The periodic traveling waves in a diffusive periodic SIR epidemic model with nonlinear incidence

Abstract

In this paper, a reaction-diffusion SIR epidemic model is proposed. It takes into account the individuals mobility, the time periodicity of the infection rate and recovery rate, and the general nonlinear incidence function, which contains a number of classical incidence functions. In our model, due to the introduction of the general nonlinear incidence function, the boundedness of the infected individuals can not be obtained, so we study the existence and nonexistence of periodic traveling wave solutions of original system with the aid of auxiliary system. The basic reproduction number and the critical wave speed are given. We obtained the existence and uniqueness of periodic traveling waves for each wave speed using the Schauder’s fixed points theorem when . The nonexistence of periodic traveling waves for two cases (i) and (ii) and are also obtained. These results generalize and improve the existing conclusions. Finally, the numerical experiments support the theoretical results. The differences of traveling wave solution between periodic system and general aperiodic coefficient system are analyzed by numerical simulations.

On the Exact analytical solution and Van der Pol like equation of infectious diseases model with the time-dependent total population

“…Several researchers have demonstrated the existence of periodic traveling wave solutions with models. For instances, a complex SIR such as considering diffusion reaction conditions [39] show the existence of a periodic solution with the reproduction number as a critical parameter. Further, using numerical simulations they show that the traveling wave solution of infected individuals is more oscillating (has periodicity) when the infection coefficient and recovery rate are time-dependent.…”

“…By using Taylor expansion, we get, (39) If we expand in ϕn=ϕ0+ϕ1+ϕ2+…then for terms of the power of zero (0-order), we have, (40) This is a non-homogeneous linear first-order differential equation. We define a function as follow,…”

Asymptotic analysis of SIR epidemic model with nonlocal diffusion and generalized nonlinear incidence functional

“…For diffusive SIR models, when the domain is bounded, quantitative properties including permanence, local and global stability of have been obtained in previous studies, [4][5][6][7][8][9][10][11][12][13] while for unbounded domains, the main focus is on the existence/nonexistence of traveling waves. 14,15 In these models, diseases spread to the whole region immediately even though the initial infection is confined to a small part of the region. This does not agree the reality that diseases spread gradually.…”

“…Simple compartmental models such as SIR models have been used to understand the principal mechanisms governing disease transmission and have been extended to reaction‐diffusion equations to estimate the asymptotic rate of spatial spread. For diffusive SIR models, when the domain is bounded, quantitative properties including permanence, local and global stability of have been obtained in previous studies, 4–13 while for unbounded domains, the main focus is on the existence/nonexistence of traveling waves 14,15 . In these models, diseases spread to the whole region immediately even though the initial infection is confined to a small part of the region.…”