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Xun T, Chen P, Wang S, Pi Y, Luo Y. A fractional order friction model. ISA Trans 2023; 142:550-561. [PMID: 37558514 DOI: 10.1016/j.isatra.2023.07.027] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2021] [Revised: 07/20/2023] [Accepted: 07/20/2023] [Indexed: 08/11/2023]
Abstract
A new dynamic model for friction is proposed based on the commonality between frictional order dynamics and friction behavior. The model can achieve accurate modeling both in the sliding and the pre-sliding region with fewer parameters. Simulation results of the FOFM are illustrated with the parameters identification using the particle swarm optimization algorithm. The simulation shows that the fractional order friction model (FOFM) can capture most of the friction behavior accurately, including the hysteresis, non-local memory, Stribeck effect, and friction lag with only 7 parameters. Experimental results are also carried out to demonstrate the superiority of the proposed FOFM in terms of modeling accuracy and complexity, comparing with the well-known GMS model.
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Affiliation(s)
- Tianzhu Xun
- School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China
| | - Pengchong Chen
- School of Electrical and Information Engineering, Zhengzhou University, Zhengzhou, 450001, China
| | - Shaohua Wang
- School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China
| | - Youguo Pi
- School of Automation Science and Engineering, South China University of Technology, Guangzhou, 510641, China
| | - Ying Luo
- School of Automation Science and Engineering, South China University of Technology, Guangzhou, 510641, China.
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2
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Khan IU, Mustafa S, Shokri A, Li S, Akgül A, Bariq A. The stability analysis of a nonlinear mathematical model for typhoid fever disease. Sci Rep 2023; 13:15284. [PMID: 37714901 PMCID: PMC10504385 DOI: 10.1038/s41598-023-42244-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2023] [Accepted: 09/07/2023] [Indexed: 09/17/2023] Open
Abstract
Typhoid fever is a contagious disease that is generally caused by bacteria known as Salmonella typhi. This disease spreads through manure contamination of food or water and infects unprotected people. In this work, our focus is to numerically examine the dynamical behavior of a typhoid fever nonlinear mathematical model. To achieve our objective, we utilize a conditionally stable Runge-Kutta scheme of order 4 (RK-4) and an unconditionally stable non-standard finite difference (NSFD) scheme to better understand the dynamical behavior of the continuous model. The primary advantage of using the NSFD scheme to solve differential equations is its capacity to discretize the continuous model while upholding crucial dynamical properties like the solutions convergence to equilibria and its positivity for all finite step sizes. Additionally, the NSFD scheme does not only address the deficiencies of the RK-4 scheme, but also provides results that are consistent with the continuous system's solutions. Our numerical results demonstrate that RK-4 scheme is dynamically reliable only for lower step size and, consequently cannot exactly retain the important features of the original continuous model. The NSFD scheme, on the other hand, is a strong and efficient method that presents an accurate portrayal of the original model. The purpose of developing the NSFD scheme for differential equations is to make sure that it is dynamically consistent, which means to discretize the continuous model while keeping significant dynamical properties including the convergence of equilibria and positivity of solutions for all step sizes. The numerical simulation also indicates that all the dynamical characteristics of the continuous model are conserved by discrete NSFD scheme. The theoretical and numerical results in the current work can be engaged as a useful tool for tracking the occurrence of typhoid fever disease.
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Affiliation(s)
- Ihsan Ullah Khan
- Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan, 29050, KPK, Pakistan
| | - Shahbaz Mustafa
- Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan, 29050, KPK, Pakistan
| | - Ali Shokri
- Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, 83111-55181, Iran
| | - Shuo Li
- School of Mathematics and Data Sciences, Changji University, Changji, 831100, Xinjiang, People's Republic of China
| | - Ali Akgül
- Department of Computer Science and Mathematics, Lebanese American University, Beirut, 5053, Lebanon
- Mathematics Research Center, Department of Mathematics, Near East University, Near East Boulevard, 99138, Nicosia, Mersin, Turkey
- Department of Mathematics, Art and Science Faculty, Siirt University, 56100, Siirt, Turkey
| | - Abdul Bariq
- Department of Mathematics, Laghman University, Mehtarlam City, Laghman, 2701, Afghanistan.
