Dynamical Models of Tuberculosis and Their Applications

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.

Mathematical Models of Tuberculosis Extension and Control of it

“…По словам автора, внесенные мо-дификации отчасти позаимствованы из работ C.Броггера (S. Brogger) [7] (раздел 2.2) и К. Ревеля [8,9] (раздел 2.3). 8 Изложение ведется по обзору в [61], оригинальные тексты авторам данного обзора недоступны. 9 Изложение ведется по обзору в [61], оригинальный текст авторам данного обзора недоступен.…”

Analysis of mathematical model of leukemia

“…To analyze the local stability in critical cases, we use the center manifold theory, as described in CastilloChavez and Song (Theorem 4.1) [1]. Center manifold theory has been used to decide the local stability of a nonhyperbolic equilibrium (linearization matrix has at least one eigenvalue with zero real part).…”

A Two Strain Tuberculosis Transmission Model With Therapy and Quarantine

“…⊓ ⊔ To determine the asymptotic stability of U 1 we make use of the Centre Manifold theory (Carr,[6]) as described in Theorem 4.1 of Castillo-Chavez and Song [9]. To use the Centre Manifold theory let us make the following change of…”