The analysis of a time delay fractional COVID-19 model via Caputo type fractional derivative.

Novel coronavirus (COVID-19), a global threat whose source is not correctly yet known, was firstly recognised in the city of Wuhan, China, in December 2019. Now, this disease has been spread out to many countries in all over the world. In this paper, we solved a time delay fractional COVID-19 SEIR epidemic model via Caputo fractional derivatives using a predictor-corrector method. We provided numerical simulations to show the nature of the diseases for different classes. We derived existence of unique global solutions to the given time delay fractional differential equations (DFDEs) under a mild Lipschitz condition using properties of a weighted norm, Mittag-Leffler functions and the Banach fixed point theorem. For the graphical simulations, we used real numerical data based on a case study of Wuhan, China, to show the nature of the projected model with respect to time variable. We performed various plots for different values of time delay and fractional order. We observed that the proposed scheme is highly emphatic and easy to implementation for the system of DFDEs.

Impact of vaccination on the dynamics of COVID-19: A mathematical study using fractional derivatives

Arrival of a new disease marks a yearlong destruction of human lives and economy in general, and if the disease turns out to be a pandemic the loss is frightening. COVID-19 is one such pandemic that has claimed millions of lives till date. There is a suffering throughout the world due to various factors associated with the pandemic, be it loss of livelihoods because of sudden shutdown of companies and lockdown, or loss of lives due to lack of medical aid and inadequate vaccination supplies. In this study, we develop a six-compartmental epidemiological model incorporating vaccination. The motivation behind the study is to analyze the significance of higher vaccination efficacy and higher rate of population getting vaccinated in controlling the rise in infectives and thereby the untimely demise of various individuals. The work begins with an ordinary differential equation model followed by stability analysis of the same, after which a fractional-order derivative model of the same is formulated and the existence of uniformly stable solution for the system is proved. In addition to this, we present the stability of the equilibria in general for the fractional model framed. The sensitivity analysis of the basic reproduction number along with its correlation with various parameters is presented. In addition to this, sensitivity of certain state variables in the fractional model with respect to different fractional orders as well with respect to different infection rate is exhibited in this work. Factors related to lockdown and usage of face shields are incorporated in the entire study, and importance of these is highlighted in the study as well. The major takeaway from the study is that mere vaccination will not suffice in eradication of the virus. The vaccine efficacy plays a major role along with other intervention included in the model. The numerical simulations are carried out in MATLAB software using ode45 and fde12. [ FROM AUTHOR] Copyright of International Journal of Biomathematics is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full . (Copyright applies to all s.)

A new fractional mathematical model to study the impact of vaccination on COVID-19 outbreaks

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. © 2022 The Author(s)

A Detailed Mathematical Analysis of the Vaccination Model for COVID-19

This study aims to structure and evaluate a new COVID-19 model which predicts vaccination effect in the Kingdom of Saudi Arabia (KSA) under Atangana-Baleanu-Caputo (ABC) fractional derivatives. On the statistical aspect, we analyze the collected statistical data of fully vaccinated people from June 01, 2021, to February 15, 2022. Then we apply the Eviews program to find the best model for predicting the vaccination against this pandemic, based on daily series data from February 16, 2022, to April 15, 2022. The results of data analysis show that the appropriate model is autoregressive integrated moving average ARIMA (1, 1, 2), and hence, a forecast about the evolution of the COVID-19 vaccination in 60 days is presented. The theoretical aspect provides equilibrium points, reproduction number R0, and biologically feasible region of the proposed model. Also, we obtain the existence and uniqueness results by using the Picard-Lindel method and the iterative scheme with the Laplace transform. On the numerical aspect, we apply the generalized scheme of the Adams-Bashforth technique in order to simulate the fractional model. Moreover, numerical simulations are performed dependent on real data of COVID-19 in KSA to show the plots of the effects of the fractional-order operator with the anticipation that the suggested model approximation will be better than that of the established traditional model. Finally, the concerned numerical simulations are compared with the exact real available date given in the statistical aspect. © 2023 Authors. All rights reserved.

