  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.
