# 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
##### References:
 [1] R. F. Baggaley, G. P. Garnet and N. M. Ferguson, Modelling the Impact of Antiretroviral Use in Resource-Poor Settings, PLoS Medicine, 3 (2006), e124. doi: 10.1371/journal.pmed.0030124. [2] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic, New York, 1979. [3] C. P. Bhunu, W. Garira and Z. Mukandavire, Modeling HIV/AIDS and Tuberculosis coinfection, Bulletin of Mathematical Biology, 71 (2009), 1745-1780. doi: 10.1007/s11538-009-9423-9. [4] S. M. Blower and T. Chou, Modeling the emergence of the 'hot zones': Tuberculosis and the amplification dynamics of drug resistance, Nature Medicine, 10 (2004), 1111-1116. doi: 10.1038/nm1102. [5] S. M. Blower, P. M. Small and P. Hopewell, Control strategies for tuberculosis epidemics: New models for old problems, Science, 273 (1996), 497-500. doi: 10.1126/science.273.5274.497. [6] S. M. Blower, A. R. McLean, T. C. Porco, P. M. Small, P. C. Hopewell, M. A. Sanchez and A. R. Moss, The intrinsic transmission dynamics of tuberculosis epidemics, Nature Medicine, 1 (1995), 815-821. doi: 10.1038/nm0895-815. [7] S. M. Blower and J. L. Gerberding, Understanding, predicting and controlling the emergence of drug-resistant tuberculosis: A theoretical framework, Journal of Molecular Medicine, 76 (1998), 624-636. doi: 10.1007/s001090050260. [8] M. W. Borgdorff, New measurable indicator for tuberculosis case detection, Emerging Infectious Diseases, 10 (2004), 1523-1528. doi: 10.3201/eid1009.040349. [9] S. Borrell and S. Gagneoux, Infectiousness, reproductive fitness and evolution of drug-resistant Mycobacterium tuberculosis, The International Journal of Tuberculosis and Lung Disease, 13 (2009), 1456-1466. [10] C. R. Braden, G. P. Morlock, C. L. Woodley, K. R. Johnson and A. C. Colombel et al., Simultaneous infection with multiple strains of Mycobacterium tuberculosis, Clinical Infectious Diseases, 33 (2001), e42-e47. doi: 10.1086/322635. [11] S. Bowong and J. Kurths, Modeling and analysis of the transmission dynamics of tuberculosis without and with seasonality, Nonlinear Dynamics, 67 (2012), 2027-2051. doi: 10.1007/s11071-011-0127-y. [12] CDC., Drug resistant tuberculosis among the homeless Boston, MMWR, 34 (1985), 429-431. [13] T. Cohen and M. Murray, Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness, Nature Medicine, 10 (2004), 1117-1121. doi: 10.1038/nm1110. [14] T. Cohen, C. Colijn, B. Finklea, A. Wright, M. Zignol, A. Pym and M. Murray, Are survey-based estimates of the burden of drug resistant TB too low? Insight from a simulation study, PLos ONE, 3 (2008), e2363. doi: 10.1371/journal.pone.0002363. [15] C. Colijin, T. Cohen, A. Ganesh and M. Murray, Spontaneous emergence of multiple drug resistance in Tuberculosis before and during therapy, PLos ONE, 6 (2011), e18327. doi: 10.1371/journal.pone.0018327. [16] C. Colijn, T. Cohen and M. Murray, Latent coeinfection and the maintenance of strain diversity, Bulletin of Mathematical Biology, 71 (2009), 247-263. doi: 10.1007/s11538-008-9361-y. [17] H. D. Costello, G. J. Caras and Snider DE Jr., Drug resistance among previously treated tuberculosis patients, a brief report. American Review of Respiratory Disease, 121 (1980), 313-316. [18] Dickman et al., Detection of multiple strains of Mycobacterium tuberculosis using MIRU-VNTR in patients with pulmonary tuberculosis in Kampala, Uganda. BMC Infectious Diseases, 10 (2010),349 http://www.biomedcentral.com/1471-2334/10/349. [19] C. Dye and M. A. Espinal, Will tuberculosis become resistant to all antibiotics? Proceedings of the Royal Society of London B, 268 (2001), 45-52. doi: 10.1098/rspb.2000.1328. [20] M. A. Espinal, The global situation of MDR-TB, Tuberculosis, 83 (2003), 44-51. doi: 