A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space-Time Reaction-Diffusion Equations on Time-Graded Meshes

A weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order alpha,), where time accuracy is of the order (2 - alpha) or (1 + alpha). To deal with this problem, we present a second-order numerical scheme for nonlinear time-space fractional reaction-diffusion equations. For spatial resolution, we employ a matrix transfer technique. Using graded meshes in time, we improve the convergence rate of the algorithm. Furthermore, some sharp error estimates that give an optimal second-order rate of convergence are presented and proven. We discuss the stability properties of the numerical scheme and elaborate on several empirical examples that corroborate our theoretical observations.

Studying of COVID-19 fractional model: Stability analysis.

This article focuses on the recent epidemic caused by COVID-19 and takes into account several measures that have been taken by governments, including complete closure, media coverage, and attention to public hygiene. It is well known that mathematical models in epidemiology have helped determine the best strategies for disease control. This motivates us to construct a fractional mathematical model that includes quarantine categories as well as government sanctions. In this article, we prove the existence and uniqueness of positive bounded solutions for the suggested model. Also, we investigate the stability of the disease-free and endemic equilibriums by using the basic reproduction number (BRN). Moreover, we investigate the stability of the considering model in the sense of Ulam-Hyers criteria. To underpin and demonstrate this study, we provide a numerical simulation, whose results are consistent with the analysis presented in this article.

The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach

COVID-19, a novel coronavirus disease, is still causing concern all over the world. Recently, researchers have been concentrating their efforts on understanding the complex dynamics of this widespread illness. Mathematics plays a big role in understanding the mechanism of the spread of this disease by modeling it and trying to find approximate solutions. In this study, we implement a new technique for an approximation of the analytic series solution called the multistep Laplace optimized decomposition method for solving fractional nonlinear systems of ordinary differential equations. The proposed method is a combination of the multistep method, the Laplace transform, and the optimized decomposition method. To show the ability and effectiveness of this method, we chose the COVID-19 model to apply the proposed technique to it. To develop the model, the Caputo-type fractional-order derivative is employed. The suggested algorithm efficacy is assessed using the fourth-order Runge-Kutta method, and when compared to it, the results show that the proposed approach has a high level of accuracy. Several representative graphs are displayed and analyzed in two dimensions to show the growth and decay in the model concerning the fractional parameter α values. The central processing unit computational time cost in finding graphical results is utilized and tabulated. From a numerical viewpoint, the archived simulations and results justify that the proposed iterative algorithm is a straightforward and appropriate tool with computational efficiency for several coronavirus disease differential model solutions. © 2022 the author(s), published by De Gruyter.

Theoretical and Numerical Analysis of Fractional Order Mathematical Model on Recent COVID-19 Model Using Singular Kernel

This study presents a fractional-order mathematical model of coronavirus. We select COVID-19 model and convert the model into fractional order. Discuss its theoretical and numerical analysis. Firstly, we investigate the existence and uniqueness results using some fixed point theorems for the proposed fractional-order COVID-19 model. Further, we provide the stability analysis with the help of the Hyers-Ulam stability. The fractional operator is used in the Caputo sense. We obtain numerical solutions using famous numerical methods and provide a graphical interpretation using adopted numerical methods. Finally, we compare the above techniques and provide observations according to the obtained solutions. © 2023, The Author(s), under exclusive licence to The National Academy of Sciences, India.

A fractional-order mathematical model based on vaccinated and infected compartments of SARS-CoV-2 with a real case study during the last stages of the epidemiological event.

