Optimal bang-bang control for variable-order dengue virus; numerical studies,Journal of Advanced Research

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Optimal bang-bang control for variable-order dengue virus; numerical studies
Journal of Advanced Research ( IF 11.4 ) Pub Date : 2021-04-02 , DOI: 10.1016/j.jare.2021.03.010
N H Sweilam _{1,

2} , S M Al-Mekhlafi _{1,

2} , S A Shatta ₃

Affiliation

Introduction

Dengue and Malaria are the most important mosquito-borne viral diseases affecting humans. Fever is transmitted between human hosts by infected female aedes mosquitoes. The modeling study of viral infections is very useful to show how the virus replicates in an infected individual and how the human antibody response acts to control that replication, which antibody playing a key role in controlling infection.

Objectives

Optimal control of a novel variable-order nonlinear model of dengue virus is studied in the present work. Bang-bang control is suggested to minimize the viral infection as well as quick clearance of the virus from the host. Necessary conditions for the control problem are given. The variable-order derivatives are given in the sense of Caputo. Moreover, the parameters of the proposed model are dependent on the same variable-order fractional power. Two numerical schemes are constructed for solving the optimality systems. Comparative studies and numerical simulations are implemented. The variable-order fractional derivative can be describe the effects of long variable memory of time dependent systems than the integer order and fractional order derivatives.

Methods

Both the nonstandard generalized fourth order Runge-Kutta and the nonstandard generalized Euler methods are presented.

Results

We have successfully applied a kind of Pontryagin’s maximum principle with bang-bang control and were able to reduce the viraemia level by adding the dose of DI particles. The nonstandard generalized fourth order Runge-Kutta method has the best results than nonstandard generalized Euler method.

Conclusion

The combination of the variable-order fractional derivative and bang-bang control in the Dengue mathematical model improves the dynamics of the model. The nonstandard generalized Euler method and the nonstandard generalized fourth order Runge-Kutta method can be used to study the variable order fractional optimal control problem simply.

中文翻译：

变阶登革热病毒的最佳爆炸控制；数值研究

介绍

登革热和疟疾是影响人类的最重要的蚊媒病毒性疾病。发烧是通过受感染的雌性伊蚊在人类宿主之间传播的。病毒感染的建模研究对于展示病毒如何在感染个体中复制以及人类抗体反应如何控制病毒复制非常有用，而哪种抗体在控制感染中发挥着关键作用。

目标

本工作研究了登革热病毒新型变阶非线性模型的最优控制。建议进行爆炸式控制，以最大程度地减少病毒感染并快速清除宿主体内的病毒。给出了控制问题的必要条件。变阶导数是在 Caputo 意义上给出的。此外，所提出模型的参数取决于相同的变阶分数幂。构建了两个数值方案来求解最优系统。进行了比较研究和数值模拟。变阶分数阶导数比整数阶和分数阶导数更能描述时间相关系统的长变量记忆效应。