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Srivastava A, Nilam. Optimal control of a fractional order SEIQR epidemic model with non-monotonic incidence and quarantine class. Comput Biol Med 2024; 178:108682. [PMID: 38861897 DOI: 10.1016/j.compbiomed.2024.108682] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/11/2024] [Revised: 04/26/2024] [Accepted: 05/30/2024] [Indexed: 06/13/2024]
Abstract
During any infectious disease outbreak, effective and timely quarantine of infected individuals, along with preventive measures by the population, is vital for controlling the spread of infection in society. Therefore, this study attempts to provide a mathematically validated approach for managing the epidemic spread by incorporating the Monod-Haldane incidence rate, which accounts for psychological effects, and a saturated quarantine rate as Holling functional type III that considers the limitation in quarantining of infected individuals into the standard Susceptible-Exposed-Infected-Quarantine-Recovered (SEIQR) model. The rate of change of each subpopulation is considered as the Caputo form of fractional derivative where the order of derivative represents the memory effects in epidemic transmission dynamics and can enhance the accuracy of disease prediction by considering the experience of individuals with previously encountered. The mathematical study of the model reveals that the solutions are well-posed, ensuring nonnegativity and boundedness within an attractive region. Further, the study identifies two equilibria, namely, disease-free (DFE) and endemic (EE); and stability analysis of equilibria is performed for local as well as global behaviours for the same. The stability behaviours of equilibria mainly depend on the basic reproduction number R0 and its alternative threshold T0 which is computed using the Next-generation matrix method. It is investigated that DFE is locally and globally asymptotic stable when R0<1. Furthermore, we show the existence of EE and investigate that it is locally and globally asymptotic stable using the Routh-Hurwitz criterion and the Lyapunov stability theorem for fractional order systems with R0>1 under certain conditions. This study also addresses a fractional optimal control problem (FOCP) using Pontryagin's maximum principle aiming to minimize the spread of infection with minimal expenditure. This approach involves introducing a time-dependent control measure, u(t), representing the behavioural response of susceptible individuals. Finally, numerical simulations are presented to support the analytical findings using the Adams Bashforth Moulton scheme in MATLAB, providing a comprehensive understanding of the proposed SEIQR model.
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Affiliation(s)
- Abhay Srivastava
- Department of Applied Mathematics, Delhi Technological University, Delhi 110042, India.
| | - Nilam
- Department of Applied Mathematics, Delhi Technological University, Delhi 110042, India.
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Mohammadi S, Hejazi SR. Lie symmetry, chaos optimal control in non-linear fractional-order diabetes mellitus, human immunodeficiency virus, migraine Parkinson's diseases models: using evolutionary algorithms. Comput Methods Biomech Biomed Engin 2024; 27:651-679. [PMID: 37068041 DOI: 10.1080/10255842.2023.2198628] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/28/2022] [Accepted: 03/23/2023] [Indexed: 04/18/2023]
Abstract
The purpose of this article is to investigate the optimal control of nonlinear fractional order chaotic models of diabetes mellitus, human immunodeficiency virus, migraine and Parkinson's diseases using genetic algorithms and particle swarm optimization. Mathematical chaotic models of nonlinear fractional order type of the above diseases were presented. Then optimal control for each of the models and numerical simulation was done using genetic algorithm and particle swarm optimization algorithm. The results of the genetic algorithm method are excellent. All the results obtained for the particle swarm optimization method show that this method is also very successful and the results are very close to the genetic algorithm method. Very low values of MSE and RMSE errors indicate that the simulation is effective and efficient. Also, Lie symmetry was calculated for the proposed models and the results were presented.
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Affiliation(s)
- Shaban Mohammadi
- Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
| | - S Reza Hejazi
- Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
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Teklu SW, Meshesha AA, Ullah S. Analysis of optimal control strategies on the fungal Tinea capitis infection fractional order model with cost-effective analysis. Sci Rep 2024; 14:1508. [PMID: 38233520 PMCID: PMC10794199 DOI: 10.1038/s41598-024-51670-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2023] [Accepted: 01/08/2024] [Indexed: 01/19/2024] Open
Abstract
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.
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Affiliation(s)
- Shewafera Wondimagegnhu Teklu
- Department of Mathematics, College of Natural and Computational Sciences, Debre Berhan University, 445, Debre Berhan, Ethiopia.
