1
|
Mediani M, Slama A, Boudaoui A, Abdeljawad T. Analysis of a stochastic SEI u I r R epidemic model incorporating the Ornstein-Uhlenbeck process. Heliyon 2024; 10:e35749. [PMID: 39224271 PMCID: PMC11367037 DOI: 10.1016/j.heliyon.2024.e35749] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2024] [Revised: 07/30/2024] [Accepted: 08/02/2024] [Indexed: 09/04/2024] Open
Abstract
This article aims to analyze a stochastic epidemic model S E I u I r R (Susceptible-exposed-undetected infected-detected infected (reported -recovered) assuming that the transmission rate at which people undetected become detected is perturbed by the Ornstein-Uhlenbeck process. Our first objective is to prove that the stochastic model has a unique positive global solution by constructing a nonnegative Lyapunov function. Afterward, we provide a sufficient criterion to prove the existence of an ergodic stationary distribution of the mode by constructing a suitable series of Lyapunov functions. Subsequently, we establish sufficient conditions for the extinction of the disease. Finally, a series of numerical simulations are carried out to illustrate the theoretical results.
Collapse
Affiliation(s)
- Mhammed Mediani
- Laboratory of Mathematics, Modeling and Applications (LaMMA), University of Adrar, Adrar, Algeria
| | - Abdeldjalil Slama
- Laboratory of Mathematics, Modeling and Applications (LaMMA), University of Adrar, Adrar, Algeria
| | - Ahmed Boudaoui
- Laboratory of Mathematics, Modeling and Applications (LaMMA), University of Adrar, Adrar, Algeria
| | - Thabet Abdeljawad
- Department of Mathematics and Sciences, Prince Sultan University Riyadh, Saudi Arabia
- Department of Medical Research, China Medical University, Taichung 40402, Taiwan
- Center for Applied Mathematics and Bioinformatics (CAMB), Gulf University for Science and Technology, Hawally, 32093, Kuwait
- Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, Medusa 0204, South Africa
| |
Collapse
|
2
|
Li S, Samreen, Ullah S, Riaz MB, Awwad FA, Teklu SW. Global dynamics and computational modeling approach for analyzing and controlling of alcohol addiction using a novel fractional and fractal-fractional modeling approach. Sci Rep 2024; 14:5065. [PMID: 38429318 PMCID: PMC11310334 DOI: 10.1038/s41598-024-54578-9] [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: 11/18/2023] [Accepted: 02/14/2024] [Indexed: 03/03/2024] Open
Abstract
In recent years, alcohol addiction has become a major public health concern and a global threat due to its potential negative health and social impacts. Beyond the health consequences, the detrimental consumption of alcohol results in substantial social and economic burdens on both individuals and society as a whole. Therefore, a proper understanding and effective control of the spread of alcohol addictive behavior has become an appealing global issue to be solved. In this study, we develop a new mathematical model of alcohol addiction with treatment class. We analyze the dynamics of the alcohol addiction model for the first time using advanced operators known as fractal-fractional operators, which incorporate two distinct fractal and fractional orders with the well-known Caputo derivative based on power law kernels. The existence and uniqueness of the newly developed fractal-fractional alcohol addiction model are shown using the Picard-Lindelöf and fixed point theories. Initially, a comprehensive qualitative analysis of the alcohol addiction fractional model is presented. The possible equilibria of the model and the threshold parameter called the reproduction number are evaluated theoretically and numerically. The boundedness and biologically feasible region for the model are derived. To assess the stability of the proposed model, the Ulam-Hyers coupled with the Ulam-Hyers-Rassias stability criteria are employed. Moreover, utilizing effecting numerical schemes, the models are solved numerically and a detailed simulation and discussion are presented. The model global dynamics are shown graphically for various values of fractional and fractal dimensions. The present study aims to provide valuable insights for the understanding the dynamics and control of alcohol addiction within a community.
