We will investigate the optimal control strategy of an SIR epidemic model with time delay in state and control variables. We use a vaccination program to minimize the number of susceptible and infected individuals and to maximize the number of recovered individuals. Existence for the optimal control is established; Pontryagin’s maximum principle is used to characterize this optimal control, and the optimality system is solved by a discretization method based on the forward and backward difference approximations. The numerical simulation is carried out using data regarding the course of influenza A (H1N1) in Morocco. The obtained results confirm the performance of the optimization strategy. 1. Introduction For a long time, infectious diseases have caused several epidemics, leaving behind them not only millions of dead and infected individuals but also severe socioeconomic consequences. Nowadays, mathematical modeling of infectious diseases is one of the most important research areas. Indeed, mathematical epidemiology has contributed to a better understanding of the dynamical behavior of infectious diseases, its impacts, and possible future predictions about its spreading. Mathematical models are used in comparing, planning, implementing, evaluating, and optimizing various detection, prevention, therapy, and control programs. Many influential results related to the development and analysis of epidemiological models have been established and can be found in many articles and books (see, e.g., [1–3]). Epidemiological models often take the form of a system of nonlinear, ordinary, and differential equations without time delay. However, for various biological reasons, the real dynamic behavior of an epidemic depends not only on its current state but also on its past history. Thus, to reflect the real behavior of some diseases, many researchers have proposed and analyzed more realistic models including delays to model different mechanisms in the dynamics of epidemics like latent period, temporary immunity and length of infection (see, e.g., [4–8] and the references therein). To the best of our knowledge, including time delay in both state and control variables in the context of an epidemic controlled model has not been studied. There have been some works (like [9, 10]) which study an optimal control problem with time delay but only in the state variable. In this paper, we will investigate the effect of a vaccination program in the case of an SIR (susceptible-infected-recovered) epidemic model with time delay in the control and the state variables. To do this,
References
[1]
R. M. Anderson and R. M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford University Press, Oxford, UK, 1992.
[2]
O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, John Wiley & Sons, Chisteter, UK, 2000.
[3]
F. Brauer and C. C. Castillo, Mathematical Models in Population Biology and Epidemiology, Springer, New York, NY, USA, 2000.
[4]
P. Van Den Driessche, “Some epidemiological models with delays,” in Differential Equations and Applications to Biology and to Industry (Claremont, CA, 1994), pp. 507–520, World Scientific, River Edge, NJ, USA, 1996.
[5]
Q. J. A. Khan and E. V. Krishnan, “An epidemic model with a time delay in transmission,” Applications of Mathematics, vol. 48, no. 3, pp. 193–203, 2003.
[6]
N. Yoshida and T. Hara, “Global stability of a delayed SIR epidemic model with density dependent birth and death rates,” Journal of Computational and Applied Mathematics, vol. 201, no. 2, pp. 339–347, 2007.
[7]
R. Xu, Z. Ma, and Z. Wang, “Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity,” Computers and Mathematics with Applications, vol. 59, no. 9, pp. 3211–3221, 2010.
[8]
X. Meng, L. Chen, and B. Wu, “A delay SIR epidemic model with pulse vaccination and incubation times,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 88–98, 2010.
[9]
G. Zaman, Y. H. Kang, and I. H. Jung, “Optimal treatment of an SIR epidemic model with time delay,” BioSystems, vol. 98, no. 1, pp. 43–50, 2009.
[10]
K. Hattaf and N. Yousfi, “Optimal control of a delayed HIV infection model with immune response using an efficient numerical method,” ISRN Biomathematics, vol. 2012, Article ID 215124, 7 pages, 2012.
[11]
G. Birkhoff and G. C. Rota, Ordinary Differential Equations, John Wiley & Sons, New York,NY, USA, 4th edition, 1989.
[12]
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, NY, USA, 1975.
[13]
D. L. Lukes, Differential Equations: Classical to Controlled, Vol. 162 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1982.
[14]
L. G？llmann, D. Kern, and H. Maurer, “Optimal control problems with delays in state and control variables subject to mixed control-state constraints,” Optimal Control Applications and Methods, vol. 30, no. 4, pp. 341–365, 2009.
[15]
K. Hattaf and N. Yousfi, “Mathematical Model of the Influenza A(H1N1) Infection,” Advanced Studies in Biology, vol. 1, no. 8, pp. 383–390, 2009.
[16]
M. El hia, O. Balatif, J. Bouyaghroumni, E. Labriji, and M. Rachik, “Optimal control applied to the spread of influenza A(H1N1),” Applied Mathematical Sciences, vol. 6, no. 82, pp. 4057–4065, 2012.
