Abstract
In this paper the optimal control strategies with two control variable of an SIR (susceptible-infected-recovered) epidemic model are introduced. The effect of dispersion of the population in a bounded habitat has been taken into consideration. The aim of this work is to minimize the infective and susceptible individuals and to maximize the total number of recovered individuals by using the possible control variables. To solve optimal control problem we use direct and indirect methods, Bernstein-Bezier parametrisation of control variable and invasive weed optimization of objective function, and adaptive critic design with echo state networks, respectively. Our results indicate that these two methods are able to solve optimal control problems.
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References
Mehrabian, A.R.: A novel numerical optimization algorithm inspired from weed colonization. Ecol. Inform. 1(4), 355–366 (2006)
Brauer, F., Castillo-Chavez, C.: Mathematical Models in Population Biology and Epidemiology. Springer, New York (2001). https://doi.org/10.1007/978-1-4614-1686-9
Farin, G.: Curves and Surfaces for Computer Aided Geometric Design — A Practical Guide. Academic Press Professional, San Diego (1993)
Ghosh, A., Das, S., Chowdhury, A., Giri, R.: An ecologically inspired direct search method for solving optimal control problems with Bezier parameterization. Eng. Appl. Artif. Intell. 24, 1195–1203 (2011)
Giri, R., Chowdhury, A., Ghosh, A., Das, S., Abraham, A., Snasel, V.: A modified invasive weed optimization algorithm for training of feed-forward neural networks. In: IEEE International Conference on Systems Man and Cybernetics, pp. 3166–3173. IEEE (2010)
Gollman, L., Kern, D., Mauer, H.: Optimal control problem with delays in state and control variables subject to mixed control-state constraints. Optim. Control Appl. Meth. 30, 341–365 (2006)
Jaeger, H.: The “Echo State” approach to analysing and training recurrent neural networks. Technical report GMD 148, German National Research Institute for Computer Science, Bonn (2001)
Kar, T.K., Batabyal, A.: Stability analysis and optimal control of an SIR epidemic model with vaccination. BioSyst. Sci. 104, 127–135 (2011)
Kirk, D.E.: Optimal Control Theory: An Introduction. Dover Publications, New York (1989)
Kmet, T., Kmetova, M.: Neural networks simulation of distributed control problems with state and control constraints. In: Villa, A.E.P., Masulli, P., Pons Rivero, A.J. (eds.) ICANN 2016. LNCS, vol. 9886, pp. 468–477. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44778-0_55
Kmet, T., Kmetova, M.: Echo state networks simulation of sir distributed control. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2017. LNCS (LNAI), vol. 10245, pp. 86–96. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59063-9_8
Mittelmann, H.D.: Solving elliptic control problems with interior point and SQP methods: control and state constraints. J. Comput. Appl. Math. 120, 175–195 (2000)
Padhi, R., Unnikrishnan, N., Wang, X., Balakrishnan, S.N.: Adaptive-critic based optimal control synthesis for distributed parameter systems. Automatica 37, 1223–1234 (2001)
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R., Mischenko, E.F.: Freshwater Ecosystems. Modelling and Simulation, Developments in Environmental Modelling. Nauka (in Russian), Moscow (1983)
Werbos, P.J.: Approximate dynamic programming for real-time control and neural modelling. In: White, D.A., Sofge, D.A. (eds.) Handbook of intelligent control: Neural Fuzzy, and Adaptive Approaches, pp. 493–525. Van Nostrand, New York (1992)
Yoshida, N., Hara, T.: Global stability of a delayed SIR epidemic model with density dependent birth and death rates. J. Comput. Appl. Math. 201, 339–347 (2007)
Zaman, G., Kang, Y.H., Jung, I.H.: Optimal treatment of an SIR epidemic model with time delay. BioSystems 98, 43–50 (2009)
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Kmet, T., Kmetova, M. (2017). Bezier Curve Parameterization Methods for Solving Optimal Control Problems of SIR Model. In: Martín-Vide, C., Neruda, R., Vega-Rodríguez, M. (eds) Theory and Practice of Natural Computing. TPNC 2017. Lecture Notes in Computer Science(), vol 10687. Springer, Cham. https://doi.org/10.1007/978-3-319-71069-3_8
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