In 2020 the world faced with a pandemic spread that affected almost everything of humans' social and health life. Regulations to decrease the epidemiological spread and studies to produce the vaccine of SARS-CoV-2 were on one side a hope to return back to the regular life, but on the other side there were also notable criticism about the vaccines itself. In this study, we established a fractional order differential equations system incorporating the vaccinated and re-infected compartments to a S I R frame to consider the expanded and detailed form as an S V I I v R model. We considered in the model some essential parameters, such as the protection rate of the vaccines, the vaccination rate, and the vaccine's lost efficacy after a certain period. We obtained the local stability of the disease-free and co-existing equilibrium points under specific conditions using the Routh-Hurwitz Criterion and the global stability in using a suitable Lyapunov function. For the numerical solutions we applied the Euler's method. The data for the simulations were taken from the World Health Organization (WHO) to illustrate numerically some scenarios that happened.

A new fractional mathematical model to study the impact of vaccination on COVID-19 outbreaks

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 targetting 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.

Clinical effects of 2-DG drug restraining SARS-CoV-2 infection: A fractional order optimal control study.

Fractional calculus is very convenient tool in modeling of an emergent infectious disease system comprising previous disease states, memory of disease patterns, profile of genetic variation etc. Significant complex behaviors of a disease system could be calibrated in a proficient manner through fractional order derivatives making the disease system more realistic than integer order model. In this study, a fractional order differential equation model is developed in micro level to gain perceptions regarding the effects of host immunological memory in dynamics of SARS-CoV-2 infection. Additionally, the possible optimal control of the infection with the help of an antiviral drug, viz. 2-DG, has been exemplified here. The fractional order optimal control would enable to employ the proper administration of the drug minimizing its systematic cost which will assist the health policy makers in generating better therapeutic measures against SARS-CoV-2 infection. Numerical simulations have advantages to visualize the dynamical effects of the immunological memory and optimal control inputs in the epidemic system.

Approximate solution for the nonlinear fractional order mathematical model

Health organizations are working to reduce the outbreak of infectious diseases with the help of several techniques so that exposure to infectious diseases can be minimized. Mathematics is also an important tool in the study of epidemiology. Mathematical modeling presents mathematical expressions and offers a clear view of how variables and interactions between variables affect the results. The objective of this work is to solve the mathematical model of MERS-CoV with the simplest, easiest and most proficient techniques considering the fractional Caputo derivative. To acquire the approximate solution, we apply the Adomian decomposition technique coupled with the Laplace transformation. Also, a convergence analysis of the method is conducted. For the comparison of the obtained results, we apply another semi-analytic technique called the homotopy perturbation method and compare the results. We also investigate the positivity and boundedness of the selected model. The dynamics and solution of the MERS-CoV compartmental mathematical fractional order model and its transmission between the human populace and the camels are investigated graphically for theta = 0.5, 0.7, 0.9, 1.0. It is seen that the recommended schemes are proficient and powerful for the given model considering the fractional Caputo derivative.

VACCINATION STRATEGY AND PSYCHOLOGICAL EFFECTS IN A FRACTIONAL ORDER EPIDEMIC MODEL

Various approaches have been made recently to understand the complex dynamics of many epidemic diseases like COVID-19. The mathematical modeling approach is one of the considerable tools to study the disease spreading pattern. In this paper we study a fractional order SIR epidemic model with nonmonotone incidence rate and vaccination involving a Caputo type fractional derivative. Existence and uniqueness results for the problem are established which means that our model is biologically and mathematically well posed. We Firstly give some preliminaries results. Then we calculate the equilibria and investigate their global stability. Finally, we present some numerical simulations to support our analytical findings. © 2022 the author(s).

A comparison and calibration of integer and fractional-order models of COVID-19 with stratified public response.

