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2016, 13(4): 741-785. doi: 10.3934/mbe.2016017

A two-strain TB model with multiple latent stages

1. 

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran, Iran

2. 

Simon A Levin Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287, United States, United States

3. 

Department of Mathematical Sciences, Montclair State University, 1 Normal Avenue, Montclair, NJ 07043

Received  November 2015 Revised  February 2016 Published  May 2016

A two-strain tuberculosis (TB) transmission model incorporating antibiotic-generated TB resistant strains and long and variable waiting periods within the latently infected class is introduced. The mathematical analysis is carried out when the waiting periods are modeled via parametrically friendly gamma distributions, a reasonable alternative to the use of exponential distributed waiting periods or to integral equations involving ``arbitrary'' distributions. The model supports a globally-asymptotically stable disease-free equilibrium when the reproduction number is less than one and an endemic equilibriums, shown to be locally asymptotically stable, or l.a.s., whenever the basic reproduction number is greater than one. Conditions for the existence and maintenance of TB resistant strains are discussed. The possibility of exogenous re-infection is added and shown to be capable of supporting multiple equilibria; a situation that increases the challenges faced by public health experts. We show that exogenous re-infection may help established resilient communities of actively-TB infected individuals that cannot be eliminated using approaches based exclusively on the ability to bring the control reproductive number just below $1$.
Citation: Azizeh Jabbari, Carlos Castillo-Chavez, Fereshteh Nazari, Baojun Song, Hossein Kheiri. A two-strain TB model with multiple latent stages. Mathematical Biosciences & Engineering, 2016, 13 (4) : 741-785. doi: 10.3934/mbe.2016017
References:
[1]

J. P. Aparicio, A. F. Capurro and C. Castillo-Chavez, Markers of disease evolution: The case of tuberculosis,, J Theor Biol, 215 (2002), 227. doi: 10.1006/jtbi.2001.2489. Google Scholar

[2]

J. P. Aparicio, A. F. Capurro and C. Castillo-Chavez, Long-term dynamics and re-emergence of tuberculosis,, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, 125 (2002), 351. doi: 10.1007/978-1-4757-3667-0_20. Google Scholar

[3]

J. P. Aparicio, A. F. Capurro and C. Castillo-Chavez, Transmission and dynamics of tuberculosis on generalized households,, J Theor Biol, 206 (2000), 327. doi: 10.1006/jtbi.2000.2129. Google Scholar

[4]

J. P. Aparicio and C. Castillo-Chavez, Mathematical modelling of tuberculosis epidemics,, Math Biosci Eng, 6 (2009), 209. doi: 10.3934/mbe.2009.6.209. Google Scholar

[5]

J. H. Bates, W. Stead and T. A. Rado, Phage type of tubercle bacilli isolated from patients with two or more sites of organ involvement,, Am Rev Respir Dis, 114 (1976), 353. Google Scholar

[6]

B. R. Bloom, Tuberculosis: Pathogenesis, Protection, and Control,, ASM Press, (1994). Google Scholar

[7]

S. M. Blower, A. R. McLean, T. C. Porco, P. M. Small, P. C. Hopwell, M. A. Sanchez and A. R. Moss, The intrinsic transmission dynamics of tuberculosis epidemics,, Nature Medicine, 1 (1995), 815. doi: 10.1038/nm0895-815. Google Scholar

[8]

F. Brauer and C. Castillo-Chavez, Mathematical Models for Communicable Diseases,, SIAM, (2013). Google Scholar

[9]

C. Castillo-Chavez, Chalenges and opportunities in mathematical and theoretical biology and medicine: foreword to volume 2 (2013) of Biomath,, Biomath, 2 (2013). doi: 10.11145/j.biomath.2013.12.319. Google Scholar

[10]

C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis,, J Math Biol, 35 (1997), 629. doi: 10.1007/s002850050069. Google Scholar

[11]

C. Castillo-Chavez and Z. Feng, Mathematical models for the disease dynamics of tuberculosis,, Advances in Mathematical Population Dynamics - Molecules, (1998), 629. Google Scholar

