PubMed İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8650
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Article Citation - WoS: 72Citation - Scopus: 78Dynamical Behaviours and Stability Analysis of a Generalized Fractional Model With a Real Case Study(Elsevier, 2023) Baleanu, D.; Arshad, S.; Jajarmi, A.; Shokat, W.; Ghassabzade, F. Akhavan; Wali, M.Introduction: Mathematical modelling is a rapidly expanding field that offers new and interesting oppor-tunities for both mathematicians and biologists. Concerning COVID-19, this powerful tool may help humans to prevent the spread of this disease, which has affected the livelihood of all people badly. Objectives: The main objective of this research is to explore an efficient mathematical model for the investigation of COVID-19 dynamics in a generalized fractional framework.Methods: The new model in this paper is formulated in the Caputo sense, employs a nonlinear time -varying transmission rate, and consists of ten population classes including susceptible, infected, diag-nosed, ailing, recognized, infected real, threatened, diagnosed recovered, healed, and extinct people. The existence of a unique solution is explored for the new model, and the associated dynamical beha-viours are discussed in terms of equilibrium points, invariant region, local and global stability, and basic reproduction number. To implement the proposed model numerically, an efficient approximation scheme is employed by the combination of Laplace transform and a successive substitution approach; besides, the corresponding convergence analysis is also investigated.Results: Numerical simulations are reported for various fractional orders, and simulation results are com-pared with a real case of COVID-19 pandemic in Italy. By using these comparisons between the simulated and measured data, we find the best value of the fractional order with minimum absolute and relative errors. Also, the impact of different parameters on the spread of viral infection is analyzed and studied.Conclusion: According to the comparative results with real data, we justify the use of fractional concepts in the mathematical modelling, for the new non-integer formalism simulates the reality more precisely than the classical framework.& COPY; 2023 The Authors. Published by Elsevier B.V. on behalf of Cairo University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Article Citation - WoS: 17Citation - Scopus: 21Nonstandard Finite Difference Method for Solving Complex-Order Fractional Burgers' Equations(Elsevier, 2020) AL-Mekhlafi, S. M.; Baleanu, D.; Sweilam, N. H.The aim of this work is to present numerical treatments to a complex order fractional nonlinear one-dimensional problem of Burgers' equations. A new parameter sigma(t) is presented in order to be consistent with the physical model problem. This parameter characterizes the existence of fractional structures in the equations. A relation between the parameter sigma(t) and the time derivative complex order is derived. An unconditionally stable numerical scheme using a kind of weighted average nonstandard finite-difference discretization is presented. Stability analysis of this method is studied. Numerical simulations are given to confirm the reliability of the proposed method. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Cairo University.Article Citation - WoS: 33Citation - Scopus: 35Mathematical Analysis for the Effect of Voluntary Vaccination on the Propagation of Corona Virus Pandemic(Elsevier, 2021) Abbas, M.; Rafiq, M.; Baleanu, D.; Ahmad, W.In this manuscript, a new nonlinear model for the rapidly spreading Corona virus disease (COVID-19) is developed. We incorporate an additional class of vaccinated humans which ascertains the impact of vaccination strategy for susceptible humans. A complete mathematical analysis of this model is conducted to predict the dynamics of Corona virus in the population. The analysis proves the effectiveness of vaccination strategy employed and helps public health services to control or to reduce the burden of corona virus pandemic. We first prove the existence and uniqueness and then boundedness and positivity of solutions. Threshold parameter for the vaccination model is computed analytically. Stability of the proposed model at fixed points is investigated analytically with the help of threshold parameter to examine epidemiological relevance of the pandemic. We apply LaSalle's invariance principle from the theory of Lyapunov function to prove the global stability of both the equilibria. Two well known numerical techniques namely Runge-Kutta method of order 4 (RK4), and the Non-Standard Finite Difference (NSFD) method are employed to solve the system of ODE's and to validate our obtained theoretical results. For different coverage levels of voluntary vaccination, we explored a complete quantitative analysis of the model. To draw our conclusions, the effect of proposed vaccination on threshold parameter is studied numerically. It is claimed that Corona virus disease could be eradicated faster if a human community selfishly adopts mandatory vaccination measures at various coverage levels with proper awareness. Finally, we have executed the joint variability of all classes to understand the effect of vaccination strategy on a disease dynamics.Article Citation - WoS: 38Citation - Scopus: 38A Hybrid Fractional Optimal Control for a Novel Coronavirus (2019-Ncov) Mathematical Model(Elsevier, 2021) AL-Mekhlafi, S. M.; Baleanu, D.; Sweilam, N. H.Introduction: Coronavirus COVID-19 pandemic is the defining global health crisis of our time and the greatest challenge we have faced since world war two. To describe this disease mathematically, we noted that COVID-19, due to uncertainties associated to the pandemic, ordinal derivatives and their associated integral operators show deficient. The fractional order differential equations models seem more consistent with this disease than the integer order models. This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence there is a growing need to study and use the fractional order differential equations. Also, optimal control theory is very important topic to control the variables in mathematical models of infectious disease. Moreover, a hybrid fractional operator which may be expressed as a linear combination of the Caputo fractional derivative and the Riemann-Liouville fractional integral is recently introduced. This new operator is more general than the operator of Caputo's fractional derivative. Numerical techniques are very important tool in this area of research because most fractional order problems do not have exact analytic solutions. Objectives: A novel fractional order Coronavirus (2019-nCov) mathematical model with modified parameters will be presented. Optimal control of the suggested model is the main objective of this work. Three control variables are presented in this model to minimize the number of infected populations. Necessary control conditions will be derived. Methods: The numerical methods used to study the fractional optimality system are the weighted average nonstandard finite difference method and the Grunwald-Letnikov nonstandard finite difference method. Results: The proposed model with a new fractional operator is presented. We have successfully applied a kind of Pontryagin's maximum principle and were able to reduce the number of infected people using the proposed numerical methods. The weighted average nonstandard finite difference method with the new operator derivative has the best results than Grunwald-Letnikov nonstandard finite difference method with the same operator. Moreover, the proposed methods with the new operator have the best results than the proposed methods with Caputo operator. Conclusions: The combination of fractional order derivative and optimal control in the Coronavirus (2019-nCov) mathematical model improves the dynamics of the model. The new operator is more general and suitable to study the optimal control of the proposed model than the Caputo operator and could be more useful for the researchers and scientists. (C) 2021 The Authors. Published by Elsevier B.V. on behalf of Cairo University.