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2016, 13(4): 741-785. doi: 10.3934/mbe.2016017

## A two-strain TB model with multiple latent stages

 1 Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran, Iran 2 Simon A Levin Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287, United States, United States 3 Department of Mathematical Sciences, Montclair State University, 1 Normal Avenue, Montclair, NJ 07043

Received  November 2015 Revised  February 2016 Published  May 2016

A two-strain tuberculosis (TB) transmission model incorporating antibiotic-generated TB resistant strains and long and variable waiting periods within the latently infected class is introduced. The mathematical analysis is carried out when the waiting periods are modeled via parametrically friendly gamma distributions, a reasonable alternative to the use of exponential distributed waiting periods or to integral equations involving arbitrary'' distributions. The model supports a globally-asymptotically stable disease-free equilibrium when the reproduction number is less than one and an endemic equilibriums, shown to be locally asymptotically stable, or l.a.s., whenever the basic reproduction number is greater than one. Conditions for the existence and maintenance of TB resistant strains are discussed. The possibility of exogenous re-infection is added and shown to be capable of supporting multiple equilibria; a situation that increases the challenges faced by public health experts. We show that exogenous re-infection may help established resilient communities of actively-TB infected individuals that cannot be eliminated using approaches based exclusively on the ability to bring the control reproductive number just below $1$.
Citation: Azizeh Jabbari, Carlos Castillo-Chavez, Fereshteh Nazari, Baojun Song, Hossein Kheiri. A two-strain TB model with multiple latent stages. Mathematical Biosciences & Engineering, 2016, 13 (4) : 741-785. doi: 10.3934/mbe.2016017
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