# The Emergence of Turing Instability and Pattern Formation in a Nonlinear Stochastic Spatiotemporal Epidemic Model with Reinfections

## TL;DR

This paper explores how patterns form in an epidemic model with reinfections, showing how randomness and nonlinear effects influence disease spread and stability.

## Contribution

The study introduces a novel analysis of Turing instability and pattern formation in a stochastic epidemic model with reinfections.

## Key findings

- Turing instability and equilibrium patterns emerge due to reinfection, saturation, and noise intensity.
- Stochastic excitation and saturation compete to influence pattern formation and system stability.
- Results are broadly applicable to noise-induced instabilities in nonlinear spatiotemporal systems.

## Abstract

Instabilities and Turing patterns in stochastic spatiotemporal systems in which a fraction of an evolving population, after undergoing a series of dynamic transitions, returns to its original state, remain largely unexplored. Adopting an epidemic model incorporating reinfections as an exemplar of such a system, we present stability and pattern-formation analyses of the stochastic reaction-diffusion equations that represent the model. Saturation effects in epidemic spread lead to nonlinear considerations, while random environmental effects motivate a stochastic term. Turing bifurcation and the emergence of equilibrium patterns are analysed with respect to three fundamental parameters - reinfection, saturation, and noise intensity. Using higher-order stability analysis and stochastic averaging, we find the Turing instability and also uncover self–organized, distinct equilibrium patterns of infection spread. Additionally, results elucidating the effects of stochastic excitation and its intensity, as well as the competing influence of saturation and reinfection on stability and pattern formation, are presented. The results are also expected to be broadly significant beyond epidemic modelling, for studies of noise-induced instabilities and morphogenesis in spatiotemporal nonlinear dynamical systems.