Application of Fractional Optimal Control Problems on Some Mathematical Bioscience

Introduction

Infectious diseases are the most important problems facing us in the present time. For example, HIV and TB cause 10% of all deaths (Misra et al, 2011; Huo et al, 2015; Hikal & Zahra, 2016). Over recent years, the occurs of infectious diseases outbreaks has increased dramatically which leads to millions of deaths and affecting international economics (Salah et al, 2015; Global Health Policy, 2014). There is an opportunity to collect initial data when a diseases outbreak. These collected data are essential to understand the behaviour of the spreading mechanism, predict the future path and adjust control strategies of diseases. Pathological limitations are one of the main reasons for impossible to collect these initial data (see e.g., Misra et al, 2011; Mukandavire et al, 2009). Consequently, the description of infectious diseases as mathematical models should be used to give a better understanding, analyzing and controlling such diseases.

One of the ways in which construct mathematical models is by differential equations of fractional order (see e.g., (Ameen & Novati, 2017; Abderrahim et al, 2018; Khalil et al, 2019). Fractional differential equations (FDEs) have been attracted attention due to their appearance in various applications in biology, physics, epidemiology, fluid mechanics, viscoelasticity, and engineering. Moreover, the fractional-order derivatives (i.e. Caputo fractional derivative) for ordinary or partial differential equations proved that it's a valuable tool for modelling of many various phenomena (see e.g., (Ali & Ameen, 2019; Arafa et al, 2014; Ali, 2019)).

This is due to the fact that the realistic modelling of a phenomenon often does not depend only on the instant time but also on the history of the previous time (memory effects) which can also be successfully achieved by using fractional calculus (where it's non-local operator, for details see Ameen, 2017, p.38-39). More precisely, the modelling of biological and biomedicine by FDEs has more advantages than the integer-order ones, in which memory effects are neglected (Sardar et al, 2015; Du et al, 2013).

Fractional optimal control problems (FOCPs) can be viewed as a generalization of classical optimal control problems (OCPs), in which the dynamics of the control system are characterized by FDEs and might include an objective function given by a fractional integration operator. The motivation to formulate and solve FOCPs depend on the reality that there are a significant number of cases in which FDEs describe the behaviour of the control systems of interest more precisely than the common integer differential equations (Ali et al, 2016).

There have been many efforts in formulating and solving FOCPs to overcome and eliminate certain diseases, for example, Rosa and Torres in (Rosa & Torres, 2018) applied fractional optimal control theory on human respiratory syncytial virus (HRSV) infection. Ding et al (Ding et al, 2011) formulated and addressed a FOCP of HIV-Immune system model and solved it. Sweilam et al (Sweilam et al, 2016) studied a FOCP of a multi-strain tuberculosis model and used numerical methods to find a solution to this problem. Bonyah et al (Bonyah et al, 2018) proposed a FOCP of human African trypanosomiasis model and use preventive and treatment mechanisms to minimize the number of a person infected with trypanosomiasis in the African society. Al Basir and Roy developed and solved a FOCP for HIV-model in (Al Basir & Roy, 2017). There are other some research conducted on diseases which formulated as a FOCPs (see e.g. Elal et al, 2016; Sweilam & AL–Mekhlafi, 2017; Rachah & Torres, 2017; Sweilam et al, 2019; Ali & Ameen, 2020; Ameen et al, 2020).