Mathematical models of tuberculosis extension and control of it (review)

K. K. Avilov ; A. A. Romanyukha

Matematičeskaâ biologiâ i bioinformatika, Tome 2 (2007) no. 2, pp. 188-318.

Résumé

The survey considers mathematical models of epidemiological processes, which determine the dynamics of tuberculosis morbidity. The first models of tuberculosis epidemiology were developed and published in the early sixties of the last century. These works and the works of seventies have formulated and described peculiarities of tuberculosis epidemiology, which are: long-lasting latent phase of the infection, very low probability of total deliverance from the infection, possibility of quick growth of the disease after introduction of infection, and dependence of probability of activation of the infection on the condition of the vehicle of the disease and duration of the latent stage. The most important sphere of application for mathematical models of tuberculosis epidemiology is estimation of efficiency of the control strategies for this disease. A new tide of interest in mathematical models of spread of tuberculosis is connected with growth of morbidity rate in developing countries because of HIV epidemics and emergence of mycobacterium strains, which are resistant to one or several medications. The models of the 80-s and 90-s are devoted to interaction of HIV infection and mycobacteria, to formation and spread of drug-resistant strains. Much attention is given to investigation of properties of the models, to estimation of parameters, and comparison with real data. During this period models became important means for working out and argumentation of activity of both national and international organizations, which are responsible for fight against this infection. For convenience sake a unified system of designation of variables and parameters is used in the survey, block schematic diagrams of the simulated processes and estimations of values of parameters are given; assumptions and presuppositions having been employed in construction of the models are discussed. The work under consideration is the first complete survey of the models of this class up to the year 2006.

Bibliographie

[1] Waaler H. T., Geser A., Andersen S., “The use of mathematical models in the study of the epidemiology of tuberculosis”, Am. J. publ. Health., 52 (1962), 1002–1013 | DOI

[2] Waaler H. T., “Cost-benefit analyses of BCG vaccination under various epidemiological situations”, Bull. int. Union Tuberc., 41 (1968), 42–52

[3] Waaler H. T., “A Dynamic Model for the Epidemiology of Tuberculosis”, American Review of Respiratory Disease, 98 (1968), 591–600

[4] Waaler H. T., Piot M. A., “The use of an epidemiological model for estimationg the effectiveness of tuberculosis control measures. Sensitivity of the effectiveness of tuberculosis control measures to the coverage of the population”, Bulletin of the World Health Organization, 41 (1969), 75–93

[5] Waaler H. T., Piot M. A., “Use of an Epidemiological Model for Estimating the Effectiveness of Tuberculosis Control Measures. Sensitivity of the Effectiveness of Tuberculosis Control Measures to the Social Time Preference”, Bulletin of the World Health Organization, 43 (1970), 1–16

[6] Waaler H. T., Gothi G. D., Baily G. V. J., Nair S. S., “Tuberculosis in rural South India. A study of possible trends and the potential impact of antituberculosis programmes”, Bulletin of the World Health Organization, 51 (1974), 263–271

[7] Brogger S., “Systems analysis in tuberculosis control: A model”, American Review of Respiratory Disease, 95:3 (1967), 4190–434

[8] ReVelle C. S., Lynn W. R., Feldmann F., “Mathematical models for the economic allocation of tuberculosis control activities in developing nations”, American Review of Respiratory Disease, 96 (1967), 893–909

[9] ReVelle C. S., The Economic Allocation of Tuberculosis Control Activities in Developing Nations, Thesis, Cornell University, 1967

[10] Hedrich A. W., “Monthly Estimates of the Child Population “Susceptible” to Measles, 1900-1931, Baltimore”, Md. Am. J. Hyg., 17 (1933), 613–636

[11] Muench H., Catalytic Models in Epidemiology, Harvard University Press, Cambridge, Mass., 1959

[12] Frost W. H., “How Much Control of Tuberculosis?”, A.J.P.H., 27 (1937), 759 | DOI

[13] Feldman F. M., “How Much Control of Tuberculosis: 1937–1957–1977?”, Am. J. Public Health Nations Health., 47:10 (1957), 1235–1241 | DOI

[14] Palmer C. E., Shaw L. W., Comstock G. W., “Community trials of BCG vaccination”, Am. Rev. Tuberc., 77 (1958), 877

[15] Sutherland I., “An Estimation of the Scope for BCG Vaccination in Preventing Tuberculosis Among Those Aged 15–19 Years in England and Wales at the present time”, Tubercle, 40 (1959), 413 | DOI

[16] Frimodt-Moller J. A., “Community-Wide tuberculosis Study in a South Indian Rural Population, 1950–1955”, Bull World Health Organ., 22 (1960), 61

