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Saleem MU, Farman M, Nisar KS, Ahmad A, Munir Z, Hincal E. Investigation and application of a classical piecewise hybrid with a fractional derivative for the epidemic model: Dynamical transmission and modeling. PLoS One 2024; 19:e0307732. [PMID: 39208269 PMCID: PMC11361697 DOI: 10.1371/journal.pone.0307732] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/04/2024] [Accepted: 07/09/2024] [Indexed: 09/04/2024] Open
Abstract
In this research, we developed an epidemic model with a combination of Atangana-Baleanu Caputo derivative and classical operators for the hybrid operator's memory effects, allowing us to observe the dynamics and treatment effects at different time phases of syphilis infection caused by sex. The developed model properties, which take into account linear growth and Lipschitz requirements relating the rate of effects within its many sub-compartments according to the equilibrium points, include positivity, unique solution, exitance, and boundedness in the feasible domain. After conducting sensitivity analysis with various parameters influencing the model for the piecewise fractional operator, the reproductive number R0 for the biological viability of the model is determined. Generalized Ulam-Hyers stability results are employed to preserve global stability. The investigated model thus has a unique solution in the specified subinterval in light of the Banach conclusion, and contraction as a consequence holds for the Atangana-Baleanu Caputo derivative with classical operators. The piecewise model that has been suggested has a maximum of one solution. For numerical solutions, piecewise fractional hybrid operators at various fractional order values are solved using the Newton polynomial interpolation method. A comparison is also made between Caputo operator and the piecewise derivative proposed operator. This work improves our knowledge of the dynamics of syphilis and offers a solid framework for assessing the effectiveness of interventions for planning and making decisions to manage the illness.
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Affiliation(s)
| | - Muhammad Farman
- Mathematics Research Center, Near East University, Nicosia, North Cyprus, Turkey
- Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
| | - Kottakkaran Sooppy Nisar
- Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj, Saudi Arabia
- Saveetha School of Engineering, SIMATS, Chennai, India
| | - Aqeel Ahmad
- Mathematics Research Center, Near East University, Nicosia, North Cyprus, Turkey
- Department of Mathematics, Ghazi University, D G Khan, Pakistan
| | - Zainab Munir
- Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan, Pakistan
| | - Evren Hincal
- Mathematics Research Center, Near East University, Nicosia, North Cyprus, Turkey
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Agrawal K, Kumar S. Wavelet Collocation Method for HIV-1/HTLV-I Co-Infection Model Using Hermite Polynomial. Adv Biol (Weinh) 2024:e2300629. [PMID: 39123297 DOI: 10.1002/adbi.202300629] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/18/2023] [Revised: 02/17/2024] [Indexed: 08/12/2024]
Abstract
In this study, the dynamic behavior of fractional order co-infection model with human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) is analyzed using operational matrix of Hermite wavelet collocation method. Also, the uniqueness and existence of solutions are calculated based on the fixed point hypothesis. For the fractional order co-infection model, its positivity and boundedness are demonstrated. Furthermore, different types of Ulam-Hyres stability are also discussed. The numerical solution of the model are obtained by using the operational matrix of the Hermite wavelet approach. This scheme is used to solve the system of nonlinear equations that are very fruitful and easy to implement. Additionally, the stability analysis of the numerical scheme is explained. The mathematical model taken in this work incorporates the biological characteristics of both HIV-1 and HTLV-I. After that all the equilibrium points of the fractional order co-infection model are found and their existence conditions are explored with the help of the Caputo derivative. The global stability of all equilibrium points of this model are determined with the help of Lyapunov functions and the LaSalle invariance principle. Convergence analysis is also discussed. Hermite wavelet operational matrix methods are more accurate and convergent than other numerical methods. Lastly, variations in model dynamics are found when examining different fractional order values. These findings will be valuable to biologists in the treatment of HIV-1/HTLV-I.
