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

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]