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Optimal Tuberculosis Prevention and Control Strategy from a Mathematical Model Based on Real Data

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Abstract

A mathematical control model for the transmission dynamics of tuberculosis (TB) in South Korea is developed on the basis of the reported active-TB and relapse-TB incidence data. In this work, optimal control theory is used to propose optimal TB prevention and control strategy and rearrange the government TB budget for the best TB elimination plan. The impact of distancing, case finding, and/or case holding controls are investigated when the number of infected and infectious individuals are minimized, while the intervention costs are kept low. The implementation of optimal control measures shows that the distancing control, such as isolation of infectious people, early TB patient detection, and educational program/campaign for healthy control, is the most effective control factor for the prevention of TB transmission in South Korea.

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References

  • Anderson BD, Moore JB (1990) Optimal control: linear quadratic methods, vol 1. Prentice Hall, Englewood Cliffs, NJ

    MATH  Google Scholar 

  • Aparicio JP, Capurro AF, Castillo-Chavez C (2000) Transmission and dynamics of tuberculosis on generalized households. J Theor Biol 206(3):327–341

    Article  Google Scholar 

  • Aparicio JP, Capurro AF, Castillo-Chvez C (2002) Markers of disease evolution: the case of tuberculosis. J Theor Biol 215(2):227–237

    Article  MathSciNet  Google Scholar 

  • Aparicio JP, Hernndez JC (2006) Preventive treatment of tuberculosis through contact tracing. Contemp Math 410:17–30

    Article  MathSciNet  MATH  Google Scholar 

  • Aparicio JP, Castillo-Chavez C (2009) Mathematical modelling of tuberculosis epidemics. Math Biosci Eng 6:209–37

    Article  MathSciNet  MATH  Google Scholar 

  • Barnes DS (1995) The making of a social disease: tuberculosis in nineteenth-century France. University of California Press, California

    Google Scholar 

  • Behncke H (2000) Optimal control of deterministic epidemics. Optim Control Appl Methods 21(6):269–285

    Article  MathSciNet  MATH  Google Scholar 

  • Blower SM, Mclean AR, Porco TC, Small PM, Hopewell PC, Sanchez MA, Moss AR (1995) The intrinsic transmission dynamics of tuberculosis epidemics. Nat Med 1(8):815–821

    Article  Google Scholar 

  • Blower SM, Small PM, Hopewell PC (1996) Control strategies for tuberculosis epidemics: new models for old problems. Science 273(5274):497–500

    Article  Google Scholar 

  • Blower SM, Gerberding JL (1998) Understanding, predicting and controlling the emergence of drug-resistant tuberculosis: a theoretical framework. J Mol Med 76(9):624–636

    Article  Google Scholar 

  • Blythe SP, Castillo-Chavez C, Palmer JS, Cheng M (1991) Toward a unified theory of sexual mixing and pair formation. Math Biosci 107(2):379–405

    Article  MATH  Google Scholar 

  • Brauer F, Castillo-Chavez C (2011) Mathematical models in population biology and epidemiology. Springer, Berlin

    MATH  Google Scholar 

  • Castillo-Chavez C, Feng Z (1997) To treat or not to treat: the case of tuberculosis. J Math Biol 35(6):629–656

    Article  MathSciNet  MATH  Google Scholar 

  • Castillo-Chavez C, Feng Z (1998) Mathematical models for the disease dynamics of tuberculosis. In: Arino O, Axelrod D, Kimmel M (eds) Advances in mathematical population dynamics: molecules, cells and man. World Scientific, Singapore, pp 629–656

    Google Scholar 

  • Castillo-Chavez C, Blower S (eds) (2002) Mathematical approaches for emerging and reemerging infectious diseases: an introduction, vol 1. Springer, Berlin.

  • CDC (2012) The social distancing law assessment template. Retrieved October 23, 2013 from http://www.cdc.gov/phlp/publications/social_distancing.html

  • Chaisson RE, Churchyard GJ (2010) Recurrent tuberculosis: relapse, reinfection, and HIV. J Infect Dis 201(5):653–655

    Article  Google Scholar 

  • Chaulet P (1983) Treatment of tuberculosis: case holding until cure. Geneva: World Health Organization, WHO/TB/83.141 Rev. 2.

  • Dye C, Scheele S, Dolin P, Pathania V, Raviglione MC (1999) Global burden of tuberculosis: estimated incidence, prevalence, and mortality by country. JAMA 282(7):677–686

    Article  Google Scholar 

  • Dye C, Watt CJ, Bleed DM, Hosseini SM, Raviglione MC (2005) Evolution of tuberculosis control and prospects for reducing tuberculosis incidence, prevalence, and deaths globally. JAMA 293(22):2767–2775

    Article  Google Scholar 

  • Feng Z, Castillo-Chavez C, Capurro AF (2000) A model for tuberculosis with exogenous reinfection. Theor Popul Biol 57(3):235–247

