American Institute of Mathematical Sciences

2015, 2015(special): 549-561. doi: 10.3934/proc.2015.0549

Optimal control for an epidemic in populations of varying size

 1 Department of Mathematics and Computer Sciences, Texas Woman's University, Denton, TX 76204 2 Department of Computer Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992 3 Centre de Recerca Matemática, Campus de Bellaterra, Edifici C, 08193 Barcelona

Received  September 2014 Revised  February 2015 Published  November 2015

For a Susceptible-Infected-Recovered (SIR) control model with varying population size, the optimal control problem of minimization of the infected individuals at a terminal time is stated and solved. Three distinctive control policies are considered, namely the vaccination of the susceptible individuals, treatment of the infected individuals and an indirect policy aimed at reduction of the transmission. Such values of the model parameters and control constraints are used, for which the optimal controls are bang-bang. We estimated the maximal possible number of switchings of these controls, which task is related to the estimation of the number of zeros of the corresponding switching functions. Different approaches of estimating the number of zeros of the switching functions are applied. The found estimates enable us to reduce the optimal control problem to a considerably simpler problem of the finite-dimensional constrained minimization.
Citation: Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Optimal control for an epidemic in populations of varying size. Conference Publications, 2015, 2015 (special) : 549-561. doi: 10.3934/proc.2015.0549
References:
 [1] M. Alkama, M. Elhia, Z. Rachik, M. Rachik and E. Labriji, Free terminal time optimal control problem of an SIR epidemic model with vaccination, International Journal of Science and Research, 3, N 5, (2014), 227-230. [2] R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, UK, 1992. [3] H. Behncke, Optimal control of deterministic epidemics, Optimal Control Applications and Methods, 21, N 6, (2000), 269-285. [4] F. Brauer, Some simple epidemic models, Mathematical Biosciences and Engineering, 3, N 1, (2006), 1-15. [5] V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, vol. 97, Springer, Heidelberg, 2008. [6] C. Castilho, Optimal control of an epidemic through educational campaigns, Electronic Journal of Differential Equations, 2006, N 125, (2006), 1-11. [7] D.J. Daley and J. Gani, Epidemic Modelling: An Introduction, Cambridge University Press, Cambridge, 1999. [8] O. Diekmann and J.A.P. Heesterbeek, Mathematical Epidemiology of Infectiuos Diseases. Model Building, Analysis and Interpretation, John Wiley & Sons, New York, 2000. [9] A.V. Dmitruk, A generalized estimate on the number of zeros for solutions of a class of linear differential equations, SIAM Journal on Control and Optimization, 30, N 5, (1992), 1087-1091. [10] M. Elhia, O. Balatif, J. Bouyaghroumni, E. Labriji and M. Rachik, Optimal control applied to the spread of influenza A (H1N1), Applied Mathematical Sciences, 6, N 82, (2012), 4057-4065. [11] H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Mathematical Biosciences and Engineering, 6, N 3, (2009), 469-492. [12] E. Grigorieva, N. Bondarenko, E. Khailov and A. Korobeinikov, Finite-Dimensional Methods for Optimal Control of Autothermal Thermophilic Aerobic Digestion, in Industrial Waste, (eds. K.-Y. Show and X. Guo), InTech, Croatia, (2012), 91-120. [13] E.V. Grigorieva, E.N. Khailov and A. Korobeinikov, Parametrization of the attainable set for a nonlinear control model of a biochemical process, Mathematical Biosciences and Engineering, 10, N 4, (2013), 1067-1094. [14] E. Grigorieva, E. Khailov and A. Korobeinikov, Optimal control for a susceptible-infected-recovered infectious disease model, Journal of Coupled Systems and Multiscale Dynamics, 1, N 3, (2013), 324-331. [15] E. Grigorieva, E. Khailov and A. Korobeinikov, An optimal control problem in HIV treatment, Discrete and Continuous Dynamical Systems, supplement volume, (2013), 311-322. [16] E.V. Grigorieva, E.N. Khailov, N.V. Bondarenko and A. Korobeinikov, Modeling and optimal control for antiretroviral therapy, Journal of Biological Systems, 22, N 2, (2014), 199-217. [17] E.V. Grigorieva and E.N. Khailov, Optimal vaccination, treatment, and priventive campaigns in regard to the SIR epidemic model, Mathematical Modelling and Natural Phenomena, 9, N 4, (2014), 105-121. [18] E. Gubar and E. Zhitkova, Decision making procedure in optimal control problem for the SIR model, Contributions to Game Theory and Management, 6, (2013), 189-199. [19] K. Hattaf and N. Yousfi, Mathematical model of the influenza A (H1N1) infection, Advanced Studies in Biology, 1, N 8, (2009), 383-390. [20] H.W. Hethcote, A Thousand and One Epidemic Models, in Frontiers in Theoretical Biology, (ed. S.A. Levin), Springer-Verlag, Berlin-Heidelberg-New York-London, (1994), 504-515. [21] H.W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42, N 4, (2000), 599-653. [22] M.J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, 2008. [23] U. Ledzewicz and H. Schättler, On optimal singular controls for a general SIR-model with vaccination and treatment, Discrete and Continuous Dynamical Systems, supplement volume, (2011), 981-990. [24] E.B. Lee and L. Marcus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, 1967. [25] M.S. Nikol'skii, On the time-optimality problem for three- and four-dimensional control systems, Proceedings of the Steklov Institute of Mathematics, 277, (2012), 184-190. [26] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, 1962. [27] G. Sansone, Equazioni Differenziali nel Campo Reale, Parte Prima, Nicola Zanichelli, Bologna, 1948. [28] H. Schättler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples, Springer, New York-Heidelberg-Dordrecht-London, 2012. [29] S.A. Vakhrameev, Bang-bang theorems and related questions, Proceedings of the Steklov Institute of Mathematics, 220, (1988), 45-108. [30] F.P. Vasil'ev, Optimization Methods, Factorial Press, Moscow, 2002. [31] T.T. Yusuf and F. Benyah, Optimal control of vaccination and treatment for an SIR epidemiological model, World Journal of Modelling and Simulation, 8, N 3, (2012), 194-204. [32] M.I. Zelikin and V.F. Borisov, Theory of Chattering Control. With Applications to Astronautics, Robotics, Economics, and Engineering, Birkhäuser, Boston, MA, 1994.

