Basil Bayati, a member of our Epidemiological Modeling team, recently published a paper in the Journal of Chemical Physics. The paper — Fractional diffusion-reaction stochastic simulations — details our work with stochastic simulations of reaction-diffusion processes for modeling physical phenomena**.** The underlying spatial process of physical phenomena, including epidemiological processes, is often assumed to be Gaussian. This assumption, however, is not necessarily valid. Examples of processes that deviate from classical diffusion range from wandering albatross to the circulation of bank notes, the latter of which has been used as a proxy for human travel. In order to accurately simulate the propagation of human diseases, we must take into account the possibly non-Gaussian character of human travel. In this article, a novel method is presented for the simulation of discrete-space, continuous-time Markov processes that are subject to fractional, heavy-tailed, diffusion. The method is based on Lie-Trotter operator splitting of the diffusion and reaction terms in the master equation. The diffusion term follows a multinomial distribution governed by a kernel that is the discretized solution of the fractional diffusion equation. The algorithm is validated and simulations are provided for the Fisher-KPP wavefront. It is shown that the wave speed is dictated by the order of the fractional derivative, where lower values result in a faster wave than in the case of classical diffusion. Since many physical processes deviate from classical diffusion, fractional diffusion methods are necessary for accurate simulations.

Learn about more about our Epidemiological Modeling program here or check out other recent EMOD publications on our blog.

**Published Paper**

Bayati, B.S. (2013) Fractional diffusion-reaction stochastic simulations. *Journal of Chemical Physics* 138, 104117.