Fractional optimal control in transmission dynamics of West Nile virus model with state and control time delay: a numerical approach

N H Sweilam,O M Saad,D G Mohamed

doi:10.1186/s13662-019-2147-8

N H Sweilam, O M Saad + Show 1 more

Open Access

https://doi.org/10.1186/s13662-019-2147-8

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Abstract

In this paper, an optimal control for a novel fractional West Nile virus model with time delay is presented. The proposed model is governed by a system of fractional delay differential equations, where the fractional derivative is defined in the Grünwald–Letnikov sense. Stability analysis of fixed points is studied. Corresponding fractional optimal control problem, with time delays in both state and control variables, is formulated and studied. Two simple numerical methods are used to study the nonlinear fractional delay optimal control problem. The methods are standard finite difference method and nonstandard finite difference method. Comparative studies are implemented, it is found that the nonstandard finite difference method is better than the standard finite difference method.

Highlights

Time delays play an important role in the modeling of real-life phenomena in different fields of applications, for example, engineering, chemical, and biological systems that are modeled by differential equations
In [13], Halanay proved a maximum principle for optimal control problems with multiple constant delays in state and control variables
We introduced Nonstandard finite difference method (NSFDM) and standard finite difference method (SFDM) to study the behavior of this optimal control problem

Summary

Introduction

Time delays play an important role in the modeling of real-life phenomena in different fields of applications, for example, engineering, chemical, and biological systems that are modeled by differential equations. Fractional time-delay optimal control in transmission dynamics of WNV model is presented as an extension of the WNV model given in [26]. Time delays in both state and control variables are formulated and studied. The aim of this paper is to study the fractional time-delay optimal control of this proposed model. Let us introduce a discrete time delay in the state variables Mi(t), S(t), and H(t), denoted by dd, dd, and dH, respectively (where dd represents the time required for the WNV transmission from mosquitos to humans and dH represents the time required for diagnosis and commencement of treatment of the infection of WNV model).

Numerical experiment

Conclusions

NSFDM 1

Methods