101
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Khan MA, Ullah S, Kumar S. A robust study on 2019-nCOV outbreaks through non-singular derivative. EUROPEAN PHYSICAL JOURNAL PLUS 2021; 136:168. [PMID: 33552828 PMCID: PMC7854889 DOI: 10.1140/epjp/s13360-021-01159-8] [Citation(s) in RCA: 29] [Impact Index Per Article: 9.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/2020] [Accepted: 01/27/2021] [Indexed: 05/18/2023]
Abstract
The new coronavirus disease is still a major panic for people all over the world. The world is grappling with the second wave of this new pandemic. Different approaches are taken into consideration to tackle this deadly disease. These approaches were suggested in the form of modeling, analysis of the data, controlling the disease spread and clinical perspectives. In all these suggested approaches, the main aim was to eliminate or decrease the infection of the coronavirus from the community. Here, in this paper, we focus on developing a new mathematical model to understand its dynamics and possible control. We formulate the model first in the integer order and then use the Atangana-Baleanu derivative concept with a non-singular kernel for its generalization. We present some of the necessary mathematical aspects of the fractional model. We use a nonlinear fractional Lyapunov function in order to present the global asymptotical stability of the model at the disease-free equilibrium. In order to solve the model numerically in the fractional case, we use an efficient modified Adams-Bashforth scheme. The resulting iterative scheme is then used to demonstrate the detailed simulation results of the ABC mathematical model to examine the importance of the memory index and model parameters on the transmission and control of COVID-19 infection.
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Affiliation(s)
- Muhammad Altaf Khan
- Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
- Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
| | - Saif Ullah
- Department of Mathematics, University of Peshawar, Peshawar, Pakistan
| | - Sunil Kumar
- Department of Mathematics, National Institute of Technology, Jamshedpur, Jharkhand India
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102
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Ahmed I, Modu GU, Yusuf A, Kumam P, Yusuf I. A mathematical model of Coronavirus Disease (COVID-19) containing asymptomatic and symptomatic classes. RESULTS IN PHYSICS 2021; 21:103776. [PMID: 33432294 PMCID: PMC7787076 DOI: 10.1016/j.rinp.2020.103776] [Citation(s) in RCA: 40] [Impact Index Per Article: 13.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/20/2020] [Revised: 12/11/2020] [Accepted: 12/16/2020] [Indexed: 05/18/2023]
Abstract
The research work in this paper attempts to describe the outbreak of Coronavirus Disease 2019 (COVID-19) with the help of a mathematical model using both the Ordinary Differential Equation (ODE) and Fractional Differential Equation. The spread of the disease has been on the increase across the globe for some time with no end in sight. The research used the data of COVID-19 cases in Nigeria for the numerical simulation which has been fitted to the model. We brought in the consideration of both asymptomatic and symptomatic infected individuals with the fact that an exposed individual is either sent to quarantine first or move to one of the infected classes with the possibility that susceptible individual can also move to quarantined class directly. It was found that the proposed model has two equilibrium points; the disease-free equilibrium point ( DFE ) and the endemic equilibrium point ( E 1 ) . Stability analysis of the equilibrium points shows ( E 0 ) is locally asymptotically stable whenever the basic reproduction number,R 0 < 1 and ( E 1 ) is globally asymptotically stable wheneverR 0 > 1 . Sensitivity analysis of the parameters in theR 0 was conducted and the profile of each state variable was also depicted using the fitted values of the parameters showing the spread of the disease. The most sensitive parameters in theR 0 are the contact rate between susceptible individuals and the rate of transfer of individuals from exposed class to symptomatically infected class. Moreover, the basic reproduction number for the data is calculated asR 0 ≈ 1.7031 . Existence and uniqueness of solution established via the technique of fixed point theorem. Also, using the least square curve fitting method together with the fminsearch function in the MATLAB optimization toolbox, we obtain the best values for some of the unknown biological parameters involved in the proposed model. Furthermore, we solved the fractional model numerically using the Atangana-Toufik numerical scheme and presenting different forms of graphical results that can be useful in minimizing the infection.
