Stochastic dynamics in epidemiological models with interconnected populations

Abstract

Epidemic models are essential tools for understanding the spread of infectious diseases and evaluating containment measures. The COVID-19 pandemic highlighted the critical role of population movement in shaping epidemic dynamics, emphasizing the need for models that incorporate mobility effects. In this work, we study disease transmission between two interconnected populations using a stochastic framework. Building on a deterministic model, we introduce a continuous-time Markov chain stochastic model and compare it with three approximations. While continuous-time Markov chains provide a natural stochastic counterpart to deterministic models, they pose challenges in scalability and implementation for large systems. To address these issues, we explore simplified approximations that retain key stochastic features while reducing computational complexity. Our analysis focuses on the impact of movement on disease persistence, particularly in source-sink scenarios where one population serves as a reservoir of infection. We show that stochastic effects can lead to extinction events absent in deterministic models, underscoring the importance of randomness in epidemic forecasting. Numerical simulations illustrate each approach, providing insights into the interplay between mobility and epidemic spread.