Fractional Order Optimal Control Research Articles

A Compartmental Approach to Modeling the Measles Disease: A Fractional Order Optimal Control Model

Measles is the most infectious disease with a high basic reproduction number (R0). For measles, it is reported that R0 lies between 12 and 18 in an endemic situation. In this paper, a fractional order mathematical model for measles disease is proposed to identify the dynamics of disease transmission following a declining memory process. In the proposed model, a fractional order differential operator is used to justify the effect and success rate of vaccination. The total population of the model is subdivided into five sub-compartments: susceptible (S), exposed (E), infected (I), vaccinated (V), and recovered (R). Here, we consider the first dose of measles vaccination and convert the model to a controlled system. Finally, we transform the control-induced model to an optimal control model using control theory. Both models are analyzed to find the stability of the system, the basic reproduction number, the optimal control input, and the adjoint equations with the boundary conditions. Also, the numerical simulation of the model is presented along with using the analytical findings. We also verify the effective role of the fractional order parameter alpha on the model dynamics and changes in the dynamical behavior of the model with R0=1.

Optimal control and bifurcation analysis of a delayed fractional-order SIRS model with general incidence rate and delayed control

A fractional-order generalized SIRS model considering incubation period is established in this paper for the transmission of emerging pathogens. The corresponding Hopf bifurcation is discussed by selecting time delay as the bifurcation parameter. In order to control the occurrence of Hopf bifurcation and achieve better dynamic behaviors, a delayed feedback control is adopted to the model. Further, the delayed fractional-order optimal control problem (DFOCP) is proposed and discussed. The parameters of the proposed model are identified through the measurement data of coronavirus disease 2019 (COVID-19). Based on the results of parameter identification, the corresponding DFOCP with delayed control is numerically solved.

Optimal control and bifurcation analysis of a delayed fractional-order epidemic model for the COVID-19 pandemic

In this paper, a delayed fractional-order epidemic model with general incidence rate and incubation period is proposed for the Corona Virus Disease 2019 (COVID-19) pandemic. The corresponding sufficient conditions are established to analyze the existence and stability of disease-free equilibrium and endemic equilibrium of the proposed model. The conditions for the existence of Hopf bifurcation are obtained by selecting the time delay as the bifurcation parameter. The control strategies for the COVID-19 pandemic are designed, and the corresponding delay fractional order optimal control problem (DFOCP) is analyzed based on the generalized Euler–Lagrange equation. The parameters of the model are identified based on the data of multiple types of the COVID-19 pandemic. Further, the effectiveness of the model in describing the trend of the COVID-19 pandemic is verified. Based on the results of parameter identification, the influence of incubation period on the COVID-19 pandemic is discussed. The forward–backward sweep method (FBSM) is adopted to numerically solve DFOCP, and the control effects under different control measures are analyzed.

Fractional order modelling and optimal control of dual active bridge converters

As the construction of new power systems centred around renewable energy gains traction, the installed capacity of new energy sources has experienced explosive growth. Research has shown that the actual external characteristics of inductors and capacitors in circuits exhibit fractional order characteristics. As a core device in dual active bridge converters, the non-integer order of inductors and capacitors has an important impact on their dynamic performance, frequency domain characteristics, and control system design. This paper conducts modelling and control research on dual active bridge converters based on fractional-order calculus theory. First, based on the definition of fractional-order calculus, according to the operating mode of the fractional-order dual active bridge converter and the characteristics of fractional-order components, The fractional-order state space average model was established, and the influence of the order of inductance and capacitance on the amplitude frequency characteristics, phase frequency characteristics and dynamic performance of the converter was analyzed. Secondly, the average model and circuit model were built on the MATLAB/Simulink simulation platform, and the simulation results under different fractional orders were compared. Finally, in order to further improve the control performance of the converter, a fractional-order P I λ control strategy is designed based on the established state space average model of the fractional-order dual active bridge converter and based on the transfer function from the control to the output of the fractional-order dual active bridge converter. The research results show that the fractional-order model can more accurately describe the actual characteristics of the converter. In addition, the fractional-order controller P I λ enables the fractional-order dual active bridge converter to obtain better robustness and improve the dynamic performance of the converter.

