Z. Naturforsch. **68a,** 677 – 685
(2013)

doi:10.5560/ZNA.2013-0051

Optimal Vaccination of an Endemic Model with Variable Infectivity and Infinite Delay

Gul
Zaman^{1},
Yasuhisa
Saito^{2}, and
Madad
Khan^{3}

^{1 }Department of Mathematics, University of Malakand, Chakdara, Dir(Lower), Khyber Pakhtunkhwa, Pakistan

^{2 }Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan

^{3 }Department of Mathematics, COMSATS Institute of Information Technology Abbottabad, Pakistan

Received February 12, 2013 / revised June 25, 2013 / published online September 18, 2013

In this work, we consider a nonlinear *SEIR* (susceptible, exposed, infectious, and removed) endemic model, which describes the dynamics of the interaction between susceptible and infected individuals in a population. The model represents the disease evolution through a system of nonlinear differential equations with variable infectivity which determines that the infectivity of an infected individual may not be constant during the time after infection. To control the spread of infection and to find a vaccination schedule for an endemic situation, we use optimal control strategies which reduce the susceptible, exposed, and infected individuals and increase the total number of recovered individuals. In order to do this, we introduce the optimal control problem with a suitable control function using an objective functional. We first show the existence of an optimal control for the control problem and then derive the
optimality system. Finally the numerical simulations of the model is identified to fit realistic measurement which shows the effectiveness of the model.

*Key words:* Variable Infectivity; Optimality; Vaccination; Numerical Simulation.

*Mathematics Subject Classification 2000:* 92D30; 34D05