| | - Abebe Addise Meshesha
- Department of Surgery, College of Health Science, Debre Berhan University, 445, Debre Berhan, Ethiopia
| | - Saif Ullah
- Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan
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Liu B, Farid S, Ullah S, Altanji M, Nawaz R, Wondimagegnhu Teklu S. Mathematical assessment of monkeypox disease with the impact of vaccination using a fractional epidemiological modeling approach. Sci Rep 2023; 13:13550. [PMID: 37599330 PMCID: PMC10440346 DOI: 10.1038/s41598-023-40745-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2023] [Accepted: 08/16/2023] [Indexed: 08/22/2023] Open
Abstract
This present paper aims to examine various epidemiological aspects of the monkeypox viral infection using a fractional-order mathematical model. Initially, the model is formulated using integer-order nonlinear differential equations. The imperfect vaccination is considered for human population in the model formulation. The proposed model is then reformulated using a fractional order derivative with power law to gain a deeper understanding of disease dynamics. The values of the model parameters are determined from the cumulative reported monkeypox cases in the United States during the period from May 10th to October 10th, 2022. Besides this, some of the demographic parameters are evaluated from the population of the literature. We establish sufficient conditions to ensure the existence and uniqueness of the model's solution in the fractional case. Furthermore, the stability of the endemic equilibrium of the fractional monkeypox model is presented. The Lyapunov function approach is used to demonstrate the global stability of the model equilibria. Moreover, the fractional order model is numerically solved using an efficient numerical technique known as the fractional Adams-Bashforth-Moulton method. The numerical simulations are conducted using estimated parameters, considering various values of the fractional order of the Caputo derivative. The finding of this study reveals the impact of various model parameters and fractional order values on the dynamics and control of monkeypox.
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Affiliation(s)
- Botao Liu
- School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People's Republic of China
| | - Samreen Farid
- Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan
| | - Saif Ullah
- Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan
| | - Mohamed Altanji
- Department of Mathematics, College of Science, King Khalid University, Abha, 61413, Saudi Arabia
| | - Rashid Nawaz
- Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan
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Teklu SW. Analysis of fractional order model on higher institution students' anxiety towards mathematics with optimal control theory. Sci Rep 2023; 13:6867. [PMID: 37106010 PMCID: PMC10140381 DOI: 10.1038/s41598-023-33961-y] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2022] [Accepted: 04/21/2023] [Indexed: 04/29/2023] Open
Abstract
Anxiety towards mathematics is the most common problem throughout nations in the world. In this study, we have mainly formulated and analyzed a Caputo fractional order mathematical model with optimal control strategies on higher institution students' anxiety towards mathematics. The non-negativity and boundedness of the fractional order dynamical system solutions have been analysed. Both the anxiety-free and anxiety endemic equilibrium points of the Caputo fractional order model are found, and the local stability analysis of the anxiety-free and anxiety endemic equilibrium points are examined. Conditions for Caputo fractional order model backward bifurcation are analyzed whenever the anxiety effective reproduction number is less than one. We have shown the global asymptotic stability of the endemic equilibrium point. Moreover, we have carried out the optimal control strategy analysis of the fractional order model. Eventually, we have established the analytical results through numerical simulations to investigate the memory effect of the fractional order derivative approach, the behavior of the model solutions and the effects of parameters on the students anxiety towards mathematics in the community. Protection and treatment of anxiety infectious students have fundamental roles to minimize and possibly to eradicate mathematics anxiety from the higher institutions.
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Benavides EM, Ordobás Gavín M, Mallaina García R, de Miguel García S, Ortíz Pinto M, Doménech Gimenez R, Gandarillas Grande A. COVID-19 dynamics in Madrid (Spain): A new convolutional model to find out the missing information during the first three waves. PLoS One 2022; 17:e0279080. [PMID: 36548226 PMCID: PMC9778560 DOI: 10.1371/journal.pone.0279080] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/15/2021] [Accepted: 11/30/2022] [Indexed: 12/24/2022] Open
Abstract
This article presents a novel mathematical model to describe the spread of an infectious disease in the presence of social and health events: it uses 15 compartments, 7 convolution integrals and 4 types of infected individuals, asymptomatic, mild, moderate and severe. A unique feature of this work is that the convolutions and the compartments have been selected to maximize the number of independent input parameters, leading to a 56-parameter model where only one had to evolve over time. The results show that 1) the proposed mathematical model is flexible and robust enough to describe the complex dynamic of the pandemic during the first three waves of the COVID-19 spread in the region of Madrid (Spain) and 2) the proposed model allows us to calculate the number of asymptomatic individuals and the number of persons who presented antibodies during the first waves. The study shows that the following results are compatible with the reported data: close to 28% of the infected individuals were asymptomatic during the three waves, close to 29% of asymptomatic individuals were detected during the subsequent waves and close to 26% of the Madrid population had antibodies at the end of the third wave. This calculated number of persons with antibodies is in great agreement with four direct measurements obtained from an independent sero-epidemiological research. In addition, six calculated curves (total number of confirmed cases, asymptomatic who are confirmed as positive, hospital admissions and discharges and intensive care units admissions) show good agreement with data from an epidemiological surveillance database.
