A combined numerical algorithm for reconstructing the mathematical model for tuberculosis transmission with control programs

References

[1] K. Avilov and A. Romanyukha, Mathematical modeling of tuberculosis propagation and patient detection, Autom. Remote Control 68 (2007), no. 9, 1604–1617. 10.1134/S0005117907090159Search in Google Scholar

[2] S. Blower, A. Mclean, T. Porco, P. Small, P. Hopewell, M. Sanchez and A. Moss, The intrinsic transmission dynamics of tuberculosis epidemics, Nature Med. 1 (1995), no. 8, 815–821. 10.1038/nm0895-815Search in Google Scholar

[3] S. Blower and T. Porco, Quantifying the intrinsic transmission dynamics of tuberculosis, Theoret. Popul. Biol. 54 (1998), no. 2, 117–132. 10.1006/tpbi.1998.1366Search in Google Scholar

[4] S. Blower, T. Porco and T. Lietman, Tuberculosis: The evolution of antibiotic resistance and the design of epidemic control strategies, Math. Models Med. Health Sci. (1998). Search in Google Scholar

[5] S. Blower, T. Porco, P. Small, Amplification dynamics: Predicting the effect of hiv on tuberculosis outbreaks, J. Aquired Immune Deficiency Syndromes 28 (2001), no. 5, 437–444. 10.1097/00042560-200112150-00005Search in Google Scholar

[6] S. Blower, P. Small and P. Hopewell, Control strategies for tuberculosis epidemics: New models for old problems, Science 273 (1996), no. 5274, 497–500. 10.1126/science.273.5274.497Search in Google Scholar PubMed

[7] L. Elsgolts, Differential Equations and the Calculus of Variations, Nauka, Moscow, 1970. Search in Google Scholar

[8] A. Ilyin, S. Kabanikhin and O. Krivorotko, Determination of the parameters of the models described by systems of nonlinear differential equations, Sib. Èlektron. Mat. Izv. 11 (2014). 10.1515/jiip-2015-0072Search in Google Scholar

[9] L. Ingber, Very fast simulated re-annealing, Math. Comput. Model. 12 (1989), no. 8, 967–973. 10.1016/0895-7177(89)90202-1Search in Google Scholar

[10] L. Ingber, Simulated annealing: Practice versus theory, Math. Comput. Model. 18 (1993), no. 11, 29–57. 10.1016/0895-7177(93)90204-CSearch in Google Scholar

[11] L. Ingber, Adaptive simulated annealing (asa): Lessons learned, Control Cybernet. 25 (1996), 33–54. Search in Google Scholar

[12] S. Kabanikhin, Inverse and Ill-Posed Problems: Theory and Applications, Walter De Gruyter, Berlin, 2011. 10.1515/9783110224016Search in Google Scholar

[13] S. Kabanikhin and O. Krivorotko, Identication of biological models described by systems of nonlinear differential equations, J. Inverse Ill-Posed Probl. 23 (2015), no. 5, 519–527. 10.1515/jiip-2015-0072Search in Google Scholar

[14] S. Kabanikhin and M. Shishlenin, Direct and iteration methods for solving inverse and ill-posed problems, Sib. Èlektron. Mat. Izv. 5 (2008), 595–608. Search in Google Scholar

[15] S. Kabanikhin, D. Voronov, A. Grodz and O. Krivorotko, Identifiability of mathematical models in medical biology, Vavilov J. Genet. Breeding 19 (2015), no. 6, 738–744. 10.18699/VJ15.097Search in Google Scholar

[16] A. Lopatin, Annealing method, Stochastic Optim. Inform. (2005), no. 1, 133–149. Search in Google Scholar

[17] H. Miao, X. Xia, A. S. Perelson and H. Wu, On identifiability of nonlinear ode models and applications in viral dynamics, SIAM Rev. 53 (2011), no. 1, 3–39. 10.1137/090757009Search in Google Scholar PubMed PubMed Central