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Affiliation(s)
- Idris Ahmed
- Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
- KMUTTFixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
| | - Goni Umar Modu
- Department of Statistics, Ramat Polytechnic Maiduguri, P. M. B 1070 Maiduguri, Borno State, Nigeria
| | - Abdullahi Yusuf
- Department of Computer Engineering, Biruni University, Istanbul 34010, Turkey
- Department of Mathematics, Federal University Dutse, Jigawa 7156, Nigeria
| | - Poom Kumam
- KMUTTFixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
- Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
| | - Ibrahim Yusuf
- Department of Mathematical Sciences, Bayero University Kano, P. M. B. 3011 Kano, Nigeria
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103
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Hincal E, Alsaadi SH. Stability analysis of fractional order model on corona transmission dynamics. CHAOS, SOLITONS, AND FRACTALS 2021; 143:110628. [PMID: 33519120 PMCID: PMC7834460 DOI: 10.1016/j.chaos.2020.110628] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/02/2020] [Revised: 12/08/2020] [Accepted: 12/26/2020] [Indexed: 06/12/2023]
Abstract
In this paper a fractional order mathematical model is constructed to study the dynamics of corona virus in Oman. The model consists of a system of eight non-linear fractional order differential equations in Caputo sense. Existence and uniqueness as well as the stability analysis of the solution of the model are given. The stability analysis is in the frame of Ulam-Hyers and generalized Ulam-Hyers criteria. Numerical simulations are given to support the theoretical results. Many informations on the dynamics of COVID -19 in Oman were obtained using this model. Also many informations on the qualitative behaviour of the model were obtained.
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Affiliation(s)
- Evren Hincal
- Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey
| | - Sultan Hamed Alsaadi
- Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey
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104
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Jahanshahi H, Munoz-Pacheco JM, Bekiros S, Alotaibi ND. A fractional-order SIRD model with time-dependent memory indexes for encompassing the multi-fractional characteristics of the COVID-19. CHAOS, SOLITONS, AND FRACTALS 2021; 143:110632. [PMID: 33519121 PMCID: PMC7832492 DOI: 10.1016/j.chaos.2020.110632] [Citation(s) in RCA: 28] [Impact Index Per Article: 9.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/11/2020] [Revised: 12/23/2020] [Accepted: 12/25/2020] [Indexed: 05/04/2023]
Abstract
COVID-19 is a novel coronavirus affecting all the world since December last year. Up to date, the spread of the outbreak continues to complicate our lives, and therefore, several research efforts from many scientific areas are proposed. Among them, mathematical models are an excellent way to understand and predict the epidemic outbreaks evolution to some extent. Due to the COVID-19 may be modeled as a non-Markovian process that follows power-law scaling features, we present a fractional-order SIRD (Susceptible-Infected-Recovered-Dead) model based on the Caputo derivative for incorporating the memory effects (long and short) in the outbreak progress. Additionally, we analyze the experimental time series of 23 countries using fractal formalism. Like previous works, we identify that the COVID-19 evolution shows various power-law exponents (no just a single one) and share some universality among geographical regions. Hence, we incorporate numerous memory indexes in the proposed model, i.e., distinct fractional-orders defined by a time-dependent function that permits us to set specific memory contributions during the evolution. This allows controlling the memory effects of more early states, e.g., before and after a quarantine decree, which could be less relevant than the contribution of more recent ones on the current state of the SIRD system. We also prove our model with Italy's real data from the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University.
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Affiliation(s)
- Hadi Jahanshahi
- Department of Mechanical Engineering, University of Manitoba, Winnipeg R3T 5V6, Canada
| | - Jesus M Munoz-Pacheco
- Faculty of Electronics Sciences, Benemerita Universidad Autonoma de Puebla, 72570 Mexico
| | - Stelios Bekiros
- European University Institute, Department of Economics, Via delle Fontanelle, 18, Florence, I-50014, Italy
- Rimini Centre for Economic Analysis (RCEA), LH3079, Wilfrid Laurier University, 75 University Ave W., ON Waterloo, N2L3C5, Canada
| | - Naif D Alotaibi
- Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia
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105
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Gnanvi JE, Salako KV, Kotanmi GB, Glèlè Kakaï R. On the reliability of predictions on Covid-19 dynamics: A systematic and critical review of modelling techniques. Infect Dis Model 2021; 6:258-272. [PMID: 33458453 PMCID: PMC7802527 DOI: 10.1016/j.idm.2020.12.008] [Citation(s) in RCA: 34] [Impact Index Per Article: 11.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2020] [Revised: 12/29/2020] [Accepted: 12/29/2020] [Indexed: 12/18/2022] Open
Abstract
