stated in [32], asymmetric kernels may induce non-positive minimal wave speed. Thus, it is significant to study the signs of minimum wave speeds, which determine whether it happens that the asymmetric kernel changes the propagation direction of traveling wave solutions. Motivated by the work of [19] for (3), we show that the signs of minimum wave speeds of (1) depend only on the number of elements in some set, which is further applied to two specific forms of K(·) (i.e., normal distribution and uniform distribution). For these two specific forms, the study of signs of minimum wave speeds is quite different from that considered in [19] for (3), because in this work for (1) we consider the influences of the asymmetric dispersal of v under the assumption that u has symmetric local diffusion, but in [19] for (3), the authors study the influences of symmetric nonlocal kernel of v when u has asymmetric nonlocal dispersal. We show that when K(·) is normal distribution or uniform distribution, the signs of minimum wave speeds depend only on µ and µ √ σ where µ ∈ R is the expectation and σ is the variance of K, which is different from the results obtained in [19] for (3). Thus, the study for the cases of normal distribution and uniform distribution in this paper is a new result to understand the influences of asymmetric dispersal on the signs of minimum wave speeds.
The rest of this paper is organized as follows. In Section 2, we give two definitions of minimum wave speeds and prove they are equivalent. Section 3 presents the existence, uniqueness, and decaying behavior of traveling wave fronts of (1). Section 4 deals with the signs of minimum wave speeds, and the results for two specific forms of K are given.

Two Definitions of Minimum Wave Speeds
In the section, we give two definitions of minimum wave speeds and prove that they are equivalent. First we state the assumptions. Assume that (A1) g(·) and h(·) are two functions in We assume that K(·) is a continuous and nonnegative function satisfying Note that we do not assume that K(·) is symmetric.

The First Definition
We denote Consider the matrix Let χ(λ) be the large one of the two eigenvalues of E(λ), namely, Theorem 1. We have the following statements about c(λ): (ii) There are two unique constants λ * R > 0 and λ * L < 0 such that and c (λ) Similarly, it holds that lim The proofs of (ii) and (iii) are similar to the counterpart in the proof of [19] (Theorem 2.1).

Theorem 2.
For sufficiently large c ∈ R, ∆ c (λ) = 0 has exactly three different positive roots and one negative root. For sufficiently small c ∈ R, ∆ c (λ) = 0 has exactly three different negative roots and one positive root.
Similarly, we have that
From (6), it follows that Then ξ 1 (c) and ξ 2 (c) are two different positive roots of ∆ c (λ) = 0, and M(c) 2 for any c > c * R . From (7) it follows that c * R C * R . Second, we prove C * R c * R . For any c > C * R , by the proof of Theorem 2 and Definition 1, ∆ c (λ) = 0 has two different positive roots in (0, λ 3 ), and we denote them by η 1 (c) and η 2 (c) with η 1 (c) < η 2 (c). Then for i = 1, 2, we have that which is a contradiction. Thus, we have that which implies that We obtain C * R c * R . Therefore, it holds that c * R = C * R , and the proof of c * L = C * L is similar.
Theorem 3 shows that the first definition for c * R and c * L in Theorem 1 (iii) is equivalent to the second definition for C * R and C * L in Definition 1. Thus, we use c * R and c * L for the minimum wave speeds in the rest of the paper.

Definition 2. A continuous function
..,m is a set containing countable points in R, and they satisfy A lower solution of (9) is defined similarly by reversing the inequality in (12).

Proof. The proof is similar to the proofs of [40] (Theorem 2.2) and [41] (Theorem 3.2), where
Schauder's fixed point theorem is applied to obtain the fixed point of F. The properties for f 1 can be studied by the method in [40] (Lemmas 2.3 and 2.4). The properties for f 2 can be obtained from [41] (Lemmas 3.3, 3.5, 3.6, and 3.7). Thus, we omit the details.
Proof. We consider the existence of a non-increasing traveling wave front satisfying (10) and the existence of a non-decreasing traveling wave front satisfying (11) to be similar. By Theorem 2 and Definition 1, when c > c * R , there are two different positive roots of ∆ c (λ) = 0 in (0, λ 3 ), and we denote them by µ 1 (c) and µ 2 (c) with µ 1 (c) < µ 2 (c). Define Consider δ satisfying where δ 0 is the constant in (A1). We definē where q is a sufficiently large constant, and L(c, δ) satisfies By the methods in [13] (Lemma 2.7) and [8] (Lemma 2.2), we can get thatΦ c (ξ) (φ c (ξ),ψ c (ξ)) is an upper solution and Φ c (ξ) (φ c (ξ), ψ c (ξ)) is a lower solution of (1). Note that Φ c (·) is non-increasing and Φ c (ξ) Φ c (ξ) for any ξ ∈ R. Then we have which implies the condition (a) in Theorem 4 holds. Recall that there is no equilibrium (u, v) of (1) satisfying 0 < u, v < 1, and then the condition (b) holds. The conditions (c) and (d) can be easily checked. By Theorem 4, (1) has a non-increasing traveling wave front (φ(x − ct), ψ(x − ct)) with c > c * R satisfying (10). Now we consider the case c = c * R . Let {c n } satisfy c n > c * R and c n → c * R as n → +∞. Then there is a sequence of non-increasing continuous functions From (13), it follows that Then (φ c * , ψ c * ) is a traveling wave front of (1) with c = c * R satisfying (10). Finally, the proof of the nonexistence of traveling wave solution with c ∈ (c * L , c * R ) is similar to the counterpart in [13] (Theorem 2.1) or [7] (Theorem 1.1).
The proof of Theorem 6 is similar to Theorem 2.2 in [13], and we give a scheme here. By a similar argument for (27) in [13], there are two constants γ > 0 and M > 0 such that Multiplying (9) by e λξ and integrating it over R, we obtain that