10.1016/S1472-9792(02)00058-6. [21] L. Esteva and C. Vargas, Influence of vertical and mechanical transmission on the dynamics of dengue disease, Math. Biosc., 167 (2000), 51-64. doi: 10.1016/S0025-5564(00)00024-9. [22] Z. Feng, M. Ianelli and F. A. Milner, A two-strain Tuberculosis model with age of infection, SIAM Journal on Applied Mathematics, 62 (2002), 1634-1656. doi: 10.1137/S003613990038205X. [23] M. L. Garcia-Garcia et. al., Clinical consequences and transmissibility of drug-resistant tuberculosis in souther Mexico, Archives of Internal Medicine, 160 (2000), 630-636. [24] M. Gomes, A. Franco and G. Medley, The reinfection threshold promotes variability in tuberculosis epidemiology and vaccine efficacy, Proceedings of the Royal Society B, 271 (2004), 617-623. doi: 10.1098/rspb.2003.2606. [25] J. K. Hale, Ordinary Differential Equations, 2nd Ed. krieger, Basel, 1980. [26] W. H. Hethcote, The mathematcs of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [27] M. C.M. Jong , O. Diekmann and J. A. P. Heesterbbeek, How does transmission of infection depend on population size? in D. Mollison (Ed.), Epidemic Models: Their Structure and Relation to Data, Cambridge University, Cambridge, 5 (1994), p. 84. [28] Y. Liu, Z. Sun, G. Sun, Q. Zhong, L. Jinag, L. Zhou, Y. Qiao and Z. Jia, Modeling Transmission of Tuberculosis with MDR and Undetected Cases, Discrete Dynamics in Nature and Society, 2011. doi: 10.1155/2011/296905. [29] S. M. Moghadas, C. S. Bowman, G. Rost and J. Wu, Population-wide emergence of antiviral resistance during pandemic influenza, PLos ONE, 3 (2008), e1839. doi: 10.1371/journal.pone.0001839. [30] E. Nardell, B. McInnes, B. Thomas and S. Weidhaas, Exogenous reinfection with tuberculosis in a shelter for the homeless, The New England Journal of Medicine, 315 (1986), 1570-1575. doi: 10.1056/NEJM198612183152502. [31] D. Okuonghae and S. E. Omosigho, Analysis of a mathematical model for tuberculosis: What could be done to increase case detection, Journal of Theoretical Biology, 269 (2011), 31-45. doi: 10.1016/j.jtbi.2010.09.044. [32] D. J. Ordway, M. G. Sonnenberg, S. A. Donahue, J. T. Belisle and I. M. Orme, Drug-resistant strains of Mycobaterium tuberculosis exhibit a range of virulence for mice, Infection and Immunity, 63 (1995), 741-743. [33] S. M. Raimundo, H. M. Yang, E. Venturino and E. Massad, Modeling the emergence of HIV-1 drug-resistance resulting from antiretroviral therapy: Insights from theoretical and numerical studies, BioSystems, 108 (2012), 1-13. doi: 10.1016/j.biosystems.2011.11.009. [34] S. M. Raimundo, H. M. Yang, R. C. Bassanezi, M. A. C. Ferreira, The attracting basins and the assessment of the transmission coefficients for HIV and M. Tuberculosis infections among women inmates, Journal of Biological Systems, 10 (2002), 61-83. [35] S. M. Raimundo, A. B. Engel, H. M. Yang and R. C. Bassanezi, An approach to estimating the transmission coefficients for AIDS and for tuberculosis using mathematical models, Systems Analysis Modelling Simulation, 43 (2003), 423-442. doi: 10.1080/02329290290027175. [36] S. M. Raimundo, E. Massad and H. M. Yang, Modelling congenital transmission of Chagas'disease, Biosystems, 99 (2010), 215-222. doi: 10.1016/j.biosystems.2009.11.005. [37] H. Rinder, K. T. Mieskes and T. Loscher, Heteroesistance in Mycobacterium tuberculsosis, The International Journal of Tuberculosis and Lung Disease, 5 (2001), 339-354. [38] P. Rodrigues, M. G. M. Gomes and C. Rebelo, Drug resistance in tuberculosis - a reinfection model, Theoretical Population Biology, 71 (2007), 196-212. doi: 10.1016/j.tpb.2006.10.004. [39] R. Sergeev, C. Colijn and T. Cohen, Models to understand the popualtion-level impact of mixed strain M. tuberculosis infections, Journal