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3
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Alrabaiah H, Din RU, Ansari KJ, Ur Rehman Irshad A, Ozdemir B. Stability and numerical analysis via non-standard finite difference scheme of a nonlinear classical and fractional order model. Results Phys 2023; 49:106536. [PMID: 37214757 PMCID: PMC10184875 DOI: 10.1016/j.rinp.2023.106536] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/12/2023] [Revised: 05/05/2023] [Accepted: 05/08/2023] [Indexed: 05/24/2023]
Abstract
In this paper, we develop a new mathematical model for an in-depth understanding of COVID-19 (Omicron variant). The mathematical study of an omicron variant of the corona virus is discussed. In this new Omicron model, we used idea of dividing infected compartment further into more classes i.e asymptomatic, symptomatic and Omicron infected compartment. Model is asymptotically locally stable whenever R0<1 and when R0≤1 at disease free equilibrium the system is globally asymptotically stable. Local stability is investigated with Jacobian matrix and with Lyapunov function global stability is analyzed. Moreover basic reduction number is calculated through next generation matrix and numerical analysis will be used to verify the model with real data. We consider also the this model under fractional order derivative. We use Grunwald-Letnikov concept to establish a numerical scheme. We use nonstandard finite difference (NSFD) scheme to simulate the results. Graphical presentations are given corresponding to classical and fractional order derivative. According to our graphical results for the model with numerical parameters, the population's risk of infection can be reduced by adhering to the WHO's suggestions, which include keeping social distances, wearing facemasks, washing one's hands, avoiding crowds, etc.
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Affiliation(s)
- Hussam Alrabaiah
- Al Ain University, Al Ain, United Arab Emirates
- Mathematics Department, Tafila Technical University, Tafila, Jordan
| | - Rahim Ud Din
- Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan
| | - Khursheed J Ansari
- Department of Mathematics, College of Science, King Khalid University, 61413, Abha, Saudi Arabia
| | - Ateeq Ur Rehman Irshad
- Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia
| | - Burhanettin Ozdemir
- Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia
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Othman A.M. Omar, Reda A. Elbarkouky, Hamdy M. Ahmed. Fractional stochastic modelling of COVID-19 under wide spread of vaccinations: Egyptian case study. Alexandria Engineering Journal 2022; 61. [ DOI: 10.1016/j.aej.2022.02.002] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
This work predicts the dynamics of the COVID-19 under widespread vaccination to anticipate the virus's current and future waves. We focused on establishing two population-based models for predictions: the fractional-order model and the fractional-order stochastic model. Based on dose efficacy, which is one of the main imposed assumptions in our study, some vaccinated people will probably be exposed to infection by the same viral wave. We validated the generated models by applying them to the current viral wave in Egypt. We assumed that the Egyptian current wave began on 10th September 2021. Using current actual data and varying our models’ fractional orders, we generate different predicted wave scenarios. The numerical solution of our models is obtained using the fractional Euler method and the fractional Euler Maruyama method. At the end, we compared the current predicted wave under a high vaccination rate with the previous viral wave. Through this comparison, the vaccination control effect is quantified.