Optimal treatment and stochastic stability on a fractional-order epidemic model incorporating media awareness

Infectious diseases have become a severe concern in our society in recent times. In this respect, conducting a qualitative investigation of infectious disease systems is very crucial. Nowadays, developing a mathematical model and its analysis for disease control has been very popular. Our present study has formulated a SIR-type epidemic model by considering a saturated incidence function with media impact. To account for the influence of memory, we have modified the model by employing a system of fractional-order differential equations of Caputo’s notion. We have determined all the system equilibrium points and discussed their stability criteria based on the basic reproduction number (R0) as the threshold parameter. It is observed that for the threshold R0<1, disease-free equilibrium becomes both locally and globally asymptotically stable. A transcritical bifurcation occurs at the threshold R0=1. Further, as the threshold R0 exceeds unity, the endemic equilibrium becomes asymptotically stable. A fractional-order optimal control problem with treatment as the control parameter is developed to reduce the impact of disease. Using Pontryagin’s Maximum Principle yields a theoretical and numerical solution to the fractional order optimal control problem. It is observed that both the media and memory effect dramatically reduces infection rates. Considering environmental variations, the stochastic stability of the endemic equilibrium point has also been studied. Sensitivity analysis is carried out to comprehend how the parameters impact the threshold value (R0). Finally, extensive numerical simulations are performed to demonstrate our theoretical examination.

Analysis of optimal control strategies on the fungal Tinea capitis infection fractional order model with cost-effective analysis

In this study, we have formulated and analyzed the Tinea capitis infection Caputo fractional order model by implementing three time-dependent control measures. In the qualitative analysis part, we investigated the following: by using the well-known Picard–Lindelöf criteria we have proved the model solutions' existence and uniqueness, using the next generation matrix approach we calculated the model basic reproduction number, we computed the model equilibrium points and investigated their stabilities, using the three time-dependent control variables (prevention measure, non-inflammatory infection treatment measure, and inflammatory infection treatment measure) and from the formulated fractional order model we re-formulated the fractional order optimal control problem. The necessary optimality conditions for the Tinea capitis fractional order optimal control problem and the existence of optimal control strategies are derived and presented by using Pontryagin’s Maximum Principle. Also, the study carried out the sensitivity and numerical analysis to investigate the most sensitive parameters and to verify the qualitative analysis results. Finally, we performed the cost-effective analysis to investigate the most cost-effective measures from the possible proposed control measures, and from the findings we can suggest that implementing prevention measures only is the most cost-effective control measure that stakeholders should consider.

Insight into the optimal control strategies on corruption dynamics using fractional order derivatives

The fractional-order design for conceptualizing corruption offers several advantages on the corruption spreading over the integer-order corruption models such as acknowledgement of the intricate dynamics of corruption, including corruption super-spreading aspects, variable corruption transmission rates, it capture the persistence of memory effects, accounting for the past trajectory of corruption epidemic, and also providing a more accurate explanation of corruption dynamics. The author of the study has formulated the fractional order model on corruption dynamics by dividing the total population into five sub-groups namely the susceptible group, the exposed group, the moderately corrupted group, the highly corrupted group, and the corruption repented group. In the qualitative analyses section the author of this study has: shown the model solutions existence and uniqueness by applying the well-known Picard–Lindelöf criteria, computed the model basic reproduction number using the next generation approach, computed the model equilibrium points and investigated their stabilities, re-formulated the corresponding fractional order optimal control problem using the proposed three time-dependent controlling variables (prevention measure, moderate corruption treatment measure and high corruption treatment measure). The necessary optimality conditions for the fractional order optimal control problem and the existence of optimal control strategies are derived and presented by applying Pontryagin's Maximum Principle. Eventually, the study demonstrated the effectiveness of the combinations of the three controlling strategies through numerical methods (simulations) and the analysis results shows that the implementation of the three time-dependent controlling strategies significantly minimizes the number of corrupted individuals in the community.