The Signs of Minimum Wave Speeds
In this section, we show how to identify the signs of c * R and c * L . Recall that Proof. The proofs of "⇐" in (i)-(v) are similar to the proof of [19] (Theorem 2.2). Now we prove "⇒" for (i)-(v). By Theorem 1, we get c * L < c * R . Then the relationships among 0, c * L , and c * R in (i)-(v) are all possible cases. By [19] (Theorem 2.2), Λ is an empty set or a closed interval without 0 in R. Then, either Λ ⊆ R + or Λ ⊆ R − . We have that the conditions of Λ in (i)-(v) contain all possible cases, which means that Λ must satisfy one of the conditions in (i)-(v). Therefore, the proofs of "⇒" can be obtained from "⇐" for (i)-(v).
Next we give two specific forms of the kernel function K(·). For each case, we show how to apply Theorem 8 to identify the signs of c * R and c * L .

Normal Distribution
Assume that K(x) satisfies that Let r = µ/ √ 2σ. When µ = 0, K(·) can be regarded as a function with parameters µ and r, namely, Corollary 1. For any fixed r satisfying K > 1, there is a constant µ * > 0 such that (i) µ > µ * ⇔ the propagation to left fails, namely, For any r satisfying K 1, it holds that c * L < 0 < c * R for any µ ∈ R.
By Theorem 8, we have proved "⇒" for (i)-(v) in Corollary 1. Note that the relationships among 0, c * L , and c * R in (i)-(v) are all possible cases; and the relationships among µ, µ * , and −µ * in (i)-(v) are also all possible cases. Thus, the proofs of "⇐" can be obtained from "⇒" for (i)-(v).

Uniform Distribution
Suppose that K(·) is given by where the constants A ∈ R + and B ∈ R − stand for the farthest distances of the movements of infectious agents during a unit time period to the right and left of x-axis, respectively. Some calculations imply that Next, we state the following lemma whose proof can be found in [42] (Lemma 5.3).
Denote r = −A/B > 0, and let z r be the constant defined in Lemma 1. It follows from Lemma 1 that When r = 1, it holds that z r = 0 and we denote When r = 1, since min{b(λ); λ ∈ R} = −β, we can simply denote K = αβ/(g (0)h (0)) < 1. We denote which implies that when r = 1, A = 2rµ r−1 and B = − 2µ r−1 . When r = 1 (i.e., µ = 0), K(·) can be regarded as a function with parameters µ and r, namely, Note that K depends only on r and is independent of µ.
Then, Λ = ∅ for µ > 0 sufficiently small. The rest of the proof is similar to the counterpart in the proof of Corollary 1.

Remark 1.
Note that the parameter r for the normal distribution in Section 4.1 is defined by µ/ √ 2σ, where σ is variance and µ is expectation. For the uniform distribution, the variance Var(K) of K is (A − B) 2 /12, and the expectation is µ = (A + B)/2. Consider Since the function r → (r − 1)/(r + 1) from R + to (−1, 1) is bijection and increasing, we can use (or with some coefficient) and µ are important parameters to describe whether an asymmetric kernel changes the signs of minimum wave speeds.

Conclusions
We studied traveling wave solutions of an epidemic model with mixed diffusion. We gave two definitions of the minimum wave speeds, and the equivalence of these two definitions was proved. The existence, decaying behavior, and uniqueness of traveling wave fronts were obtained. We also presented how to identify the signs of minimum wave speeds and apply them to two specific forms of the kernel function, namely, normal distribution and uniform distribution. Our study indicates that in these two scenarios, the asymmetric nonlocal kernel may induce non-positive minimal wave speed and standing wave solution whose wave speed is zero. However, for general dispersal kernel K(·) with the expectation µ and the variance σ, it is unknown whether the parameters µ √ σ and µ can determine the signs of minimum wave speeds, and this interesting question will be the topic of future research.