Since the emergence of the novel 2019 coronavirus pandemic in December 2019 (COVID-19), numerous modellers have used diverse techniques to assess the dynamics of transmission of the disease, predict its future course and determine the impact of different control measures. In this study, we conducted a global systematic literature review to summarize trends in the modelling techniques used for Covid-19 from January 1st, 2020 to November 30th, 2020. We further examined the accuracy and precision of predictions by comparing predicted and observed values for cumulative cases and deaths as well as uncertainties of these predictions. From an initial 4311 peer-reviewed articles and preprints found with our defined keywords, 242 were fully analysed. Most studies were done on Asian (78.93%) and European (59.09%) countries. Most of them used compartmental models (namely SIR and SEIR) (46.1%) and statistical models (growth models and time series) (31.8%) while few used artificial intelligence (6.7%), Bayesian approach (4.7%), Network models (2.3%) and Agent-based models (1.3%). For the number of cumulative cases, the ratio of the predicted over the observed values and the ratio of the amplitude of confidence interval (CI) or credibility interval (CrI) of predictions and the central value were on average larger than 1 indicating cases of inaccurate and imprecise predictions, and large variation across predictions. There was no clear difference among models used for these two ratios. In 75% of predictions that provided CI or CrI, observed values fall within the 95% CI or CrI of the cumulative cases predicted. Only 3.7% of the studies predicted the cumulative number of deaths. For 70% of the predictions, the ratio of predicted over observed cumulative deaths was less or close to 1. Also, the Bayesian model made predictions closer to reality than classical statistical models, although these differences are only suggestive due to the small number of predictions within our dataset (9 in total). In addition, we found a significant negative correlation (rho = - 0.56, p = 0.021) between this ratio and the length (in days) of the period covered by the modelling, suggesting that the longer the period covered by the model the likely more accurate the estimates tend to be. Our findings suggest that while predictions made by the different models are useful to understand the pandemic course and guide policy-making, some were relatively accurate and precise while other not.
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Affiliation(s)
- Janyce Eunice Gnanvi
- Laboratoire de Biomathématiques et d’Estimations Forestières, Université d’Abomey-Calavi, 04 BP 1525, Cotonou, Benin
| | - Kolawolé Valère Salako
- Laboratoire de Biomathématiques et d’Estimations Forestières, Université d’Abomey-Calavi, 04 BP 1525, Cotonou, Benin
| | - Gaëtan Brezesky Kotanmi
- Laboratoire de Biomathématiques et d’Estimations Forestières, Université d’Abomey-Calavi, 04 BP 1525, Cotonou, Benin
| | - Romain Glèlè Kakaï
- Laboratoire de Biomathématiques et d’Estimations Forestières, Université d’Abomey-Calavi, 04 BP 1525, Cotonou, Benin
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106
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Askari Q, Younas I. Political Optimizer Based Feedforward Neural Network for Classification and Function Approximation. Neural Process Lett 2021. [DOI: 10.1007/s11063-020-10406-5] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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107
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Lü X, Hui HW, Liu FF, Bai YL. Stability and optimal control strategies for a novel epidemic model of COVID-19. NONLINEAR DYNAMICS 2021; 106:1491-1507. [PMID: 34054221 PMCID: PMC8148406 DOI: 10.1007/s11071-021-06524-x] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/05/2021] [Accepted: 05/06/2021] [Indexed: 05/09/2023]
Abstract
In this paper, a novel two-stage epidemic model with a dynamic control strategy is proposed to describe the spread of Corona Virus Disease 2019 (COVID-19) in China. Combined with local epidemic control policies, an epidemic model with a traceability process is established. We aim to investigate the appropriate control strategies to minimize the control cost and ensure the normal operation of society under the premise of containing the epidemic. This work mainly includes: (i) propose the concept about the first and the second waves of COVID-19, as well as study the case data and regularity of four cities; (ii) derive the existence and stability of the equilibrium, the parameter sensitivity of the model, and the existence of the optimal control strategy; (iii) carry out the numerical simulation associated with the theoretical results and construct a dynamic control strategy and verify its feasibility.
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Affiliation(s)
- Xing Lü
- Department of Mathematics, Beijing Jiaotong University, Beijing, 100044 China
| | - Hong-wen Hui
- School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing, 100083 China
| | - Fei-fei Liu
- School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing, 100083 China
| | - Ya-li Bai
- School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing, 100083 China
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108
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Babaei A, Ahmadi M, Jafari H, Liya A. A mathematical model to examine the effect of quarantine on the spread of coronavirus. CHAOS, SOLITONS, AND FRACTALS 2021; 142:110418. [PMID: 33288973 PMCID: PMC7703523 DOI: 10.1016/j.chaos.2020.110418] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2020] [Revised: 10/30/2020] [Accepted: 10/31/2020] [Indexed: 05/04/2023]
Abstract
In this study, we propose a mathematical model about the spread of novel coronavirus. This model is a system of fractional order differential equations in Caputo's sense. The aim is to explain the virus transmission and to investigate the impact of quarantine on decreasing the prevalence rate of the virus in the environment. The unique solvability of the presented COVID-19 model is proved. Also, the equilibrium points and the reproduction number of the proposed model are discussed in two cases with and without considering the quarantine factor. Using the Adams-Bashforth-Moulton predictor-corrector method, some numerical simulations are implemented to survey the behavior of the considered model.
