Abstract
This article deals with optimal control applied to vaccination and treatment strategies for an SIRS epidemic model with logistic growth and delay. The delay is incorporated into the model in order to modeled the latent period or incubation period. The existence for the optimal control pair is also proved. Pontryagin’s maximum principle with delay is used to characterize these optimal controls. The optimality system is derived and then solved numerically using an algorithm based on the forward and backward difference approximation.
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References
Abta A, Kaddar A, Talibi AH (2011) A comparison of delayed SIR and SEIR epidemic models. Nlinear Anal Model Control 16(2):181–190
Abta A, Kaddar A, Talibi AH (2012) Global stability for delay SIR and SEIR epidemic models with saturated incidence rates. Electron J Differ Equ 23:1–13
Capasso V, Serio G (1978) A generalization of the Kermack–Mckendrick deterministic epidemic model. Math Biosci 42:41–61
Driessche P, Watmough J (2000) A simple SIS epidemic model with a backward bifurcation. J Math Biol 40:525–540
Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, New York
Göllmann L, Kern D, Maurer H (2009) Optimal control problems with delays in state and control variables subject to mixed control-state constraints. Optim Control Appl Methods 30(4):341–365
Hattaf K, Lashari AA, Louartassi Y, Yousfi N (2013) A delayed SIR epidemic model with general incidence rate. Electron J Qual Theory Differ Equ 3:1–9
Hattaf K, Yousfi N (2012) Optimal control of a delayed HIV infection model with immune response using an efficient numerical method. ISRN Biomath. doi:10.5402/2012/215124
Kar TK, Batabyal A (2011) Stability analysis and optimal control of an SIR epidemic model with vaccination. BioSystems 104:127–135
Kermack M, Mckendrick A (1927) Contributions to the mathematical theory of epidemic model. Proc Roy Soc A 115:700–721
Laarabi H, Labriji E, Rachik M, Kaddar A (2012) Optimal control of an epidemic model with a saturated incidence rate. Nonlinear Anal Model Control 17(4):448–459
Lashari AA, Hattaf K, Zaman G, Li XZ (2013) Backward bifurcation and optimal control of a vector borne disease. Appl Math Inf Sci 7(1):301–309
Lashari AA, Zaman G (2012) Optimal control of a vector borne disease with horizontal transmission. Nonlinear Anal Real World Appl 13:203–212
Lee KS, Lashari AA (2014) Stability analysis and optimal control of pine wilt disease with horizontal transmission in vector population. Appl Math Comput 226:793–804
Lenhart S, Workman JT (2007) Optimal control applied to biological models. Mathematical and computational biology series. Chapman and Hall/CRC, London
Lukes DL (1982) Differential equations: classical to controlled. Math Sci Eng 162. Academic Press, New York
Mena-Lorca J, Hethcote HW (1992) Dynamic models of infectious disease as regulations of population sizes. J Math Biol 30:693–716
Zaman G, Kang YH, Jung IH (2009) Optimal treatment of an SIR epidemic model with time delay. BioSystems 98(1):43–50
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The authors thank the editor and the anonymous referees for very helpful suggestions and comments that helped us to improve the paper.
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Laarabi, H., Abta, A. & Hattaf, K. Optimal Control of a Delayed SIRS Epidemic Model with Vaccination and Treatment. Acta Biotheor 63, 87–97 (2015). https://doi.org/10.1007/s10441-015-9244-1
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DOI: https://doi.org/10.1007/s10441-015-9244-1