Traveling waves for an epidemic patchy model with bilinear incidence.

Area of Science:

• Mathematical epidemiology
• Dynamical systems
• Partial differential equations

Background:

• Epidemic models are crucial for understanding disease spread.
• Patchy environments introduce spatial heterogeneity in disease dynamics.
• Traveling wave solutions represent the spatial spread of epidemics.

Purpose of the Study:

• To investigate the existence and non-existence of traveling wave solutions in epidemic models with bilinear incidence in a periodic patchy environment.
• To determine the minimal wave speed ($c^{\ast }$) for bounded traveling wave solutions.
• To analyze the impact of environmental heterogeneity and model parameters on epidemic spread and final size.

Main Methods:

• Analysis of traveling wave solutions for a reaction-diffusion system.
• Derivation of conditions for the existence and non-existence of solutions based on wave speed ($c$) and the basic reproduction number ($R_0$).
• Numerical simulations to explore parameter effects on epidemic final size.

Main Results:

• A minimal wave speed $c^{\ast }$ is established for bounded traveling wave solutions when $R_0>1$ and $c\ge c^{\ast }$.
• Traveling wave solutions do not exist if $R_0\le 1$ or if $R_0>1$ and $c\in (0,c_{}^{\ast })$.
• Heterogeneity in transmission and removal rates increases $c^{\ast }$, while heterogeneity in diffusion decreases it.

Conclusions:

• The existence of traveling epidemic waves is contingent on wave speed relative to a critical threshold and the basic reproduction number.
• Environmental heterogeneity significantly influences epidemic spread dynamics and the critical wave speed.
• Findings provide insights for disease control strategies by highlighting the impact of parameter heterogeneity on epidemic outcomes.