Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.
Academy of Scientific Research &Technology, 101 Al Kasr El Aini, Cairo, Egypt.
J Adv Res. 2021 Apr 2;32:37-44. doi: 10.1016/j.jare.2021.03.010. eCollection 2021 Sep.
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.
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.
Both the nonstandard generalized fourth order Runge-Kutta and the nonstandard generalized Euler methods are presented.
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.
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.
登革热和疟疾是影响人类的最重要的两种蚊媒病毒病。受感染的雌性伊蚊在人类宿主之间传播发热。病毒感染的建模研究非常有用，可以显示病毒在感染个体中的复制方式，以及人体抗体反应如何控制这种复制，抗体在控制感染方面起着关键作用。
本研究对登革病毒新型变阶非线性模型进行最优控制。提出了Bang-bang 控制，以最大限度地减少病毒感染，并迅速从宿主中清除病毒。给出了控制问题的必要条件。给出了变阶导数的 Caputo 意义。此外，所提出模型的参数依赖于相同的变阶分数幂。构建了两种数值方案来求解最优系统。进行了比较研究和数值模拟。与整数阶和分数阶导数相比，变阶分数导数可以描述时变系统的长时间变量记忆的影响。
提出了非标准广义四阶龙格库塔法和非标准广义欧拉法。
我们成功地应用了一种具有 Bang-bang 控制的 Pontryagin 极大值原理，并能够通过添加 DI 颗粒的剂量来降低病毒血症水平。非标准广义四阶龙格库塔法比非标准广义欧拉法具有更好的结果。
在登革热数学模型中，变阶分数导数和 Bang-bang 控制的结合改善了模型的动力学。非标准广义欧拉法和非标准广义四阶龙格库塔法可用于简单地研究变阶分数最优控制问题。
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