Spatial propagation in an epidemic model with nonlocal diffusion: The influences of initial data and dispersals
- PDF / 364,501 Bytes
- 30 Pages / 612 x 792 pts (letter) Page_size
- 37 Downloads / 201 Views
https://doi.org/10.1007/s11425-020-1740-1
. ARTICLES .
Spatial propagation in an epidemic model with nonlocal diffusion: The influences of initial data and dispersals Wen-Bing Xu1,2 , Wan-Tong Li2,∗ & Shigui Ruan3 1Academy
of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China; 3Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA
2School
Email: [email protected], [email protected], [email protected] Received March 20, 2020; accepted July 12, 2020
Abstract
This paper studies an epidemic model with nonlocal dispersals. We focus on the influences of initial
data and nonlocal dispersals on its spatial propagation. Here, initial data stand for the spatial concentrations of the infectious agent and the infectious human population when the epidemic breaks out and the nonlocal dispersals mean their diffusion strategies. Two types of initial data decaying to zero exponentially or faster are considered. For the first type, we show that spreading speeds are two constants whose signs change with the number of elements in some set. Moreover, we find an interesting phenomenon: the asymmetry of nonlocal dispersals can influence the propagating directions of the solutions and the stability of steady states. For the second type, we show that the spreading speed is decreasing with respect to the exponentially decaying rate of initial data, and further, its minimum value coincides with the spreading speed for the first type. In addition, we give some results about the nonexistence of traveling wave solutions and the monotone property of the solutions. Finally, some applications are presented to illustrate the theoretical results. Keywords MSC(2010)
nonlocal dispersal, epidemic model, spreading speed, initial data, dispersal kernel 35C07, 35K57, 92D25
Citation: Xu W-B, Li W-T, Ruan S G. Spatial propagation in an epidemic model with nonlocal diffusion: The influences of initial data and dispersals. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-0201740-1
1
Introduction
To model the spread of cholera in the European Mediterranean regions in 1973, Capasso and Maddalena [8, 9] proposed a system of two parabolic differential equations to describe a positive feedback interaction between the concentration of bacteria and the infectious human population; namely, the high concentration of bacteria leads to the large infection rate of the human population and once infected the human population increases the growth rate of bacteria. Capasso and Kunisch [7] and Capasso and Wilson [10] also applied this mechanism to model other epidemics with fecal-oral transmission (such as * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
math.scichina.com
link.springer.com
2
Xu W-B et al.
Sci China Math
typhoid fever and hepatitis A). In these studies, the spatial movements of the infectious agent and the infectious human host are described by the
Data Loading...