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Affiliation(s)
- A Babaei
- Department of Applied Mathematics, University of Mazandaran, Babolsar, Iran
| | - M Ahmadi
- Department of Applied Mathematics, University of Mazandaran, Babolsar, Iran
| | - H Jafari
- Department of Applied Mathematics, University of Mazandaran, Babolsar, Iran
- Department of Mathematical Sciences, University of South Africa, UNISA0003, South Africa
- Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 110122, Taiwan
| | - A Liya
- Department of Applied Mathematics, University of Mazandaran, Babolsar, Iran
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109
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Rafiq M, Macías-Díaz JE, Raza A, Ahmed N. Design of a nonlinear model for the propagation of COVID-19 and its efficient nonstandard computational implementation. APPLIED MATHEMATICAL MODELLING 2021; 89:1835-1846. [PMID: 32982020 PMCID: PMC7506502 DOI: 10.1016/j.apm.2020.08.082] [Citation(s) in RCA: 21] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2020] [Revised: 08/14/2020] [Accepted: 08/26/2020] [Indexed: 05/03/2023]
Abstract
In this manuscript, we develop a mathematical model to describe the spreading of an epidemic disease in a human population. The emphasis in this work will be on the study of the propagation of the coronavirus disease (COVID-19). Various epidemiologically relevant assumptions will be imposed upon the problem, and a coupled system of first-order ordinary differential equations will be obtained. The model adopts the form of a nonlinear susceptible-exposed-infected-quarantined-recovered system, and we investigate it both analytically and numerically. Analytically, we obtain the equilibrium points in the presence and absence of the coronavirus. We also calculate the reproduction number and provide conditions that guarantee the local and global asymptotic stability of the equilibria. To that end, various tools from analysis will be employed, including Volterra-type Lyapunov functions, LaSalle's invariance principle and the Routh-Hurwitz criterion. To simulate computationally the dynamics of propagation of the disease, we propose a nonstandard finite-difference scheme to approximate the solutions of the mathematical model. A thorough analysis of the discrete model is provided in this work, including the consistency and the stability analyses, along with the capability of the discrete model to preserve the equilibria of the continuous system. Among other interesting results, our numerical simulations confirm the stability properties of the equilibrium points.
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Affiliation(s)
- Muhammad Rafiq
- Faculty of Engineering, University of Central Punjab, Lahore, Pakistan
| | - J E Macías-Díaz
- Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20131, Mexico
| | - Ali Raza
- Department of Mathematics, National College of Business Administration and Economics Lahore, Pakistan
| | - Nauman Ahmed
- Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
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110
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Aba Oud MA, Ali A, Alrabaiah H, Ullah S, Khan MA, Islam S. A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load. ADVANCES IN DIFFERENCE EQUATIONS 2021; 2021:106. [PMID: 33613668 DOI: 10.1186/s13662-020-03162-2] [Citation(s) in RCA: 26] [Impact Index Per Article: 8.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2020] [Accepted: 01/31/2021] [Indexed: 05/22/2023]
Abstract
COVID-19 or coronavirus is a newly emerged infectious disease that started in Wuhan, China, in December 2019 and spread worldwide very quickly. Although the recovery rate is greater than the death rate, the COVID-19 infection is becoming very harmful for the human community and causing financial loses to their economy. No proper vaccine for this infection has been introduced in the market in order to treat the infected people. Various approaches have been implemented recently to study the dynamics of this novel infection. Mathematical models are one of the effective tools in this regard to understand the transmission patterns of COVID-19. In the present paper, we formulate a fractional epidemic model in the Caputo sense with the consideration of quarantine, isolation, and environmental impacts to examine the dynamics of the COVID-19 outbreak. The fractional models are quite useful for understanding better the disease epidemics as well as capture the memory and nonlocality effects. First, we construct the model in ordinary differential equations and further consider the Caputo operator to formulate its fractional derivative. We present some of the necessary mathematical analysis for the fractional model. Furthermore, the model is fitted to the reported cases in Pakistan, one of the epicenters of COVID-19 in Asia. The estimated value of the important threshold parameter of the model, known as the basic reproduction number, is evaluated theoretically and numerically. Based on the real fitted parameters, we obtained R 0 ≈ 1.50 . Finally, an efficient numerical scheme of Adams-Moulton type is used in order to simulate the fractional model. The impact of some of the key model parameters on the disease dynamics and its elimination are shown graphically for various values of noninteger order of the Caputo derivative. We conclude that the use of fractional epidemic model provides a better understanding and biologically more insights about the disease dynamics.