Complex Dynamics and Fractional-Order Optimal Control of an Epidemic Model with Saturated Treatment and Incidence

In this study, we have developed a novel SIR epidemic model by incorporating fractional-order differential equations and utilizing saturated-type functions to describe both disease incidence and treatment. The intricate dynamical characteristics of the proposed model, encompassing the determination of the conditions for the existence of all possible feasible equilibria with their local and global stability criteria, are investigated thoroughly. The model undergoes backward bifurcation with respect to the parameter representing the side effects due to treatment. This phenomenon emphasizes the critical role of treatment control parameters in shaping epidemic outcomes. In addition, to understand the optimal role of the treatment in mitigating the disease prevalence and minimizing the associated cost, we investigated a fractional-order optimal control problem. To further visualize the analytical results, we have conducted simulation works considering feasible parameter values for the model. Finally, we have employed local and global sensitivity analysis techniques to identify the factors that have the greatest potential to reduce the impact of the disease.

A comprehensive review on fractional-order optimal control problem and its solution

Abstract This article presents a comprehensive literature survey on fractional-order optimal control problems. Fractional-order differential equation is extensively used nowadays to model real-world systems accurately, which exhibit fractal dimensions, memory effects, as well as chaotic behaviour. These versatile features attract engineers to concentrate more on this, and it is widely used in the broad domain of science and technology. The mentioned numerical tools take the necessary optimal conditions into account, which makes it a two-point boundary value problem of non-integer order. In this review article, some numerical approaches for the approximation have been stated for obtaining the solution to fractional optimal control problems (FOCPs). Here, few numerical approaches including Grunwald-Letnikov approximation, Adams type predictor-corrector method, generalized Euler’s method, Caputo-Fabrizio method Bernoulli and Legendre polynomials method, Legendre operational method, and Ritz’s and Jacobi’s method are treated as an advanced method to obtain the solution of FOCP. Fractional delayed optimal control is selected for our investigation. It refers to a type of control problem where the control action is delayed by a fractional amount of time. In other words, the control input at a given time depends not only on the current state of the system but also on its past state at fractional times. The fractional delayed optimal control problem is formulated as an optimization problem that seeks to minimize a cost function subject to a set of constraints that represent the dynamics of the system and the fractional delay in the control input. The solution to this problem typically involves the use of fractional polynomials types, i.e. Chebyshev and Bassel polynomials.

A fractional-order tuberculosis model with efficient and cost-effective optimal control interventions

This study is concerned with applying fractional calculus in modelling tuberculosis (TB) transmission dynamics. A new six-dimensional fractional-order mathematical model in the sense of the Caputo derivative operator is formulated to capture memory effects in the spread process of tuberculosis. Some key features such as vaccination, forces of TB infection, and exogenous re-infection by two actively infectious classes are considered. Banach fixed point theory is employed to prove the existence and uniqueness of a solution for the fractional-order model. Equilibrium points and effective reproduction number of the model are determined. Analysis reveals that the model exhibits backward bifurcation where stable TB-free and TB-present equilibrium points co-exist when the effective reproduction number is below unity. Some sensitive parameters of the model are identified to help design intervention measures against the disease. Hence, four time-dependent controls, namely advocacy effort against infection and re-infection, vaccination control, prophylaxis against latent infection, and treatment control, are considered to form a fractional-order optimal control model. The controls are characterized using Pontryagin’s maximum principle. Efficiency and cost-effectiveness assessments are conducted to quantify the most efficient and cost-effective intervention plan to minimize TB spread in the population. Behaviours of the fractional-order TB model at varying values of the order of the Caputo derivative are extensively explored to show memory effects in the transmission dynamics of the disease.

Hybrid fractional-order optimal control problem for immuno-chemotherapy with gene therapy and time-delay: numerical treatments

ABSTRACT This paper investigates the optimal control problem for the tumor-immune system with time-delay in the presence of gene therapy and immuno chemotherapy. Riemann-Liouville fractional order integrals and Caputo fractional order derivatives are combined to form a hybrid fractional order operator. For consistency with the physical model problem, a new parameter is presented. A stability and bifurcation analysis of the proposed model is performed. The positivity, boundedness, and existence of optimal control for the proposed model are discussed. Grünwald-Letnikov nonstandard finite difference method with the discretization of the hybrid fractional order operator is constructed to solve the proposed model. Examples and comparative studies are presented to demonstrate the simplicity of the approximation approaches and the applicability of the utilized methods.