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Owoyemi AE, Sulaiman IM, Kumar P, Govindaraj V, Mamat M. Some novel mathematical analysis on the fractional-order 2019-nCoV dynamical model. Math Methods Appl Sci 2022; 46:MMA8772. [PMID: 36714679 PMCID: PMC9874666 DOI: 10.1002/mma.8772] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/13/2021] [Revised: 07/13/2022] [Accepted: 09/21/2022] [Indexed: 06/18/2023]
Abstract
Since December 2019, the whole world has been facing the big challenge of Covid-19 or 2019-nCoV. Some nations have controlled or are controlling the spread of this virus strongly, but some countries are in big trouble because of their poor control strategies. Nowadays, mathematical models are very effective tools to simulate outbreaks of this virus. In this research, we analyze a fractional-order model of Covid-19 in terms of the Caputo fractional derivative. First, we generalize an integer-order model to a fractional sense, and then, we check the stability of equilibrium points. To check the dynamics of Covid-19, we plot several graphs on the time scale of daily and monthly cases. The main goal of this content is to show the effectiveness of fractional-order models as compared to integer-order dynamics.
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Affiliation(s)
| | - Ibrahim Mohammed Sulaiman
- Institute of Strategic Industrial Decision Modelling (ISIDM), School of Quantitative SciencesUniversiti Utara MalaysiaSintok06010KedahMalaysia
| | - Pushpendra Kumar
- Institute for the Future of KnowledgeUniversity of JohannesburgPO Box 524Auckland Park2006South Africa
| | - Venkatesan Govindaraj
- Department of MathematicsNational Institute of Technology PuducherryKaraikal609609India
| | - Mustafa Mamat
- Faculty of Informatics and ComputingUniversiti Sultan Zainal AbidinKuala TerengganuMalaysia
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Vellappandi M, Kumar P, Govindaraj V. A case study of 2019-nCoV in Russia using integer and fractional order derivatives. Math Methods Appl Sci 2022; 46:MMA8736. [PMID: 36247230 PMCID: PMC9538883 DOI: 10.1002/mma.8736] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/08/2022] [Revised: 08/27/2022] [Accepted: 09/08/2022] [Indexed: 06/16/2023]
Abstract
In this article, we define a mathematical model to analyze the outbreaks of the most deadly disease of the decade named 2019-nCoV by using integer and fractional order derivatives. For the case study, the real data of Russia is taken to perform novel parameter estimation by using the Trust Region Reflective (TRR) algorithm. First, we define an integer order model and then generalize it by using fractional derivatives. A novel optimal control problem is derived to see the impact of possible preventive measures against the spread of 2019-nCoV. We implement the forward-backward sweep method to numerically solve our proposed model and control problem. A number of graphs have been plotted to see the impact of the proposed control practically. The Russian data-based parameter estimation along with the proposal of a mathematical model in the sense of Caputo fractional derivative that contains the memory term in the system are the main novel features of this study.
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Affiliation(s)
- M. Vellappandi
- Department of MathematicsNational Institute of Technology PuducherryKaraikalIndia
| | - Pushpendra Kumar
- Department of MathematicsNational Institute of Technology PuducherryKaraikalIndia
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Tsvetkov VP, Mikheev SA, Tsvetkov IV, Derbov VL, Gusev AA, Vinitsky SI. Modeling the multifractal dynamics of COVID-19 pandemic. Chaos Solitons Fractals 2022; 161:112301. [PMID: 35755146 PMCID: PMC9212712 DOI: 10.1016/j.chaos.2022.112301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/28/2021] [Revised: 05/23/2022] [Accepted: 06/02/2022] [Indexed: 06/15/2023]
Abstract
To describe the COVID-19 pandemic, we propose to use a mathematical model of multifractal dynamics, which is alternative to other models and free of their shortcomings. It is based on the fractal properties of pandemics only and allows describing their time behavior using no hypotheses and assumptions about the structure of the disease process. The model is applied to describe the dynamics of the COVID-19 pandemic from day 1 to day 699 from the beginning of the pandemic. The calculated parameters of the model accurately determine the parameters of the trend and the large jump in daily diseases in this time interval. Within the framework of this model and finite-difference parametric nonlinear equations of the reduced SIR (Susceptible-Infected-Removed) model, the fractal dimensions of various segments of daily incidence in the world and variations in the main reproduction number of COVID-19 were calculated based on the data of COVID-19 world statistics.
