Turing-Type Instabilities and Pattern Formation Induced by Saturation Effects and Randomness in Nonlinear, Diffusive Epidemic Spread

The COVID-19 pandemic has reinvigorated mathematical analysis of epidemic spread dynamics. We analytically investigate a partial differential equation (PDE) based, compartmental model of spatiotemporal epidemic spread, incorporating nonlinear infection forces accounting for saturation effects in the infection transmission mechanism. Using higher-order perturbation analysis and computing the local Lyapunov exponent, we find the emergence of dynamic instabilities induced both by the saturation parameter and stochastic environmental forces driving the epidemic spread. Notably, a second-order perturbation is found to be essential to uncover the noise-induced instabilities since they are not observed under first-order perturbations. We also analyze the effects of saturation and noise on such instabilities. Finally, using numerical, stationary solutions of the governing PDEs, we study the formation of spatial patterns of infection spread corresponding to the instabilities. We find the emergence of diffusion-driven patterns in the deterministic case and noise-induced patterns in cases when diffusion alone does not induce steady-state patterns.