| When immune cells detect foreign molecules, they secrete soluble factors that attract other immune cells to the site of the infection. In this thesis, I study numerical solutions to a model of this behavior proposed by Thomas Kepler. The soluble factors are governed by a system of reaction-diffusion equations with sources that are centered on the cells. The motion of the cells is stochastic, but biased toward the gradient of the soluble factors. 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 developed a numerical method for solving the reaction-diffusion stochastic system based on a first order splitting scheme. This method makes use of known first order schemes for solving the diffusion, the reaction and the stochastic differential equations separately. The domain is discretized using finite elements and the diffusion is solved using a backward Euler scheme combined with multigrid. The stochastic differential equations are solved using a Milstein scheme, making use of the fact the noise is commutative. |
