Turing Instability and Pattern Formation in Stochastic Epidemic Dynamics With Time Delay

Date of Award

8-15-2026

Degree Name

M.S. in Mechanical Engineering

Department

Department of Mechanical and Aerospace Engineering

Advisor/Chair

Subramanian Ramakrishnan

Abstract

This thesis investigates the Turing instability and spatial pattern formation in a stochastic reaction-diffusion partial differential equation (PDE) model of epidemic spread that incorporates time delay and nonlinear infection dynamics. The PDE model utilizes a compartmental framework comprising susceptible and infected populations, and a nonlinear function to describe infection transmission. Time delay accounts for incubation effects and delayed response in disease transmission, while stochasticity is incorporated through multiplicative noise arising intrinsically from the discreteness and randomness of person-to-person contact and infection events. A linear stability analysis of the homogeneous steady state is carried out, accounting for diffusion, time delay, and intrinsic (multiplicative) noise. The effects of the multiplicative noise term are analyzed using Novikov’s theorem. The analysis: (1) demonstrates that the interplay between nonlinearity, stochastic fluctuations, and time delay significantly influences the stability of the system, triggering Turing instabilities and corresponding self-organized, spatial pattern formation in equilibrium, and (2) identifies regions in parameter space that support Turing bifurcations. In particular, diffusion-driven instability leads to the formation of spatially heterogeneous structures in the infected population density. In addition, the numerical simulation results presented illustrate the effects of varying noise intensity and time delay on spatial pattern formation. The results show that increasing the intensity of multiplicative noise enhances irregularity and disrupts coherent spatial structures, whereas increasing the time delay duration tends to stabilize the system and reduce spatial variations. Overall, the thesis contributes to a better understanding of the critical roles of nonlinearity, stochasticity, and time delay, as well as their complex interactions, in epidemic spread dynamics. In addition, the results provide insights into the dynamic mechanisms underlying spatial pattern formation in spatiotemporal epidemic spread.

Keywords

Applied Mathematics, Epidemiology, Mathematics, Mechanical Engineering

Rights Statement

Copyright 2026, author

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