    Article  MATH  Google Scholar 

  • Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, New York

    Book  MATH  Google Scholar 

  • Jung E, Lenhart S, Feng Z (2002) Optimal control of treatments in a two-strain tuberculosis model. Discret Contin Dyn Syst Ser B 2(4):473–482

    Article  MathSciNet  MATH  Google Scholar 

  • Jung E, Lenhart S, Protopopescu V, Babbs CF (2005) Optimal control theory applied to a difference equation model for cardiopulmonary resuscitation. Math Models Methods Appl Sci 15(10):1519–1531

    Article  MathSciNet  MATH  Google Scholar 

  • Jung E, Babbs CF, Lenhart S, Protopopescu VA (2006) Optimal strategy for cardiopulmonary resuscitation with continuous chest compression. Acad Emerg Med 13(7):715–721

    Article  Google Scholar 

  • Jung E, Lenhart S, Protopopescu V, Babbs C (2008) Optimal control applied to a thoraco-abdominal CPR model. Math Med Biol 25(2):157–170

    Article  MATH  Google Scholar 

  • Jung E, Iwami S, Rakeuchi Y, Jo TC (2009) Optimal control strategy for prevention of avian influenza pandemic. J Theor Biol 260(2):220–229

    Article  MathSciNet  Google Scholar 

  • Korea Centers for Disease Control and Prevention (2013). Retrieved October 23, 2013 from http://www.cdc.go.kr/CDC/info/CdcKrInfo0302.jsp?menuIds=HOME001-MNU0004-MNU0739-MNU0038

  • Korean Statistical Information Service. http://www.kosis.kr/abroad/abroad_01List.jsp

  • Lambert ML, Hasker E, Deun AV, Roberfroid D, Boelaert M, Van der Stuyft P (2003) Recurrence in tuberculosis: relapse or reinfection? Lancet Infect Dis 3(5):282–287

    Article  Google Scholar 

  • Lee S, Jung E, Castillo-Chavez C (2010) Optimal control intervention strategies in low-and high-risk problem drinking populations. Socioecon Plan Sci 44(4):258–265

    Article  Google Scholar 

  • Lee S, Chowell G, Castillo-Chvez C (2010) Optimal control for pandemic influenza: the role of limited antiviral treatment and isolation. J Theor Biol 265(2):136–150

    Article  MathSciNet  Google Scholar 

  • Lenhart SM, Workman JT (2007) Optimal control applied to biological models, vol 15. CRC, New York

    MATH  Google Scholar 

  • Lukes D (1982) Differential equations: classical to controlled. Mathematics in science and engineering, vol 162. Elsevier, Amsterdam.

  • National Discovery for Science Leaders (NDSL) (2011). Retrieved October 23, 2013 from http://report.ndsl.kr/repDetail.do?cn=TRKO201300000533

  • National Library of Medicine Visual Culture and Public Health Posters (2011). Retrieved October 23, 2013 from www.nlm.nih.gov/exhibition/visualculture/tuberculosis.html

  • Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes.

  • Porco TC, Blower SM (1998) Quantifying the intrinsic transmission dynamics of tuberculosis. Theor Popul Biol 54(2):117–132

    Article  MATH  Google Scholar 

  • Rowthorn RE, Laxminarayan R, Gilligan CA (2009) Optimal control of epidemics in metapopulations. J R Soc Interface 6(41):1135–1144

    Article  Google Scholar 

  • Song B, Castillo-Chavez C, Aparicio JP (2002) Tuberculosis models with fast and slow dynamics: the role of close and casual contacts. Math Biosci 180(1):187–205

    Article  MathSciNet  MATH  Google Scholar 

  • Styblo K, Enarson DA, Association RNT (1991) Selected papers: epidemiology of tuberculosis : epidemiology of tuberculosis in HIV prevalent countries. Royal Netherlands Tuberculosis Association, Saint Maurice

    Google Scholar 

  • Waaler H, Geser A, Andersen S (1962) The use of mathematical models in the study of the epidemiology of tuberculosis. Am J Public Health Nations Health 52(6):1002–1013

    Article  Google Scholar 

  • Whang S, Choi S, Jung E (2011) A dynamic model for tuberculosis transmission and optimal treatment strategies in South Korea. JTheor Biol 279(1):120–131

    Article  MathSciNet  Google Scholar 

  • WHO (2011) Global tuberculosis control, http://www.who.int/tb/publications/global_report/2011/gtbr11_full

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Acknowledgments

The research works of Jung and Choi are supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2012R1A2A2A01011725). Jung’s work is also resulted from the Konkuk University research support program.

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Authors and Affiliations

  1. Department of Mathematics, Konkuk University, Seoul , 143-701, Republic of Korea

    Sunhwa Choi & Eunok Jung

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  1. Sunhwa Choi
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  2. Eunok Jung
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Correspondence to Eunok Jung.

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Choi, S., Jung, E. Optimal Tuberculosis Prevention and Control Strategy from a Mathematical Model Based on Real Data. Bull Math Biol 76, 1566–1589 (2014). https://doi.org/10.1007/s11538-014-9962-6

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  • DOI: https://doi.org/10.1007/s11538-014-9962-6

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