Impact of latent delay and environment infection on tuberculosis dynamics in a population.

In this paper we explore the impact of latency delay and infection by Mycobacterium tuberculosis in the environment on the spread of tuberculosis in a population. We first derive a delay differential equation model with environmental indirect transmission. We address the well-posenedness and identify the basic reproduction number R0 of the model. We then discuss the equilibria and their stability in terms of the composite threshold parameter R0 which determine whether or not the tuberculosis will go extinct of persist in the popolaiton: the disease free equilibrium is globally stable if R0<1, and it becomes unstable if R0>1. In the latter case, there exists a unique endemic equilibrium, which is locally asymptotically stable when τ is sufficiently small; furthermore, we obtain the conditions for the existence of Hopf bifurcation around the endemic equilibrium. The condition implies that the interplay of the latency delay and infection of Mycobacterium tuberculosis in the environment may contribute not only to the TB's persistence but also the way it persists: either as an constant pattern (endemic equilibrium) or as a periodic pattern (oscillation around the endemic equilibrium). We also discuss the epidemiological implication of the mathematical results.