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References:
 [1] M. Alkama, M. Elhia, Z. Rachik, M. Rachik and E. Labriji, Free terminal time optimal control problem of an SIR epidemic model with vaccination, International Journal of Science and Research, 3, N 5, (2014), 227-230. [2] R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, UK, 1992. [3] H. Behncke, Optimal control of deterministic epidemics, Optimal Control Applications and Methods, 21, N 6, (2000), 269-285. [4] F. Brauer, Some simple epidemic models, Mathematical Biosciences and Engineering, 3, N 1, (2006), 1-15. [5] V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, vol. 97, Springer, Heidelberg, 2008. [6] C. Castilho, Optimal control of an epidemic through educational campaigns, Electronic Journal of Differential Equations, 2006, N 125, (2006), 1-11. [7] D.J. Daley and J. Gani, Epidemic Modelling: An Introduction, Cambridge University Press, Cambridge, 1999. [8] O. Diekmann and J.A.P. Heesterbeek, Mathematical Epidemiology of Infectiuos Diseases. Model Building, Analysis and Interpretation, John Wiley & Sons, New York, 2000. [9] A.V. Dmitruk, A generalized estimate on the number of zeros for solutions of a class of linear differential equations, SIAM Journal on Control and Optimization, 30, N 5, (1992), 1087-1091. [10] M. Elhia, O. Balatif, J. Bouyaghroumni, E. Labriji and M. Rachik, Optimal control applied to the spread of influenza A (H1N1), Applied Mathematical Sciences, 6, N 82, (2012), 4057-4065. [11] H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Mathematical Biosciences and Engineering, 6, N 3, (2009), 469-492. [12] E. Grigorieva, N. Bondarenko, E. Khailov and A. Korobeinikov, Finite-Dimensional Methods for Optimal Control of Autothermal Thermophilic Aerobic Digestion, in Industrial Waste, (eds. K.-Y. Show and X. Guo), InTech, Croatia, (2012), 91-120. [13] E.V. Grigorieva, E.N. Khailov and A. Korobeinikov, Parametrization of the attainable set for a nonlinear control model of a biochemical process, Mathematical Biosciences and Engineering, 10, N 4, (2013), 1067-1094. [14] E. Grigorieva, E. Khailov and A. Korobeinikov, Optimal control for a susceptible-infected-recovered infectious disease model, Journal of Coupled Systems and Multiscale Dynamics, 1, N 3, (2013), 324-331. [15] E. Grigorieva, E. Khailov and A. Korobeinikov, An optimal control problem in HIV treatment, Discrete and Continuous Dynamical Systems, supplement volume, (2013), 311-322. [16] E.V. Grigorieva, E.N. Khailov, N.V. Bondarenko and A. Korobeinikov, Modeling and optimal control for antiretroviral therapy, Journal of Biological Systems, 22, N 2, (2014), 199-217. [17] E.V. Grigorieva and E.N. Khailov, Optimal vaccination, treatment, and priventive campaigns in regard to the SIR epidemic model, Mathematical Modelling and Natural Phenomena, 9, N 4, (2014), 105-121. [18] E. Gubar and E. Zhitkova, Decision making procedure in optimal control problem for the SIR model, Contributions to Game Theory and Management, 6, (2013), 189-199. [19] K. Hattaf and N. Yousfi, Mathematical model of the influenza A (H1N1) infection, Advanced Studies in Biology, 1, N 8, (2009), 383-390. [20] H.W. Hethcote, A Thousand and One Epidemic Models, in Frontiers in Theoretical Biology, (ed. S.A. Levin), Springer-Verlag, Berlin-Heidelberg-New York-London, (1994), 504-515. [21] H.W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42, N 4, (2000), 599-653. [22] M.J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, 2008. [23] U. Ledzewicz and H. Schättler, On optimal singular controls for a general SIR-model with vaccination and treatment, Discrete and Continuous Dynamical Systems, supplement volume, (2011), 981-990. [24] E.B. Lee and L. Marcus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, 1967. [25] M.S. Nikol'skii, On the time-optimality problem for three- and four-dimensional control systems, Proceedings of the Steklov Institute of Mathematics, 277, (2012), 184-190. [26] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, 1962. [27] G. Sansone, Equazioni Differenziali nel Campo Reale, Parte Prima, Nicola Zanichelli, Bologna, 1948. [28] H. Schättler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples, Springer, New York-Heidelberg-Dordrecht-London, 2012. [29] S.A. Vakhrameev, Bang-bang theorems and related questions, Proceedings of the Steklov Institute of Mathematics, 220, (1988), 45-108. [30] F.P. Vasil'ev, Optimization Methods, Factorial Press, Moscow, 2002. [31] T.T. Yusuf and F. Benyah, Optimal control of vaccination and treatment for an SIR epidemiological model, World Journal of Modelling and Simulation, 8, N 3, (2012), 194-204. [32] M.I. Zelikin and V.F. Borisov, Theory of Chattering Control. With Applications to Astronautics, Robotics, Economics, and Engineering, Birkhäuser, Boston, MA, 1994.
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