Stochastic diffusion using mean-field limits to approximate master equations

Stochastic diffusion is the noisy process through which dynamics like epidemics, or agents like animal species, disperse over a larger area. These processes are increasingly important to better prepare for pandemics and as species ranges shift in response to climate change. Unfortunately, modelling is mostly done with expensive computational simulations or inaccurate deterministic tools that ignore the randomness of dispersal. We introduce 'mean-FLAME' models, tracking stochastic dispersion using approximate master equations to follow the probability distribution over all possible states of an area of interest, up to states active enough to be approximated using a mean-field model. In the limit where we track all states, this approach is locally exact, and in the other limit collapses to traditional deterministic models. In predator-prey systems, we show that tracking a handful of states around key absorbing states is sufficient to accurately model extinction. In disease models, we show that classic mean-field approaches underestimate the heterogeneity of epidemics. And in nonlinear dispersal models, we show that deterministic tools fail to capture the speed of spatial diffusion. These effects are all important for marginal areas that are close to unsuitable for diffusion, like the edge of a species range or epidemics in small populations.