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

Shuang-Ming Wang

TL;DR

The paper addresses the minimal wave speed for time-periodic traveling waves in a diffusive Kermack–McKendrick epidemic model with seasonality and nonlocal latent-period interactions. It develops a perturbation-based approach that introduces an auxiliary linear equation and constructs upper and lower solutions on truncated intervals to enable a comparison argument in a non-autonomous, nonlocal setting. The main result proves the nonexistence of time-periodic traveling waves for speeds $c$ in $(0,c^*)$ when the basic reproduction number satisfies $R_0>1$, thereby establishing the critical speed $c^*$ as the minimal wave speed. This work not only resolves an open problem but also provides a methodological framework potentially applicable to other incubation-delay epidemic systems and to analyses of asymptotic spreading in time-periodic environments.

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]

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

TL;DR

The paper addresses the minimal wave speed for time-periodic traveling waves in a diffusive Kermack–McKendrick epidemic model with seasonality and nonlocal latent-period interactions. It develops a perturbation-based approach that introduces an auxiliary linear equation and constructs upper and lower solutions on truncated intervals to enable a comparison argument in a non-autonomous, nonlocal setting. The main result proves the nonexistence of time-periodic traveling waves for speeds in when the basic reproduction number satisfies , thereby establishing the critical speed as the minimal wave speed. This work not only resolves an open problem but also provides a methodological framework potentially applicable to other incubation-delay epidemic systems and to analyses of asymptotic spreading in time-periodic environments.

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 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]
Paper Structure (3 sections, 1 theorem, 20 equations)

This paper contains 3 sections, 1 theorem, 20 equations.

Table of Contents

  1. Introduction
  2. The nonexistence of periodic traveling waves
  3. Conclusions

Key Result

Theorem 2.1

Assume that $R_0>1$. For all $c\in(0,c^{*})$, system RWA admits no time-periodic traveling wave solution with wave speed $c$ that satisfies asymptotic boundary condition BC.

Theorems & Definitions (2)

  • Theorem 2.1
  • proof