×

Optimal campaign in the smoking dynamics. (English) Zbl 1207.92051

Summary: We present optimal campaigns in the smoking dynamics. Assuming that the giving up smoking model is described by the simplified PLSQ (potential-light-smoker-quit smoker) model, we consider two possible control variables in the form of education and treatment campaigns oriented to decrease the attitude towards smoking. In order to do this we minimize the number of light (occasional) and persistent smokers and maximize the number of quit smokers in a community. We first show the existence of an optimal control for the control problem and then derive the optimality system by using the Pontryagin maximum principle. Finally numerical results of real epidemic are presented to show the applicability and efficiency of this approach.

MSC:

92D99 Genetics and population dynamics
92D25 Population dynamics (general)
49N90 Applications of optimal control and differential games
92C99 Physiological, cellular and medical topics
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] C. Castilho, “Optimal control of an epidemic through educational campaigns,” Electronic Journal of Differential Equations, vol. 2006, pp. 1-11, 2006. · Zbl 1108.92035
[2] N. G. Becker and D. N. Starczak, “Optimal vaccination strategies for a community of households,” Mathematical Biosciences, vol. 139, no. 2, pp. 117-132, 1997. · Zbl 0881.92028 · doi:10.1016/S0025-5564(96)00139-3
[3] M. L. Brandeau, G. S. Zaric, and A. Richter, “Resource allocation for control of infectious diseases in multiple independent populations: beyond cost-effectiveness analysis,” Journal of Health Economics, vol. 22, no. 4, pp. 575-598, 2003. · doi:10.1016/S0167-6296(03)00043-2
[4] K. Wickwire, “Mathematical models for the control of pests and infectious diseases: a survey,” Theoretical Population Biology, vol. 11, no. 2, pp. 182-238, 1977. · Zbl 0356.92001 · doi:10.1016/0040-5809(77)90025-9
[5] K. R. Fister, S. Lenhart, and J. S. McNally, “Optimizing chemotherapy in an HIV model,” Electronic Journal of Differential Equations, vol. 1998, no. 32, pp. 1-12, 1998. · Zbl 1068.92503
[6] D. Kirschner, S. Lenhart, and S. Serbin, “Optimal control of the chemotherapy of HIV,” Journal of Mathematical Biology, vol. 35, no. 7, pp. 775-792, 1997. · Zbl 0876.92016 · doi:10.1007/s002850050076
[7] H. R. Joshi, “Optimal control of an HIV immunology model,” Optimal Control Applications and Methods, vol. 23, no. 4, pp. 199-213, 2002. · Zbl 1072.92509 · doi:10.1002/oca.710
[8] G. Zaman, Y. H. Kang, and I. H. Jung, “Stability analysis and optimal vaccination of an SIR epidemic model,” BioSystems, vol. 93, no. 3, pp. 240-249, 2008. · doi:10.1016/j.biosystems.2008.05.004
[9] Y. H. Kang, S. Lenhart, and V. Protopopescu, “Optimal control of parameters and input functions for nonlinear systems,” Houston Journal of Mathematics, vol. 33, no. 4, pp. 1231-1256, 2007. · Zbl 1187.49003
[10] K. Dietz and D. Schenzle, “Mathematical models for infectious disease statistics,” in A Celebrationf of Statistics, ISI Centenary, A. C. Atkinson and S. E. Fienberg, Eds., pp. 167-204, Springer, New York, NY, USA, 1985. · Zbl 0586.92017
[11] C. Castillo-Garsow, G. Jordan-Salivia, and A. Rodriguez Herrera, “Mathematical models for the dynamics of tobacco use, recovery, and relapse,” Technical Report Series BU-1505-M, Cornell University, Ithaca, NY, USA, 2000.
[12] O. Sharomi and A. B. Gumel, “Curtailing smoking dynamics: a mathematical modeling approach,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 475-499, 2008. · Zbl 1261.92023 · doi:10.1016/j.amc.2007.05.012
[13] G. Zaman, “Qualitative behavior of giving up smoking model,” Bulletin of the Malaysian Mathematical Sciences Society. In press. · Zbl 1221.92067
[14] A. J. Arenas, J. A. Moraño, and J. C. Cortés, “Non-standard numerical method for a mathematical model of RSV epidemiological transmission,” Computers and Mathematics with Applications, vol. 56, no. 3, pp. 670-678, 2008. · Zbl 1155.92337 · doi:10.1016/j.camwa.2008.01.010
[15] G. Zaman, Y. H. Kang, and I. H. Jung, “Optimal treatment of an SIR epidemic model with time delay,” BioSystems, vol. 98, no. 1, pp. 43-50, 2009. · doi:10.1016/j.biosystems.2009.05.006
[16] G. Birkhoff and G. C. Rota, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 4th edition, 1989. · Zbl 0183.35601
[17] I. K. Morton and L. S. Nancy, Dynamics Optimization: The Calculus of Variations and Optimal Control in Economics and Management, Elsevier Science, New York, NY, USA, 2000.
[18] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Mathematical and Computational Biology Series, Chapman & Hall/CRC, London, UK, 2007. · Zbl 1291.92010
[19] American National Institute of Drug Abuse, “Cigarettes and Other Nicotine Products,” http://www.nida.nih.gov/pdf/infofacts/Nicotine04.pdf.
[20] National Cancer Information Center, “Death statistics based on vital registration,” February 2006, http://www.ncc.re.kr/.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.