Numerical Fractional Optimal Control of Respiratory Syncytial Virus Infection in Octave/MATLAB

In this article, we develop a simple mathematical GNU Octave/MATLAB code that is easy to modify for the simulation of mathematical models governed by fractional-order differential equations, and for the resolution of fractional-order optimal control problems through Pontryagin’s maximum principle (indirect approach to optimal control). For this purpose, a fractional-order model for the respiratory syncytial virus (RSV) infection is considered. The model is an improvement of one first proposed by the authors in 2018. The initial value problem associated with the RSV infection fractional model is numerically solved using Garrapa’s fde12 solver and two simple methods coded here in Octave/MATLAB: the fractional forward Euler’s method and the predict-evaluate-correct-evaluate (PECE) method of Adams–Bashforth–Moulton. A fractional optimal control problem is then formulated having treatment as the control. The fractional Pontryagin maximum principle is used to characterize the fractional optimal control and the extremals of the problem are determined numerically through the implementation of the forward-backward PECE method. The implemented algorithms are available on GitHub and, at the end of the paper, in appendixes, both for the uncontrolled initial value problem as well as for the fractional optimal control problem, using the free GNU Octave computing software and assuring compatibility with MATLAB.

Numerical Method for Solving Fractional Order Optimal Control Problems with Free and Non-Free Terminal Time

The optimal control theory in mathematics aims to study the finding of control for a dynamic system over time, where an objective function is optimized. It has a broad range of applications in engineering, operations research, and science. The main purpose of this study is to provide numerical algorithms for two cases of optimal control problems of fractional order that involve fractional order derivatives with free and non-free terminal time. In addition to comparing the numerical results for three test problems with exact solutions of these problems, various computer simulations are also introduced.

Theoretical Analysis of a COVID-19 CF-Fractional Model to Optimally Control the Spread of Pandemic

In this manuscript, we formulate a mathematical model of the deadly COVID-19 pandemic to understand the dynamic behavior of COVID-19. For the dynamic study, a new SEIAPHR fractional model was purposed in which infectious individuals were divided into three sub-compartments. The purpose is to construct a more reliable and realistic model for a complete mathematical and computational analysis and design of different control strategies for the proposed Caputo–Fabrizio fractional model. We prove the existence and uniqueness of solutions by employing well-known theorems of fractional calculus and functional analyses. The positivity and boundedness of the solutions are proved using the fractional-order properties of the Laplace transformation. The basic reproduction number for the model is computed using a next-generation technique to handle the future dynamics of the pandemic. The local–global stability of the model was also investigated at each equilibrium point. We propose basic fixed controls through manipulation of quarantine rates and formulate an optimal control problem to find the best controls (quarantine rates) employed on infected, asymptomatic, and “superspreader” humans, respectively, to restrict the spread of the disease. For the numerical solution of the fractional model, a computationally efficient Adams–Bashforth method is presented. A fractional-order optimal control problem and the associated optimality conditions of Pontryagin maximum principle are discussed in order to optimally reduce the number of infected, asymptomatic, and superspreader humans. The obtained numerical results are discussed and shown through graphs.

Transmission dynamics of monkeypox virus with treatment and vaccination controls: a fractional order mathematical approach

Presently, monkeypox virus infection has spread worldwide in the ongoing outbreak that began in the UK. To study the transmission dynamics of monkeypox, we formulate here a seven-compartmental (five compartments for the human population and two compartments for animals or rodents) fractional-order mathematical model. The existence and uniqueness of the solution of the proposed fractional order model are examined here. The basic reproduction number for humans () and animals () are obtained through the next-generation matrix approach. Depending on the values of and , we observed that the fractional order model has three equilibria, namely, monkeypox-free equilibrium, animal-free endemic equilibrium, and endemic equilibrium. Also, the stability of all equilibria is checked in this present article. We found that the model goes through transcritical bifurcation at for any values of and at for . Best of our knowledge, this is the first work where the fractional order optimal control for monkeypox is formulated and solved considering vaccination and treatment controls. Several feasible parameter values are used in the simulations to visualize and verify the findings, from which the results show that fractional order is more appropriate. Finally, parameters involved in the expression of and are scaled using the sensitivity index approach.

