Large number of endemic equilibria for disease transmission models in patchy environment

We show that disease transmission models in a spatially heterogeneous environment can have a large number of coexisting endemic equilibria. A general compartmental model is considered to describe the spread of an infectious disease in a population distributed over several patches. For disconnected regions, many boundary equilibria may exist with mixed disease free and endemic components, but these steady states usually disappear in the presence of spatial dispersal. However, if backward bifurcations can occur in the regions, some partially endemic equilibria of the disconnected system move into the interior of the nonnegative cone and persist with the introduction of mobility between the patches. We provide a mathematical procedure that precisely describes in terms of the local reproduction numbers and the connectivity network of the patches, whether a steady state of the disconnected system is preserved or ceases to exist for low volumes of travel. Our results are illustrated on a patchy HIV transmission model with subthreshold endemic equilibria and backward bifurcation. We demonstrate the rich dynamical behavior (i.e., creation and destruction of steady states) and the presence of multiple stable endemic equilibria for various connection networks

Biomath Forum Backward Bifurcation in SIVS Model with Immigration of Non-Infectives

Abstract—This paper investigates a simple SIVS (susceptible–infected–vaccinated–susceptible) disease transmission model with immigration of susceptible and vaccinated individuals. We show global stability results for the model, and give an explicit condition for the existence of backward bifurcation and multiple endemic equilibria. We examine in detail how the structure of the bifurcation diagram depends on the immigration. Keywords-vaccination model with immigration; backward bifurcation; stability analysis I