Cox process representation and inference for stochastic reaction-diffusion processes

Complex behaviour in many systems arises from the stochastic interactions of spatially distributed particles or agents. Stochastic reaction-diffusion processes are widely used to model such behaviour in disciplines ranging from biology to the social sciences, yet they are notoriously difficult to simulate and calibrate to observational data. Here we use ideas from statistical physics and machine learning to provide a solution to the inverse problem of learning a stochastic reactiondiffusion process from data. Our solution relies on a nontrivial connection between stochastic reaction-diffusion processes and spatiotemporal Cox processes, a well-studied class of models from computational statistics. This connection leads to an efficient and flexible algorithm for parameter inference and model selection. Our approach shows excellent accuracy on numeric and real data examples from systems biology and epidemiology. Our work provides both insights into spatiotemporal stochastic systems, and a practical solution to a longstanding problem in computational modelling. Many complex behaviours in several disciplines originate from a common mechanism: the dynamics of locally interacting, spatially distributed agents. Examples arise at all spatial scales and in a wide range of scientific fields, from microscopic interactions of low-abundance molecules within cells, to ecological and epidemic phenomena at the continental scale. Frequently, stochasticity and spatial heterogeneity play a crucial role in determining the process dynamics and the emergence of collective behaviour [1]-[8]. Stochastic reaction-diffusion processes (SRDPs) constitute a convenient mathematical framework to model such systems. SRDPs were originally introduced in statistical physics [10, 11] to describe the collective behaviour of populations of point-wise agents performing Brownian diffusion in space and stochastically interacting with other, nearby agents according to predefined rules. The flexibility afforded by the local interaction rules has led to a wide application of SRDPs in many different scientific disciplines where complex spatiotemporal behaviours arise, from molecular biology [4, 9, 12], to ecology [13], to the social sciences [14]. Despite their popularity, SRDPs pose considerable challenges, as analytical computations are only possible for a handful of systems [8].