Multiobjective approach to optimal control for a dengue transmission model
Abstract
During the last decades, the global prevalence of dengue progressed dramatically. It is a disease which is now endemic in more than one hundred countries of Africa, America, Asia and the Western Pacific. This study addresses a mathematical model for the dengue disease transmission and finding the most effective ways of controlling the disease. The model is described by a system of ordinary differential equations representing human and vector dynamics. Multiobjective optimization is applied to find the optimal control strategies, considering the simultaneous minimization of infected humans and costs due to insecticide application. The obtained results show that multiobjective optimization is an effective tool for finding the optimal control. The set of trade-off solutions encompasses a whole range of optimal scenarios, providing valuable information about the dynamics of infection transmissions. The results are discussed for different values of model parameters.During the last decades, the global prevalence of dengue progressed dramatically.It is a disease which is now endemic in more than one hundred countries of Africa,America, Asia and the Western Pacific. This study addresses a mathematical modelfor the dengue disease transmission and finding the most effective waysof controlling the disease. The model is described by a system of ordinarydifferential equations representing human and vector dynamics. Multiobjectiveoptimization is applied to find the optimal control strategies, consideringthe simultaneous minimization of infected humans and costs due to insecticide application.The obtained results show that multiobjective optimization is an effective toolfor finding the optimal control. The set of trade-off solutions encompassesa whole range of optimal scenarios, providing valuable information about thedynamics of infection transmissions. The results are discussed for differentvalues of model parameters.References
V. J. Bowman Jr., On the relationship of the Chebyshev norm and the efficient frontier of multiple criteria objectives, Lecture Notes in Economics and Mathematical Systems, 130 (1976), 76–86.
I. Das, On characterizing the knee of the Pareto curve based on normal-boundary intersection, Structural Optimization, 18 (1999), 107–115.
M. Derouich, A. Boutayeb and E. Twizell, A model of dengue fever, Biomedical Engineering Online, 2 (2003), 1–10.
G. J. Devine, E. Z. Perea, G. F. Killen, J. D. Stancil, S. J. Clark and A. C. Morrison, Using adult mosquitoes to transfer insecticides to aedes aegypti larval habitats, in Proceeding of the National Academy of Sciences of the United States of America, 2009.
Y. Y. Haimes, L. S. Lasdon and D. A. Wismer, On a bicriterion formulation of the problems of integrated system identification and system optimization, IEEE Transactions on Systems, Man and Cybernetics, 1 (1971), 296–297.
V. N. Kolokoltsov and V. P. Maslov, The Bellman differential equation and the Pontryagin maximum principle for multicriterial optimization problems. Dokl. Akad. Nauk 324 (1992), no. 1, 29–34; translation in Russian Acad. Sci. Dokl. Math. 45 (1992), no. 3, 522–527 (1993).
V. N. Kolokoltsov and V. P. Maslov, Idempotent analysis and its applications, translation of Idempotent analysis and its application in optimal control (Russian), “Nauka” Moscow, 1994, translated by V. E. Nazaikinskii, Mathematics and its Applications, 401, Kluwer Acad. Publ., Dordrecht, 1997.
V. P. Maslov, New differential equation for the dynamics of the Pareto sets. In: Idempotency, J. Gunawardena (Editor), Publications of the Newton Institute, 11, Cambridge Univ. Press, Cambridge, 1998, 322–330.
A. Messac, A. Ismail-Yahaya and C. A. Mattson, The normalized normal constraint method for generating the Pareto frontier, Struct. Multidiscip. Optim. 25 (2003), 86–98.
A. Messac and C. Mattson, Normal constraint method with guarantee of even representation of complete Pareto frontier, AIAA Journal, 42 (2004), 2101–2111.
K. Miettinen, Nonlinear multiobjective optimization, vol. 12 of International Series in Operations Research and Management Science, Kluwer Academic Publishers, Boston, MA, 1999.
Ministério da Saúde de Cabo Verde, Dengue, http://www.minsaude.gov.cv, 2012.
M. Otero, N. Schweigmann and H. G. Solari, A stochastic spatial dynamical model for aedes aegypti, Bulletin of Mathematical Biology, 70 (2008), 1297–1325.
A. Pascoletti and P. Serafini, Scalarizing vector optimization problems, Journal of Optimization Theory and Applications, 42 (1984), 499–524.
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The mathematical theory of optimal processes, International series of monographs in pure and applied mathematics, Pergamon Press, New York, USA, 1964.
H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Bioeconomic perspectives to an optimal control dengue model, International Journal of Computer Mathematics, 90 (2013), 2126–2136.
H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Dengue in Cape Verde: vector control and vaccination, Mathematical Population Studies, 20 (2013), 208–223.
H. S. Rodrigues, M. T. T. Monteiro, D. F. M. Torres and A. Zinober, Dengue disease, basic reproduction number and control, International Journal of Computer Mathematics, 89 (2012), 334-346.
R. Verma, Multiobjective fractional programming problems and second order generalized hybrid invexity frameworks, Statistics, Optimization & Information Computing 2 (2014), 280–304.
Q. Zhang, A. Zhou and Y. Jin, RM-MEDA: A regularity model-based multiobjective estimation of distribution algorithm, IEEE Transactions on Evolutionary Computation, 12 (2008), 41–63.
E. Zitzler and L. Thiele, Multiobjective optimization using evolutionary algorithms – A comparative case study, in Proceedings of the Conference on Parallel Problem Solving From Nature, Lecture Notes in Computer Science, 1498 (1998), 292–301.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
