# American Institute of Mathematical Sciences

January  2017, 14(1): 263-275. doi: 10.3934/mbe.2017017

## Sufficient optimality conditions for a class of epidemic problems with control on the boundary

 Faculty of Math and Computer Sciences, University of Lodz, Banacha 22, 90-238 Lodz, Poland

Received  October 22, 2015 Accepted  April 22, 2016 Published  October 2016

In earlier paper of V. Capasso et al it is considered a simply model of controlling an epidemic, which is described by three functionals and systems of two PDE equations having the feedback operator on the boundary. Necessary optimality conditions and two gradient-type algorithms are derived. This paper constructs dual dynamic programming method to derive sufficient optimality conditions for optimal solution as well $\varepsilon$-optimality conditions in terms of dual dynamic inequalities. Approximate optimality and numerical calculations are presented too.

Citation: Miniak-Górecka Alicja, Nowakowski Andrzej. Sufficient optimality conditions for a class of epidemic problems with control on the boundary. Mathematical Biosciences & Engineering, 2017, 14 (1) : 263-275. doi: 10.3934/mbe.2017017
##### References:
 [1] V. Arnautu, V. Barbu and V. Capasso, Controlling the spread of a class of epidemics, Appl. Math. Optim., 20 (1989), 297-317.  doi: 10.1007/BF01447658.  Google Scholar [2] V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces Science+Business Media, Springer 2012. doi: 10.1007/978-94-007-2247-7.  Google Scholar [3] V. Capasso, Mathematical Structures of Epidemic Systems Lect. Notes in Biomath., 97 Springer 2008.  Google Scholar [4] V. Capasso and K. Kunisch, A reaction-diffusion system arising in modelling man-environment diseases, Quart. Appl. Math., 46 (1988), 431-450.   Google Scholar [5] E. Galewska and A. Nowakowski, A dual dynamic programming for multidimensional elliptic optimal control problems, Numer. Funct. Anal. Optim., 27 (2006), 279-289.  doi: 10.1080/01630560600698160.  Google Scholar [6] W. Hao and A. Friedman, The LDL-HDL profile determines the risk of atherosclerosis: A mathematical model PLoS ONE 9 (2014), e90497. doi: 10.1371/journal.pone.0090497.  Google Scholar [7] A. Miniak-Górecka, Construction of Computational Method for $\varepsilon$-Optimal Solutions Shape Optimization Problems PhD thesis, 2015. Google Scholar [8] A. Nowakowski, The dual dynamic programming, Proc. Amer. Math. Soc., 116 (1992), 1089-1096.  doi: 10.1090/S0002-9939-1992-1102860-3.  Google Scholar [9] A. Nowakowski, Sufficient optimality conditions for Dirichlet boundary control of wave equations, SIAM J. Control Optim., 47 (2008), 92-110.  doi: 10.1137/050644008.  Google Scholar [10] I. Nowakowska and A. Nowakowski, A dual dynamic programming for minimax optimal control problems governed by parabolic equation, Optimization, 60 (2011), 347-363.  doi: 10.1080/02331930903104390.  Google Scholar [11] A. Nowakowski and J. Sokołowski, On dual dynamic programming in shape control, Commun. Pure Appl. Anal., 11 (2012), 2473-2485.  doi: 10.3934/cpaa.2012.11.2473.  Google Scholar

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##### References:
 [1] V. Arnautu, V. Barbu and V. Capasso, Controlling the spread of a class of epidemics, Appl. Math. Optim., 20 (1989), 297-317.  doi: 10.1007/BF01447658.  Google Scholar [2] V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces Science+Business Media, Springer 2012. doi: 10.1007/978-94-007-2247-7.  Google Scholar [3] V. Capasso, Mathematical Structures of Epidemic Systems Lect. Notes in Biomath., 97 Springer 2008.  Google Scholar [4] V. Capasso and K. Kunisch, A reaction-diffusion system arising in modelling man-environment diseases, Quart. Appl. Math., 46 (1988), 431-450.   Google Scholar [5] E. Galewska and A. Nowakowski, A dual dynamic programming for multidimensional elliptic optimal control problems, Numer. Funct. Anal. Optim., 27 (2006), 279-289.  doi: 10.1080/01630560600698160.  Google Scholar [6] W. Hao and A. Friedman, The LDL-HDL profile determines the risk of atherosclerosis: A mathematical model PLoS ONE 9 (2014), e90497. doi: 10.1371/journal.pone.0090497.  Google Scholar [7] A. Miniak-Górecka, Construction of Computational Method for $\varepsilon$-Optimal Solutions Shape Optimization Problems PhD thesis, 2015. Google Scholar [8] A. Nowakowski, The dual dynamic programming, Proc. Amer. Math. Soc., 116 (1992), 1089-1096.  doi: 10.1090/S0002-9939-1992-1102860-3.  Google Scholar [9] A. Nowakowski, Sufficient optimality conditions for Dirichlet boundary control of wave equations, SIAM J. Control Optim., 47 (2008), 92-110.  doi: 10.1137/050644008.  Google Scholar [10] I. Nowakowska and A. Nowakowski, A dual dynamic programming for minimax optimal control problems governed by parabolic equation, Optimization, 60 (2011), 347-363.  doi: 10.1080/02331930903104390.  Google Scholar [11] A. Nowakowski and J. Sokołowski, On dual dynamic programming in shape control, Commun. Pure Appl. Anal., 11 (2012), 2473-2485.  doi: 10.3934/cpaa.2012.11.2473.  Google Scholar
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