Qualitative Analysis of Infectious Disease Spread in a Spatial SIR Model

Abstract

Interpersonal interactions and population mobility play a crucial role in the spread of infectious diseases. However, the classical SIR (Susceptible–Infected–Recovered) model assumes homogeneous mixing and neglects spatial movement. To address this limitation, we investigate disease dynamics using a reaction–diffusion spatial SIR model. We combine analytical and numerical approaches by deriving the basic reproduction ratio (R_0 ) using the Next Generation Matrix method, determining equilibrium points, and conducting local stability analysis through Jacobian linearization and diffusion-dependent characteristic equations. We further examine spatial disease propagation through traveling wave analysis and derive the minimum wave speed. Numerical simulations based on the Forward Time Centered Space (FTCS) finite difference scheme are performed to validate the analytical results and illustrate spatiotemporal disease dynamics. The results show that R_0 serves as the threshold parameter governing disease persistence. Specifically, R_0 increases with the transmission rate (β) and birth rate (Λ), and decreases with the natural death rate (μ) and recovery rate (γ). The model admits a disease-free equilibrium (E_0 ) and an endemic equilibrium (E_1 ). When R_0<1, E_0 is locally asymptotically stable, indicating disease eradication. Conversely, when R_0>1, E_1 becomes locally asymptotically stable, implying disease persistence. Traveling wave analysis shows that disease invasion occurs only when R_0>1, with a minimum wave speed of c = 2√(d_2 (γ + μ)(R₀- 1) ). FTCS simulations confirm the analytical predictions and demonstrate that increased mobility of infected individuals accelerates spatial disease spread.