The minimal wave speed of time-periodic traveling waves arising from a diffusive Kermack-McKendrick model with seasonality and nonlocal delayed interactions

TL;DR

This study proves the minimal wave speed for time-periodic traveling waves in a seasonally affected, nonlocal delayed epidemic model, confirming a previously open problem and advancing understanding of wave propagation in such systems.

Contribution

The paper establishes the critical wave speed for traveling waves in a non-autonomous, nonlocal delayed epidemic model, solving an open problem from prior research.

Findings

Confirmed the critical wave speed as the minimal speed for traveling waves.

Developed a new method using upper and lower solutions for non-autonomous systems.

Resolved an open problem in the analysis of epidemic wave propagation.

Abstract

This paper is concerned with the non-existence of time-periodic traveling wave solution with speed less than the critical speed for diffusive Kermack-McKendrick epidemic model incorporating seasonality and nonlocal interactions induced by latent period. By a technical construction of upper and lower solutions on truncated intervals for an auxiliary linear equation, we overcome the challenges arising from the coupling of nonlocal delay and the fact that the system is non-autonomous. Thus the critical value $c^*$ defined in [S.-M. Wang et al., Nonlinear Anal. Real World Appl., 55 (2020) 103117] is confirmed as the minimal wave speed of time-periodic traveling waves. We have completely solved the open problem [S.-M. Wang et al., Nonlinear Anal. Real World Appl., 55 (2020) 103117]