The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries

Abstract

We consider an epidemic model with nonlocal diffusion and free boundaries, which describes the evolution of an infectious agents with nonlocal diffusion and the infected humans without diffusion, where humans get infected by the agents, and infected humans in return contribute to the growth of the agents. The model can be viewed as a nonlocal version of the free boundary model studied by Ahn, Beak and Lin [2], with its origin tracing back to Capasso et al. [5,6]. We prove that the problem has a unique solution defined for all t > 0, and its long-time dynamical behaviour is governed by a spreading-vanishing dichotomy. Sharp criteria for spreading and vanishing are also obtained, which reveal significant differences from the local diffusion model in [2]. Depending on the choice of the kernel function in the nonlocal diffusion operator, it is expected that the nonlocal model here may have accelerated spreading, which would contrast sharply to the model of [2], where the spreading has finite speed whenever spreading happens [33].proposed by Capasso and Paveri-Fontana [6] to describe the cholera epidemic which spread in the European Mediterranean regions in 1973. Here a, b, c are all positive constants, u(t) and v(t), respectively, stand for the average population concentration of the infectious agents and the infective humans in the infected area at time t.

Propagation, diffusion and free boundaries

“…Some of the above results for (5.2) have been extended to systems of equations with nonlocal diffusion with free boundary. For example, similar results to Theorems 5.2-5.4 have been obtained for a West Nile virus model in [36], and analogous results have been obtained for weak competition and weak predatorprey models in [39], and for other epidemic models in [88,89]. In [27], similar results to Theorems 5.5 and 5.6 have also been obtained for the epidemic model of [88].…”

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

“…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].…”

“…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…”

“…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 ).…”

“…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.…”