Spreading speeds of epidemic models with nonlocal delays

Guo Lin; Shuxia Pan; Xiang-Ping Yan; Guo Lin; Shuxia Pan; Xiang-Ping Yan

doi:10.3934/mbe.2019380

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 7562-7588. doi: 10.3934/mbe.2019380

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Spreading speeds of epidemic models with nonlocal delays

1.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
2.
School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China
3.
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China

Received: 20 February 2019 Accepted: 13 August 2019 Published: 20 August 2019

We estimate the spreading speeds in diffusive epidemic models with nonlocal delays, nonlinear incidence rate and constant recruitment rate. The purpose is to model the process that the infective invades the habitat of the susceptible, and they coexist eventually. In order to focus on our idea, a system with a nonlinear incidence rate is firstly studied, which implies a saturation level of the infective individuals and monotone incidence rate. When the initial value of the infective has nonempty compact support, we prove the rough spreading speed that equals the minimal wave speed of traveling wave solutions in the known results. Then for a general (nonmonotone) incidence rate, we obtain the spreading speeds by constructing auxiliary systems admitting a monotone incidence rate, and prove the convergence of solutions on any compact spatial interval. Furthermore, some numerical examples are given to estimate the invasion speed and show the nontrivial effect of time delay and spatial nonlocality, which implies that the stronger spatial nonlocality leads to larger spreading speeds.
- auxiliary equation,
- nonmonotone system,
- asymptotic spreading
Citation: Guo Lin, Shuxia Pan, Xiang-Ping Yan. Spreading speeds of epidemic models with nonlocal delays[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7562-7588. doi: 10.3934/mbe.2019380

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Abstract

We estimate the spreading speeds in diffusive epidemic models with nonlocal delays, nonlinear incidence rate and constant recruitment rate. The purpose is to model the process that the infective invades the habitat of the susceptible, and they coexist eventually. In order to focus on our idea, a system with a nonlinear incidence rate is firstly studied, which implies a saturation level of the infective individuals and monotone incidence rate. When the initial value of the infective has nonempty compact support, we prove the rough spreading speed that equals the minimal wave speed of traveling wave solutions in the known results. Then for a general (nonmonotone) incidence rate, we obtain the spreading speeds by constructing auxiliary systems admitting a monotone incidence rate, and prove the convergence of solutions on any compact spatial interval. Furthermore, some numerical examples are given to estimate the invasion speed and show the nontrivial effect of time delay and spatial nonlocality, which implies that the stronger spatial nonlocality leads to larger spreading speeds.

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