Space and Disease: Insights from Interacting Particle Systems

There is a long history of modeling diseases, starting with the simple epidemic models by Kermack and McKendrick in the 1920s and 30s. These models provide useful insights into the dynamics of epidemics, such as conditions under which an epidemic will spread. These early models are formulated within the mathematical framework of systems of ordinary differential equations, and are thus deterministic and assume that interactions are global. The assumptions of deterministic and global dynamics are often a good first approximation but leave out important characteristics of epidemics, namely that encounters of susceptible and infected individuals are random and typically only occur over relatively short distances. A framework to describe stochastic and local interactions is provided by a class of stochastic models, known as interacting particle systems. These models were introduced in the 1970s to describe the emergent properties of stochastic and local interactions of individual agents in d-dimensional space. In this presentation, we will give a brief historical overview of the types of disease models that have been studied mathematically rigorously within the framework of interacting particle systems, and present more recent results on interactions between hosts and disease agents, with a particular emphasis on the consequences of stochasticity and local interactions.