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Affiliation(s)
- V P Tsvetkov
- Tver State University, 33, Zhelyabova St., Tver 170100, Russia
| | - S A Mikheev
- Tver State University, 33, Zhelyabova St., Tver 170100, Russia
| | - I V Tsvetkov
- Tver State University, 33, Zhelyabova St., Tver 170100, Russia
| | - V L Derbov
- N.G. Chernyshevsky Saratov National Research State University, Saratov, Russia
| | - A A Gusev
- Joint Institute for Nuclear Research, Dubna, Russia
| | - S I Vinitsky
- Joint Institute for Nuclear Research, Dubna, Russia
- Peoples' Friendship University of Russia (RUDN University), 117198 Moscow, Russia
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8
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Khan MA, Atangana A. Mathematical modeling and analysis of COVID-19: A study of new variant Omicron. Physica A 2022; 599:127452. [PMID: 35498561 PMCID: PMC9040451 DOI: 10.1016/j.physa.2022.127452] [Citation(s) in RCA: 22] [Impact Index Per Article: 11.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2022] [Revised: 03/31/2022] [Indexed: 05/09/2023]
Abstract
We construct a new mathematical model to better understand the novel coronavirus (omicron variant). We briefly present the modeling of COVID-19 with the omicron variant and present their mathematical results. We study that the Omicron model is locally asymptotically stable if the basic reproduction number R 0 < 1 , while for R 0 ≤ 1 , the model at the disease-free equilibrium is globally asymptotically stable. We extend the model to the second-order differential equations to study the possible occurrence of the layers(waves). We then extend the model to a fractional stochastic version and studied its numerical results. The real data for the period ranging from November 1, 2021, to January 23, 2022, from South Africa are considered to obtain the realistic values of the model parameters. The basic reproduction number for the suggested data is found to be approximate R 0 ≈ 2 . 1107 which is very close to the actual basic reproduction in South Africa. We perform the global sensitivity analysis using the PRCC method to investigate the most influential parameters that increase or decrease R 0 . We use the new numerical scheme recently reported for the solution of piecewise fractional differential equations to present the numerical simulation of the model. Some graphical results for the model with sensitive parameters are given which indicate that the infection in the population can be minimized by following the recommendations of the world health organizations (WHO), such as social distances, using facemasks, washing hands, avoiding gathering, etc.
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Affiliation(s)
- Muhammad Altaf Khan
- Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa
| | - Abdon Atangana
- Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa
- Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
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9
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Tang B, Xia F, Bragazzi NL, McCarthy Z, Wang X, He S, Sun X, Tang S, Xiao Y, Wu J. Lessons drawn from China and South Korea for managing COVID-19 epidemic: Insights from a comparative modeling study. ISA Trans 2022; 124:164-175. [PMID: 35164963 PMCID: PMC8713134 DOI: 10.1016/j.isatra.2021.12.004] [Citation(s) in RCA: 16] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2020] [Revised: 12/04/2021] [Accepted: 12/05/2021] [Indexed: 05/21/2023]
Abstract
We conducted a comparative study of the COVID-19 epidemic in three different settings: mainland China, the Guangdong province of China and South Korea, by formulating two disease transmission dynamics models which incorporate epidemic characteristics and setting-specific interventions, and fitting the models to multi-source data to identify initial and effective reproduction numbers and evaluate effectiveness of interventions. We estimated the initial basic reproduction number for South Korea, the Guangdong province and mainland China as 2.6 (95% confidence interval (CI): (2.5, 2.7)), 3.0 (95%CI: (2.6, 3.3)) and 3.8 (95%CI: (3.5,4.2)), respectively, given a serial interval with mean of 5 days with standard deviation of 3 days. We found that the effective reproduction number for the Guangdong province and mainland China has fallen below the threshold 1 since February 8th and 18th respectively, while the effective reproduction number for South Korea remains high until March 2nd Moreover our model-based analysis shows that the COVID-19 epidemics in South Korean is almost under control with the cumulative confirmed cases tending to be stable as of April 14th. Through sensitivity analysis, we show that a coherent and integrated approach with stringent public health interventions is the key to the success of containing the epidemic in China and especially its provinces outside its epicenter. In comparison, we find that the extremely high detection rate is the key factor determining the success in controlling the COVID-19 epidemics in South Korea. The experience of outbreak control in mainland China and South Korea should be a guiding reference for the rest of the world.
