Optimal Control Model for Vaccination Against H1N1 Flu

Pablo Amauri Carvalho Araujo e Souza, Claudia Mazza Dias, Edilson Fernandes de Arruda

Abstract


This paper introduces a mathematical model to describe the dynamics of the spread of H1N1 flu in a human population. The model is comprised of a system of ordinary differential equations that involve susceptible, exposed, infected and recovered/immune individuals. The distinguishing feature in the proposed model with respect to other models in the literature is that it takes into account the possibility of infection due to immunity loss over time. The acquired immunity comes from self-recovery or via vaccination. Furthermore, the proposed model strives to find an optimal vaccination strategy by means of an optimal control problem and Pontryagin’s Maximum Principle.


Keywords


Optimal control; Mathematical modeling; Pontryagin’s maximum principle; H1N1 flu; Vaccination

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References


ARRUDA, E. F.; DIAS, C. M.; MAGALHÃES, C. V. M.; PASTORE, D. H.; THOMÉ, R. C. A.; YANG, H. M. Na optimal control approach to HIV immunology. Applied Mathematics, New York, v. 6, p. 1115–1130, 2015.

BANKER, M. G.; WILSON, H. K. Pandemic H1N1 influenza lessons from the southern Hemisphere. Eurosurveillance, Stockholm, v. 14, n. 42, p. 19370, 2009.

BURDEN, R. L.; FAIRES, J. D. Numerical analysis. 2. ed. Massachusetts: Cengage Learning, 2008.

DIAS, A. C.; ARRUDA, E. F.; SOUZA, P. A. C. A. A study for computational cost optimization for influenza flu including vaccination’s impact. In: CONGRESS OF NUMERICAL METHODS IN ENGINEERING, 2015, Portugal. Proceedings [. . . ].Lisboa: CMNE, 2015. p. 1-16.

DIAS, C. M.; ARRUDA, E. F. Computational cost optimization for influenza A (H1N1) epidemic model. In: INTERNATIONAL CONGRESS ON ENV. MODELLING AND SOFTWARE, 7., 2014, San Diego. Proceedings [. . . ].San Diego: IEMSS, 2014. p. 1 - 8.

FATMAWATI, H.,TASMAN, H., An optimal control strategy to reduce the spread of malaria resistance. Mathematical Biosciences, New York, 262, 73-79, 2015.

FLEMING,W., RISHEL, R., Deterministic and stochastic optimal control.New York, Springer-Verlag, 1975.

HELAL, Z. A., REHBOCK, V., LOXTON, R., Modelling and optimal control of blood glucose levels in the human body. Journal of Industrial and Management Optimization, United States, v. 11, n. 4, p. 1149-1164, 2015.

JABERI-DOURAK, M., MOGHADAS, S. M., Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences and Engineering, Springfield, v. 11, n. 5, p. 1045-1063, 2014.

KERMACK, W. O., MCKENDRIC, A. G., A contribution to the mathematical theory of epidemics. Royal Soc. London Proc. Series., London, A115, p. 700-721, 1927.

KIM, S., LEE, J., JUNG, E., Mathematical model of transmission dynamics and optimal control strategies for 2009 A/H1N1 influenza in the Republic of Korea. Journal of Theoretical Biology, London, v. 412, p. 74-85, 2017.

KIRK, D., Optimal control theory: an introduction. New Jersey: Prentice-Hall, 1970.

LARSON, R. C.; TEYTELMAN, A. Modeling the effects of H1N1 influenza vaccine distribution in the United States. Value in Health, New York, v. 15, p. 158-166, 2012.

LEE, J.; KIN, J. ; KWON, H-D. Optimal control of an influenza model with seasonal forcing and age-dependent transmission rates. Journal of Theoretical Biology, Amsterdam, v. 317, p. 310-320, 2013.

LEE, S.; CHOWELL, G. Exploring optimal control strategies in seasonally varying flu-like epidemics. Journal of Theoretical Biology, Amsterdam, v. 412, p. 36-47, 2017.

LENHART, S.;WOKMAN, J. T. Optimal control Applied to Biological Models. [London]: Chapman & Hall, 2007.

MAGALHÃES, S. R. S.; VEIGA, R. D.; SÁFADI, T.; CIRILLO, M. A.; MAGINI, M. Avaliação de métodos para comparação de modelos de regressão por simulação de dados. Semina: Ciências Exatas e Tecnológicas, Londrina, v. 25, n. 2, p. 117-122, 2004.

MALIK, T., IMRAN, M., JAYRAMAN, R., Optimal control with multiple human papillomavirus vaccines. Journal of Theoretical Biology, London, v. 393, p. 179-193, 2016.

MATRAJT, L. ; HALLORAN, M. E.; LONGINI JUNIOR, I. M. Optimal vaccine allocation for the early mitigation of pandemic influenza. PLOS Computational Biology, San Francisco, v. 9, n. 3, 2013.

NANNYONGA, B.; MWANGA, G. G.; LUBOOBI, L. S. An optimal control problem for ovine brucellosis with culling. Journal Of Biological Dynamics, Abingdon, v. 9, p. 198-214, 2015.

OLIVEIRA, W. K.; HAGE, E. C.; PENNA, G. O.; KUCHENBECKER, R. S.; SANTOS, H. B.; ARAÚJO, W. N. Pandemic H1N1 influenza in Brazil: analysis of the first 34,506 notified cases of influenza-like illness with severe acute respiratory infection (SARI). Euro Surveill, Saint-Maurice, v. 14, n. 42, p. 1-6, 2009.

PASTORE, D. H.; THOMÉ, R. C. A.; DIAS, C. M.; ARRUDA, E. F.; YANG, Y. M. A model for interactions be- tween immune cells and HIV considering drug treatments. Computational and Applied Mathematics, Petrópolis, v. 37, p. 282-295, 2018.

TAI, C.; KESHWANI, D. R.; VOLTAN, D. S.; KUHAR, P. S.; ENGE, A. J. Optimal Control Strategy for Fed-Batch Enzymatic Hydrolysis of Lignocellulosic Biomass Based on Epidemic Modeling. Biotechnology and Bioengineering, New York, v. 112, p. 1376-1382, 2015.

TCHUENCHE, J. M.; KHANIS, S. A.; AUGUSTO, F. B.; MPESHE, S. C. Optimal Control and Sensitivity Analysis of an Influenza Model with Treatment and vaccination. Acta Biotheoretica, Leiden, v. 59, n. 1, 2011.

ZHOU, X.; GUO, Z. Analysis of an influenza A (H1N1) epidemic model with vaccination. Arabian Journal of Mathematics, Berlin, v. 1, p. 267-282, 2012.




DOI: http://dx.doi.org/10.5433/1679-0375.2020v41n1p105

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Semin., Ciênc. Exatas Tecnol.
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E-ISSN: 16790375
DOI: 10.5433/1679-0375
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