[12]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math Biosci Eng, 1 (2004), 361. doi: 10.3934/mbe.2004.1.361. Google Scholar

[13]

C. Y. Chiang and L. W. Riley, Exogenous reinfection in tuberculosis,, Lancet Infect Dis, 5 (2005), 629. doi: 10.1016/S1473-3099(05)70240-1. Google Scholar

[14]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math Biosci, 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[15]

Z. Feng, C. Castillo-Chavez and A. F. Capurro, A model for tuberculosis with exogenous reinfection,, Theor Popul Biol, 57 (2000), 235. doi: 10.1006/tpbi.2000.1451. Google Scholar

[16]

Z. Feng, W. Huang and C. Castillo-Chavez, On the role of variable latent periods in mathematical models for tuberculosis,, Journal of Dynamics and Differential Equations, 13 (2001), 425. doi: 10.1023/A:1016688209771. Google Scholar

[17]

Z. Feng, D. Xu and H. Zhao, Epidemiological models with non-exponentially distributed disease stages and applications to disease control,, Bulletin of Mathematical Biology, 69 (2007), 1511. doi: 10.1007/s11538-006-9174-9. Google Scholar

[18]

, Antibiotic-resistant Diseases Pose 'Apocalyptic' Threat, Top Expert Says,, 2013. Available from: , (). Google Scholar

[19]

, Guidelines on the Management of Latent Tuberculosis Infection,, 2015. Available from: , (). Google Scholar

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H. M. Hethcote, Qualitative analysis for communicable disease models,, Math Biosc, 28 (1976), 335. doi: 10.1016/0025-5564(76)90132-2. Google Scholar

[21]

H. M. Hethcote, The Mathematics of infectious diseases,, SIAM Rev, 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar

[22]

J. M. Hyman and J. Li, An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations,, Mathematical Biosciences, 167 (2000), 65. doi: 10.1016/S0025-5564(00)00025-0. Google Scholar

[23]

E. Ibargüen-Mondragón and L. Esteva, On the interactions of sensitive and resistant Mycobacterium tuberculosis to antibiotics,, Math Biosc, 246 (2013), 84. doi: 10.1016/j.mbs.2013.08.005. Google Scholar

[24]

V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems,, Marcel Dekker Inc, 41 (1989). Google Scholar

[25]

M. L. Lambert, E. Hasker, A. Van Deun, D. Roberfroid, M. Boelaert and P. Van Der Stuyft, Recurrence in tuberculosis: Relapse or reinfection?,, Lancet Infect Dis, 3 (2003), 282. doi: 10.1016/S1473-3099(03)00607-8. Google Scholar

[26]

E. Nardell, B. Mc Innis, B. Thomas and S. Weidhaas, Exogenous reinfection with tuberculosis in a shelter for the homeless,, N Engl J Med, 315 (1986), 1570. doi: 10.1056/NEJM198612183152502. Google Scholar

[27]

E. Oldfield and X. Feng, Resistance-resistant antibiotics,, Trends in Pharmacological Sciences, 35 (2014), 664. doi: 10.1016/j.tips.2014.10.007. Google Scholar

[28]

T. C. Porco and S. M. Blower, Quantifying the intrinsic transmission dynamics of tuberculosis,, Theoretical Population Biology, 54 (1998), 117. doi: 10.1006/tpbi.1998.1366. Google Scholar

[29]

J. W. Raleigh and R. H. Wichelhausen, Exogenous reinfection with mycobacterium tuberculosis confirmed by phage typing,, Am Rev Respir Dis, 108 (1973), 639. Google Scholar

[30]

J. W. Raleigh, R. H. Wichelhausen, T. A. Rado and J. H. Bates, Evidence for infection by two distinct strains of mycobacterium tuberculosis in pulmonary tuberculosis: Report of 9 cases,, Am Rev Respir Dis, 112 (1975), 497. Google Scholar

[31]