[17] Ferebee S. H., “An epidemiological model of tuberculosis in the United States”, Bulletin of the National Tuberculosis Association, 4 (1967), 7

[18] Narain R., Nair N. N., Naganna K., et al., “Problems in defining a “case” of pulmonary tuberculosis in prevalence surveys”, Bulletin of the World Health Organization, 39 (1968), 701

[19] “National Tuberculosis Institute, Bangalore. Tuberculosis in a rural population of South India: a five-year epidemiological study”, Bulletin of the World Health Organization, 51 (1974), 473–488

[20] Azuma Y., “A simple simulation model of tuberculosis epidemiology for use without large-scale computers”, Bulletin of the World Health Organization, 52 (1975), 313–322

[21] Styblo K., Bumgarner J. R., Tuberculosis can be controlled with existing technologies: evidence, Tuberculosis Surveillance Research Unit Progress Report, The Hague, 1991, 60–72

[22] The Stop TB Strategy, World Health Organization (WHO/HTM/STB/2006.37), 2006

[23] Blower S. M., McLean A. R., Porco T. C., Small P. M., Hopewell P. C., Sanchez M. A., Moss A. R., “The intrinsic transmission dynamics of tuberculosis epidemics”, Nature Medcine, 1:8 (1995), 815–821 | DOI

[24] Anderson R., Mei R., Infektsionnye bolezni cheloveka. Dinamika i Kontrol, Mir, “Nauchnyi mir”, Moskva, 2004, 784 pp. (Per. s angl.)

[25] Blower S. M., Small P. M., Hopewell P. C., “Control strategies for tuberculosis: new models for old problems”, Science, 273 (1996), 497–500 | DOI

[26] Lietman T., Porco T., Blower S., “Leprosy and tuberculosis: the epidemiological consequences of cross-immunity”, American Journal of Public Health, 87:12 (1997), 1923 | DOI

[27] Lechat M. F., “The torments and blessings of the leprosy epidemiometric model”, Lepr. Rev., 52 (1981), 187–196

[28] Abel L., Mallet A., Demenais F., Booney G. E., “Modeling the age-of-onset function in segregation analysis: a casual scheme for leprosy”, Genet. Epidemiol., 6 (1989), 501–516 | DOI

[29] Porco T. C., Blower S. M., “Quantifying the Intrinsic Transmission Dynamics of Tuberculosis”, Theoretical Population Biology, 54 (1998), 117–132 | DOI | Zbl

[30] Blower S. M., Gerberding J. L., “Understanding, predicting and controlling the emergence of drug-resistant tuberculosis: a theoretical framework”, J. Mol. Med., 76 (1998), 624–636 | DOI

[31] Blower S., Porco T., Lietman T., “Tuberculosis: The Evolution of Antibiotic Resistance and the Design of Epidemic Control Strategies”: Horn, Simonett, Webb, Mathematical Models in Medical and Health Sciences, Vanderbilt Press, USA, 1998 | MR | Zbl

[32] Ziv E., Daley C. L., Blower S. M., “Early Therapy for Latent Tuberculosis Infection”, American Journal of Epidemiology, 153:4 (2001), 381–385 | DOI | MR

[33] Lietman T., Blower S. M., “Potential Impact of Tuberculosis Vaccines as Epidemic Control Agents”, Clinical Infectious Diseases, 30, Suppl 3 (2000), S316–s322 | DOI

[34] Ziv E., Daley C. L., Blower S., “Potential Public Health Impact of New Tuberculosis Vaccines”, Emerging Infectious Diseases, 10:9 (2004), 1529–1535

[35] Porco T. C., Small P. M., Blower S. M., “Amplification Dynamics: Predicting the Effect of HIV on Tuberculosis Outbreaks”, Journal of Aquired Immune Deficiency Syndromes, 28 (2001), 437–444

[36] Blower S. M., Chou T., “Modeling the emergence of the ‘hot zones’: tuberculosis and the amplification dynamics of drug resistance”, Nature Medcine, 10:10 (2004), 1111–1116 | DOI | MR

[37] Feng Z., Castillo-Chavez C., Capurro A. F., “A Model for Tuberculosis with Exogenous Reinfection”, Theoretical Population Biology, 57 (2000), 235–247 | DOI | Zbl

[38] Blower S. M., Daley C. L., “Problems and solutions for the Stop TB partnership”, The Lancet Infectious Diseases, 2 (2002), 374–376 | DOI

[39] Sutherland I., Švandová E., “Endogenous reactivation and exogenous infection their relative importance with regard to the development of non-primary tuberculosis”, Bulletin of the World Health Organization, 47 (1972), 123

