Summary
Epidemic spread dynamics driven by nonlinearity and noise is a focal research topic post the COVID-19 pandemic. Turing-type instabilities can drastically alter epidemic trajectories leading to unique spread patterns. We study instabilities in epidemic spread using a compartmental, stochastic partial differential equation (SPDE) model, accounting for (1) nonlinear saturation effects and (2) environmental effects represented by additive noise. From second-order perturbations of the disease-free equilibrium we analytically derive a hierarchy of moment equations using Novikov’s theorem. Judiciously truncating this infinite hierarchy of moment equations, we analyze stability by computing the maximal Lyapunov exponent against the wave number that characterizes the perturbations. We analyze stability with respect to the saturation parameter and the noise intensity. Additionally, obtaining stationary solutions of the SPDE, we find distinct spatial patterns of infection spread consistent with the stability results. Specifically, we observe the emergence of noise-induced patterns in cases where diffusion and saturation are insufficient to trigger instabilities. The results are expected to have broader significance beyond epidemic dynamics since reaction-diffusion systems represent a variety of phenomena in the natural sciences and engineering.