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Affiliation(s)
- Biao Tang
- The Interdisciplinary Research Center for Mathematics and Life Sciences, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China; Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada, M3J 1P3
| | - Fan Xia
- The Interdisciplinary Research Center for Mathematics and Life Sciences, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China; School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
| | - Nicola Luigi Bragazzi
- Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada, M3J 1P3
| | - Zachary McCarthy
- Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada, M3J 1P3; Fields-CQAM Laboratory of Mathematics for Public Health, York University, Toronto, Ontario, Canada, M3J 1P3
| | - Xia Wang
- School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710119, People's Republic of China
| | - Sha He
- School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710119, People's Republic of China
| | - Xiaodan Sun
- The Interdisciplinary Research Center for Mathematics and Life Sciences, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China; School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
| | - Sanyi Tang
- School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710119, People's Republic of China
| | - Yanni Xiao
- The Interdisciplinary Research Center for Mathematics and Life Sciences, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China; School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
| | - Jianhong Wu
- The Interdisciplinary Research Center for Mathematics and Life Sciences, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China; Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada, M3J 1P3; Fields-CQAM Laboratory of Mathematics for Public Health, York University, Toronto, Ontario, Canada, M3J 1P3
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Kumar P, Govindaraj V, Erturk VS, Mohamed MS. Effects of greenhouse gases and hypoxia on the population of aquatic species: a fractional mathematical model. Adv Contin Discret Model 2022; 2022:31. [PMID: 35450200 PMCID: PMC9010246 DOI: 10.1186/s13662-022-03679-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/22/2021] [Accepted: 12/30/2021] [Indexed: 11/10/2022]
Abstract
Study of ecosystems has always been an interesting topic in the view of real-world dynamics. In this paper, we propose a fractional-order nonlinear mathematical model to describe the prelude of deteriorating quality of water cause of greenhouse gases on the population of aquatic animals. In the proposed system, we recall that greenhouse gases raise the temperature of water, and because of this reason, the dissolved oxygen level goes down, and also the rate of circulation of disintegrated oxygen by the aquatic animals rises, which causes a decrement in the density of aquatic species. We use a generalized form of the Caputo fractional derivative to describe the dynamics of the proposed problem. We also investigate equilibrium points of the given fractional-order model and discuss the asymptotic stability of the equilibria of the proposed autonomous model. We recall some important results to prove the existence of a unique solution of the model. For finding the numerical solution of the established fractional-order system, we apply a generalized predictor-corrector technique in the sense of proposed derivative and also justify the stability of the method. To express the novelty of the simulated results, we perform a number of graphs at various fractional-order cases. The given study is fully novel and useful for understanding the proposed real-world phenomena.
