Comparison of an Infinite Dimensional Model for Parasitic Diseases with a Related 2-Dimensional System

A model for parasitic diseases which is described by an infinite system of ordinary differential equations is compared to a related two-dimensional system It is shown that depending on a bifurcation parameter κ the infinite dimensional system can have solutions which asymptotically grow exponentially in time or a unique endemic steady state. The threshold values for κ and the results for the bifurcation branches of exponential solutions and steady states are comparable to similar results for the related two-dimensional system. Finally it is shown that invariant distributions for the infinite dimensional model are overdispersed; i.e., the variance to mean ratio is larger than 1.