# American Institute of Mathematical Sciences

2014, 11(4): 971-993. doi: 10.3934/mbe.2014.11.971

## Theoretical assessment of the relative incidences of sensitive and resistant tuberculosis epidemic in presence of drug treatment

 1 Faculdade de Medicina da Universidade de São Paulo, Disciplina de Informática Médica and LIM01-HCFMUSP, Rua Teodoro Sampaio, 115, CEP: 05405-000, São Paulo, SP, Brazil 2 Universidade de Campinas, IMECC, DMA, Praça Sérgio Buarque de Holanda, 651, Campinas, SP, Brazil 3 Università di Torino, Dipartimento di Matematica "Giuseppe Peano", Via Carlo Alberto 10, 10123 Torino, Italy

Received  August 2013 Revised  October 2013 Published  March 2014

Despite the availability of effective treatment, tuberculosis (TB) remains a major global cause of mortality. Multidrug-resistant tuberculosis (MDR-TB) is a form of TB that is resistant to at least two drugs used for the treatment of TB, and originally is developed when a case of drug-susceptible TB is improperly or incompletely treated. This work is concerned with a mathematical model to evaluate the effect of MDR-TB on TB epidemic and its control. The model assessing the transmission dynamics of both drug-sensitive and drug-resistant TB includes slow TB (cases that result from endogenous reactivation of susceptible and resistant latent infections). We identify the steady states of the model to analyse their stability. We establish threshold conditions for possible scenarios: elimination of sensitive and resistant strains and coexistence of both. We find that the effective reproductive number is composed of two critical values, relative reproductive number for drug-sensitive and drug-resistant strains. Our results imply that the potential for the spreading of the drug-resistant strain should be evaluated within the context of several others factors. We have also found that even the considerably less fit drug-resistant strains can lead to a high MDR-TB incidence, because the treatment is less effective against them.
Citation: Silvia Martorano Raimundo, Hyun Mo Yang, Ezio Venturino. Theoretical assessment of the relative incidences of sensitive and resistant tuberculosis epidemic in presence of drug treatment. Mathematical Biosciences & Engineering, 2014, 11 (4) : 971-993. doi: 10.3934/mbe.2014.11.971
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##### References:
 [1] Urszula Ledzewicz, Shuo Wang, Heinz Schättler, Nicolas André, Marie Amélie Heng, Eddy Pasquier. On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach. Mathematical Biosciences & Engineering, 2017, 14 (1) : 217-235. doi: 10.3934/mbe.2017014 [2] Oluwaseun Sharomi, Chandra N. Podder, Abba B. Gumel, Baojun Song. Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment. Mathematical Biosciences & Engineering, 2008, 5 (1) : 145-174. doi: 10.3934/mbe.2008.5.145 [3] Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129 [4] Cristian Tomasetti, Doron Levy. An elementary approach to modeling drug resistance in cancer. Mathematical Biosciences & Engineering, 2010, 7 (4) : 905-918. doi: 10.3934/mbe.2010.7.905 [5] Avner Friedman, Najat Ziyadi, Khalid Boushaba. A model of drug resistance with infection by health care workers. Mathematical Biosciences & Engineering, 2010, 7 (4) : 779-792. doi: 10.3934/mbe.2010.7.779 [6] Tao Feng, Zhipeng Qiu. Global analysis of a stochastic TB model with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2923-2939. doi: 10.3934/dcdsb.2018292 [7] Zhenzhen Chen, Sze-Bi Hsu, Ya-Tang Yang. The continuous morbidostat: A chemostat with controlled drug application to select for drug resistance mutants. Communications on Pure & Applied Analysis, 2020, 19 (1) : 203-220. doi: 10.3934/cpaa.2020011 [8] Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227 [9] Ami B. Shah, Katarzyna A. Rejniak, Jana L. Gevertz. Limiting the development of anti-cancer drug resistance in a spatial model of micrometastases. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1185-1206. doi: 10.3934/mbe.2016038 [10] Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639 [11] Timothy C. Reluga, Jan Medlock. Resistance mechanisms matter in SIR models. Mathematical Biosciences & Engineering, 2007, 4 (3) : 553-563. doi: 10.3934/mbe.2007.4.553 [12] Elena Fimmel, Yury S. Semenov, Alexander S. Bratus. On optimal and suboptimal treatment strategies for a mathematical model of leukemia. Mathematical Biosciences & Engineering, 2013, 10 (1) : 151-165. doi: 10.3934/mbe.2013.10.151 [13] Helen Moore, Weiqing Gu. A mathematical model for treatment-resistant mutations of HIV. Mathematical Biosciences & Engineering, 2005, 2 (2) : 363-380. doi: 10.3934/mbe.2005.2.363 [14] Marek Bodnar, Monika Joanna Piotrowska, Urszula Foryś. Gompertz model with delays and treatment: Mathematical analysis. Mathematical Biosciences & Engineering, 2013, 10 (3) : 551-563. doi: 10.3934/mbe.2013.10.551 [15] Abba B. Gumel, Baojun Song. Existence of multiple-stable equilibria for a multi-drug-resistant model of mycobacterium tuberculosis. Mathematical Biosciences & Engineering, 2008, 5 (3) : 437-455. doi: 10.3934/mbe.2008.5.437 [16] Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006 [17] Paul Georgescu, Hong Zhang, Daniel Maxin. The global stability of coexisting equilibria for three models of mutualism. Mathematical Biosciences & Engineering, 2016, 13 (1) : 101-118. doi: 10.3934/mbe.2016.13.101 [18] J.C. Arciero, T.L. Jackson, D.E. Kirschner. A mathematical model of tumor-immune evasion and siRNA treatment. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 39-58. doi: 10.3934/dcdsb.2004.4.39 [19] Alacia M. Voth, John G. Alford, Edward W. Swim. Mathematical modeling of continuous and intermittent androgen suppression for the treatment of advanced prostate cancer. Mathematical Biosciences & Engineering, 2017, 14 (3) : 777-804. doi: 10.3934/mbe.2017043 [20] Lisette dePillis, Trevor Caldwell, Elizabeth Sarapata, Heather Williams. Mathematical modeling of regulatory T cell effects on renal cell carcinoma treatment. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 915-943. doi: 10.3934/dcdsb.2013.18.915

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