The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries

In this paper, we examine an epidemic model which is described by a system of two equations with nonlocal diffusion on the equation for the infectious agents \begin{document}$ u $\end{document} , while no dispersal is assumed in the other equation for the infective humans \begin{document}$ v $\end{document} . The underlying spatial region \begin{document}$ [g(t), h(t)] $\end{document} (i.e., the infected region) is assumed to change with time, governed by a set of free boundary conditions. In the recent work [ 33 ], such a model was considered where the growth rate of \begin{document}$ u $\end{document} due to the contribution from \begin{document}$ v $\end{document} is given by \begin{document}$ cv $\end{document} for some positive constant \begin{document}$ c $\end{document} . Here this term is replaced by a nonlocal reaction function of \begin{document}$ v $\end{document} in the form \begin{document}$ c\int_{g(t)}^{h(t)}K(x-y)v(t,y)dy $\end{document} with a suitable kernel function \begin{document}$ K $\end{document} , to represent the nonlocal effect of \begin{document}$ v $\end{document} on the growth of \begin{document}$ u $\end{document} . We first show that this problem has a unique solution for all \begin{document}$ t>0 $\end{document} , and then we show that its longtime behaviour is determined by a spreading-vanishing dichotomy, which indicates that the long-time dynamics of the model is not vastly altered by this change of the term \begin{document}$ cv $\end{document} . We also obtain sharp criteria for spreading and vanishing, which reveal that changes do occur in these criteria from the earlier model in [ 33 ] where the term \begin{document}$ cv $\end{document} was used; in particular, small nonlocal dispersal rate of \begin{document}$ u $\end{document} alone no longer guarantees successful spreading of the disease as in the model of [ 33 ].


(Communicated by Junping Shi)
Abstract. In this paper, we examine an epidemic model which is described by a system of two equations with nonlocal diffusion on the equation for the infectious agents u, while no dispersal is assumed in the other equation for the infective humans v. The underlying spatial region [g(t), h(t)] (i.e., the infected region) is assumed to change with time, governed by a set of free boundary conditions. In the recent work [33], such a model was considered where the growth rate of u due to the contribution from v is given by cv for some positive constant c. Here this term is replaced by a nonlocal reaction function of v in the form c h(t) g(t) K(x − y)v(t, y)dy with a suitable kernel function K, to represent the nonlocal effect of v on the growth of u. We first show that this problem has a unique solution for all t > 0, and then we show that its longtime behaviour is determined by a spreading-vanishing dichotomy, which indicates that the longtime dynamics of the model is not vastly altered by this change of the term cv. We also obtain sharp criteria for spreading and vanishing, which reveal that changes do occur in these criteria from the earlier model in [33] where the term cv was used; in particular, small nonlocal dispersal rate of u alone no longer guarantees successful spreading of the disease as in the model of [33].
Note that in (1.1) and (1.2), the dispersal of the infectious agents is assumed to follow the rules of random diffusion, which is not ideal in many practical situations. This kind of dispersal may be alternatively described by a nonlocal diffusion operator of the form which can capture short-range as well as long-range factors in the dispersal process by choosing the kernel function J properly [2,24].
Recently, Cao et al. [4] proposed a nonlocal version of the free boundary model in [10], and extended many basic results of [10] to the corresponding nonlocal model. More recently, Du et al. [9] investigated the spreading speed of the nonlocal model in [4], and demonstrated that, depending on the choice of the kernel function J, the spreading speed of the nonlocal model can be finite or infinite, contrasting sharply to the spreading speed determined by the local diffusion model, which is always finite.
Motivated by the work [4], some related models with nonlocal diffusion and free boundaries have been considered in several recent works (see, for example, [13,23,33]). In particular, Zhao et al. [33] studied a nonlocal version of (1.2), which has the form (1.5) Here the kernel function J : R → R is assumed to satisfy As before, G satisfies (G1)-(G2). They showed that, for (1.5), a spreading-vanishing dichotomy similar to that in [1] for the corresponding local model (1.2) still holds. But the criteria for spreading and vanishing is different from that in [1]; more precisely, it was shown in [33] that (i) If R 0 ≤ 1, then vanishing happens for all admissible initial data (u 0 , v 0 ).
Since R 0 is independent of d, from (ii) above we immediately see that, when R 0 > 1, for all small d > 0, R 0 ≥ 1 + d a and hence spreading happens regardless of the initial data (u 0 , v 0 ).
Note that in (1.1), (1.2) and (1.5), the infective agents u at a spatial point x is assumed to depend only on the infective humans v at spatial location x. In reality, it should also depend on the infective humans v at some spatial neighbourhood of x. Such a consideration was included in the model in [5], where instead of cv, the growth rate of infectious agents due to the infective humans is described by with K(x, y) representing the transfer kernel of infectious agents produced by the infective humans at y and made available at x.
In this paper, we would like to examine the effect of such a change on the dynamics of (1.5). We assume that K(x, y) depends only on the distance between x and y, and so we may write K = K(x − y). The modified (1.5) has the following form: where the parameters a, b, c, d, µ and h 0 are positive constants, J(x) and K(x) satisfy (J), and the initial functions u 0 (x) and v 0 (x) satisfy (1.6).
The main results of this paper are the following theorems.
Theorem 1.2 (Spreading-vanishing dichotomy). Let the conditions of Theorem 1.1 hold and (u, v, g, h) be the unique solution of (1.7). Then one of the following alternatives must happen: Let us recall that R 0 is given by (1.3).