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Affiliation(s)
- Pushpendra Kumar
- Department of Mathematics, National Institute of Technology Puducherry, Karaikal, 609609 India
| | - V Govindaraj
- Department of Mathematics, National Institute of Technology Puducherry, Karaikal, 609609 India
| | - Vedat Suat Erturk
- Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayis University, Atakum, 55200 Samsun Turkey
| | - Mohamed S Mohamed
- Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif, 21944 Saudi Arabia
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Ebubeogu AF, Ozigbu CE, Maswadi K, Seixas A, Ofem P, Conserve DF. Predicting the number of COVID-19 infections and deaths in USA. Global Health 2022; 18:37. [PMID: 35346262 PMCID: PMC8959784 DOI: 10.1186/s12992-022-00827-3] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2022] [Accepted: 03/03/2022] [Indexed: 12/20/2022] Open
Abstract
Background Uncertainties surrounding the 2019 novel coronavirus (COVID-19) remain a major global health challenge and requires attention. Researchers and medical experts have made remarkable efforts to reduce the number of cases and prevent future outbreaks through vaccines and other measures. However, there is little evidence on how severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection entropy can be applied in predicting the possible number of infections and deaths. In addition, more studies on how the COVID-19 infection density contributes to the rise in infections are needed. This study demonstrates how the SARS-COV-2 daily infection entropy can be applied in predicting the number of infections within a given period. In addition, the infection density within a given population attributes to an increase in the number of COVID-19 cases and, consequently, the new variants. Results Using the COVID-19 initial data reported by Johns Hopkins University, World Health Organization (WHO) and Global Initiative on Sharing All Influenza Data (GISAID), the result shows that the original SAR-COV-2 strain has R0<1 with an initial infection growth rate entropy of 9.11 bits for the United States (U.S.). At close proximity, the average infection time for an infected individual to infect others within a susceptible population is approximately 7 minutes. Assuming no vaccines were available, in the U.S., the number of infections could range between 41,220,199 and 82,440,398 in late March 2022 with approximately, 1,211,036 deaths. However, with the available vaccines, nearly 48 Million COVID-19 cases and 706, 437 deaths have been prevented. Conclusion The proposed technique will contribute to the ongoing investigation of the COVID-19 pandemic and a blueprint to address the uncertainties surrounding the pandemic.
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Pushpendra Kumar, Vedat Suat Erturk, Kottakkaran Sooppy Nisar, Wasim Jamshed, Mohamed S. Mohamed. Fractional dynamics of 2019-nCOV in Spain at different transmission rate with an idea of optimal control problem formulation. Alexandria Engineering Journal 2022; 61. [ DOI: 10.1016/j.aej.2021.07.005] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2021] [Revised: 04/18/2021] [Accepted: 07/10/2021] [Indexed: 05/24/2023]
Abstract
In this article, we studied the fractional dynamics of the most dangerous deathly disease which outbreaks have been recorded all over the world, called 2019-nCOV or COVID-19. We used the numerical values of the given parameters based on the real data of the 2019-nCOV cases in Spain for the time duration of 25 February to 9 October 2020. We performed our observations with the help of the Atangana-Baleanu (AB) non-integer order derivative. We analysed the optimal control problem in a fractional sense for giving the information on all necessary health care issues. We applied the Predictor-Corrector method to do the important graphical simulations. Also, we provided the analysis related to the existence of a unique solution and the stability of the proposed scheme. The aim and the main contribution of this research is to analyse the structure of novel coronavirus in Spain at different transmission rate and to indicate the danger of this deathly disease for future with the introduction of some optimal controls and health care measures.
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Zeb A, Kumar P, Erturk VS, Sitthiwirattham T. A new study on two different vaccinated fractional-order COVID-19 models via numerical algorithms. J King Saud Univ Sci 2022; 34:101914. [PMID: 35194351 PMCID: PMC8851876 DOI: 10.1016/j.jksus.2022.101914] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/05/2021] [Revised: 09/27/2021] [Accepted: 02/14/2022] [Indexed: 05/24/2023]
Abstract
The main purpose of this paper is to provide new vaccinated models of COVID-19 in the sense of Caputo-Fabrizio and new generalized Caputo-type fractional derivatives. The formulation of the given models is presented including an exhaustive study of the model dynamics such as positivity, boundedness of the solutions and local stability analysis. Furthermore, the unique solution existence for the proposed fractional order models is discussed via fixed point theory. Numerical solutions are also derived by using two-steps Adams-Bashforth algorithm for Caputo-Fabrizio operator, and modified Predictor-Corrector method for generalised Caputo fractional derivative. Our analysis allow to show that the given fractional-order models exemplify the dynamics of COVID-19 much better than the classical ones. Also, the analysis on the convergence and stability for the proposed methods are performed. By this study, we see that how the vaccine availability plays an important role in the control of COVID-19 infection.