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Affiliation(s)
- Mohammed A Aba Oud
- Department of Mathematics and Statistics, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Kingdom of Saudi Arabia
| | - Aatif Ali
- Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, 23100 Mardan, Pakistan
| | - Hussam Alrabaiah
- College of Engineering, Al Ain University, Al Ain, United Arab Emirates
- Mathematics Department, Tafila Technical University, Tafila, Jordan
| | - Saif Ullah
- Department of Mathematics, University of Peshawar, Peshawar, 25000 Pakistan
| | - Muhammad Altaf Khan
- Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
- Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
| | - Saeed Islam
- Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, 23100 Mardan, Pakistan
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111
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Abdulmajid S, Hassan AS. Analysis of time delayed Rabies model in human and dog populations with controls. AFRIKA MATEMATIKA 2021; 32:1067-1085. [PMCID: PMC7862050 DOI: 10.1007/s13370-021-00882-w] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/10/2019] [Accepted: 01/23/2021] [Indexed: 08/29/2023]
Abstract
Rabies is a fatal zoonotic disease caused by a virus through bites or saliva of an infected animal. Dogs are the main reservoir of rabies and responsible for most cases in humans worldwide. In this article, a delay differential equations model for assessing the effects of controls and time delay as incubation period on the transmission dynamics of rabies in human and dog populations is formulated and analyzed. Analysis from the model show that there is a locally and globally asymptotic stable disease-free equilibrium whenever a certain epidemiological threshold, the control reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_v$$\end{document} R v , is less than unity. Furthermore, the model has a unique endemic equilibrium when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_v$$\end{document} R v exceed unity which is also locally and globally asymptotically stable for all delays. Time delay is found to have influence on the endemicity of rabies. Vaccination of humans and dogs coupled with annual crop of puppies are imposed to curtail the spread of rabies in the populations. Sensitivity analysis on the number of infected humans and dogs revealed that increasing dog vaccination rate and decreasing annual birth of puppies are more effective in human populations. However in dog populations, the vaccination and birth control of puppies, have equal effective measures for rabies control. Numerical experiments are conducted to illustrate the theoretical results and control strategies.
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Affiliation(s)
- Shafiu Abdulmajid
- Department of Mathematical Sciences, College of Physical and Pharmaceutical Sciences, Bayero University Kano, P.M.B. 3011 Kano, Nigeria
| | - Adamu Shitu Hassan
- Department of Mathematical Sciences, College of Physical and Pharmaceutical Sciences, Bayero University Kano, P.M.B. 3011 Kano, Nigeria
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112
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Aba Oud MA, Ali A, Alrabaiah H, Ullah S, Khan MA, Islam S. A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load. ADVANCES IN DIFFERENCE EQUATIONS 2021; 2021:106. [PMID: 33613668 PMCID: PMC7877321 DOI: 10.1186/s13662-021-03265-4] [Citation(s) in RCA: 40] [Impact Index Per Article: 13.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2020] [Accepted: 01/31/2021] [Indexed: 05/11/2023]
Abstract
COVID-19 or coronavirus is a newly emerged infectious disease that started in Wuhan, China, in December 2019 and spread worldwide very quickly. Although the recovery rate is greater than the death rate, the COVID-19 infection is becoming very harmful for the human community and causing financial loses to their economy. No proper vaccine for this infection has been introduced in the market in order to treat the infected people. Various approaches have been implemented recently to study the dynamics of this novel infection. Mathematical models are one of the effective tools in this regard to understand the transmission patterns of COVID-19. In the present paper, we formulate a fractional epidemic model in the Caputo sense with the consideration of quarantine, isolation, and environmental impacts to examine the dynamics of the COVID-19 outbreak. The fractional models are quite useful for understanding better the disease epidemics as well as capture the memory and nonlocality effects. First, we construct the model in ordinary differential equations and further consider the Caputo operator to formulate its fractional derivative. We present some of the necessary mathematical analysis for the fractional model. Furthermore, the model is fitted to the reported cases in Pakistan, one of the epicenters of COVID-19 in Asia. The estimated value of the important threshold parameter of the model, known as the basic reproduction number, is evaluated theoretically and numerically. Based on the real fitted parameters, we obtained R 0 ≈ 1.50 . Finally, an efficient numerical scheme of Adams-Moulton type is used in order to simulate the fractional model. The impact of some of the key model parameters on the disease dynamics and its elimination are shown graphically for various values of noninteger order of the Caputo derivative. We conclude that the use of fractional epidemic model provides a better understanding and biologically more insights about the disease dynamics.