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Affiliation(s)
- Khushbu Agrawal
- Department of Mathematics, National Institute of Technology, Jamshedpur, Jharkhand, 831014, India
| | - Sunil Kumar
- Department of Mathematics, National Institute of Technology, Jamshedpur, Jharkhand, 831014, India
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A simulation of undiagnosed population and excess mortality during the COVID-19 pandemic. RESULTS IN CONTROL AND OPTIMIZATION 2023; 12:100262. [PMCID: PMC10290741 DOI: 10.1016/j.rico.2023.100262] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2023] [Revised: 06/19/2023] [Accepted: 06/20/2023] [Indexed: 06/21/2024]
Abstract
Whereas the extent of outbreak of COVID-19 is usually accessed via the number of reported cases and the number of patients succumbed to the disease, the officially recorded overall excess mortality numbers during the pandemic waves, which are significant and often followed the rise and fall of the pandemic waves, put a question mark on the above methodology. Gradually it has been recognized that estimating the size of the undiagnosed population (which includes asymptomatic cases and symptomatic cases but not reported) is also crucial. Here we used the classical mathematical SEIR model having an additional compartment, that is the undiagnosed group in addition to the susceptible, exposed, diagnosed, recovered and deceased groups, to link the undiagnosed COVID-19 cases to the reported excess mortality numbers and thereby try to know the actual size of the disease outbreak. The developed model wase successfully applied to relevant COVID-19 waves in USA (initial months of 2020), South Africa (mid of 2021) and Russia (2020–21) when a large discrepancy between the reported COVID-19 mortality and the overall excess mortality had been noticed.
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A new fractional mathematical model to study the impact of vaccination on COVID-19 outbreaks. DECISION ANALYTICS JOURNAL 2023; 6:100156. [PMCID: PMC9789549 DOI: 10.1016/j.dajour.2022.100156] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/05/2022] [Revised: 12/16/2022] [Accepted: 12/22/2022] [Indexed: 01/23/2024]
Abstract
This study proposes a new fractional mathematical model to study the impact of vaccination on COVID-19 outbreaks by categorizing infected people into non-vaccinated, first dose-vaccinated, and second dose-vaccinated groups and exploring the transmission dynamics of the disease outbreaks. We present a non-linear integer order mathematical model of COVID-19 dynamics and modify it by introducing Caputo fractional derivative operator. We start by proving the good state of the model and then calculating its reproduction number. The Caputo fractional-order model is discretized by applying a reliable numerical technique. The model is proven to be stable. The classical model is fitted to the corresponding cumulative number of daily reported cases during the vaccination regime in India between 01 August 2021 and 21 July 2022. We explore the sensitivities of the reproduction number with respect to the model parameters. It is shown that the effective transmission rate and the recovery rate of unvaccinated infected individuals are the most sensitive parameters that drive the transmission dynamics of the pandemic in the population. Numerical simulations are used to demonstrate the applicability of the proposed fractional mathematical model via the memory index at different values of 0 . 7 , 0 . 8 , 0 . 9 and 1. We discuss the epidemiological significance of the findings and provide perspectives on future health policy tendencies. For instance, efforts targeting a decrease in the transmission rate and an increase in the recovery rate of non-vaccinated infected individuals are required to ensure virus-free population. This can be achieved if the population strictly adhere to precautionary measures, and prompt and adequate treatment is provided for non-vaccinated infectious individuals. Also, given the ongoing community spread of COVID-19 in India and almost the pandemic-affected countries worldwide, the need to scale up the effort of mass vaccination policy cannot be overemphasized in order to reduce the number of unvaccinated infections with a view to halting the transmission dynamics of the disease in the population.