Remark 1.
For the nonlocal diffusion model (1.5), we see from [33,Theorem 1.3] that if R 0 > 1 then spreading happens to (1.5) for all small d > 0, regardless of the initial data. But, due to the introduction of the nonlocal effect involving K, we found from Theorem 1.3 that the epidemic modelled by (1.7) will not always spread for small d > 0; instead, the initial data play a role through h 0 or through (u 0 , v 0 ). Biologically, this means that the nonlocal effect may decrease the chance of epidemic disease spreading, compared with the case that there is no nonlocal effect. We should emphasise that l * is determined by an eigenvalue problem involving d. In [25], the local diffusion model (1.2) with the term cv replaced by h(t) g(t) K(x − y)v(t, y)dy was investigated, and similar results to [1] was obtained.
The rest of this paper is devoted to the proof of Theorems 1.1, 1.2 and 1.3, which consists of the next section. The approach follows that of [33], but considerable changes are needed, since the nonlocal reaction term c dy causes a number of difficulties. For example, new techniques are required to treat the associated linearised eigenvalue problem of (1.7), as well as the associated auxiliary fixed boundary problems, as both are significantly different from the corresponding problems for (1.5).
2. Proof of the main results. Throughout this section, we always assume that J(x) and K(x) satisfy (J), and G satisfies (G1) and (G2). For h 0 > 0, (u 0 , v 0 ) satisfying (1.6), and any given T > 0, we introduce the following notations: We start with three comparison results.
The existence and uniqueness of solutions to the problem (1.7) can be done in a similar fashion as in [33]. We only list the main steps in the proof.
The proof of Theorem 1.1. For any given T > 0, (g * , h * ) ∈ G T × H T and v * ∈ X v0 T , it follows from Then we can apply the same argument in the proof of [33, Lemma 2.2] to obtain that the problem and then define t x as in Step 1 of the proof of [33, Lemma 2.2], but with (g, h) replaced by (g * , h * ). To mark the difference, we denote t x by t * x . Now, for each x ∈ (g * (T ), h * (T )), we consider the initial value problem (2.4) By the Fundamental Theorem of ODEs and some simple comparison argument, it can be easily shown that (2.4) has a unique solution v * (t, x), and it is continuous and satisfies Therefore v * ∈ X v0 T .

(2.5)
Then we can define the mapping F(v * , g * , h * ) = ( v * , g * , h * ). Let Using the corresponding argument of [4], we can show that there exists some sufficiently small T 0 = T 0 (µ, A, h 0 , 0 , u 0 , J) > 0 such that ( v * , g * , h * ) ∈ Σ T for any T ∈ (0, T 0 ], which implies that F(Σ T ) ⊂ Σ T for T ∈ (0, By this fact, we can follow the approach of the proof of [33, Theorem 1.1] to show F is a contraction mapping for sufficiently small T ∈ (0, T 0 ], and hence for such t, F has a unique fixed point in Σ T , which clearly is a solution of (1.7) for t ∈ [0, T ]. Similar to Steps 3 and 4 in the proof of [33, Theorem 1.1], we can show that this is the unique solution of (1.7) and it can be extended uniquely to all t > 0.
is nonnegative, and (u(t, x), v(t, x)) as well as (u t (t, x), v t (t, x)) are continuous in Ω 0 and satisfy Proof. This follows from a simple variation of the argument in the proof of Lemma 2.1. We omit the details.
then the unique solution (u, v, g, h) of (1.7) satisfies Proof. By using Lemma 2.1, we can argue as in [33] to prove this lemma. We omit the details.
The following result is a direct consequence of the above comparison principle (Lemma 2.3), where to stress the dependence on the parameter µ, we use (u µ , v µ , g µ , h µ ) to denote the solution of problem (1.7).
It is easily seen that h(t) is monotonically increasing and g(t) is monotonically decreasing. Therefore Clearly θ ≤ 0 is equivalent to R 0 ≤ 1.
By the comparison principle (Lemma 2.1), we have
For all T 0 ≤ s ≤ t, by G(z) z < G (0) for z > 0 and Fubini theorem, we have Hence, we have K(x − y)v(T 0 , y)dy dx then we can get that h ∞ − g ∞ < ∞ by letting t → ∞. This fact and (2.7) implies that vanishing happens.
(i) We will prove this conclusion by the following two steps.
Arguing as the proof of (2.10), we can show that there exists a positive constant c 1 such that w(x) > c 1 in [l 1 , l 2 ]. Let For x ∈ [l 1 , l 2 ], it follows from this inequality that K(x − y)G(w(y))dy = 0.
We end this paper with some details about footnote 1 in the introduction. So suppose (G2) is replaced by (G2) G(z)/z is nonincreasing for z > 0 and strictly decreasing in a neighbourhood of Σ := {z > 0 : G(z)/z = ab/c} in the case G (0) > ab/c.
We now explain the changes we have to make to the results proved in this paper. Firstly we note that the strict monotonicity near Σ is necessary to guarantee the uniqueness of K 1 determined by G(K 1 )/K 1 = ab/c.
Secondly we note that under (G2) , in Lemma 2.5 part (i), the same arguments can be used to prove there are a maximal and a minimal positive solution, say (W , Z) and (W , Z), respectively, but the uniqueness result is lost in general. The nonexistence result in part (ii) still holds as the proof is not affected. Corollary 2 should be modified accordingly.
The rest of the paper is not affected by the changes from (G2) to (G2) .