Propagation dynamics for an influenza transmission model

This paper characterizes how human movement influences the propagation of influenza by analyzing traveling wave solutions within a spatial transmission framework, where the dynamics are governed by the basic reproduction number $R_{0}$ and the critical wave speed $c_{*}$ . We first demonstrate the existence of such wave solutions when $R_{0} > 1$ and the wave speed c exceeds $c_{*}$ , employing a constructive approach based on upper and lower solutions and applying Schauder’s fixed-point theorem. The asymptotic boundary behavior of traveling wave solutions at +∞ is derived by constructing an appropriate Lyapunov functional. In the critical case $c = c_{*}$ , we establish the existence of traveling fronts using Arzelà–Ascoli’s theorem in combination with asymptotic spreading speed estimates. To investigate wave nonexistence, we consider two scenarios: (i) $R_{0}>1$ but $c < c_{*}$ , and (ii) $R_{0}<1$ , and derive contradictions by comparing with suitable auxiliary solutions. To validate our theoretical findings, we conduct numerical simulations and explore how varying levels of human mobility and diffusion among infected individuals affect the minimal wave speed $c_{*}$ .