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Affiliation(s)
- Efrén M. Benavides
- Department of Fluid Mechanics and Aersospace Propulsion, Universidad Politécnica de Madrid, Madrid, Spain
- * E-mail:
| | - María Ordobás Gavín
- Epidemiology Department, Directorate General of Public Health, Madrid Regional Health Authority, Madrid, Spain
| | - Raúl Mallaina García
- Strategic Planning Department, Directorate of Integrated Healthcare Process, Foundation on Innovation and Research in Primary Care Foundation FIIBAP, Madrid, Spain
| | - Sara de Miguel García
- Epidemiology Department, Directorate General of Public Health, Madrid Regional Health Authority, Madrid, Spain
| | - Maira Ortíz Pinto
- Epidemiology Department, Directorate General of Public Health, Madrid Regional Health Authority, Madrid, Spain
| | - Ramón Doménech Gimenez
- Epidemiology Department, Directorate General of Public Health, Madrid Regional Health Authority, Madrid, Spain
| | - Ana Gandarillas Grande
- Epidemiology Department, Directorate General of Public Health, Madrid Regional Health Authority, Madrid, Spain
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Different strategies to confront maize streak disease based on fractional optimal control formulation. CHAOS SOLITONS & FRACTALS 2022. [DOI: 10.1016/j.chaos.2022.112699] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/06/2023]
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El Hadj Moussa Y, Boudaoui A, Ullah S, Muzammil K, Riaz MB. Application of fractional optimal control theory for the mitigating of novel coronavirus in Algeria. RESULTS IN PHYSICS 2022; 39:105651. [PMID: 35668848 PMCID: PMC9161688 DOI: 10.1016/j.rinp.2022.105651] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/15/2022] [Revised: 05/23/2022] [Accepted: 05/23/2022] [Indexed: 06/15/2023]
Abstract
In this paper, we investigate the dynamics of novel coronavirus infection (COVID-19) using a fractional mathematical model in Caputo sense. Based on the spread of COVID-19 virus observed in Algeria, we formulate the model by dividing the infected population into two sub-classes namely the reported and unreported infective individuals. The existence and uniqueness of the model solution are given by using the well-known Picard-Lindelöf approach. The basic reproduction number R 0 is obtained and its value is estimated from the actual cases reported in Algeria. The model equilibriums and their stability analysis are analyzed. The impact of various constant control parameters is depicted for integer and fractional values of α . Further, we perform the sensitivity analysis showing the most sensitive parameters of the model versus R 0 to predict the incidence of the infection in the population. Further, based on the sensitivity analysis, the Caputo model with constant controls is extended to time-dependent variable controls in order obtain a fractional optimal control problem. The associated four time-dependent control variables are considered for the prevention, treatment, testing and vaccination. The fractional optimality condition for the control COVID-19 transmission model is presented. The existence of the Caputo optimal control model is studied and necessary condition for optimality in the Caputo case is derived from Pontryagin's Maximum Principle. Finally, the effectiveness of the proposed control strategies are demonstrated through numerical simulations. The graphical results revealed that the implantation of time-dependent controls significantly reduces the number of infective cases and are useful in mitigating the infection.
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Affiliation(s)
| | - Ahmed Boudaoui
- Laboratory of Mathematics Modeling and Applications, University of Adrar, Algeria
| | - Saif Ullah
- Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa, Pakistan
| | - Khursheed Muzammil
- Department of Public Health, CAMS, Khamis Mushait Campus, King Khalid University, Abha, Kingdom of Saudi Arabia
| | - Muhammad Bilal Riaz
- Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowskiego St., 90-924 Lodz, Poland
- Department of Mathematics, University of Management and Technology, C-II Johar Town, 54770 Lahore, Pakistan
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Seddighi Chaharborj S, Seddighi Chaharborj S, Hassanzadeh Asl J, Phang PS. Controlling of pandemic COVID-19 using optimal control theory. RESULTS IN PHYSICS 2021; 26:104311. [PMID: 34094820 PMCID: PMC8168522 DOI: 10.1016/j.rinp.2021.104311] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/04/2021] [Revised: 05/03/2021] [Accepted: 05/06/2021] [Indexed: 06/12/2023]
Abstract
In 2019, a new infectious disease called pandemic COVID-19 began to spread from Wuhan, China. In spite of the efforts to stop the disease, being out of the control of the governments it spread rapidly all over the world. From then on, much research has been done in the world with the aim of controlling this contagious disease. A mathematical model for modeling the spread of COVID-19 and also controlling the spread of the disease has been presented in this paper. We find the disease-free equilibrium points as trivial equilibrium (TE), virus absenteeism equilibrium (VAE) and virus incidence equilibrium (VIE) for the proposed model; and at the trivial equilibrium point for the presented dynamic system we obtain the Jacobian matrix so as to be used in finding the largest eigenvalue. Radius spectral method has been used for finding the reproductive number. In the following, by adding a controller to the model and also using the theory of optimal control, we can improve the performance of the model. We must have a correct understanding of the system i.e. how it works, the various variables affecting the system, and the interaction of the variables on each other. To search for the optimal values, we need to use an appropriate optimization method. Given the limitations and needs of the problem, the aim of the optimization is to find the best solutions, to find conditions that result in the maximum of susceptiblity, the minimum of infection, and optimal quarantination.
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Affiliation(s)
| | | | - Jalal Hassanzadeh Asl
- Department of Mathematics, Faculty of Science, Tabriz Branch, Islamic Azad University, Tabriz, Iran
| | - Pei See Phang
- Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM, Malaysia
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