Abstract
In this chapter we describe several models for tuberculosis (TB) . The disease is endemic in many areas of the world. The models in this chapter will be extensions of the standard SIR or SEIR type of endemic models presented in Chap. 3. Depending on the typical characteristics of a specific disease, various modifications of the standard models will be considered.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Aparicio, J. P., and C. Castillo-Chavez (2009) Mathematical modelling of tuberculosis epidemics, Math. Biosc. Eng. 6: 209–237.
Blower, S. M., and T. Chou (2004) Modeling the emergence of the ‘hot zones’: tuberculosis and the amplification dynamics of drug resistance, Nature Medicine 10: 1111–1116.
Blower, S. M., P.M. Small, and P.C. Hopewell (1996) Control strategies for tuberculosis epidemics: new models for old problems, Science 273: 497–500.
Castillo-Chavez, C., and Z. Feng (1997) To treat or not to treat: the case of tuberculosis. J. Math. Biol. 35: 629–656.
Castillo-Chavez, C., and Z. Feng (1998) Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math.Biosc. 151: 135–154.
CDC, Tuberculosis treatment (2014) http://www.cdc.gov/tb/topic/treatment/.
Cohen, T., and M. Murray (2004) Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness Nature Medicine 10: 1117–1121.
Dye, C., and B.G. Williams (2000) Criteria for the control of drug-resistant tuberculosis, Proc. Nat. Acad. Sci. 97: 8180–8185.
Feng, Z., C. Castillo-Chavez, and A.F. Capurro (2000) A model for tuberculosis with exogenous reinfection, Theor. Pop. Biol. 57: 235–247.
Feng, Z., W. Huang, and C. Castillo-Chavez (2001) On the role of variable latent periods in mathematical models for tuberculosis, J. Dyn. Diff. Eq. 13: 435–452.
Feng, Z., M. Iannelli, and F.A. Milner (2002) A two-strain tuberculosis model with age of infection, SIAM J. Appl. Math. 62: 1634–1656.
Fister, K. R., S. Lenhart, and J.S. McNally (1998) Optimizing chemotherapy in an HIV model, Electronic J. Diff. Eq. 32: 1–12.
Fleming, W., and R. Rishel (1975) Deterministic and Stochastic Optimal Control, Springer.
Jung, E., S. Lenhart, and Z. Feng (2002) Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems Series B 2: 473–482.
Kirschner, D., S. Lenhart, and S. Serbin (1997) Optimal control of the chemotherapy of HIV J. Math. Biol. 35: 775–792.
Levin, S. and D. Pimentel (1981) Selection of intermediate rates of increase in parasite-host systems, Am. Naturalist 117: 308–315.
Murphy, B. M., B.H. Singer, and D. Kirschner (2003) On treatment of tuberculosis in heterogeneous populations, J. Theor. Biol. 223: 391–404.
Nowak, M.A., and R.M. May (1994) Superinfection and the evolution of parasite virulence, Proc. Roy. Soc. London, Series B: Biological Sciences 255: 81–89.
Pontryagin, L.S. (1987) Mathematical Theory of Optimal Processes, CRC Press.
Porco, T.C., P.M. Small, and S.M. Blower (2001) Amplification dynamics: predicting the effect of HIV on tuberculosis outbreaks, J. Acquired Immune Deficiency Syndromes 28: 437–444.
Rodrigues, P., M.G.M. Gomes, and C. Rebelo (2007) Drug resistance in tuberculosis, a reinfection model, Theor. Pop. Biol. 71: 196–212.
Roeger, L-I.W., Z. Feng, and C. Castillo-Chavez (2009) Modeling TB and HIV co-infections Math. Biosc. Eng. 6: 815–837.
Singer, B.H., and D.E. Kirschner (2004) Influence of backward bifurcation on interpretation of \({\mathcal R}_0\) in a model of epidemic tuberculosis with reinfection, Math. Biosc. Eng. 1: 81–93.
Styblo, K. (1991) Selected papers. vol. 24, Epidemiology of tuberculosis, Hague, The Netherlands: Royal Netherlands Tuberculosis Association.
WHO. Global tuberculosis report 2013 http://www.who.int/tb/publications/global_report/.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Science+Business Media, LLC, part of Springer Nature
About this chapter
Cite this chapter
Brauer, F., Castillo-Chavez, C., Feng, Z. (2019). Models for Tuberculosis. In: Mathematical Models in Epidemiology. Texts in Applied Mathematics, vol 69. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9828-9_7
Download citation
DOI: https://doi.org/10.1007/978-1-4939-9828-9_7
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-9826-5
Online ISBN: 978-1-4939-9828-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)