  This is a truncation of the second year group project at Imperial college
london. In this paper, we consider a semilinear reaction diffusion system of
SIR model which involves the birth rate and the death rate. We first prove the
non-negativity and global existence theorem to ensure that the model makes
sense. We prove the uniform convergence of the infection-free solution and
study an example that separable solutions can be computed. We also focus on the
steady state solution, which we prove the non-uniqueness of the solution and
investigate the regularity of the general solution. In the end we also
introduce an interesting phenomenon, which is called the Turing instability
caused by the diffusion in the model.
