Stochastic diffusion using mean-field limits to approximate master equations

Stochastic diffusion is the noisy and uncertain process through which dynamics like epidemics, or agents like animal species, disperse over a larger area. Understanding these processes is becoming increasingly important as we attempt to better prepare for potential pandemics and as species ranges shift in response to climate change. Unfortunately, modeling of stochastic diffusion is mostly done through inaccurate deterministic tools that fail to capture the random nature of dispersal or else through expensive computational simulations. In particular, standard tools fail to fully capture the heterogeneity of the area over which this diffusion occurs. Rural areas with low population density require different epidemic models than urban areas; likewise, the edges of a species range require us to explicitly track low integer numbers of individuals rather than vague averages. In this work, we introduce a series of new tools called "mean-FLAME" models that track stochastic dispersion using approximate master equations that explicitly follow the probability distribution of an area of interest over all of its possible states, up to states that are active enough to be approximated using a mean-field model. In one limit, this approach is locally exact if we explicitly track enough states, and in the other limit collapses back to traditional deterministic models if we track no state explicitly. Applying this approach, we show how deterministic tools fail to capture the uncertainty around the speed of nonlinear dynamical processes. This is especially true for marginal areas that are close to unsuitable for diffusion, like the edge of a species range or epidemics in small populations. Capturing the uncertainty in such areas is key to producing accurate forecasts and guiding potential interventions.