Traveling waves for a nonlocal delayed reaction-diffusion SIR epidemic model with demography effects

Abstract

Considering the importance of time delay phenomenon in disease transmission and the interdependence of time delay and spatial location, a reaction-diffusion SIR epidemic model with nonlocal time delay (or time-space time delay) is proposed and the travelling wave solutions are discussed. Specifically, we define the basic reproduction number ${\mathcal{R}}_0$ and the critical wave speed $c^{\ast }$. For every wave speed $c\ge c^{\ast }$, the existence of travelling wave solutions is studied by using the upper–lower solutions, the fixed-point theorem and some limit techniques when ${\mathcal{R}}_0>1$. The nonexistence of travelling waves when ${\mathcal{R}}_0>1$ for any $0<c<c^{\ast }$ or ${\mathcal{R}}_0<1$ for any c>0 is also proved. Finally, the influence of time-delay on disease propagation, especially the critical wave speed, is discussed through analysis and simulations.

Keywords:

• Epidemic model
• nonlocal delayed
• travelling wave solution
• critical wave speed
• asymptotic boundary conditions

Mathematics Subject Classifications:

• 35B40
• 35C07
• 92D30
• 35K57

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by a grant from the Young Scientist Funds of Natural Science Foundation of Xinjiang Uygur Autonomous Region [grant number 2022D01C63] and Natural Science Foundation of China [grant number 12271421 and 12301638].