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Affiliation(s)
- Mohammed A. Aba Oud
- Department of Mathematics and Statistics, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Kingdom of Saudi Arabia
| | - Aatif Ali
- Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, 23100 Mardan, Pakistan
| | - Hussam Alrabaiah
- College of Engineering, Al Ain University, Al Ain, United Arab Emirates
- Mathematics Department, Tafila Technical University, Tafila, Jordan
| | - Saif Ullah
- Department of Mathematics, University of Peshawar, Peshawar, 25000 Pakistan
| | - Muhammad Altaf Khan
- Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
- Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
| | - Saeed Islam
- Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, 23100 Mardan, Pakistan
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113
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Thakur NK, Srivastava SC, Ojha A. Dynamical Study of an Eco-Epidemiological Delay Model for Plankton System with Toxicity. IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY. TRANSACTION A, SCIENCE 2021; 45:283-304. [PMID: 33424195 PMCID: PMC7781835 DOI: 10.1007/s40995-020-01042-8] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/07/2020] [Accepted: 11/23/2020] [Indexed: 01/31/2023]
Abstract
In this paper, we analyze the complexity of an eco-epidemiological model for phytoplankton-zooplankton system in presence of toxicity and time delay. Holling type II function response is incorporated to address the predation rate as well as toxic substance distribution in zooplankton. It is also presumed that infected phytoplankton does recover from the viral infection. In the absence of time delay, stability and Hopf-bifurcation conditions are investigated to explore the system dynamics around all the possible equilibrium points. Further, in the presence of time delay, conditions for local stability are derived around the interior equilibria and the properties of the periodic solution are obtained by applying normal form theory and central manifold arguments. Computational simulation is performed to illustrate our theoretical findings. It is explored that system dynamics is very sensitive corresponding to carrying capacity and toxin liberation rate and able to generate chaos. Further, it is observed that time delay in the viral infection process can destabilize the phytoplankton density whereas zooplankton density remains in its old state. Incorporation of time delay also gives the scenario of double Hopf-bifurcation. Some control parameters are discussed to stabilize system dynamics. The effect of time delay on (i) growth rate of susceptible phytoplankton shows the extinction and double Hopf-bifurcation in the zooplankton population, (ii) a sufficiently large value of carrying capacity stabilizes the chaotic dynamics or makes the whole system chaotic with further increment.
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Affiliation(s)
- Nilesh Kumar Thakur
- Department of Mathematics, National Institute of Technology Raipur, Raipur, CG 492010 India
| | | | - Archana Ojha
- Department of Mathematics, National Institute of Technology Raipur, Raipur, CG 492010 India
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114
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Khan MA, Ali NHM, Hamid NNA. A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid. ADVANCES IN DIFFERENCE EQUATIONS 2020; 2020:598. [DOI: 10.1186/s13662-020-03061-6] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2020] [Accepted: 10/18/2020] [Indexed: 09/02/2023]
Abstract
Abstract
In this article, a new explicit group iterative scheme is developed for the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid. The proposed scheme is based on the high-order compact Crank–Nicolson finite difference method. The resulting scheme consists of three-level finite difference approximations. The stability and convergence of the proposed method are studied using the matrix energy method. Finally, some numerical examples are provided to show the accuracy of the proposed method.
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115
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Cheema TN, Raja MAZ, Ahmad I, Naz S, Ilyas H, Shoaib M. Intelligent computing with Levenberg-Marquardt artificial neural networks for nonlinear system of COVID-19 epidemic model for future generation disease control. EUROPEAN PHYSICAL JOURNAL PLUS 2020; 135:932. [PMID: 33251082 PMCID: PMC7682771 DOI: 10.1140/epjp/s13360-020-00910-x] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2020] [Accepted: 11/02/2020] [Indexed: 05/05/2023]
Abstract
The aim of this work is to design an intelligent computing paradigm through Levenberg-Marquardt artificial neural networks (LMANNs) for solving the mathematical model of Corona virus disease 19 (COVID-19) propagation via human to human interaction. The model is represented with systems of nonlinear ordinary differential equations represented with susceptible, exposed, symptomatic and infectious, super spreaders, infection but asymptomatic, hospitalized, recovery and fatality classes, and reference dataset of the COVID-19 model is generated by exploiting the strength of explicit Runge-Kutta numerical method for metropolitans of China and Pakistan including Wuhan, Karachi, Lahore, Rawalpindi and Faisalabad. The created dataset is arbitrary used for training, validation and testing processes for each cyclic update in Levenberg-Marquardt backpropagation for numerical treatment of the dynamics of COVID-19 model. The effectiveness and reliable performance of the design LMANNs are endorsed on the basis of assessments of achieved accuracy in terms of mean squared error based merit functions, error histograms and regression studies.
