Mathematical Biosciences and Engineering (MBE)

Dynamical Models of Tuberculosis and Their Applications

Pages: 361 - 404, Volume 1, Issue 2, September 2004      doi:10.3934/mbe.2004.1.361

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Carlos Castillo-Chavez - Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287, United States (email)
Baojun Song - Department of Mathematical Sciences, Montclair State University, Upper Montclair, NJ 07043, United States (email)

Abstract: The reemergence of tuberculosis (TB) from the 1980s to the early 1990s instigated extensive researches on the mechanisms behind the transmission dynamics of TB epidemics. This article provides a detailed review of the work on the dynamics and control of TB. The earliest mathematical models describing the TB dynamics appeared in the 1960s and focused on the prediction and control strategies using simulation approaches. Most recently developed models not only pay attention to simulations but also take care of dynamical analysis using modern knowledge of dynamical systems. Questions addressed by these models mainly concentrate on TB control strategies, optimal vaccination policies, approaches toward the elimination of TB in the U.S.A., TB co-infection with HIV/AIDS, drug-resistant TB, responses of the immune system, impacts of demography, the role of public transportation systems, and the impact of contact patterns. Model formulations involve a variety of mathematical areas, such as ODEs (Ordinary Differential Equations) (both autonomous and non-autonomous systems), PDEs (Partial Differential Equations), system of difference equations, system of integro-differential equations, Markov chain model, and simulation models.

Keywords:  Tuberculosis, dynamical models, prevention and control, global dynamics, bifurcation, HIV, disease emergence and reemergence.
Mathematics Subject Classification:  92D30.

Available Online: July 2004.