Optimal Vaccination Campaigns Using Stochastic SIR Model and Multiobjective Impulsive Control
DOI:
https://doi.org/10.5540/tcam.2021.022.02.00179Keywords:
Planning of vaccination campaigns, Multiobjective optimization, Impulsive control, Stochastic SIRAbstract
A multiobjective impulsive control scheme is proposed to give answers on how optimal vaccination campaigns should be implemented, regarding two conflicting targets: making the total number of infecteds small and the vaccination campaign as handy as possible.
In this paper, a stochastic SIR model is used to better depict the characteristics of a disease in practical terms, where little influences may lead to sudden and unpredictable changes in the behavior of transmissions. This model is extended to analyze the effects of impulsive vaccinations in two phases: the transient regime control, taking into account the necessity to reduce the number of infected individuals to an acceptable level in a finite time, and the permanent regime control, that will act with fixed vaccinations to avoid another outbreak. A parallel version of NSGA-II is used as the multiobjective optimization machinery, considering both the probability of eradication and the vaccination campaign costs. The final result using the proposed framework nondominated policies that can guide for public managers to decide which is the best procedure to be taken depending on the present situation.
References
References
W. Kermack and A. McKendrick, “A contribution to the mathematical theory of epidemics,” Proceedings of the Royal Society of London Series A Mathemat- ical and Physical Sciences, vol. 115, pp. 700–721, 1927.
R. Cardoso and R. Takahashi, “Solving impulsive control problems by discrete- time dynamic optimization methods,” Tendências em Matemática Aplicada e Computacional, vol. 9, no. 1, pp. 21–30, 2008.
A. Da Cruz, R. Cardoso, and R. Takahashi, “Multi-objective design with a stochastic validation of vaccination campaigns,” in IFAC Workshop on Control Applications of Optimization, vol. 7, pp. 289–294, 2009.
A. da Cruz, R. Cardoso, and R. Takahashi, “Multiobjective synthesis of robust vaccination policies,” Applied Soft Computing, vol. 50, pp. 34–47, 2017.
T. Bäck, Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms. Oxford University Press, USA, 1996.
W.R.EspositoandC.A.Floudas,“Deterministicglobaloptimizationinnonlin- ear optimal control problems,” Journal of Global Optimization, vol. 17, no. 1-4, pp. 97–126, 2000.
T. Britton, “Stochastic epidemic models: a survey,” Mathematical biosciences, vol. 225, no. 1, pp. 24–35, 2010.
R. Kuske, L. Gordillo, and P. Greenwood, “Sustained oscillations via coherence resonance in sir,” Journal of theoretical biology, vol. 245, no. 3, pp. 459–469, 2007.
C. DENG and H. GAO, “Stability of svir system with random perturbations,” International Journal of Biomathematics, vol. 5, no. 04, 2012.
F. Ball and O. Lyne, “Optimal vaccination policies for stochastic epidemics among a population of households,” Mathematical Biosciences, vol. 177, pp. 333–354, 2002.
E. Tornatore, S. Buccellato, and P. Vetro, “On a stochastic disease model with vaccination,” Rendiconti del Circolo Matematico di Palermo, vol. 55, no. 2, pp. 223–240, 2006.
H. Hethcote, “The mathematics of infectious diseases,” Preprints of the IFAC Workshop on Control Applications of Optimisation, vol. 42, no. 4, pp. 599–653, 2000.
R. M. Anderson and R. M. May, “Infectious diseases of humans: dynamics and control,” 1992.
F. Brauer, P. Driessche, and J. Wu, Mathematical Epidemiology (Lecture Notes in Mathematics). Springer, 2008.
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential
Equations (Applications is Mathematics). Springer, 1992.
D. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM review, pp. 525–546, 2001.
R. Y. Rubinstein and D. P. Kroese, Simulation and the Monte Carlo method, vol. 707. Wiley-interscience, 2008.
S. Plotkin, W. Orenstein, and P. Offit, Vaccines, vol. 707. Elsevier Health Sciences, 5th ed., 2008.
A. da Cruz, R. Cardoso, and R. Takahashi, “Multiobjective dynamic opti- mization of vaccination campaigns using convex quadratic approximation local search,” in Evolutionary Multi-Criterion Optimization, pp. 404–417, Springer, 2011.
A. C. S. Dusse and R. T. N. Cardoso, “Using a stochastic SIR model to de- sign optimal vaccination campaigns via multiobjective optimization,” vol. 1, BIOMAT, 2019.
K. Adam and R. T. N. Cardoso, “Optimal multiobjective pulse vaccination campaigns in stochastic sir model,” vol. 1, SBMAC, 2019.
C. A. C. Coello, G. B. Lamont, and D. A. V. Veldhuizen, Evolutionary Algo- rithms for Solving Multi-Objective Problems. Springer, 2007.
D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning. Addison Wesley Longman, Inc., 1989.
K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiob- jective genetic algorithm: Nsga-ii,” Evolutionary Computation, IEEE Transactions on, vol. 6, no. 2, pp. 182–197, 2002.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.