Optimal Control of an Epidemic Model of Leptospirosis with Nonlinear Saturated Incidences
Syed Farasat Sadiq
Department of Mathematics, Islamia College University, Peshawar Khyber Pakhtunkhwa, Pakistan.
Muhammad Altaf Khan *
Department of Mathematics, Abdul Wali Khan, University Mardan, Khyber Pakhtunkhwa, Pakistan.
Saeed Islam
Department of Mathematics, Abdul Wali Khan, University Mardan, Khyber Pakhtunkhwa, Pakistan.
Gul Zaman
Department of Mathematics, University of Malakand, Chakdara Dir lower, Khyber Pakhtunkhwa, Pakistan
II Hyo Jung
Department of Mathematics, Pusan National University, Busan 609-735, South Korea
Sher Afzal Khan
Department of Computer Sciences, Abdul Wali Khan, University Mardan, Khyber Pakhtunkhwa, Pakistan.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we consider a leptospirosis epidemic model with non-linear saturated incidence by applying the optimal control techniques to eradicate the infection in the human population. Leptospirosis is the disease which effects human as well as Cattle. Our aim is to find such optimal control techniques for the eradication of leptospira in the host population. To do this, we define three control variables, one for human and the second and third one for Vector population. First we find the existence of the control problem and then we characterize the optimal control problem by using the well-known method of Pontryagin’s Maximum Principle. The numerical simulations of both the system are solving by using backward Runge-kutta order four schemes. Finally, the numerical results of both the systems are presented for comparison.
Keywords: Leptospirosis, Pontryagin’s maximum principle, optimal control, numerical simulations.