Analysis of tinea capitis epidemic fractional order model with optimal control theory

Tinea Capitis is a common worldwide fungal infectious disease. This paper investigates the transmission dynamics of tinea captis infection using Caputo fractional derivative approach. The qualitative analysis part of the proposed fractional order model investigates: the model solutions existence and uniqueness by using the well-known Picard–Lindelöf criteria, the basic reproduction number using the next generation matrix approach, the model equilibrium points and their stabilities. The study re-formulates the fractional order optimal control problem with the three time-dependent controlling variables such as prevention measure, acute infection treatment measure and chronic infection treatment measure and also investigates the necessary optimality conditions and the existence of optimal control strategies by applying Pontryagin's Maximum Principle. Finally, the paper examines the effectiveness of the combinations of the three controlling strategies through numerical simulations and the results indicate that the simultaneous implementation of the three time-dependent controlling strategies significantly minimizes the number of tinea capitis infected individuals in the community.

Clinical effects of 2-DG drug restraining SARS-CoV-2 infection: A fractional order optimal control study.

Fractional calculus is very convenient tool in modeling of an emergent infectious disease system comprising previous disease states, memory of disease patterns, profile of genetic variation etc. Significant complex behaviors of a disease system could be calibrated in a proficient manner through fractional order derivatives making the disease system more realistic than integer order model. In this study, a fractional order differential equation model is developed in micro level to gain perceptions regarding the effects of host immunological memory in dynamics of SARS-CoV-2 infection. Additionally, the possible optimal control of the infection with the help of an antiviral drug, viz. 2-DG, has been exemplified here. The fractional order optimal control would enable to employ the proper administration of the drug minimizing its systematic cost which will assist the health policy makers in generating better therapeutic measures against SARS-CoV-2 infection. Numerical simulations have advantages to visualize the dynamical effects of the immunological memory and optimal control inputs in the epidemic system.

An application of a fuzzy system for solving time delay fractional optimal control problems with Atangana–Baleanu derivative

AbstractIn this article, a new approach based on fuzzy systems is used for solving time delay fractional order optimal control problems. The fractional derivatives are considered in the Atangana–Baleanu sense that is a new derivative with the nonsingular and nonlocal kernel. By means of the calculus of variations and the formula for fractional integration by parts, the necessary optimality conditions associated to the time delay problem is derived. In order to solve the obtained optimality system, the solution of the system is first approximated by fuzzy solutions with adjustable parameters. The optimality system is then reduced to an unconstrained optimization problem by using appropriate error function. A learning algorithm is also presented to achieve the parameters of these fuzzy solutions. The efficiency and accuracy of the proposed approach are assessed through some illustrative examples of the time delay fractional optimal control problems.

Stability Analysis and Optimal Control of a Fractional Cholera Epidemic Model

In this paper, a fractional model for the transmission dynamics of cholera was developed. In invariant regions of the model, solutions were generated. Disease-free and endemic equilibrium points were obtained. The basic reproduction number was evaluated, and the sensitivity analysis was performed. Under the support of Pontryagin’s maximum principle, the fractional order optimal control was obtained. Furthermore, an optimal strategy was discussed, which minimized the total number of infected individuals and the costs associated with control. Treatment, vaccination, and awareness programs were regarded as three means to reduce the number of infected. Finally, numerical simulations and cost-effectiveness analysis were presented to show the result that the best strategy was the combination of treatment and awareness programs.

Numerical, Approximate Solutions, and Optimal Control on the Deathly Lassa Hemorrhagic Fever Disease in Pregnant Women

This paper is devoted to the model of Lassa hemorrhagic fever (LHF) disease in pregnant women. This disease is a biocidal fever and epidemic. LHF disease in pregnant women has negative impacts that were initially appeared in Africa. In the present study, we find an approximate solution to the fractional-order model that describes the fatal LHF disease. Laplace transforms coupled with the Adomian decomposition method (ADM) are applied. In addition, the fractional-order LHF model is numerically simulated in terms of a varied fractional order. Furthermore, a fractional order optimal control for the LHF model is studied.