Traveling waves of nonlocal delayed disease models: critical wave speed and propagation speed

ABSTRACT

In this paper, we investigate the traveling wave solutions of diffusive disease models with a general incidence rate, nonlocal interaction and transmission delay. We prove that a positive traveling wave solution exists if the wave speed is bigger than a threshold value and does not exist if the wave speed is smaller than this value. We also investigate the dependence of this critical wave speed on the diffusion coefficient of the infected population and average transmission delay. In the critical case when the wave speed equals the threshold value, we obtain the existence of nontrivial traveling waves without nonlocal interaction or transmission delay. We further develop numerical methods to simulate traveling wave solutions and estimate disease propagation speed. It is observed from numerical simulations that disease propagation speed is strictly less than the critical wave speed.

KEYWORDS:

• Traveling wave
• nonlocal interaction
• disease transmission delay
• critical wave speed
• disease propagation speed

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 92D30
• 35K57
• 34B40

Acknowledgments

We would like to thank two anonymous referees for their careful reading and helpful suggestions which led to an improvement to our original manuscript.

Disclosure statement

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

Additional information

Funding

H. Shu was partially supported by National Natural Science Foundation of China [No.11971285, No. 11601392], Shaanxi Hundred-Talent Program, and the Fundamental Research Funds for the Central Universities [No. GK201902005].