Collapse
Affiliation(s)
- Shuo Li
- School of Mathematics and Data Sciences, Changji University, Changji, 831100, Xinjiang, People's Republic of China
| | - Samreen
- Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa, Pakistan
| | - Saif Ullah
- Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan
| | - Muhammad Bilal Riaz
- IT4Innovations, VSB -Technical University of Ostrava, Ostrava, Czech Republic
- Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon
| | - Fuad A Awwad
- Department of Quantitative analysis, College of Business Administration, King Saud University, P.O. Box 71115, Riyadh, 11587, Saudi Arabia
| | - Shewafera Wondimagegnhu Teklu
- Department of Mathematics, College of Natural and Computational Scieces, Debre Berhan University, 445, Debre Berhan, Ethiopia.
| |
Collapse
|
3
|
Alzubaidi AM, Othman HA, Ullah S, Ahmad N, Alam MM. Analysis of Monkeypox viral infection with human to animal transmission via a fractional and Fractal-fractional operators with power law kernel. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:6666-6690. [PMID: 37161123 DOI: 10.3934/mbe.2023287] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
Monkeypox (MPX) is a global public health concern. This infectious disease affects people all over the world, not just those in West and Central Africa. Various approaches have been used to study epidemiology, the source of infection, and patterns of transmission of MPX. In this article, we analyze the dynamics of MPX using a fractional mathematical model with a power law kernel. The human-to-animal transmission is considered in the model formulation. The fractional model is further reformulated via a generalized fractal-fractional differential operator in the Caputo sense. The basic mathematical including the existence and uniqueness of both fractional and fractal-fractional problems are provided using fixed points theorems. A numerical scheme for the proposed model is obtained using an efficient iterative method. Moreover, detailed simulation results are shown for different fractional orders in the first stage. Finally, a number of graphical results of fractal-fractional MPX transmission models are presented showing the combined effect of fractal and fractional orders on model dynamics. The resulting simulations conclude that the new fractal-fractional operator added more biological insight into the dynamics of illness.
Collapse
Affiliation(s)
- Alia M Alzubaidi
- Department of Mathematics, AL-Qunfudhah University college, Umm Al-Qura University, Saudi Arabia
| | - Hakeem A Othman
- Department of Mathematics, AL-Qunfudhah University college, Umm Al-Qura University, Saudi Arabia
| | - Saif Ullah
- Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa, Pakistan
| | - Nisar Ahmad
- Institute of Numerical Sciences Kohat University of Science and Technology (KUST) Kohat, Pakistan
| | - Mohammad Mahtab Alam
- Department of Basic Medical Sciences, College of Applied Medical Science, King Khalid University, Abha 61421, Saudi Arabia
| |
Collapse
|
4
|
G M V, P RB. Vaccination control measures of an epidemic model with long-term memristive effect. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2023; 419:114738. [PMID: 36000087 PMCID: PMC9388295 DOI: 10.1016/j.cam.2022.114738] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/25/2022] [Revised: 08/06/2022] [Indexed: 06/15/2023]
Abstract
COVID-19 is a drastic air-way tract infection that set off a global pandemic recently. Most infected people with mild and moderate symptoms have recovered with naturally acquired immunity. In the interim, the defensive mechanism of vaccines helps to suppress the viral complications of the pathogenic spread. Besides effective vaccination, vaccine breakthrough infections occurred rapidly due to noxious exposure to contagions. This paper proposes a new epidemiological control model in terms of Atangana Baleanu Caputo (ABC) type fractional order differ integrals for the reported cases of COVID-19 outburst. The qualitative theoretical and numerical analysis of the aforesaid mathematical model in terms of three compartments namely susceptible, vaccinated, and infected population are exhibited through non-linear functional analysis. The hysteresis kernel involved in AB integral inherits the long-term memory of the dynamical trajectory of the epidemics. Hyer-Ulam's stability of the system is studied by the dichotomy operator. The most effective approximate solution is derived by numerical interpolation to our proposed model. An extensive analysis of the vigorous vaccination and the proportion of vaccinated individuals are explored through graphical simulations. The efficacious enforcement of this vaccination control mechanism will mitigate the contagious spread and severity.
