The evolutionary dynamics of a spatial multi-strain host-pathogen system with cross-immunity

We considered a Susceptible-Infective-Recovered-Susceptible (SIRS) model with strain mutation and cross-immunity in a non-spatial model and a lattice-structured model, where all individuals can reproduce if the space/resources allow. In the lattice-structured model, both the host reproduction and pathogen transmission processes are assumed to interact with next nearest neighbors, and the model was analyzed by an improved pair approximation (IPA). A family of correlated equations of pair approximation and mean-field were presented. We show the phase diagram of the coexistence and extinction which were obtained from parameterization by measuring the basic reproduction numbers of the strains during their infection processes. The qualitative results of the pair approximation model are similar to that of the corresponding non-spatial model. Furthermore, the spatial model predicts coexistence over a wider range of parameters than the non-spatial model. In particular, when the strain evolution tends to a larger basic reproduction number, the correlated spatial approximation could predict better than the mean-field approximation