of Theoretical Biology, 280 (2011), 88-100. doi: 10.1016/j.jtbi.2011.04.011. [40] O. Sharomi and A. B. Gumel, Dynamical analysis of a multi-strain model of HIV in the presence of antiretroviral drugs, Journal of Biological Dynamics, 2 (2008), 323-345. doi: 10.1080/17513750701775599. [41] P. M. Small, R. W. Shafer and P. C. Hopewell et al, Exogenous reinfection with multidrug-resistant Mycobacterium tuberculosis in patients with advanced HIV infection, The New England Journal of Medicine, 328 (1993), 1137-1144. doi: 10.1056/NEJM199304223281601. [42] DE Jr. Snider, G. D. Kelly, G. M. Cauthen, N. J. Thompson and J. O. Kilburn, Infection and disease among contacts of tuberculosis cases with drug resistant and drug susceptible bacilli, The American Review of Respiratory Disease, 132 (1985), 125-132. [43] I. H. Spicknall, B. Foxman, C. F. Marrs and J. N. S. Eisenberg, A modeling framework for the evolution and spread of antibiotic resistance: literature review and model categorization, Am J Epidemiol, 178 (2013), 508-520. doi: 10.1093/aje/kwt017. [44] L. Teixeira et al, Infection and disease among household contacts of patients with multidrug-resistant tuberculosis, The International Journal of Tuberculosis and Lung Disease, 5 (2001), 321-328. [45] 2011/2012 Tuberculosis Global Facts, Progress WHO Global Tuberculosis Control Report, 2011, http://www.who.int/tb/publications/2011/factsheet_tb_2011.pdf (accessed in September, 2012). [46] World Health Organization, Anti-tuberculosis drug resistance in the world. Prevalence and trends, WHO/CDS/TB/2000/.278 The WHO/IUATLD Global Project on Anti-Tuberculosis Drug Resistance Surveillance. Report 2. World Health Organization, Geneva, Switzerland (2000). [47] http://www.who.int/mediacentre/factsheets/fs104/en/index.html, Tuberculosis, Fact sheet N. 104, March 2012. (accessed in September, 2012).

show all references

##### References:
 [1] R. F. Baggaley, G. P. Garnet and N. M. Ferguson, Modelling the Impact of Antiretroviral Use in Resource-Poor Settings, PLoS Medicine, 3 (2006), e124. doi: 10.1371/journal.pmed.0030124. [2] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic, New York, 1979. [3] C. P. Bhunu, W. Garira and Z. Mukandavire, Modeling HIV/AIDS and Tuberculosis coinfection, Bulletin of Mathematical Biology, 71 (2009), 1745-1780. doi: 10.1007/s11538-009-9423-9. [4] S. M. Blower and T. Chou, Modeling the emergence of the 'hot zones': Tuberculosis and the amplification dynamics of drug resistance, Nature Medicine, 10 (2004), 1111-1116. doi: 10.1038/nm1102. [5] S. M. Blower, P. M. Small and P. Hopewell, Control strategies for tuberculosis epidemics: New models for old problems, Science, 273 (1996), 497-500. doi: 10.1126/science.273.5274.497. [6] S. M. Blower, A. R. McLean, T. C. Porco, P. M. Small, P. C. Hopewell, M. A. Sanchez and A. R. Moss, The intrinsic transmission dynamics of tuberculosis epidemics, Nature Medicine, 1 (1995), 815-821. doi: 10.1038/nm0895-815. [7] S. M. Blower and J. L. Gerberding, Understanding, predicting and controlling the emergence of drug-resistant tuberculosis: A theoretical framework, Journal of Molecular Medicine, 76 (1998), 624-636. doi: 10.1007/s001090050260. [8] M. W. Borgdorff, New measurable indicator for tuberculosis case detection, Emerging Infectious Diseases, 10 (2004), 1523-1528. doi: 10.3201/eid1009.040349. [9] S. Borrell and S. Gagneoux, Infectiousness, reproductive fitness and evolution of drug-resistant Mycobacterium tuberculosis, The International Journal of Tuberculosis and Lung Disease, 13 (2009), 1456-1466. [10] C. R. Braden, G. P. Morlock, C. L. Woodley, K. R. Johnson and A. C. Colombel et al., Simultaneous infection with multiple strains of Mycobacterium tuberculosis, Clinical Infectious