A two-strain TB model with multiplelatent stages

Azizeh  Jabbari; Carlos  Castillo-Chavez; Fereshteh  Nazari; Baojun Song; Hossein  Kheiri

doi:10.3934/mbe.2016017

Article Contents

2016, Volume 13, Issue 4: 741-785. Doi: 10.3934/mbe.2016017 open access

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A two-strain TB model with multiple latent stages

1.
Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz
2.
Simon A Levin Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287
3.
Department of Mathematical Sciences, Montclair State University, 1 Normal Avenue, Montclair, NJ 07043

Received: October 31, 2015

Revised: January 31, 2016

Published: April 2016

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Abstract

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 $1$.

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Mathematics Subject Classification: Primary: 92D25, 92D30; Secondary: 92C60.

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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), 227-237.doi: 10.1006/jtbi.2001.2489.
[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, Springer-Verlag. Edited by Sally Blower, Carlos Castillo-Chavez, Denise Kirschner, Pauline van den Driessche and Abdul-Aziz Yakubu, 125 (2002), 351-360.doi: 10.1007/978-1-4757-3667-0_20.
[3]	J. P. Aparicio, A. F. Capurro and C. Castillo-Chavez, Transmission and dynamics of tuberculosis on generalized households, J Theor Biol, 206 (2000), 327-341.doi: 10.1006/jtbi.2000.2129.
[4]	J. P. Aparicio and C. Castillo-Chavez, Mathematical modelling of tuberculosis epidemics, Math Biosci Eng, 6 (2009), 209-237.doi: 10.3934/mbe.2009.6.209.
[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), 353-358.
[6]	B. R. Bloom, Tuberculosis: Pathogenesis, Protection, and Control, ASM Press, Washington, D.C., 1994.
[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), 815-821.doi: 10.1038/nm0895-815.
[8]	F. Brauer and C. Castillo-Chavez, Mathematical Models for Communicable Diseases, SIAM, 2013.
[9]	C. Castillo-Chavez, Chalenges and opportunities in mathematical and theoretical biology and medicine: foreword to volume 2 (2013) of Biomath, Biomath, 2 (2013), 1312319, 2pp.doi: 10.11145/j.biomath.2013.12.319.
[10]	C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis, J Math Biol, 35 (1997), 629-656.doi: 10.1007/s002850050069.
[11]	C. Castillo-Chavez and Z. Feng, Mathematical models for the disease dynamics of tuberculosis, Advances in Mathematical Population Dynamics - Molecules, Cells, and Man O. Arino, D. Axelrod, M. Kimmel, (eds), World Scientific Press, (1998), 629-656.
[12]	C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math Biosci Eng, 1 (2004), 361-404.doi: 10.3934/mbe.2004.1.361.
[13]	C. Y. Chiang and L. W. Riley, Exogenous reinfection in tuberculosis, Lancet Infect Dis, 5 (2005), 629-636.doi: 10.1016/S1473-3099(05)70240-1.
[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), 29-48.doi: 10.1016/S0025-5564(02)00108-6.
[15]	Z. Feng, C. Castillo-Chavez and A. F. Capurro, A model for tuberculosis with exogenous reinfection, Theor Popul Biol, 57 ( 2000), 235-247.doi: 10.1006/tpbi.2000.1451.
[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), 425-452.doi: 10.1023/A:1016688209771.
[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), 1511-1536.doi: 10.1007/s11538-006-9174-9.
[18]	Antibiotic-resistant Diseases Pose 'Apocalyptic' Threat, Top Expert Says, 2013. Available from: http://www.theguardian.com/society/2013/jan/23/antibiotic-resistant-diseases-apocalyptic-threat,
[19]	Guidelines on the Management of Latent Tuberculosis Infection, 2015. Available from: http://apps.who.int/medicinedocs/documents/s21682en/s21682en.pdf.
[20]	H. M. Hethcote, Qualitative analysis for communicable disease models, Math Biosc, 28 (1976), 335-356.doi: 10.1016/0025-5564(76)90132-2.
[21]	H. M. Hethcote, The Mathematics of infectious diseases, SIAM Rev, 42 (2000), 599-653.doi: 10.1137/S0036144500371907.
[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), 65-86.doi: 10.1016/S0025-5564(00)00025-0.
[23]	E. Ibargüen-Mondragón and L. Esteva, On the interactions of sensitive and resistant Mycobacterium tuberculosis to antibiotics, Math Biosc, 246 (2013), 84-93.doi: 10.1016/j.mbs.2013.08.005.
[24]	V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker Inc, New York and Basel, 41, 1989.
[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), 282-287.doi: 10.1016/S1473-3099(03)00607-8.
[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), 1570-1575.doi: 10.1056/NEJM198612183152502.
[27]	E. Oldfield and X. Feng, Resistance-resistant antibiotics, Trends in Pharmacological Sciences, 35 (2014), 664-674.doi: 10.1016/j.tips.2014.10.007.
[28]	T. C. Porco and S. M. Blower, Quantifying the intrinsic transmission dynamics of tuberculosis, Theoretical Population Biology, 54 (1998), 117-132.doi: 10.1006/tpbi.1998.1366.
[29]	J. W. Raleigh and R. H. Wichelhausen, Exogenous reinfection with mycobacterium tuberculosis confirmed by phage typing, Am Rev Respir Dis, 108 (1973), 639-642.
[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), 497-503.
[31]	M. Raviglione, Drug-Resistant TB Surveillance and Response, Global Tuberculosis Report 2014, 2014. Available from: http://www.who.int/tb/publications/global_report/gtbr14_supplement_web_v3.pdf.
[32]	L. W. Roeger, Z. Feng and C. Castillo-Chavez, Modeling TB and HIV co-infections, Math Biosci Eng, 6 (2009), 815-837.doi: 10.3934/mbe.2009.6.815.
[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), 1176-1178.doi: 10.3201/eid1211.051207.
[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), 1137-1144.
[35]	H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995.
[36]	B. Song, Dynamical Epidemic Models and Their Applications, Thesis (Ph.D.)-Cornell University, 2002.
[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), 187-205.doi: 10.1016/S0025-5564(02)00112-8.
[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, S. Blower, P. van den Driessche, D. Kirschner, A. A. Yakubu), Springer, New York, IMA, 126 (2002), 275-294.doi: 10.1007/978-1-4613-0065-6_16.
[39]	W. W. Stead, The pathogenesis of pulmonary tuberculosis among older persons, Am Rev Respir Dis, 91 (1965), 811-22.
[40]	T. C. Porco and S. M. Blower, Quantifying the intrinsic transmission dynamics of tuberculosis, Theoretical Population Biology, 54 (1998), 117-132.doi: 10.1006/tpbi.1998.1366.
[41]	X. Wang, Backward Bifurcation in a Mathematical Model for Tuberculosis with Loss of Immunity, Ph.D. Thesis, Purdue University, 2005.
[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), 304-330.doi: 10.1142/9789814304900_0021.
[43]	Global Tuberculosis Control: Who Report 2010, 2010. Available from: http://reliefweb.int/sites/reliefweb.int/files/resources/F530290AD0279399C12577D8003E9D65-Full_Report.pdf.