M. Raviglione, Drug-Resistant TB Surveillance and Response, Global Tuberculosis Report 2014,, 2014. Available from: , (). Google Scholar

[32]

L. W. Roeger, Z. Feng and C. Castillo-Chavez, Modeling TB and HIV co-infections,, Math Biosci Eng, 6 (2009), 815. doi: 10.3934/mbe.2009.6.815. Google Scholar

[33]

G. Shen, Z. Xue, X. Shen, B. Sun, X. Gui, M. Shen, J. Mei and Q. Gao, Recurrent tuberculosis and exogenous reinfection, Shanghai, China,, Emerging Infectious Disease, 12 (2006), 1176. doi: 10.3201/eid1211.051207. Google Scholar

[34]

P. M. Small, R. W. Shafer, P. C. Hopewell, P. C. Singh, M. J. Murphy, E. Desmond , M. F. Sierra and G. K. Schoolnik, Exogenous reinfection with multidrug-resistant mycobacterium tuberculosis in patients wit advanced HIV infection,, N Engl J Med, 328 (1993), 1137. Google Scholar

[35]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, American Mathematical Society, (1995). Google Scholar

[36]

B. Song, Dynamical Epidemic Models and Their Applications,, Thesis (Ph.D.)-Cornell University, (2002). Google Scholar

[37]

B. Song, C. Castillo-Chavez and J. P. Aparicio, Tuberculosis models with fast and slow dynamics: The role of close and casual contacts,, Mathematical Biosciences, 180 (2002), 187. doi: 10.1016/S0025-5564(02)00112-8. Google Scholar

[38]

B. Song, C. Castillo-Chavez and J. P. Aparicio, Global dynamics of tuberculosis models with density dependent demography,, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models Methods and Theory (eds. C. Castillo-Chavez, 126 (2002), 275. doi: 10.1007/978-1-4613-0065-6_16. Google Scholar

[39]

W. W. Stead, The pathogenesis of pulmonary tuberculosis among older persons,, Am Rev Respir Dis, 91 (1965), 811. Google Scholar

[40]

T. C. Porco and S. M. Blower, Quantifying the intrinsic transmission dynamics of tuberculosis,, Theoretical Population Biology, 54 (1998), 117. doi: 10.1006/tpbi.1998.1366. Google Scholar

[41]

X. Wang, Backward Bifurcation in a Mathematical Model for Tuberculosis with Loss of Immunity,, Ph.D. Thesis, (2005). Google Scholar

[42]

X. Wang, Z. Feng, J. P. Aparicio and C. Castillo-Chavez, On the dynamics of reinfection: The case of tuberculosis, BIOMAT 2009,, International Symposium on Mathematical and Computational Biology, (2010), 304. doi: 10.1142/9789814304900_0021. Google Scholar

[43]

, Global Tuberculosis Control: Who Report 2010, 2010., Available from: , (). Google Scholar

show all references

References:
[1]

J. P. Aparicio, A. F. Capurro and C. Castillo-Chavez, Markers of disease evolution: The case of tuberculosis,, J Theor Biol, 215 (2002), 227. doi: 10.1006/jtbi.2001.2489. Google Scholar

[2]

J. P. Aparicio, A. F. Capurro and C. Castillo-Chavez, Long-term dynamics and re-emergence of tuberculosis,, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, 125 (2002), 351. doi: 10.1007/978-1-4757-3667-0_20. Google Scholar

[3]

J. P. Aparicio, A. F. Capurro and C. Castillo-Chavez, Transmission and dynamics of tuberculosis on generalized households,, J Theor Biol, 206 (2000), 327. doi: 10.1006/jtbi.2000.2129. Google Scholar

[4]

J. P. Aparicio and C. Castillo-Chavez, Mathematical modelling of tuberculosis epidemics,, Math Biosci Eng, 6 (2009), 209. doi: 10.3934/mbe.2009.6.209. Google Scholar

[5]

J. H. Bates, W. Stead and T. A. Rado, Phage type of tubercle bacilli isolated from patients with two or more sites of organ involvement,, Am Rev Respir Dis, 114 (1976), 353. Google Scholar