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Affiliation(s)
| | - Muhammad Asif Zahoor Raja
- Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Sect. 3, Douliou, Yunlin, 64002 Taiwan R.O.C
- Department of Electrical and Computer Engineering, COMSATS University Islamabad, Attock Campus, Attock, 43600 Pakistan
| | - Iftikhar Ahmad
- Department of Mathematics, University of Gujrat, Gujrat, 50700 Pakistan
| | - Shafaq Naz
- Department of Mathematics, University of Gujrat, Gujrat, 50700 Pakistan
| | - Hira Ilyas
- Department of Mathematics, University of Gujrat, Gujrat, 50700 Pakistan
| | - Muhammad Shoaib
- Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock, 43600 Pakistan
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116
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Estimating the Depth of Anesthesia During the Induction by a Novel Adaptive Neuro-Fuzzy Inference System: A Case Study. Neural Process Lett 2020. [DOI: 10.1007/s11063-020-10369-7] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023]
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117
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Ghanbari B. On the modeling of the interaction between tumor growth and the immune system using some new fractional and fractional-fractal operators. ADVANCES IN DIFFERENCE EQUATIONS 2020; 2020:585. [PMID: 33101402 PMCID: PMC7570433 DOI: 10.1186/s13662-020-03040-x] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2020] [Accepted: 10/08/2020] [Indexed: 05/04/2023]
Abstract
Humans are always exposed to the threat of infectious diseases. It has been proven that there is a direct link between the strength or weakness of the immune system and the spread of infectious diseases such as tuberculosis, hepatitis, AIDS, and Covid-19 as soon as the immune system has no the power to fight infections and infectious diseases. Moreover, it has been proven that mathematical modeling is a great tool to accurately describe complex biological phenomena. In the recent literature, we can easily find that these effective tools provide important contributions to our understanding and analysis of such problems such as tumor growth. This is indeed one of the main reasons for the need to study computational models of how the immune system interacts with other factors involved. To this end, in this paper, we present some new approximate solutions to a computational formulation that models the interaction between tumor growth and the immune system with several fractional and fractal operators. The operators used in this model are the Liouville-Caputo, Caputo-Fabrizio, and Atangana-Baleanu-Caputo in both fractional and fractal-fractional senses. The existence and uniqueness of the solution in each of these cases is also verified. To complete our analysis, we include numerous numerical simulations to show the behavior of tumors. These diagrams help us explain mathematical results and better describe related biological concepts. In many cases the approximate results obtained have a chaotic structure, which justifies the complexity of unpredictable and uncontrollable behavior of cancerous tumors. As a result, the newly implemented operators certainly open new research windows in further computational models arising in the modeling of different diseases. It is confirmed that similar problems in the field can be also be modeled by the approaches employed in this paper.
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Affiliation(s)
- Behzad Ghanbari
- Department of Engineering Science, Kermanshah University of Technology, Kermanshah, Iran
- Department of Mathematics, Faculty of Engineering and Natural Sciences, Bahçeşehir University, 34349 Istanbul, Turkey
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118
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A New Fixed-Time Stability Criterion and Its Application to Synchronization Control of Memristor-Based Fuzzy Inertial Neural Networks with Proportional Delay. Neural Process Lett 2020. [DOI: 10.1007/s11063-020-10305-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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119
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Abdel-Aty AH, Khater MMA, Dutta H, Bouslimi J, Omri M. Computational solutions of the HIV-1 infection of CD4 + T-cells fractional mathematical model that causes acquired immunodeficiency syndrome (AIDS) with the effect of antiviral drug therapy. CHAOS, SOLITONS, AND FRACTALS 2020; 139:110092. [PMID: 32834626 PMCID: PMC7375326 DOI: 10.1016/j.chaos.2020.110092] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/17/2020] [Revised: 06/26/2020] [Accepted: 07/04/2020] [Indexed: 05/23/2023]
Abstract
This paper investigates the exact traveling wave solutions of the fractional model of the human immunodeficiency virus (HIV-1) infection for CD4 + T-cells. This model also treats with the effect of antiviral drug therapy. These solutions calculate both the boundary and initial conditions that allow employing the septic-B-spline scheme which is one of the most recent schemes in the numerical field. We use the obtained computational solutions via the modified Khater, the extended simplest equation, and sech-tanh methods through Atangana-Baleanu derivative operator. The comparison between the exact and numerical evaluated solutions is illustrated by some distinct sketches. The functioning of our numerical method is tested under three computational obtained solutions.
