Disease emergence in deterministic and stochastic models for host and pathogen

The emergence of new infectious diseases often occurs because of the evolution of a new pathogen. The generation time for pathogens is often much shorter than the generation time for hosts, and therefore, virulence evolution in the pathogen population occurs much faster than the development of resistance or immunity in the host population. This results in a disease outbreak in the host population. In this investigation, we develop a model for a host-pathogen system, where the genetics of the pathogen are included. We use this model to study the evolutionary dynamics of the pathogen. The pathogen evolution is determined by a single autosomal locus (haploid or diploid) for which there are n different alleles. The host population is subdivided into susceptible and infected individuals. The infected host population governs the growth of the pathogen population. A special case is studied in detail, where the pathogen population obeys the Hardy-Weinberg law. We define the basic reproduction number for the host-pathogen system and investigate local and global stability of the disease-free equilibrium. It is shown that the initial proportions of the pathogen genotypes and the basic reproduction number determine whether a new infectious disease will emerge. In addition, a stochastic model is developed based on the deterministic formulation. The dynamics of the deterministic and stochastic models are compared through extensive numerical simulations. Stochastic variability can lead to the emergence of pathogenic genotypes different from those predicted by the deterministic model.