[6]

B. R. Bloom, Tuberculosis: Pathogenesis, Protection, and Control,, ASM Press, (1994). Google Scholar

[7]

S. M. Blower, A. R. McLean, T. C. Porco, P. M. Small, P. C. Hopwell, M. A. Sanchez and A. R. Moss, The intrinsic transmission dynamics of tuberculosis epidemics,, Nature Medicine, 1 (1995), 815. doi: 10.1038/nm0895-815. Google Scholar

[8]

F. Brauer and C. Castillo-Chavez, Mathematical Models for Communicable Diseases,, SIAM, (2013). Google Scholar

[9]

C. Castillo-Chavez, Chalenges and opportunities in mathematical and theoretical biology and medicine: foreword to volume 2 (2013) of Biomath,, Biomath, 2 (2013). doi: 10.11145/j.biomath.2013.12.319. Google Scholar

[10]

C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis,, J Math Biol, 35 (1997), 629. doi: 10.1007/s002850050069. Google Scholar

[11]

C. Castillo-Chavez and Z. Feng, Mathematical models for the disease dynamics of tuberculosis,, Advances in Mathematical Population Dynamics - Molecules, (1998), 629. Google Scholar

[12]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math Biosci Eng, 1 (2004), 361. doi: 10.3934/mbe.2004.1.361. Google Scholar

[13]

C. Y. Chiang and L. W. Riley, Exogenous reinfection in tuberculosis,, Lancet Infect Dis, 5 (2005), 629. doi: 10.1016/S1473-3099(05)70240-1. Google Scholar

[14]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math Biosci, 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[15]

Z. Feng, C. Castillo-Chavez and A. F. Capurro, A model for tuberculosis with exogenous reinfection,, Theor Popul Biol, 57 (2000), 235. doi: 10.1006/tpbi.2000.1451. Google Scholar

[16]

Z. Feng, W. Huang and C. Castillo-Chavez, On the role of variable latent periods in mathematical models for tuberculosis,, Journal of Dynamics and Differential Equations, 13 (2001), 425. doi: 10.1023/A:1016688209771. Google Scholar

[17]

Z. Feng, D. Xu and H. Zhao, Epidemiological models with non-exponentially distributed disease stages and applications to disease control,, Bulletin of Mathematical Biology, 69 (2007), 1511. doi: 10.1007/s11538-006-9174-9. Google Scholar

[18]

, Antibiotic-resistant Diseases Pose 'Apocalyptic' Threat, Top Expert Says,, 2013. Available from: , (). Google Scholar

[19]

, Guidelines on the Management of Latent Tuberculosis Infection,, 2015. Available from: , (). Google Scholar

[20]

H. M. Hethcote, Qualitative analysis for communicable disease models,, Math Biosc, 28 (1976), 335. doi: 10.1016/0025-5564(76)90132-2. Google Scholar

[21]

H. M. Hethcote, The Mathematics of infectious diseases,, SIAM Rev, 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar

[22]

J. M. Hyman and J. Li, An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations,, Mathematical Biosciences, 167 (2000), 65. doi: 10.1016/S0025-5564(00)00025-0. Google Scholar

[23]

E. Ibargüen-Mondragón and L. Esteva, On the interactions of sensitive and resistant Mycobacterium tuberculosis to antibiotics,, Math Biosc, 246 (2013), 84. doi: 10.1016/j.mbs.2013.08.005. Google Scholar

[24]

V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems,, Marcel Dekker Inc, 41 (1989). Google Scholar

[25]

M. L. Lambert, E. Hasker, A. Van Deun, D. Roberfroid, M. Boelaert and P. Van Der Stuyft, Recurrence in tuberculosis: Relapse or reinfection?,, Lancet Infect Dis, 3 (2003), 282. doi: 10.1016/S1473-3099(03)00607-8. Google Scholar

[26]

