A Note on Some Qualitative Properties of a Tuberculosis Differential Equation Model with a Time Delay

• Published in: Differential equations and dynamical systems Vol. 23; no. 2; pp. 181 - 194
• Main Author: Okuonghae, D.
• Format: Journal Article
• Language: English
• Published: India Springer India 01.04.2015
• Subjects: Computer Science; Engineering; Mathematics; Mathematics and Statistics; Original Research; Tuberculosis; Treatment; Fast latency; Mathematical model; Self cure; Delay
• ISSN: 0971-3514; 0974-6870
• DOI: 10.1007/s12591-013-0190-6

Tuberculosis is a global disease with much significant impact. Well over two million deaths are recorded each year due to Mycobacterium tuberculosis . The work by Colijn et al. (J Theor Biol 247, 765–774, 2007 ) examined two mathematical models (delay differential equation and network models) that explored the role of the contact structure of the population and found that in declining epidemics, localized outbreaks may occur as a result of contact heterogeneities even in the absence of host or strain variability. In this work, we carry out several qualitative studies (including the use of Sturm sequences for obtaining conditions on bifurcations) on the differential equation model with a time delay in Ref. Colijn et al., J Theor Biol 247, 765–774 ( 2007 ), and explicitly obtain threshold conditions for disease eradication as well as existence of bifurcations depending on a critical delay value which characterize the duration of fast latency. Further analysis showed the existence of a transcritical bifurcation when the reproduction number was less than one with the disease-free equilibrium co-existing with just one endemic state that was shown to be stable; the period of fast latency and the reinfection progression rate were shown to be very significant in the dynamics of the disease viz a viz the individuals who progressed to active TB within 5 years of their infection.