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Gunatilake T, Miller SA. Adapting a Physical Earthquake-Aftershock Model to Simulate the Spread of COVID-19. INTERNATIONAL JOURNAL OF ENVIRONMENTAL RESEARCH AND PUBLIC HEALTH 2022; 19:16527. [PMID: 36554410 PMCID: PMC9778620 DOI: 10.3390/ijerph192416527] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/13/2022] [Revised: 11/26/2022] [Accepted: 12/05/2022] [Indexed: 06/17/2023]
Abstract
There exists a need for a simple, deterministic, scalable, and accurate model that captures the dominant physics of pandemic propagation. We propose such a model by adapting a physical earthquake/aftershock model to COVID-19. The aftershock model revealed the physical basis for the statistical Epidemic Type Aftershock Sequence (ETAS) model as a highly non-linear diffusion process, thus permitting a grafting of the underlying physical equations into a formulation for calculating infection pressure propagation in a pandemic-type model. Our model shows that the COVID-19 pandemic propagates through an analogous porous media with hydraulic properties approximating beach sand and water. Model results show good correlations with reported cumulative infections for all cases studied. In alphabetical order, these include Austria, Belgium, Brazil, France, Germany, Italy, New Zealand, Melbourne (AU), Spain, Sweden, Switzerland, the UK, and the USA. Importantly, the model is predominantly controlled by one parameter (α), which modulates the societal recovery from the spread of the virus. The obtained recovery times for the different pandemic waves vary considerably from country to country and are reflected in the temporal evolution of registered infections. These results provide an intuition-based approach to designing and implementing mitigation measures, with predictive capabilities for various mitigation scenarios.
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Affiliation(s)
- Thanushika Gunatilake
- Center for Hydrogeology and Geothermics (CHYN), University of Neuchâtel, 2000 Neuchâtel, Switzerland
- Swiss Seismological Service (SED), ETH Zürich, 8092 Zürich, Switzerland
| | - Stephen A. Miller
- Center for Hydrogeology and Geothermics (CHYN), University of Neuchâtel, 2000 Neuchâtel, Switzerland
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El-Saka HAA, Obaya I, Lee S, Jang B. Fractional model for Middle East respiratory syndrome coronavirus on a complex heterogeneous network. Sci Rep 2022; 12:20706. [PMID: 36456670 PMCID: PMC9713123 DOI: 10.1038/s41598-022-24814-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/23/2022] [Accepted: 11/21/2022] [Indexed: 12/05/2022] Open
Abstract
In this paper, we present a new fractional epidemiological model on a heterogeneous network to investigate Middle East respiratory syndrome (MERS-CoV), which is caused by a virus in the coronavirus family. We also consider the development of equations for the camel population, given that it is the primary animal source of the virus, as well as direct human interaction with this population. The model is configured in an SIS form for both the human population and the camel population. We study the equilibrium positions of the system and the conditions for the existence of each of them, as well as the local stability of each equilibrium position. Then, we provide some numerical examples that compare real data and numerical results.
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Affiliation(s)
- H. A. A. El-Saka
- grid.462079.e0000 0004 4699 2981Mathematics Department, Faculty of Science, Damietta University, New Damietta, 34517 Egypt
| | - Ibrahim Obaya
- Basic Science Department, Nile Higher Institute for Engineering and Technology, Mansoura, Egypt ,grid.10251.370000000103426662Department of Mathematics, Faculty of Science, Mansoura University, P.O. Box 64, Mansoura, 35516 Egypt
| | - Seyeon Lee
- grid.419553.f0000 0004 0500 6567Division of Industrial Mathematics, National Institute for Mathematical Sciences, Daejeon, 34047 Republic of Korea ,grid.42687.3f0000 0004 0381 814XDepartment of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), Ulsan, 44919 Republic of Korea
| | - Bongsoo Jang
- grid.42687.3f0000 0004 0381 814XDepartment of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), Ulsan, 44919 Republic of Korea
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Denu D, Kermausuor S. Analysis of a Fractional-Order COVID-19 Epidemic Model with Lockdown. Vaccines (Basel) 2022; 10:1773. [PMID: 36366284 PMCID: PMC9693277 DOI: 10.3390/vaccines10111773] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2022] [Revised: 10/20/2022] [Accepted: 10/20/2022] [Indexed: 03/27/2024] Open
Abstract
The outbreak of the coronavirus disease (COVID-19) has caused a lot of disruptions around the world. In an attempt to control the spread of the disease among the population, several measures such as lockdown, and mask mandates, amongst others, were implemented by many governments in their countries. To understand the effectiveness of these measures in controlling the disease, several mathematical models have been proposed in the literature. In this paper, we study a mathematical model of the coronavirus disease with lockdown by employing the Caputo fractional-order derivative. We establish the existence and uniqueness of the solution to the model. We also study the local and global stability of the disease-free equilibrium and endemic equilibrium solutions. By using the residual power series method, we obtain a fractional power series approximation of the analytic solution. Finally, to show the accuracy of the theoretical results, we provide some numerical and graphical results.
