93D30, 34G20, 92D20.

Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

Influence of backward bifurcation on interpretation of $R_0$ in a model of epidemic tuberculosis with reinfection

1. Department of Microbiology and Immunology, The University of Michigan Medical School, Ann Arbor, MI 48109-0620

## Abstract    Related pages

There is signiﬁcant disagreement in the epidemiological literature regarding the extent to which reinfection of latently infected individuals contributes to the dynamics of tuberculosis (TB) epidemics. In this study we present an epidemiological model of Mycobacterium tuberculosis infection that includes the process of reinfection. Using analysis and numerical simulations, we observe the eﬀect that varying levels of reinfection has on the qualitative dynamics of the TB epidemic. We examine cases of the model both with and without treatment of actively infected individuals. Next, we consider a variation of the model describing a heterogeneous population, stratiﬁed by susceptibility to TB infection. Results show that a threshold level of reinfection exists in all cases of the model. Beyond this threshold, the dynamics of the model are described by a backward bifurcation. Uncertainty analysis of the parameters shows that this threshold is too high to be attained in a realistic epidemic. However, we show that even for sub-threshold levels of reinfection, including reinfection in the model changes dynamic behavior of the model. In particular, when reinfection is present the basic reproductive number, $R_0$, does not accurately describe the severity of an epidemic.
Figure/Table
Supplementary
Article Metrics

Citation: Benjamin H. Singer, Denise E. Kirschner. Influence of backward bifurcation on interpretation of $R_0$ in a model of epidemic tuberculosis with reinfection. Mathematical Biosciences and Engineering, 2004, 1(1): 81-93. doi: 10.3934/mbe.2004.1.81

• 1. Nicolas Bacaër, Rachid Ouifki, Carel Pretorius, Robin Wood, Brian Williams, Modeling the joint epidemics of TB and HIV in a South African township, Journal of Mathematical Biology, 2008, 57, 4, 557, 10.1007/s00285-008-0177-z
• 2. Frank M. Hilker, Population collapse to extinction: the catastrophic combination of parasitism and Allee effect, Journal of Biological Dynamics, 2010, 4, 1, 86, 10.1080/17513750903026429
• 3. David J. Gerberry, Practical aspects of backward bifurcation in a mathematical model for tuberculosis, Journal of Theoretical Biology, 2016, 388, 15, 10.1016/j.jtbi.2015.10.003
• 4. D.P. Moualeu, A. Nana Yakam, S. Bowong, A. Temgoua, Analysis of a tuberculosis model with undetected and lost-sight cases, Communications in Nonlinear Science and Numerical Simulation, 2016, 41, 48, 10.1016/j.cnsns.2016.04.012
• 5. Prasanta Kumar Mondal, T. K. Kar, Optimal treatment control and bifurcation analysis of a tuberculosis model with effect of multiple re-infections, International Journal of Dynamics and Control, 2017, 5, 2, 367, 10.1007/s40435-015-0176-z
• 6. Hyun M Yang, Silvia M Raimundo, Assessing the effects of multiple infections and long latency in the dynamics of tuberculosis, Theoretical Biology and Medical Modelling, 2010, 7, 1, 10.1186/1742-4682-7-41
• 7. Rajivganthi Chinnathambi, Fathalla A. Rihan, Hebatallah J. Alsakaji, A fractional‐order model with time delay for tuberculosis with endogenous reactivation and exogenous reinfections, Mathematical Methods in the Applied Sciences, 2019, 10.1002/mma.5676
• 8. Fred Brauer, Carlos Castillo-Chavez, Zhilan Feng, , Mathematical Models in Epidemiology, 2019, Chapter 7, 249, 10.1007/978-1-4939-9828-9_7