Equilibrium properties of the spatial SIS model as a point pattern dynamics - How is infection distributed over space?

We revisit the classical epidemiological SIS model as a stochastic point pattern dynamics with special focus on its spatial distribution at equilibrium. In this model, each point on a continuous space is either susceptible S or infectious I, and infection occurs with an infection kernel as a function of distance from I to S. This stochastic process has been mathematically described by the hierarchical dynamics of the probabilities that a point, a pair made by two points, and a triplet made by three points, etc., is in a specific configuration of status. Using a simple closure thereby triplet probabilities that appear in the dynamics are approximated, we show that the average singlet probabilities and the pair probabilities that describe spatial distributions of Ss and Is at equilibrium can be explicitly derived using the infection kernel; Is are spatially clustered in the same order of the infection kernel. The results highlight the advantage of point pattern approach to model spatial population dynamics in general ecology where local interactions among individuals likely depend on distance between them.