Instabilities and self-organization in spatiotemporal epidemic dynamics driven by nonlinearity and noise.

Theoretical analysis of epidemic dynamics has attracted significant attention in the aftermath
of the COVID-19 pandemic. In this article, we study dynamic instabilities in a spatiotemporal
compartmental epidemic model represented by a stochastic system of coupled partial differential
equations (SPDE). Saturation effects in infection spread-anchored in physical considerations-lead
to strong nonlinearities in the SPDE. Our goal is to study the onset of dynamic, Turing-type
instabilities, and the concomitant emergence of steady-state patterns under the interplay between
three critical model parameters-the saturation parameter, the noise intensity, and the transmission
rate. Employing a second-order perturbation analysis to investigate stability, we uncover both
diffusion-driven and noise-induced instabilities and corresponding self-organized distinct patterns
of infection spread in the steady state. We also analyze the effects of the saturation parameter
and the transmission rate on the instabilities and the pattern formation. In summary, our results
indicate that the nuanced interplay between the three parameters considered has a profound effect
on the emergence of dynamical instabilities and therefore on pattern formation in the steady state.
Moreover, due to the central role played by the Turing phenomenon in pattern formation in a variety
of biological dynamic systems, the results are expected to have broader significance beyond epidemic
dynamics.