Diseases, 33 (2001), e42-e47. doi: 10.1086/322635. [11] S. Bowong and J. Kurths, Modeling and analysis of the transmission dynamics of tuberculosis without and with seasonality, Nonlinear Dynamics, 67 (2012), 2027-2051. doi: 10.1007/s11071-011-0127-y. [12] CDC., Drug resistant tuberculosis among the homeless Boston, MMWR, 34 (1985), 429-431. [13] T. Cohen and M. Murray, Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness, Nature Medicine, 10 (2004), 1117-1121. doi: 10.1038/nm1110. [14] T. Cohen, C. Colijn, B. Finklea, A. Wright, M. Zignol, A. Pym and M. Murray, Are survey-based estimates of the burden of drug resistant TB too low? Insight from a simulation study, PLos ONE, 3 (2008), e2363. doi: 10.1371/journal.pone.0002363. [15] C. Colijin, T. Cohen, A. Ganesh and M. Murray, Spontaneous emergence of multiple drug resistance in Tuberculosis before and during therapy, PLos ONE, 6 (2011), e18327. doi: 10.1371/journal.pone.0018327. [16] C. Colijn, T. Cohen and M. Murray, Latent coeinfection and the maintenance of strain diversity, Bulletin of Mathematical Biology, 71 (2009), 247-263. doi: 10.1007/s11538-008-9361-y. [17] H. D. Costello, G. J. Caras and Snider DE Jr., Drug resistance among previously treated tuberculosis patients, a brief report. American Review of Respiratory Disease, 121 (1980), 313-316. [18] Dickman et al., Detection of multiple strains of Mycobacterium tuberculosis using MIRU-VNTR in patients with pulmonary tuberculosis in Kampala, Uganda. BMC Infectious Diseases, 10 (2010),349 http://www.biomedcentral.com/1471-2334/10/349. [19] C. Dye and M. A. Espinal, Will tuberculosis become resistant to all antibiotics? Proceedings of the Royal Society of London B, 268 (2001), 45-52. doi: 10.1098/rspb.2000.1328. [20] M. A. Espinal, The global situation of MDR-TB, Tuberculosis, 83 (2003), 44-51. doi: 10.1016/S1472-9792(02)00058-6. [21] L. Esteva and C. Vargas, Influence of vertical and mechanical transmission on the dynamics of dengue disease, Math. Biosc., 167 (2000), 51-64. doi: 10.1016/S0025-5564(00)00024-9. [22] Z. Feng, M. Ianelli and F. A. Milner, A two-strain Tuberculosis model with age of infection, SIAM Journal on Applied Mathematics, 62 (2002), 1634-1656. doi: 10.1137/S003613990038205X. [23] M. L. Garcia-Garcia et. al., Clinical consequences and transmissibility of drug-resistant tuberculosis in souther Mexico, Archives of Internal Medicine, 160 (2000), 630-636. [24] M. Gomes, A. Franco and G. Medley, The reinfection threshold promotes variability in tuberculosis epidemiology and vaccine efficacy, Proceedings of the Royal Society B, 271 (2004), 617-623. doi: 10.1098/rspb.2003.2606. [25] J. K. Hale, Ordinary Differential Equations, 2nd Ed. krieger, Basel, 1980. [26] W. H. Hethcote, The mathematcs of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [27] M. C.M. Jong , O. Diekmann and J. A. P. Heesterbbeek, How does transmission of infection depend on population size? in D. Mollison (Ed.), Epidemic Models: Their Structure and Relation to Data, Cambridge University, Cambridge, 5 (1994), p. 84. [28] Y. Liu, Z. Sun, G. Sun, Q. Zhong, L. Jinag, L. Zhou, Y. Qiao and Z. Jia, Modeling Transmission of Tuberculosis with MDR and Undetected Cases, Discrete Dynamics in Nature and Society, 2011. doi: 10.1155/2011/296905. [29] S. M. Moghadas, C. S. Bowman, G. Rost and J. Wu, Population-wide emergence of antiviral resistance during pandemic influenza, PLos ONE, 3 (2008), e1839. doi: 10.1371/journal.pone.0001839. [30] E. Nardell, B. McInnes, B. Thomas and S. Weidhaas, Exogenous reinfection with tuberculosis in a shelter for the homeless, The New England Journal of Medicine, 315 (1986), 1570-1575. doi: 10.1056/NEJM198612183152502. [31] D. Okuonghae and S. E. Omosigho, Analysis of a