Abstract
We develop a dynamical model to understand the underlying dynamics of TUBERCULOSIS infection at population level. The model, which integrates the treatment of individuals, the infections of latent and recovery individuals, is rigorously analyzed to acquire insight into its dynamical features. The phenomenon resulted due to the exogenous infection of TUBERCULOSIS disease. The mathematical analysis reveals that the model exhibits a backward bifurcation when TB treatment remains of infected class. It is shown that, in the absence of treatment, the model has a disease-free equilibrium (DEF) which is globally asymptotically stable (GAS) and the associated reproduction threshold is less than unity. Further, the model has a unique endemic equilibrium (EEP), for a special case, whenever the associated reproduction threshold quantity exceeds unity. For a special case, the EEP is GAS using the central manifold theorem of Castillo-Chavez.
Highlights
A differential equation which describes some physical process is often called a mathematical model of the process
We develop a dynamical model to understand the underlying dynamics of TUBERCULOSIS infection at population level
In the absence of treatment, the model has a disease-free equilibrium (DEF) which is globally asymptotically stable (GAS) and the associated reproduction threshold is less than unity
Summary
A differential equation which describes some physical process is often called a mathematical model of the process. Among them the SIR model is widely used in epidemiology and public health to compute number of individuals in each category of the population and to explain the change in the number of people needing medical attention during an epidemic as well as evaluate policies effectively during the endemic Tuberculosis [1]. According to the 2004 WHO report “Global Tuberculosis Control” [12], there were 8.8 million new cases of TB worldwide in 2002, with close to 2 million TB-related deaths, more than any other infectious diseases. It has seen that the rate of infected Mycobacterium tuberculosis (MTB) will reduce day by day, influencing on Bacilli Calmette-Guerin (BCG) vaccines which were first used in 1921 medically in USA and some TB treatment therapies. We have formulated the transmission dynamics of Tuberculosis in the presence of treatment and investigated its role in the dynamics of the disease
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