[40] Sutherland I., Švandová E., Radhakrishna S., “The development of clinical tuberculosis following infection with tubercle bacilli”, Tubercle, 63 (1982), 255–268 | DOI

[41] Vynnycky E., An Investigation of the Transmission Dynamics of M. tuberculosis, PhD thesis, University of London, 1996

[42] Vynnycky E., Fine P. E. M., “The natural history of tuberculosis: the implications of agedependent risks of disease and the role of reinfection”, Epidemiol. Infect., 119 (1997), 183–201 | DOI

[43] Vynnycky E., Fine P. E. M., “Lifetime Risks, Incubation Period, and Serial Interval of Tuberculosis”, American Journal of Epidemiology, 152:3 (2000), 247–263 | DOI

[44] Vynnycky E., Nagelkerke N., Borgdorff M. W., van Soolingen D., van Embden J. D., Fine P. E. M., “The effect of age and study duration on the relationship between ‘clustering’ of DNA fingerprint patterns and the proportion of tuberculosis disease attributable to recent transmission”, Epidemiol Infect., 126:1 (2001), 43–62

[45] Vynnycky E., Borgdorff M. W., van Soolingen D., Fine P. E. M., “Annual Mycobacterium tuberculosis infection risk and interpretation of clustering statistics”, Emerging Infectious diseases, 9:2 (2003), 176–183

[46] Holm J., Development of tuberculous infection to tuberculous disease, TSRU Progress Report, KNCV, The Hague, The Netherlands, 1969

[47] Vynnycky E., Fine P. E. M., “The annual risk if infection with Mycobacterium tuberculosis in England and Wales since 1901”, Int. J. Tuberc. Lung Dis., 1 (1997), 389

[48] Sutherland I., The ten-year incidence of clinical tuberculosis following “conversion” in 2550 individuals aged 14 to 19 years, TSRU Progress Report, KNCV, The Hague, The Netherlands, 1968

[49] Castillo-Chavez C., Feng Z., “To treat or not to treat: the case of tuberculosis”, J. Math. Biol., 35 (1997), 629–656 | DOI | MR | Zbl

[50] Castillo-Chavez C., Feng Z., “Global stability of an age-structure model for TB and its applications to optimal vaccination strategies”, Mathematical Biosciences, 151 (1998), 135–154 | DOI | Zbl

[51] Castillo-Chavez C., Feng Z., “Mathemetical models for the disease dynamics of tuberculosis”, Advances in Mathematical Population Dynamics Molecules, Cells, and Man, eds. O. Arino, D. Axelrod, M. Kimmel, World Scientifc Press, 1998, 629–656 | MR

[52] Mena-Lorca J., Velasco-Hernandez J. X., Castillo-Chavez C., “Density-dependent dynamics and superinfection in an epidemic model”, IMA Journal of Mathematics Applied in Medicine and Biology, 16 (1999), 307–317 | DOI | Zbl

[53] Aparicio J. P., Capurro A. F., Castillo-Chavez C., “Transmission and Dynamics of Tuberculosis on Generalized Households”, J. theor. Biol., 206 (2000), 327–341 | DOI

[54] Aparicio J. P., Capurro A. F., Castillo-Chavez C., On the fall and rise of tuberculosis, Technical Report Series, BU-1477-M, Department of Biometrics, Cornell University, 2000

[55] Feng Z., Huang W., Castillo-Chavez C., “On the role of variable latent periods in mathematical models for tuberculosis”, Journal of Dynamics and Differential Equations, 13:2 (2001), 425–452 | DOI | MR | Zbl

[56] Song B., Castillo-Chavez C., Aparicio J. P., “Tuberculosis models with fast and slow dynamics: the role of close and casual contacts”, Mathematical Biosciences, 180 (2002), 187–205 | DOI | MR | Zbl

[57] Castillo-Chavez C., Song B., An Overview of Dynamical Models of Tuberculosis, Technical Report of BSCB, Cornell University, Ithaca, BU-1607-M, 2002

[58] Song B., Castillo-Chavez C., Aparicio J. P., “Global Dynamics of Tuberculosis Models with Density Dependent Demography”, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, IMA Volume, 126, eds. Castillo-Chavez C., Pauline van den Driessche P., Denise Kirschner D. and Yakubu A. A., Springer-Veralg, Berlin, Heidelberg, New York, 2002, 275–294 | MR | Zbl

[59] Aparicio J. P., Capurro A. F., Castillo-Chavez C., “Markers of Disease Evolution: The Case of Tuberculosis”, J. theor. Biol., 215 (2002), 227–237 | DOI | MR

