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On the role of variable latent periods in mathematical models for tuberculosis

Abstract:

The qualitative behaviors of a system of ordinary differential equations and a system of differential-integral equations, which model the dynamics of disease transmission for tuberculosis (TB), have been studied. It has been shown that the dynamics of both models are governed by a reproductive number. All solutions converge to the origin (the disease-free equilibrium) when this reproductive number is less than or equal to the critical value one. The disease-free equilibrium is unstable and there exists a unique positive (endemic) equilibrium if the reproductive number exceeds one. Moreover, the positive equilibrium is stable. Our results show that the qualitative behaviors pbkp_redicted by the model with arbitrarily distributed latent stage are similar to those given by the TB model with an exponentially distributed period of latency. © 2001 Plenum Publishing Corporation.