mathematical model for tuberculosis: What could be done to increase case detection, Journal of Theoretical Biology, 269 (2011), 31-45. doi: 10.1016/j.jtbi.2010.09.044. [32] D. J. Ordway, M. G. Sonnenberg, S. A. Donahue, J. T. Belisle and I. M. Orme, Drug-resistant strains of Mycobaterium tuberculosis exhibit a range of virulence for mice, Infection and Immunity, 63 (1995), 741-743. [33] S. M. Raimundo, H. M. Yang, E. Venturino and E. Massad, Modeling the emergence of HIV-1 drug-resistance resulting from antiretroviral therapy: Insights from theoretical and numerical studies, BioSystems, 108 (2012), 1-13. doi: 10.1016/j.biosystems.2011.11.009. [34] S. M. Raimundo, H. M. Yang, R. C. Bassanezi, M. A. C. Ferreira, The attracting basins and the assessment of the transmission coefficients for HIV and M. Tuberculosis infections among women inmates, Journal of Biological Systems, 10 (2002), 61-83. [35] S. M. Raimundo, A. B. Engel, H. M. Yang and R. C. Bassanezi, An approach to estimating the transmission coefficients for AIDS and for tuberculosis using mathematical models, Systems Analysis Modelling Simulation, 43 (2003), 423-442. doi: 10.1080/02329290290027175. [36] S. M. Raimundo, E. Massad and H. M. Yang, Modelling congenital transmission of Chagas'disease, Biosystems, 99 (2010), 215-222. doi: 10.1016/j.biosystems.2009.11.005. [37] H. Rinder, K. T. Mieskes and T. Loscher, Heteroesistance in Mycobacterium tuberculsosis, The International Journal of Tuberculosis and Lung Disease, 5 (2001), 339-354. [38] P. Rodrigues, M. G. M. Gomes and C. Rebelo, Drug resistance in tuberculosis - a reinfection model, Theoretical Population Biology, 71 (2007), 196-212. doi: 10.1016/j.tpb.2006.10.004. [39] R. Sergeev, C. Colijn and T. Cohen, Models to understand the popualtion-level impact of mixed strain M. tuberculosis infections, Journal of Theoretical Biology, 280 (2011), 88-100. doi: 10.1016/j.jtbi.2011.04.011. [40] O. Sharomi and A. B. Gumel, Dynamical analysis of a multi-strain model of HIV in the presence of antiretroviral drugs, Journal of Biological Dynamics, 2 (2008), 323-345. doi: 10.1080/17513750701775599. [41] P. M. Small, R. W. Shafer and P. C. Hopewell et al, Exogenous reinfection with multidrug-resistant Mycobacterium tuberculosis in patients with advanced HIV infection, The New England Journal of Medicine, 328 (1993), 1137-1144. doi: 10.1056/NEJM199304223281601. [42] DE Jr. Snider, G. D. Kelly, G. M. Cauthen, N. J. Thompson and J. O. Kilburn, Infection and disease among contacts of tuberculosis cases with drug resistant and drug susceptible bacilli, The American Review of Respiratory Disease, 132 (1985), 125-132. [43] I. H. Spicknall, B. Foxman, C. F. Marrs and J. N. S. Eisenberg, A modeling framework for the evolution and spread of antibiotic resistance: literature review and model categorization, Am J Epidemiol, 178 (2013), 508-520. doi: 10.1093/aje/kwt017. [44] L. Teixeira et al, Infection and disease among household contacts of patients with multidrug-resistant tuberculosis, The International Journal of Tuberculosis and Lung Disease, 5 (2001), 321-328. [45] 2011/2012 Tuberculosis Global Facts, Progress WHO Global Tuberculosis Control Report, 2011, http://www.who.int/tb/publications/2011/factsheet_tb_2011.pdf (accessed in September, 2012). [46] World Health Organization, Anti-tuberculosis drug resistance in the world. Prevalence and trends, WHO/CDS/TB/2000/.278 The WHO/IUATLD Global Project on Anti-Tuberculosis Drug Resistance Surveillance. Report 2. World Health Organization, Geneva, Switzerland (2000). [47] http://www.who.int/mediacentre/factsheets/fs104/en/index.html, Tuberculosis, Fact sheet N. 104, March 2012. (accessed in September, 2012).
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