Stability of a Two-Strain Tuberculosis Model with General Contact Rate

A two-strain tuberculosis model with general contact rate which allows tuberculosis patients with the drug-sensitive Mycobacterium tuberculosis strain to be treated is presented. The model includes both drug-sensitive and drug-resistant strains. A detailed qualitative analysis about positivity, boundedness, existence, uniqueness and global stability of the equilibria of this model is carried out. Analytical results of the model show that the quantities and , which represent the basic reproduction numbers of the sensitive and resistant strains, respectively, provide the threshold conditions which determine the competitive outcomes of the two strains. Numerical simulations are also conducted to confirm and extend the analytic results. 1. Introduction Tuberculosis (TB), an infectious disease caused by Mycobacterium tuberculosis (M. tuberculosis), is one of the world's leading causes of infectious mortality. According to the World Health Organization(WHO), one third of the world's population is infected with M. tuberculosis, leading to between two and three millions death each year. At present, about 95% of the estimated 8 million new cases of TB occurring each year are in developing countries, where 80% occur among people between the ages of 15 and 59 years [1]. For the time being, TB is becoming a world-wide problem. War, famine, homelessness, and a lack of medical care all contribute to the increasing incidence of tuberculosis among disadvantaged persons. Since TB is easily transmissible between persons, the increase in TB in any segment of the population represents a threat to all segments of the population. Sub-Saharan Africa remains the epicenter of the epidemic, but India, China, Indonesia, Bangladesh and Pakistan together account for more than half of the cases in the world [2]. TB was assumed to be on its way out in developed countries until the number of TB cases began to increase in the 1980s. With this return, we face the paradox of a well-known bacteria, fully treatable with efficient and affordable drugs according to internationally recommended guidelines, which yet causes increasing human suffering and death. Active TB cases may be pulmonary or extrapulmonary, but pulmonary cases are more infectious and form the bulk of most cases of active TB. The usual symptoms of active TB include fatigue, high fever, night sweats, loss of appetite, and a cough, but confirmation of active TB requires a positive sputum culture. Extrapulmonary accounts for between 5% and 30% of the total cases and may affect any part of the body. M. tuberculosis droplets are