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Affiliation(s)
- Abdel-Haleem Abdel-Aty
- Department of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha, 61922, Saudi Arabia
- Physics Department, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt
| | - Mostafa M A Khater
- Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang, 212013, China
- Department of Mathematics, Obour Institutes, Cairo, 11828, Egypt
| | - Hemen Dutta
- Department of Mathematics, Faculty of Science, Gauhati University, Guwahati 781014, India
| | - Jamel Bouslimi
- Department of Engineering Physics and Instrumentation, National Institute of Applied Sciences and Technology, Carthage University, Tunisia
- Physics Department, Faculty of Science, Taif University, PO.Box 888, 21974 Taif, Saudi Arabia
| | - M Omri
- Deanship of Scientific Research, King Abdulaziz University, Jeddah, Saudi Arabia
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120
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Hematpour N, Ahadpour S. Execution examination of chaotic S-box dependent on improved PSO algorithm. Neural Comput Appl 2020. [DOI: 10.1007/s00521-020-05304-9] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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121
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Synthesis of recurrent neural dynamics for monotone inclusion with application to Bayesian inference. Neural Netw 2020; 131:231-241. [PMID: 32818873 DOI: 10.1016/j.neunet.2020.07.037] [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: 02/23/2020] [Revised: 07/06/2020] [Accepted: 07/31/2020] [Indexed: 11/22/2022]
Abstract
We propose a top-down approach to construct recurrent neural circuit dynamics for the mathematical problem of monotone inclusion (MoI). MoI in a general optimization framework that encompasses a wide range of contemporary problems, including Bayesian inference and Markov decision making. We show that in a recurrent neural circuit/network with Poisson neurons, each neuron's firing curve can be understood as a proximal operator of a local objective function, while the overall circuit dynamics constitutes an operator-splitting system of ordinary differential equations whose equilibrium point corresponds to the solution of the MoI problem. Our analysis thus establishes that neural circuits are a substrate for solving a broad class of computational tasks. In this regard, we provide an explicit synthesis procedure for building neural circuits for specific MoI problems and demonstrate it for the specific case of Bayesian inference and sparse neural coding.
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122
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Leng S, Aihara K. Common stochastic inputs induce neuronal transient synchronization with partial reset. Neural Netw 2020; 128:13-21. [PMID: 32380402 DOI: 10.1016/j.neunet.2020.04.019] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/04/2019] [Revised: 04/10/2020] [Accepted: 04/20/2020] [Indexed: 10/24/2022]
Abstract
Neuronal synchronization plays important roles in information encoding and transmission in the brain. Mathematical models of neurons have been widely used to simulate synchronization behavior and analyze its mechanisms. Common stochastic inputs are considered to be effective in facilitating synchronization. However, the mechanisms of how partial reset affects neuronal synchronization are still not well understood. In this paper, the synchronization of Stein's model neurons with partial reset is studied. The differences in synchronization mechanisms between neurons with full reset and those with partial reset are analyzed, and the findings lead to the novel concept of transient synchronization. Furthermore, it is proven analytically that due to common stochastic inputs, Stein's model neurons with different initial membrane potentials and partial reset achieve transient synchronization with probability 1. Additionally, a systematic numerical analysis is performed to explore the similarities and differences between full reset and partial reset regarding model parameters, synchronization time, and desynchronization behavior. Thus, partial reset is a powerful and flexible tool that facilitates neuronal synchronization while reserving the possibility of desynchronization. Our analysis also provides an alternative approach to analyze neurons of the integrate-and-fire family and a theoretical complement implying possible information encoding mechanisms in the brain.
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Affiliation(s)
- Siyang Leng
- Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan.
| | - Kazuyuki Aihara
- Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan; International Research Center for Neurointelligence (IRCN), University of Tokyo, Tokyo 113-0033, Japan.
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123
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Abstract
In this paper, a stochastic SIRC epidemic model for Influenza A is proposed and investigated. First, we prove that the system exists a unique global positive solution. Second, the extinction of the disease is explored and the sufficient conditions for extinction of the disease are derived. And then the existence of a unique ergodic stationary distribution of the positive solutions for the system is discussed by constructing stochastic Lyapunov function. Furthermore, numerical simulations are employed to illustrate the theoretical results. Finally, we give some further discussions about the system.
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