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Affiliation(s)
- Anwar Zeb
- Department of Mathematics, COMSATS University Islamabad, Abbottabad 22060, K.P.K, Pakistan
| | - Pushpendra Kumar
- Department of Mathematics and Statistics, School of Basic and Applied Sciences, Central University of Punjab, Bathinda, Punjab, 151001, India
| | - Vedat Suat Erturk
- Department of Mathematics, Ondokuz Mayis University, Atakum-55200, Samsun, Turkey
| | - Thanin Sitthiwirattham
- Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand
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Danane J, Hammouch Z, Allali K, Rashid S, Singh J. A fractional-order model of coronavirus disease 2019 (COVID-19) with governmental action and individual reaction. Math Methods Appl Sci 2021; 46:MMA7759. [PMID: 34908634 PMCID: PMC8661979 DOI: 10.1002/mma.7759] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/29/2021] [Revised: 04/05/2021] [Accepted: 04/13/2021] [Indexed: 05/06/2023]
Abstract
The deadly coronavirus disease 2019 (COVID-19) has recently affected each corner of the world. Many governments of different countries have imposed strict measures in order to reduce the severity of the infection. In this present paper, we will study a mathematical model describing COVID-19 dynamics taking into account the government action and the individuals reaction. To this end, we will suggest a system of seven fractional deferential equations (FDEs) that describe the interaction between the classical susceptible, exposed, infectious, and removed (SEIR) individuals along with the government action and individual reaction involvement. Both human-to-human and zoonotic transmissions are considered in the model. The well-posedness of the FDEs model is established in terms of existence, positivity, and boundedness. The basic reproduction number (BRN) is found via the new generation matrix method. Different numerical simulations were carried out by taking into account real reported data from Wuhan, China. It was shown that the governmental action and the individuals' risk awareness reduce effectively the infection spread. Moreover, it was established that with the fractional derivative, the infection converges more quickly to its steady state.
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Affiliation(s)
- Jaouad Danane
- Laboratory of Systems Modeling and Analysis for Decision Support, National School of Applied SciencesHassan I UniversityBerrechidMorocco
| | - Zakia Hammouch
- Division of Applied MathematicsThu Dau Mot UniversityThu Dau MotBinh Duong ProvinceVietnam
- Department of Medical ResearchChina Medical University HospitalTaichungTaiwan
- Département des Sciences, École Normale supéieureMoulay Ismail University of MeknésMeknesMorocco
| | - Karam Allali
- Laboratory of Mathematics and Applications, Faculty of Sciences and Techniques MohammediaUniversity Hassan II‐CasablancaMohammediaMorocco
| | - Saima Rashid
- Department of MathematicsGovernment College UniversityFaisalabadPakistan
| | - Jagdev Singh
- Department of MathematicsJECRC UniversityJaipurRajasthanIndia
- Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of SciencesKing Abdulaziz UniversityJeddahSaudi Arabia
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15
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Kumar P, Erturk VS, Murillo-Arcila M, Banerjee R, Manickam A. A case study of 2019-nCOV cases in Argentina with the real data based on daily cases from March 03, 2020 to March 29, 2021 using classical and fractional derivatives. Adv Differ Equ 2021; 2021:341. [PMID: 34306044 PMCID: PMC8290213 DOI: 10.1186/s13662-021-03499-2] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/16/2021] [Accepted: 07/07/2021] [Indexed: 05/31/2023]
Abstract
In this study, our aim is to explore the dynamics of COVID-19 or 2019-nCOV in Argentina considering the parameter values based on the real data of this virus from March 03, 2020 to March 29, 2021 which is a data range of more than one complete year. We propose a Atangana-Baleanu type fractional-order model and simulate it by using predictor-corrector (P-C) method. First we introduce the biological nature of this virus in theoretical way and then formulate a mathematical model to define its dynamics. We use a well-known effective optimization scheme based on the renowned trust-region-reflective (TRR) method to perform the model calibration. We have plotted the real cases of COVID-19 and compared our integer-order model with the simulated data along with the calculation of basic reproductive number. Concerning fractional-order simulations, first we prove the existence and uniqueness of solution and then write the solution along with the stability of the given P-C method. A number of graphs at various fractional-order values are simulated to predict the future dynamics of the virus in Argentina which is the main contribution of this paper.
