Global dynamics of multi-group SEI animal disease models with indirect transmission

A challenge to multi-group epidemic models in mathematical epidemiology is the exploration of global dynamics. Here we formulate multi-group SEI animal disease models with indirect transmission via contaminated water. Under biologically motivated assumptions, the basic reproduction number R0 is derived and established as a sharp threshold that completely determines the global dynamics of the system. In particular, we prove that if R0<1, the disease-free equilibrium is globally asymptotically stable, and the disease dies out; whereas if R0>1, then the endemic equilibrium is globally asymptotically stable and thus unique, and the disease persists in all groups. Since the weight matrix for weighted digraphs may be reducible, the afore-mentioned approach is not directly applicable to our model. For the proofs we utilize the classical method of Lyapunov, graph-theoretic results developed recently and a new combinatorial identity. Since the multiple transmission pathways may correspond to the real world, the obtained results are of biological significance and possible generalizations of the model are also discussed.