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ISRN Applied Mathematics
Volume 2013 (2013), Article ID 706848, 4 pages
http://dx.doi.org/10.1155/2013/706848
Research Article

Optimal Control Strategy for SEIR with Latent Period and a Saturated Incidence Rate

1Department of Mathematics and Computer Science, Faculty of Sciences El Jadida, Chouaib Doukkali University, P.O. Box 20, El Jadida, Morocco
2Department of Mathematics and Computer Science, Faculty of Sciences Ben M’sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca, Morocco

Received 24 April 2013; Accepted 22 May 2013

Academic Editors: Q. Song and Y. Wang

Copyright © 2013 Abta Abdelhadi and Laarabi Hassan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We propose an SEIR epidemic model with latent period and a modified saturated incidence rate. This work investigates the fundamental role of the vaccination strategies to reduce the number of susceptible, exposed, and infected individuals and increase the number of recovered individuals. The existence of the optimal control of the nonlinear model is also proved. The optimality system is derived and then solved numerically using a competitive Gauss-Seidel-like implicit difference method.

1. Introduction

Epidemiological models with latent or incubation period have been studied by many authors because many diseases have a latent or incubation period, during which the individual is said to be infected but not infectious. This period can be modeled by introducing an exposed class [1]. Therefore, it is an important subject to determine the optimal vaccination strategies for the models which take into account the incubation period.

In this paper, our aim is to set up an optimal control problem related to the SEIR epidemic model. The dynamics of this model are governed by the following equations [2, 3]: where is the number of the susceptible individuals, is the number of exposed individuals, is the number of infected individuals, is the number of the recovered individuals, is the recruitment rate of the population, is the natural death of the population, is the death rate due to disease, is the transmission rate, and are the parameter that measure the inhibitory effect, is the recovery rate of the infective individuals, and is the rate at which exposed individuals become infectious. Thus is the mean latent period.

Now we introduce one control which represents the percentage of susceptible individuals being vaccinated per unit of time. Hence, (1) becomes In addition, for biological reasons, we assume that the initial data for system (2) satisfy The rest of the paper is organized as follows. In Section 2, we use Pontryagin's maximum principle to investigate analysis of control strategies and to determine the necessary conditions for the optimal control of the disease. Mathematical results are illustrated by numerical simulations in Section 3. Finally, we summarize our work and propose the future focuses.

2. The Optimal Control Problem

The optimal control problem is to minimize the objective (cost) functional given by subject to the differential equations (2), where the first tree terms in the functional objective represent benefit of , , and populations that we wish to reduce, and the parameters , , and are positive constants to keep a balance in the size of , , and , respectively. We use in the second term in the functional objective (as it is customary) the quadratic term , where is a positive weight parameter which is associated with the control , and the square of the control variable reflects the severity of the side effects of the vaccination.

Our target is to minimize the objective functional defined in (4) by decreasing the number of infected, exposed, and susceptible individuals and increasing the number of recovered individuals by using possible minimal control variables . In other words, the control variable represents the percentage of susceptible individuals being vaccinated per unit of time and is the control set defined by

2.1. Existence of an Optimal Control

For the existence of an optimal control we use the result in Lukes [4], and we obtain the following theorem.

Theorem 1. There exists a control function so that

Proof. To prove the existence of an optimal control it is easy to verify that(1)the set of controls and corresponding state variables is nonempty,(2)the admissible set is convex and closed,(3)the right hand side of the state system (2) is bounded by a linear function in the state and control variables,(4)the integrand of the objective functional is convex on , (5)there exist constants and , and such that the integrand of the objective functional satisfies and positive numbers and such that . The result follows directly from [5].

2.2. Characterization of the Optimal Control

Before characterizing the optimal control, we first define the Lagrangian for the optimal control problem (2) and (4) by and the Hamiltonian for the control problem by where , , , and are the adjoint functions to be determined suitably. Next, by applying Pontryagin’s maximum principle [6] to the Hamiltonian, we obtain the following theorem.

Theorem 2. Given an optimal control and solutions , , , and of the corresponding state system (2) and (4), there exists adjoint variables , , , and that satisfy where and with transversality conditions Furthermore, the optimal control is given by

Proof. Using the Pontryagin’s maximum principle we obtain the adjoint equations and transversality conditions such that and by using the optimality conditions we find Using the property of the control space, we obtain So the optimal control is characterized as Therefore, using the characterization of the optimal control, we have the following optimality system: with , , , , , , , and .

3. Numerical Simulations

In this section, we solve numerically the optimality system (16) using the Gauss-Seidel-like implicit finite-difference method developed by Gumel et al. [7], and we use in this simulation the parameter values given in Table 1.

tab1
Table 1: Values of the parameters.

Figure 1 shows that a significant difference in the number of susceptible, exposed, infected, and recovered individuals with and without control from the twenty days of vaccination, after that it begins to go to the stable state.

fig1
Figure 1: Evolution of different classes of individuals with or without control.

4. Conclusions and Future Research

We have presented in this paper the SEIR model with latent period and a modified saturated incidence rate. Our aim is to outline the steps in setting up an optimal control problem, so we presented an efficient numerical method based on optimal control to identify the best vaccination strategy of SEIR model. Our numerical results show that the optimal vaccination strategies for the diseases have a latent period to reduce the number of susceptible, exposed, and infected individuals and increase the number of recovered after twenty days of vaccination.

In future research, we determine the optimal control strategies for the delayed SIR model and compare it with that presented in this work. It is an important subject to study these two types of modeling the incubation period.

References

  1. H. W. Hethcote, H. W. Stech, and P. van den Driessche, “Periodicity and stability in epidemic models: a survey,” in Differential Equations and Applications in Ecology, Epidemics, and Population Problems, S. N. Busenberg and K. L. Cooke, Eds., p. 6582, Academic Press, New York, NY, USA, 1981.
  2. A. Kaddar, A. Abta, and H. T. Alaoui, “A comparison of delayed SIR and SEIR epidemic models,” Nonlinear Analysis: Modelling and Control, vol. 16, no. 2, pp. 181–190, 2011. View at Scopus
  3. A. Abta, A. Kaddar, and H. T. Alaoui, “Global stability for delay SIR and SEIR epidemic models with saturated incidence rates,” Electronic Journal of Differential Equations, vol. 2012, no. 23, pp. 1–13, 2012. View at Scopus
  4. D. L. Lukes, Differential Equations: Classical to Controlled, vol. 162 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1982.
  5. W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, NY, USA, 1975.
  6. I. K. Morton and L. S. Nancy, Dynamics Optimization the Calculus of Variations and Optimal Control in Economics and Management, Elsevier Science, Amsterdam, The Netherlands, 2000.
  7. A. B. Gumel, P. N. Shivakumar, and B. M. Sahai, “A mathematical model for the dynamics of HIV-1 during the typical course of infection,” in Proceedings of the 3rd World Congress of Nonlinear Analysts, vol. 47, pp. 2073–2083, 2001.