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Affiliation(s)
- Pushpendra Kumar
- Department of Mathematics and Statistics, School of Basic and Applied Sciences, Central University of Punjab, Bathinda, Punjab 151001 India
| | - Vedat Suat Erturk
- Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayis University, Atakum, 55200 Samsun, Turkey
| | - Marina Murillo-Arcila
- Instituto Universitario de Matematica Pura y Aplicada, Universitat Politècnica de València, 46022 Valencia, Spain
| | - Ramashis Banerjee
- Department of Electrical Engineering, National Institute of Technology, Silchar, India
| | - A. Manickam
- School of Advanced Sciences & Languages, Department of Mathematics, VIT Bhopal University, Kottri Kalan (Village), 466 114 Sehore (District), Madhya Pradesh India
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16
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Kumar P, Erturk VS, Yusuf A, Nisar KS, Abdelwahab SF. A study on canine distemper virus (CDV) and rabies epidemics in the red fox population via fractional derivatives. Results in Physics 2021; 25:104281. [DOI: 10.1016/j.rinp.2021.104281] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/08/2023]
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17
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Kumar P, Erturk VS, Murillo-Arcila M. A new fractional mathematical modelling of COVID-19 with the availability of vaccine. Results Phys 2021; 24:104213. [PMID: 33898210 PMCID: PMC8057057 DOI: 10.1016/j.rinp.2021.104213] [Citation(s) in RCA: 24] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/09/2021] [Revised: 03/11/2021] [Accepted: 04/13/2021] [Indexed: 05/10/2023]
Abstract
The most dangerous disease of this decade novel coronavirus or COVID-19 is yet not over. The whole world is facing this threat and trying to stand together to defeat this pandemic. Many countries have defeated this virus by their strong control strategies and many are still trying to do so. To date, some countries have prepared a vaccine against this virus but not in an enough amount. In this research article, we proposed a new SEIRS dynamical model by including the vaccine rate. First we formulate the model with integer order and after that we generalize it in Atangana-Baleanu derivative sense. The high motivation to apply Atangana-Baleanu fractional derivative on our model is to explore the dynamics of the model more clearly. We provide the analysis of the existence of solution for the given fractional SEIRS model. We use the famous Predictor-Corrector algorithm to derive the solution of the model. Also, the analysis for the stability of the given algorithm is established. We simulate number of graphs to see the role of vaccine on the dynamics of the population. For practical simulations, we use the parameter values which are based on real data of Spain. The main motivation or aim of this research study is to justify the role of vaccine in this tough time of COVID-19. A clear role of vaccine at this crucial time can be realized by this study.
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Affiliation(s)
- Pushpendra Kumar
- Department of Mathematics and Statistics, School of Basic and Applied Sciences, Central University of Punjab, Bathinda, Punjab 151001, India
| | - Vedat Suat Erturk
- Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayis University, Atakum 55200, Samsun, Turkey
| | - Marina Murillo-Arcila
- Instituto Universitario de Matematica Pura y Aplicada, Universitat Politècnica de València, 46022 Valencia, Spain
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