Traveling wave solutions in a two-group SIR epidemic model with constant recruitment

• Published in: Journal of mathematical biology Vol. 77; no. 6-7; pp. 1871 - 1915
• Main Authors: Zhao, Lin, Wang, Zhi-Cheng, Ruan, Shigui
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2018; Springer Nature B.V
• Subjects: Applications of Mathematics; Basic Reproduction Number - statistics & numerical data; Communicable Diseases - epidemiology; Communicable Diseases - transmission; Computer Simulation; Disease Susceptibility - epidemiology; Epidemics; Epidemics - statistics & numerical data; Humans; Mathematical and Computational Biology; Mathematical Concepts; Mathematical models; Mathematics; Mathematics and Statistics; Models, Biological; Nonlinear Dynamics; Recruitment; Reproduction; Spatio-Temporal Analysis; Time lag; Travel - statistics & numerical data; Traveling waves; Basic reproduction number; 92D30; Two-group epidemic model; Time delay; 35B40; 35K57; Traveling wave solutions; Constant recruitment; 35C07
• ISSN: 0303-6812; 1432-1416
• DOI: 10.1007/s00285-018-1227-9

Host heterogeneity can be modeled by using multi-group structures in the population. In this paper we investigate the existence and nonexistence of traveling waves of a two-group SIR epidemic model with time delay and constant recruitment and show that the existence of traveling waves is determined by the basic reproduction number R 0 . More specifically, we prove that (i) when the basic reproduction number R 0 > 1 , there exists a minimal wave speed c ∗ > 0 , such that for each c ≥ c ∗ the system admits a nontrivial traveling wave solution with wave speed c and for c < c ∗ there exists no nontrivial traveling wave satisfying the system; (ii) when R 0 ≤ 1 , the system admits no nontrivial traveling waves. Finally, we present some numerical simulations to show the existence of traveling waves of the system.