[60] Aparicio J. P., Capurro A. F., Castillo-Chavez C., “Longterm Dynamics and Reemergence of Tuberculosis”, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, eds. Sally Blower, Carlos Castillo-Chavez, Denise Kirschner, Pauline van den Driessche and Abdul-Aziz Yak, Springer-Verlag, 2002, 351–360 | MR | Zbl

[61] Castillo-Chavez C., Song B., “Dynamic models of tuberculosis and their applications”, Mathematical Biosciences and Engineering, 1:2 (2004), 361–404 | MR | Zbl

[62] Feng Z., Ianelli M., Milner F. A., “A Two-Strain Tuberculosis ModelWith Age of Infection”, SIAM J. Appl. Math., 62:5 (2002), 1634–1656 | DOI | MR | Zbl

[63] Dye C., Garnett G. P., Sleeman K., Williams B. G., “Prospects for worldwide tuberculosis control under the WHO DOTS strategy”, The Lancet., 352 (1998), 1886–1891 | DOI

[64] Baltussen R., Floyd K., Dye C., “Cost effectiveness analysis of strategies for tuberculosis control in developing countries”, BMJ, 331:7529 (2005), 1364 | DOI

[65] Williams B. G., Granich R., Chauhan L. S., Dharmshaktu N. S., Dye C., “The impact of HIV AIDS on the control of tuberculosis in India”, PNAS, 102:27 (2005), 9619–9624 | DOI

[66] Stover J., TB-HIV spreadsheet model. A model for illustrating the effects of the HIV epidemics on tuberculosis, Futures Group International, the POLICY project, Washington, DC, 1998

[67] Styblo K., Broekmans J. F., Borgdorff M. W., Expected decrease in tuberculosis incidence during the elimination phase. How to determine its trend?, Tuberculosis Surveillance and Research Unit Progress Report, Royal Netherlands Tuberculosis Association (KNCV), The Hague, 1997

[68] Dye C., Williams B. G., “Criteria for the control of drug-resistant tuberculosis”, Proc. Natl. Acad. Sci. USA, 97:14 (2000), 8180–8185 | DOI

[69] Dye C., Espinal M.A., “Will tuberculosis become resistant to all antibiotics?”, Proc. R. Soc. Lond. B. Biol. Sci., 268 (2001), 45–52 | DOI

[70] Espinal M., Simonsen L., Laszlo A., et al., Anti-Tuberculosis Drug Resistance in the World, Report No 2, World Health Organization (WHO/CDS/TB/2000.278), Geneva, 2000

[71] Currie C. S., Williams B. G., Cheng R. C., Dye C., “Tuberculosis epidemics driven by HIV: is prevention better than cure?”, AIDS, 17:17 (2003), 2501–2508 | DOI

[72] Currie C. S., Floyd K., Williams B. G., Dye C., “Cost, affordability and cost-effectiveness of strategies to control tuberculosis in countries with high HIV prevalence”, BMC Public Health, 5 (2005), 130 | DOI

[73] Murray C. J. L., Salomon J. A., “Modeling the impact of global tuberculosis control strategies”, Proc. Natl. Acad. Sci. USA, 95 (1998), 13881–13886 | DOI

[74] Murray C. J. L., Salomon J. A., Using mathematical models to evaluate global tuberculosis control strategies, Harvard Center for Population and Development Studies, Cambridge, MA, 1998 | Zbl

[75] Cohen T., Murray M., “Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness”, Nature Medicine, 10:10 (2004), 1117–1121 | DOI

[76] Resch S. C., Salomon J. A., Murray M., Weinstein M. C., “Cost-effectiveness of treating multidrug-resistant tuberculosis”, PLoS Medicine, 3:7 (2006), 1048–1057 | DOI

[77] Salomon J. A., Lloyd-Smith J. O., Getz W. M., Resch S., Sánchez M. S., Porco T. C., Borgdorff M. W., “Prospects for advancing tuberculosis control efforts through novel therapies”, PloS Medicine, 3:8 (2006), 1302–1309 | DOI

[78] Regiony Rossii. Osnovnye kharakteristiki sub'ektov RF, Statisticheskii sbornik, Ros-stat, Moskva, 2004

[79] Perelman M. I., Marchuk G. I., Borisov S. E., Kazennukh B. Ya., Avilov K. K., Karkach A. S., Romanyukha A. A., “Tuberculosis epidemiology in Russia: the mathematical model and data analysis”, Russ. J. Numer. Anal. Math. Modelling, 19:4 (2004), 305–314 | DOI | MR | Zbl

[80] Avilov K. K., Romanyukha A. A., “Matematicheskoe modelirovanie protsessov rasprostraneniya tuberkuleza i vyyavleniya bolnykh”, Avtomatika i telemekhanika, 2007, no. 9 (to appear) | MR | Zbl