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A shape optimization problem concerning the regional control of a class of spatially structured epidemics: sufficiency conditions. (English) Zbl 1446.37081

Aguiar, Maira (ed.) et al., Current trends in dynamical systems in biology and natural sciences. Selected contributions presented at the ninth international workshop of dynamical systems applied to biology and natural sciences, DSABNS, Turin, Italy, February 7–9, 2018. Cham: Springer. SEMA SIMAI Springer Ser. 21, 165-183 (2020).
Summary: In a series of papers by the first two authors a stabilization problem had been considered for an epidemic model, described by a reaction-diffusion system, including a feedback operator. This paper deals with a related optimal control problem based on functional (shape functional) and sanitation controls; a dual dynamic programming method is constructed for deriving sufficient conditions for an optimal solution as well as \(\epsilon \)-optimality conditions in terms of dual dynamic inequalities. Approximate optimality and a related conceptual algorithm are presented as well.
For the entire collection see [Zbl 1445.92001].

MSC:

37N25 Dynamical systems in biology
92B05 General biology and biomathematics
92D30 Epidemiology
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[1] Aniţa, S., Capasso, V.: A stabilizability problem for a reaction diffusion system modelling a class of spatially structured epidemic systems. Nonlinear Anal. Real World Appl. 3, 453-464 (2002) · Zbl 1020.92027 · doi:10.1016/S1468-1218(01)00025-6
[2] Aniţa, S., Capasso, V.: A stabilization strategy for a reaction-diffusion system modelling a class of spatially structured epidemic systems (think globally, act locally). Nonlinear Anal. Real World Appl. 10, 2026-2035 (2009) · Zbl 1163.91510 · doi:10.1016/j.nonrwa.2008.03.009
[3] Aniţa, S., Capasso, V.: Stabilization of reaction-diffusion system modeling a class of spatially structured epidemic systems via feedback control. Nonlinear Anal. Real World Appl. 13, 725-735 (2012) · Zbl 1238.93081 · doi:10.1016/j.nonrwa.2011.08.012
[4] Arnăutu, V., Barbu, V., Capasso, V.: Controlling the spread of a class of epidemics. Appl. Math. Optim. 20, 297-317 (1989) · Zbl 0691.49024 · doi:10.1007/BF01447658
[5] Babak, P.: Nonlocal initial problem for coupled reaction-diffusion systems and their applications. Nonlinear Anal. Real World Appl. 8, 980-986 (2007) · Zbl 1138.35346 · doi:10.1016/j.nonrwa.2006.05.001
[6] Babak, P.: Nonlocal problem involving spatial structure for coupled reaction-diffusion systems. Appl. Math. Comput. 185, 449-463 (2007) · Zbl 1114.35098
[7] Capasso, V.: Global solution for a diffusive nonlinear deterministic epidemic model. SIAM J. Appl. Math. 35, 274-284 (1978) · Zbl 0415.92018 · doi:10.1137/0135022
[8] Capasso, V.: Asymptotic stability for an integro-differential reaction-diffusion system. J. Math. Anal. Appl. 103, 575-588 (1984) · Zbl 0595.45020 · doi:10.1016/0022-247X(84)90147-1
[9] Capasso, V.: Mathematical Structures of Epidemic Systems (2nd revised printing). Lecture Notes in Biomathematics, vol. 97. Springer, Heidelberg (2009)
[10] Capasso, V., Kunisch, K.: A reaction-diffusion system arising in modelling man-environment diseases. Quart. Appl. Math. 46, 431-450 (1988) · Zbl 0704.35069 · doi:10.1090/qam/963580
[11] Capasso, V., Maddalena, L.: Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases. J. Math. Biol. 13, 173-184 (1981) · Zbl 0468.92016 · doi:10.1007/BF00275212
[12] Capasso, V., Paveri-Fontana, S.L.: A mathematical model for the 1973 cholera epidemic in the European Mediterranean region. Rev. Epidemiol. Sante Publique 27, 121-132 (1979). Errata Corrige, 28 (1980)
[13] Codeço, C.T.: Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir. BMC Infect. Dis. 1, 1-14 (2001) · doi:10.1186/1471-2334-1-1
[14] Delfour, M.C., Zolesio, J.P.: Shapes and Geometries: Analysis, Differential Calculus and Optimization, Advances in Design and Control. SIAM, Philadelphia (2001) · Zbl 1002.49029
[15] Galewska, E., Nowakowski, A.: A dual dynamic programming for multidimensional elliptic optimal control problems. Numer. Funct. Anal. Optim. 27, 279-289 (2006) · Zbl 1130.49020 · doi:10.1080/01630560600698160
[16] Hebrard, P., Henrot, A.: Optimal shape and position of the actuators for the stabilization of a string. Syst. Control Lett. 48, 199-209 (2003) · Zbl 1134.93399 · doi:10.1016/S0167-6911(02)00265-7
[17] Hebrard, P., Henrot, A.: Spillover phenomenon in the optimal locations of actuators. SIAM J. Control Optim. 44, 349-366 (2005). · Zbl 1083.49002 · doi:10.1137/S0363012903436247
[18] Henrot, A., Pierre, M.: Variation et optimisation de formes: une analyse geométrique (French). Mathématiques et Applications, vol. 48. Springer, Berlin (2005) · Zbl 1098.49001
[19] Münch, A.: Optimal design of the support of the control for the 2-D wave equation: a numerical method. Int. J. Numer. Anal. Model. 5, 331-351 (2008) · Zbl 1242.49091
[20] Münch, A.: Optimal location of the support of the control for the 1-D wave equation: numerical investigations. Comput. Optim. Appl. 42, 443-470 (2009) · Zbl 1208.49052 · doi:10.1007/s10589-007-9133-x
[21] Näsell, I., Hirsch, W.M.: The transmission dynamics of schistosomiasis. Comm. Pure Appl. Math. 26, 395-453 (1973) · Zbl 0253.92004 · doi:10.1002/cpa.3160260402
[22] Nowakowska, I., Nowakowski, A.: A dual dynamic programming for minimax optimal control problems governed by parabolic equation. Optimization 60, 347-363 (2011) · Zbl 1231.49007 · doi:10.1080/02331930903104390
[23] Nowakowski, A.: The dual dynamic programming. Proc. Amer. Math. Soc. 116, 1089-1096 (1992) · Zbl 0769.49024 · doi:10.1090/S0002-9939-1992-1102860-3
[24] Nowakowski, A.: Sufficient optimality conditions for Dirichlet boundary control of wave equations. SIAM J. Control Optim. 47, 92-110 (2008) · Zbl 1162.49031 · doi:10.1137/050644008
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