Operator Splitting for an Immunology Model Using Reaction-Diffusion Equations with Stochastic Source Terms

When immune cells detect foreign molecules, they secrete soluble factors that attract other immune cells to the site of the infection. In this paper, I study numerical solutions to a model of this behavior proposed by Kepler. In this model the soluble factors are governed by a system of reaction-diffusion equations with sources that are centered on the cells. The motion of the model cells is a Langevin process that is biased toward the gradient of the soluble factors. I have shown that the solution to this system exists for all time and remains positive, the supremum is a priori bounded and the derivatives are bounded for finite time. I have also developed a first order split scheme for solving the reaction-diffusion stochastic system. This allows us to make use of known first order schemes for solving the diffusion, the reaction and the stochastic differential equations separately.