  Infectious diseases are practically represented by models with multiple
states and complex transition rules corresponding to, for example, birth,
death, infection, recovery, disease progression, and quarantine. In addition,
networks underlying infection events are often much more complex than described
by meanfield equations or regular lattices. In models with simple transition
rules such as the SIS and SIR models, heterogeneous contact rates are known to
decrease epidemic thresholds. We analyze steady states of various multi-state
disease propagation models with heterogeneous contact rates. In many models,
heterogeneity simply decreases epidemic thresholds. However, in models with
competing pathogens and mutation, coexistence of different pathogens for small
infection rates requires network-independent conditions in addition to
heterogeneity in contact rates. Furthermore, models without spontaneous
neighbor-independent state transitions, such as cyclically competing species,
do not show heterogeneity effects.
