Long-time dynamics of an epidemic model with nonlocal diffusion and free boundaries

Ting-Ying Chang; Yihong Du; Ting-Ying Chang; Yihong Du

doi:10.3934/era.2022016

Electronic Research Archive

2022, Volume 30, Issue 1: 289-313. doi: 10.3934/era.2022016

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Long-time dynamics of an epidemic model with nonlocal diffusion and free boundaries

Ting-Ying Chang ,
Yihong Du ^,

School of Science and Technology, University of New England, Armidale, NSW 2351, Australia

Received: 26 February 2021 Revised: 13 September 2021 Accepted: 15 November 2021 Published: 04 January 2022

In this paper, we consider a reaction-diffusion epidemic model with nonlocal diffusion and free boundaries, which generalises the free-boundary epidemic model by Zhao et al. ^[1] by including spatial mobility of the infective host population. We obtain a rather complete description of the long-time dynamics of the model. For the reproduction number $ R_0 $ arising from the corresponding ODE model, we establish its relationship to the spreading-vanishing dichotomy via an associated eigenvalue problem. If $ R_0 \le 1 $, we prove that the epidemic vanishes eventually. On the other hand, if $ R_0 > 1 $, we show that either spreading or vanishing may occur depending on its initial size. In the case of spreading, we make use of recent general results by Du and Ni ^[2] to show that finite speed or accelerated spreading occurs depending on whether a threshold condition is satisfied by the kernel functions in the nonlocal diffusion operators. In particular, the rate of accelerated spreading is determined for a general class of kernel functions. Our results indicate that, with all other factors fixed, the chance of successful spreading of the disease is increased when the mobility of the infective host is decreased, reaching a maximum when such mobility is 0 (which is the situation considered by Zhao et al. ^[1]).
- nonlocal diffusion,
- free boundary,
- spreading speed,
- accelerated spreading
Citation: Ting-Ying Chang, Yihong Du. Long-time dynamics of an epidemic model with nonlocal diffusion and free boundaries[J]. Electronic Research Archive, 2022, 30(1): 289-313. doi: 10.3934/era.2022016

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Abstract

In this paper, we consider a reaction-diffusion epidemic model with nonlocal diffusion and free boundaries, which generalises the free-boundary epidemic model by Zhao et al. ^[1] by including spatial mobility of the infective host population. We obtain a rather complete description of the long-time dynamics of the model. For the reproduction number $ R_0 $ arising from the corresponding ODE model, we establish its relationship to the spreading-vanishing dichotomy via an associated eigenvalue problem. If $ R_0 \le 1 $, we prove that the epidemic vanishes eventually. On the other hand, if $ R_0 > 1 $, we show that either spreading or vanishing may occur depending on its initial size. In the case of spreading, we make use of recent general results by Du and Ni ^[2] to show that finite speed or accelerated spreading occurs depending on whether a threshold condition is satisfied by the kernel functions in the nonlocal diffusion operators. In particular, the rate of accelerated spreading is determined for a general class of kernel functions. Our results indicate that, with all other factors fixed, the chance of successful spreading of the disease is increased when the mobility of the infective host is decreased, reaching a maximum when such mobility is 0 (which is the situation considered by Zhao et al. ^[1]).

References

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