Original Article

Single and Multiobjective optimal control of epidemic models involving vaccination and treatment

Abstract

Introduction: A rigorous multiobjective optimal control strategy (that does not require the use of weighting functions) of the epidemic models that consider vaccination and treatment strategies is presented. Modifications of the standard susceptible-infectious-removed, susceptible-exposed-infectious-removed, and the modified susceptible-infectious-removed models are dynamically optimized to minimize the number of infected individuals while, controlling the rate at which the individuals are vaccinated and treated.

Method: The optimization program, Pyomo , where the differential equations are automatically converted to a Nonlinear Program using the orthogonal collocation method is used for performing the dynamic optimization calculations. The Lagrange-Radau quadrature with three collocation points and 10 finite elements are chosen. The resulting nonlinear optimization problem was solved using the solver BARON 19.3, accessed through the Pyomo-GAMS27.2 interface. 

Results: The computational results how that the multiobjective optimal control profiles generated by this strategy are very similar to those produced when weighting functions are used. 

Conclusion: The main conclusion of this work is to demonstrate that one can perform a rigorous dynamic optimization of epidemic models without the use of weighting functions that have the potential to produce some uncertainty and doubt in the results, in addition to dealing with unnecessary additional variables.

1. Ross, R. Report on the Prevention of Malaria in Mauritius. London: Waterlow and Sons (1908).
2. Macdonald, G. The Epidemiology and Control of Malaria. London: Oxford University Press (1957)
3. Garrett-Jones C 1964 Prognosis for interruption of malaria transmission through assessment of the mosquito's vectorial capacity. Nature 204, 1173–1175.
4. Garrett-Jones C, G.R. Shidrawi; 1969 Malaria vectorial capacity of a population of Anopheles gambiae: an exercise in epidemiological entomology. Bull. World Health Organ. 40,531.
5. Bailey N.T. J., The Mathematical Approach to Biology and Medicine. Wiley: New York, 1967.
6. Bailey N.T.J., The Mathematical theory of Infectious Diseases and its Applications (2nd edn). Charles Griffin & Co: London, 1975.
7. MacMahon B, Pugh TF. Epidemiology: Principles and Methods. Little Brown: Boston, 1970.
8. Capasso V., Mathematical Structures of Epidemic Systems. Springer: Berlin, 1993.
9. Frauenthal J.C. Mathematical Modeling in Epidemiology. Springer: Berlin, 1981.
10. Anderson R.M., Infectious Diseases of Humans: Dynamics and Control. University Press: Oxford, 1992.
11. Muhsam H.V., Models for infectious diseases. In Data Handling in Epidemiology, Holland WW(ed.). University Press, :Oxford, 1970.
12. Diekmann O, Heesterbeek JAP. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley: New York, 2000.
13. Felippe de Souza J.A.M., Yoneyama T. Control of Dengue using genetic algorithms for optimizing the investment in educational campaigns. Proceedings of the ECC 93, Groningen, The Netherlands, June 1993; 1994}1998.
14. Felippe de Souza J.A.M., Yoneyama T. Optimization of investment policies in the control of mosquito borne diseases. Proceedings of the ACC, Chicago, Illinois, USA, June, 1992.
15. Felippe de Souza, M. A. L. Caetano and Takashi Yonoyama, Optimal control theory applied to the anti-viral treatment of AIDS, Decision and Control, 2000, Proceedings of the 39th IEEE Conference V. 5, IEEE, (2000), 4839–4844.
16. Esteva L., C. Vargas, Coexistence of different serotypes of dengue virus, J. Math. Biol. 46 (2003) 31–47.
17. Esteva, L., C. Vargas, A model for dengue disease with variable human population, J. Math. Biol. 38 (1999) 220–240.
18. Esteva, L., C. Vargas, Analysis of a dengue disease transmission model, Math. Biosci. 150 (2) (1998) 131–151.
19. Esteva, L., C. Vargas, Influence of vertical and mechanical transmission on the dynamics of dengue disease, Math. Biosci. 167 (1)(2000) 51–64.
20. Andraud, M., Hens, N., Marais, C. & Beutels, P. Dynamic epidemiological models for dengue transmission: a systematic review of structural approaches. PloS one 7, e49085 (2012).
21. Wonham, M. J., de-Camino-Beck, T. & Lewis, M. A. An epidemiological model for West Nile virus: invasion analysis and control applications. Proceedings of the Royal Society of London B: Biological Sciences 271, 501-507 (2004).
22. H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, Journal of Mathematical Biology, 29 (1991), 271–287.
23. H. W. Hethcote, A thousand and one epidemic models, “Frontiers in Theoretical Biology” (ed. S. A. Levin), Springer-Verlag, (1994) 504–515.
24. H. W. Hethcote, H. W. Stech and P. van den Driessche, Periodicity and stability in epidemic models: A survey, “Differential Equations and Applications in Ecology, Epidemics, and Population Problems,” Academic Press (1981), 65–82.
25. H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599–653.
26. Hem Raj Joshi, Optimal control of an HIV immunology model, Optimal Control Applications & Methods, 23 (2002), 199–213.
27. E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis,Discrete and Continuous Dynamical Systems. Series B, 2 (2002), 473–482.
28. Fred Brauer, Some simple epidemic models, Math. BioSci. Eng., 3 (2006), 1–15.
29. Gaff H., and E. Schaefer, Optimal Control applied to Vaccination and treatment strategies for various epidemiological models Mathematical Biosciences and engineering Volume 6, Number 3, July 2009
30. Flores-Tlacuahuac, A. Pilar Morales and Martin Riveral Toledo; Multiobjective Nonlinear model predictive control of a class of chemical reactors. I & EC research; 5891-5899, 2012.
31. Sridhar, L. N. Multiobjective optimization and nonlinear model predictive control of the continuous fermentation process involving Saccharomyces Cerevisiae, Biofuels; https://doi.org/10.1080/17597269.2019.1674000 ISSN: 1759-7269 (Print) 1759-7277. 2019
32. Hart, William E., Carl D. Laird, Jean-Paul Watson, David L. Woodruff, Gabriel A. Hackebeil, Bethany L. Nicholson, and John D. Siirola. Pyomo – Optimization Modeling in Python. Second Edition. Vol. 67. Springer, 2017.
33. Biegler, L. T. An overview of simultaneous strategies for dynamic optimization. Chem. Eng. Process. Process Intensif. 46, 1043–105 (2007).
34. Tawarmalani, M. and N. V. Sahinidis, A polyhedral branch-and-cut approach to global optimization, Mathematical Programming, 103(2), 225-249, 2005
35. Bussieck M.R., Meeraus A. (2004) General Algebraic Modeling System (GAMS). In: Kallrath J. (eds) Modeling Languages in Mathematical Optimization. Applied Optimization, vol 88. Springer, Boston, MA
36. Miettinen, Kaisa, M., Nonlinear Multiobjective Optimization; Kluwers international series, 1999
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IssueVol 7 No 1 (2021) QRcode
SectionOriginal Article(s)
Published2021-05-15
DOI https://doi.org/10.18502/jbe.v7i1.6292
Keywords
Epidemic Optimal contro Global optimization Model predictive control

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1.
Sridhar L. Single and Multiobjective optimal control of epidemic models involving vaccination and treatment. jbe. 7(1):25-35.