E. Nardell, B. Mc Innis, B. Thomas and S. Weidhaas, Exogenous reinfection with tuberculosis in a shelter for the homeless,, N Engl J Med, 315 (1986), 1570. doi: 10.1056/NEJM198612183152502. Google Scholar

[27]

E. Oldfield and X. Feng, Resistance-resistant antibiotics,, Trends in Pharmacological Sciences, 35 (2014), 664. doi: 10.1016/j.tips.2014.10.007. Google Scholar

[28]

T. C. Porco and S. M. Blower, Quantifying the intrinsic transmission dynamics of tuberculosis,, Theoretical Population Biology, 54 (1998), 117. doi: 10.1006/tpbi.1998.1366. Google Scholar

[29]

J. W. Raleigh and R. H. Wichelhausen, Exogenous reinfection with mycobacterium tuberculosis confirmed by phage typing,, Am Rev Respir Dis, 108 (1973), 639. Google Scholar

[30]

J. W. Raleigh, R. H. Wichelhausen, T. A. Rado and J. H. Bates, Evidence for infection by two distinct strains of mycobacterium tuberculosis in pulmonary tuberculosis: Report of 9 cases,, Am Rev Respir Dis, 112 (1975), 497. Google Scholar

[31]

M. Raviglione, Drug-Resistant TB Surveillance and Response, Global Tuberculosis Report 2014,, 2014. Available from: , (). Google Scholar

[32]

L. W. Roeger, Z. Feng and C. Castillo-Chavez, Modeling TB and HIV co-infections,, Math Biosci Eng, 6 (2009), 815. doi: 10.3934/mbe.2009.6.815. Google Scholar

[33]

G. Shen, Z. Xue, X. Shen, B. Sun, X. Gui, M. Shen, J. Mei and Q. Gao, Recurrent tuberculosis and exogenous reinfection, Shanghai, China,, Emerging Infectious Disease, 12 (2006), 1176. doi: 10.3201/eid1211.051207. Google Scholar

[34]

P. M. Small, R. W. Shafer, P. C. Hopewell, P. C. Singh, M. J. Murphy, E. Desmond , M. F. Sierra and G. K. Schoolnik, Exogenous reinfection with multidrug-resistant mycobacterium tuberculosis in patients wit advanced HIV infection,, N Engl J Med, 328 (1993), 1137. Google Scholar

[35]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, American Mathematical Society, (1995). Google Scholar

[36]

B. Song, Dynamical Epidemic Models and Their Applications,, Thesis (Ph.D.)-Cornell University, (2002). Google Scholar

[37]

B. Song, C. Castillo-Chavez and J. P. Aparicio, Tuberculosis models with fast and slow dynamics: The role of close and casual contacts,, Mathematical Biosciences, 180 (2002), 187. doi: 10.1016/S0025-5564(02)00112-8. Google Scholar

[38]

B. Song, C. Castillo-Chavez and J. P. Aparicio, Global dynamics of tuberculosis models with density dependent demography,, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models Methods and Theory (eds. C. Castillo-Chavez, 126 (2002), 275. doi: 10.1007/978-1-4613-0065-6_16. Google Scholar

[39]

W. W. Stead, The pathogenesis of pulmonary tuberculosis among older persons,, Am Rev Respir Dis, 91 (1965), 811. Google Scholar

[40]

T. C. Porco and S. M. Blower, Quantifying the intrinsic transmission dynamics of tuberculosis,, Theoretical Population Biology, 54 (1998), 117. doi: 10.1006/tpbi.1998.1366. Google Scholar

[41]

X. Wang, Backward Bifurcation in a Mathematical Model for Tuberculosis with Loss of Immunity,, Ph.D. Thesis, (2005). Google Scholar

[42]

X. Wang, Z. Feng, J. P. Aparicio and C. Castillo-Chavez, On the dynamics of reinfection: The case of tuberculosis, BIOMAT 2009,, International Symposium on Mathematical and Computational Biology, (2010), 304. doi: 10.1142/9789814304900_0021. Google Scholar

[43]

, Global Tuberculosis Control: Who Report 2010, 2010., Available from: , (). Google Scholar

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