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Affiliation(s)
- Dawit Denu
- Department of Mathematical Sciences, Georgia Southern University, Savannah, GA 31419, USA
| | - Seth Kermausuor
- Department of Mathematics and Computer Science, Alabama State University, Montgomery, AL 36101, USA
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Hamed AM. Modeling of corona virus and its application in confocal microscopy. BENI-SUEF UNIVERSITY JOURNAL OF BASIC AND APPLIED SCIENCES 2022; 11:98. [PMID: 35992223 PMCID: PMC9375097 DOI: 10.1186/s43088-022-00276-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/16/2022] [Accepted: 07/25/2022] [Indexed: 11/30/2022] Open
Abstract
Background The proposal of spiky apertures showed resolution improvement compared with the circular apertures. Three models of corona virus are given. The 1st model consists of uniform circular aperture provided with 8 spikes while the 2nd model has 16 spikes for the same uniform circular aperture. The 3rd model has circular linear distribution with 8 spikes. Results The Normalized Point Spread Function (PSF) or the impulse response is computed for the three models using fast Fourier transform technique. In addition, the autocorrelation function corresponding to these apertures is calculated and compared with that corresponding to the ordinary circular and conic apertures. Coronavirus image is used as an object in the formation of images using confocal scanning laser microscope provided with suggested models. The fabricated MATLAB code is used to compute and plot all images and line plots. Conclusions The PSF plots are computed from Eqs. (8) and (12) using MATLAB code showing narrower cutoff in the PSF for spiky aperture compared with that corresponding to the uniform circular aperture and modulated linear and quadratic apertures. Hence, I reached resolution improvement in the case of spiky aperture.
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Siddiq R, Hameed MN, Zaheer MH, Khan MB, Uddin Z. Rydberg energies and transition probabilities of Li I for np–ms (m ≤ 5) transitions. BENI-SUEF UNIVERSITY JOURNAL OF BASIC AND APPLIED SCIENCES 2022; 11:42. [PMID: 35372590 PMCID: PMC8962285 DOI: 10.1186/s43088-022-00224-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/26/2021] [Accepted: 03/08/2022] [Indexed: 11/26/2022] Open
Abstract
Background Mathematical modeling provides grounds for understanding scientific systems theoretically. It serves as a guide for experimentalists in determining directions of investigation. Recently, the Covid-19 pandemic has caused disturbances in almost every walk of life. Scientists have played their role and have continued research on the effects of the pandemic. Various mathematical models have been used in different branches of science (Djilali et al. in Phys Scr 96 12 124016, 2021; Math Biosci Eng 18(6):8245–8256, 2021; Zeb et al. in Alex Eng J 61(7):5649–5665). Well-established mathematical models give results close to those obtained by experiments. The Weakest Bound Electron Potential Model is one such model, which explains hydrogen-like atoms and ions. This model has been used extensively for hydrogen-like atoms and ions to calculate energies of Rydberg levels and ionization energies. This model has been used extensively for hydrogen-like atoms and ions to calculate energies of Rydberg levels and ionization energies. Results This paper presents the energies of the Rydberg series, 2s2ns, and 2s2np of Li I, calculated using WBEPM. The energies are used to calculate transition probabilities from np to 2s, 3s, 4s, and 5s levels. The transition probabilities are compared with corresponding values in published data where available. The agreement with known values is good; most of the transition probabilities calculated in this work are new. A computer program was developed to find the value of the dipole matrix element. The calculations were further verified by calculating the lifetimes of some low-lying levels. Conclusions Four series of Li I have been studied, and energies of the Rydberg levels in the series were calculated. The energies then are used to calculate transition probabilities from np to ms transitions, where m = 2, 3, 4, & 5 and n = 1–15. The results are compared where available. An excellent agreement with previously published data shows the reliability of calculations. Most of the transition probabilities are new.
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