Global stability for a tuberculosis model with isolation and incomplete treatment
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Global stability for a tuberculosis model with isolation and incomplete treatment Jinhui Zhang · Guangting Feng
Received: 16 November 2013 / Revised: 27 June 2014 / Accepted: 2 August 2014 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2014
Abstract This paper deals with the global analysis of a dynamical model for the spread of tuberculosis with isolation and incomplete treatment. The model exhibits the traditional threshold behavior. We prove that when the basic reproductive number is less than unity, the disease-free equilibrium is globally asymptotically stable. When the basic reproductive number is greater than unity, the disease-free equilibrium is unstable and a unique endemic equilibrium exists which is locally asymptotically stable and globally asymptotically stable when the disease-induced death rate is equal to zero. The stability of disease-free equilibrium is derived by using Lyapunov stability theory and LaSalle’s invariant set theorem. The global stability of endemic equilibrium is proved by generalized Dulac–Bendixson criterion when the disease-induced death rate is equal to zero. Numerical simulations support our analytical results. Keywords criterion
Basic reproductive number · Global stability · Generalized Dulac–Bendixson
Mathematics Subject Classification
92D30 · 34D23
Communicated by Maria do Rosario de Pinho. J. Zhang (B) School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China e-mail: [email protected] G. Feng School of Mathematics and Statistics, Hubei University of Education, Wuhan 430205, China e-mail: [email protected]
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J. Zhang, G. Feng
1 Introduction Tuberculosis (TB) is a disease caused by infection with Mycobacterium tuberculosis, which most frequently affects the lungs (pulmonary TB). It is one of the most common infectious diseases with about 2 billion people (one-third of the world’s population) currently infected. About 9 million new cases of active disease develop each year, resulting in two million deaths, mostly in developing countries. There were an estimated 8.7 million incident cases of TB globally in 2011 (World Health Organization 2012). TB infection remains a serious public health challenge in China. According to the WHO estimates, China has the world’s second largest TB epidemic accounting for 12 % of global cases, only after India, with more than 1.3 million new cases of TB being reported every year. From the website of National Health and Family Planning Commission of the PRC (2001–2013), we know that over the period 2003–2013 TB was the second largest cause of death among China’s 39 notifiable communicable diseases, after HIV/AIDS (National Health and Family Planning Commission of the PRC 2003–2014). Fortunately, TB is treatable and curable. TB patients can be treated and can recover using antibiotics, though TB requires much longer periods of treatment than many other diseases (typically 6–9 months) for complete removal of the causative bacteria (Palomino et al. 2009). The treatment is usually
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