Collapse
Affiliation(s)
- Vijayalakshmi G M
- Department of Mathematics, Vel Tech Rangarajan Dr Sagunthala R & D Institute of Science and Technology, Avadi, Tamilnadu 600062, India
| | - Roselyn Besi P
- Department of Mathematics, Kanchi Sri Magalakshmi Arts and Science College for women, Kanchipuram, Tamilnadu-631603, India
| |
Collapse
|
5
|
A Stochastic Mathematical Model for Understanding the COVID-19 Infection Using Real Data. Symmetry (Basel) 2022. [DOI: 10.3390/sym14122521] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/03/2022] Open
Abstract
Natural symmetry exists in several phenomena in physics, chemistry, and biology. Incorporating these symmetries in the differential equations used to characterize these processes is thus a valid modeling assumption. The present study investigates COVID-19 infection through the stochastic model. We consider the real infection data of COVID-19 in Saudi Arabia and present its detailed mathematical results. We first present the existence and uniqueness of the deterministic model and later study the dynamical properties of the deterministic model and determine the global asymptotic stability of the system for R0≤1. We then study the dynamic properties of the stochastic model and present its global unique solution for the model. We further study the extinction of the stochastic model. Further, we use the nonlinear least-square fitting technique to fit the data to the model for the deterministic and stochastic case and the estimated basic reproduction number is R0≈1.1367. We show that the stochastic model provides a good fitting to the real data. We use the numerical approach to solve the stochastic system by presenting the results graphically. The sensitive parameters that significantly impact the model dynamics and reduce the number of infected cases in the future are shown graphically.
Collapse
|
6
|
Vijayalakshmi G, P RB. A fractal fractional order vaccination model of COVID-19 pandemic using Adam’s moulton analysis. RESULTS IN CONTROL AND OPTIMIZATION 2022. [PMCID: PMC9187878 DOI: 10.1016/j.rico.2022.100144] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
Abstract
The pandemic caused by coronaviruses (SARS-COV-2) is a zoonotic disease targeting the respiratory tract of active humans. Few mild symptoms of fever and tiredness get cured without any medicinal aid, whereas some severe symptoms of dry cough with breathing illness led to perceived risk of secondary transmission. This paper studies the effectiveness of vaccination in Covid-19 pandemic disease by modelling three compartments susceptible, vaccinated and infected (SVI) of Atangana Baleanu of Caputo (ABC) type derivatives in non-integer order. The disease dynamics is analysed and its stability is performed. Numerical approximation is derived using Adam’s Moulton method and simulated to forecast the results for controllability of pandemic spread.
Collapse
|
7
|
Vijayalakshmi GM, Roselyn Besi P. ABC Fractional Order Vaccination Model for Covid-19 with Self-Protective Measures. INTERNATIONAL JOURNAL OF APPLIED AND COMPUTATIONAL MATHEMATICS 2022; 8:130. [PMID: 35578713 PMCID: PMC9096354 DOI: 10.1007/s40819-022-01316-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Accepted: 04/02/2022] [Indexed: 11/06/2022]
Abstract
A mathematical model delineating the control strategies in transference of Covid-19 pandemic is examined through Atangana-Baleanu Caputo type fractional derivatives. The total count of people under observation is classified into Susceptible, Vaccinated, Infected and Protected groups (SVIP). The designed model studies the efficiency of vaccination and personal precautions incorporated qualitatively by every individual via fixed point theorem. Stability of the system has been investigated with spectral characterisation of Ulam Hyer's kind. Numerical interpolation has been derived by Adam's semi-analytical technique and we have approximated the solution. We have proved the theoretical analysis through graphical simulations that vaccination and self protective interventions are the significant role to decrease the contagious expansion of the virus among the people in process.
Collapse
Affiliation(s)
- G. M. Vijayalakshmi
- Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R & D Institute of Science and Technology, Avadi, Tamilnadu 600062 India
| | - P. Roselyn Besi
- PG and Research Department of Mathematics, Auxilium College, Vellore, Tamilnadu 632006 India
| |
Collapse
|