Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population

In this paper, a general mathematical model is proposed with detailed justifications to describe the spread of a disease with latency in a heterogeneous host population which includes many existing ones as special cases. For a simpler version that assumes an identical natural death rate for all groups, and with a gamma distribution for the latency, the model is shown to demonstrate the global threshold dynamics in terms of the basic reproduction number R0R0 of the model: if R0≤1R0≤1, the disease-free equilibrium is globally asymptotically stable in the positive orthant, whereas if R0>1R0>1, a unique endemic equilibrium exists and is globally asymptotically stable in the interior of the positive orthant. The proofs of the main results make use of the theory of non-negative matrices, persistence theory in dynamical systems, Lyapunov functions and a subtle grouping technique in estimating the derivatives of Lyapunov functions guided by graph theory, which was recently developed and applied by several authors to some relateted epidemic models.