[18] C. Murray and J. Salomon, Modeling the impact of global tuberculosis control strategies, Proc. Natl. Acad. Sci. USA 95 (1998), no. 23, 13881–13886. 10.1073/pnas.95.23.13881Search in Google Scholar PubMed PubMed Central

[19] C. Murray and J. Salomon, Using Mathematical Models to Evaluate Global Tuberculosis Control Strategies, Harvard Center for Population and Development Studies, Camridge, 1998. Search in Google Scholar

[20] A. Neubauer and O. Scherzer, A convergence rate result for a steepest descent method and a minimal error method for the solution of nonlinear ill-posed problems, Z. Anal. Anwend. 14 (1995), no. 2, 369–377. 10.4171/ZAA/679Search in Google Scholar

[21] M. Perelman, G. Marchuk, S. Borisov, B. Kazennukh, K. Avilov, A. Karkach and A. Romanyukha, Tuberculosis epidemiology in russia: The mathematical model and data analysis, Russian J. Numer. Anal. Math. Modelling 19 (2004), no. 4, 305–314. 10.1515/1569398041974905Search in Google Scholar

[22] K. Stỳblo and J. Bumgarner, Tuberculosis can be controlled with existing technologies: Evidence, Tuberculosis Surveillance Res. Unit Prog. Rep. 2 (1991), 60–72. Search in Google Scholar

[23] J. Trauer, J. Denholm and E. McBryde, Construction of a mathematical model for tuberculosis transmission in highly endemic regions of the asia-pacific, J. Theoret. Biol. 358 (2014), 74–84. 10.1016/j.jtbi.2014.05.023Search in Google Scholar PubMed

[24] V. Vasin, On the convergence of gradient-type methods for nonlinear equations, Dokl. Math. 57 (1998), no. 2, 173–175. Search in Google Scholar

[25] E. Vynnycky and P. Fine, The natural history of tuberculosis: The implications of age-dependent risks of disease and the role of reinfection, Epidemiology Infection 119 (1997), no. 2, 183–201. 10.1017/S0950268897007917Search in Google Scholar PubMed PubMed Central

[26] H. Waaler, A dynamic model for the epidemiology of tuberculosis 1, 2, Amer. Rev. Respiratory Disease 98 (1968), no. 4, 591–600. 10.1016/S0041-3879(68)80031-5Search in Google Scholar

[27] H. Waaler, Cost-benefit analysis of bcg-vaccination under various epidemiological situations, Bull. Internat. Union Tuberculosis 41 (1968), 42–52. Search in Google Scholar

[28] H. Waaler, A. Geser and S. Andersen, The use of mathematical models in the study of the epidemiology of tuberculosis, Amer. J. Public Health Nations Health 52 (1962), no. 6, 1002–1013. 10.2105/AJPH.52.6.1002Search in Google Scholar PubMed PubMed Central

[29] H. Waaler, G. Goth, G. Baily and S. Nair, Tuberculosis in rural south india. a study of possible trends and the potential impact of antituberculosis programmes, Bull. World Health Org. 51 (1974), no. 3, 263–271. Search in Google Scholar

[30] H. Waaler and M. Piot, The use of an epidemiological model for estimating the effectiveness of tuberculosis control measures: Sensitivity of the effectiveness of tuberculosis control measures to the coverage of the population, Bull. World Health Org. 41 (1969), no. 1, 75–93. Search in Google Scholar

[31] H. Waaler and M. Piot, The 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, Bull. World Health Org. 43 (1970), 1–16. Search in Google Scholar

[32] A. Zhiglyavskii and A. Zhilinskas, Methods of Searching the Global Extremum, Nauka, Moscow, 1991. Search in Google Scholar

[33] Birth rate, mortality and natural change of the population on the subjects of the russian federation, http://www.gks.ru/. Search in Google Scholar

[34] Number of the resident population (people) at 1 january 1990-2010 year in the russian federation, http://www.fedstat.ru/. Search in Google Scholar

[35] Tuberculosis in the Russian Federation in 2008. Analytical review of tuberculosis statistical indicators used in the Russian Federation, Moscow, 2009. Search in Google Scholar