The spread of SARS-CoV-2 in the Canadian province of Ontario has resulted in millions of infections and tens of thousands of deaths to date. Correspondingly, the implementation of modeling to inform public health policies has proven to be exceptionally important. In this work, we expand a previous model of the spread of SARS-CoV-2 in Ontario, "Modeling the impact of a public response on the COVID-19 pandemic in Ontario, " to include the discretized, Caputo fractional derivative in the susceptible compartment. We perform identifiability and sensitivity analysis on both the integer-order and fractional-order SEIRD model and contrast the quality of the fits. We note that both methods produce fits of similar qualitative strength, though the inclusion of the fractional derivative operator quantitatively improves the fits by almost 27% corroborating the appropriateness of fractional operators for the purposes of phenomenological disease forecasting. In contrasting the fit procedures, we note potential simplifications for future study. Finally, we use all four models to provide an estimate of the time-dependent basic reproduction number for the spread of SARS-CoV-2 in Ontario between January 2020 and February 2021.

(k,ψ)-Hilfer Nonlocal Integro-Multi-Point Boundary Value Problems for Fractional Differential Equations and Inclusions

In this paper, we establish existence and uniqueness results for single-valued as well as multi-valued (k,ψ)-Hilfer boundary value problems of order in (1,2], subject to nonlocal integro-multi-point boundary conditions. In the single-valued case, we use Banach and Krasnosel’skiĭ fixed point theorems as well as a Leray–Schauder nonlinear alternative to derive the existence and uniqueness results. For the multi-valued problem, we prove two existence results for the convex and non-convex nature of the multi-valued map involved in a problem by applying a Leray–Schauder nonlinear alternative for multi-valued maps, and a Covitz–Nadler fixed point theorem for multi-valued contractions, respectively. Numerical examples are presented for illustration of all the obtained results.

Application of fractional optimal control theory for the mitigating of novel coronavirus in Algeria.

In this paper, we investigate the dynamics of novel coronavirus infection (COVID-19) using a fractional mathematical model in Caputo sense. Based on the spread of COVID-19 virus observed in Algeria, we formulate the model by dividing the infected population into two sub-classes namely the reported and unreported infective individuals. The existence and uniqueness of the model solution are given by using the well-known Picard-Lindelöf approach. The basic reproduction number R 0 is obtained and its value is estimated from the actual cases reported in Algeria. The model equilibriums and their stability analysis are analyzed. The impact of various constant control parameters is depicted for integer and fractional values of α . Further, we perform the sensitivity analysis showing the most sensitive parameters of the model versus R 0 to predict the incidence of the infection in the population. Further, based on the sensitivity analysis, the Caputo model with constant controls is extended to time-dependent variable controls in order obtain a fractional optimal control problem. The associated four time-dependent control variables are considered for the prevention, treatment, testing and vaccination. The fractional optimality condition for the control COVID-19 transmission model is presented. The existence of the Caputo optimal control model is studied and necessary condition for optimality in the Caputo case is derived from Pontryagin's Maximum Principle. Finally, the effectiveness of the proposed control strategies are demonstrated through numerical simulations. The graphical results revealed that the implantation of time-dependent controls significantly reduces the number of infective cases and are useful in mitigating the infection.

A fractional-order delay differential SEIR model forCOVID-19: Stability analysis and Hopf bifurcation

In this manuscript, we study a fractional order time delay SEIR model of COVID-19 disease. Some conditions on stability and Hopf bifurcation have been derived for the model by using Laplace transformation. Further numerical simulation has been carried out for the purpose of better understanding of our results. © CSP - Cambridge, UK, I&S - Florida, USA, 2022

Dynamical Analysis of Nutrient-Phytoplankton-Zooplankton Model with Viral Disease in Phytoplankton Species under Atangana-Baleanu-Caputo Derivative

A mathematical model of the nutrient-phytoplankton-zooplankton associated with viral infection in phytoplankton under the Atangana-Baleanu derivative in Caputo sense is investigated in this study. We prove the theoretical results for the existence and uniqueness of the solutions by using Banach’s and Sadovskii’s fixed point theorems. The notion of various Ulam’s stability is used to guarantee the context of the stability analysis. Furthermore, the equilibrium points and the basic reproduction numbers for the proposed model are provided. The Adams type predictor-corrector algorithm has been applied for the theoretical confirmation to establish the approximate solutions. A variety of numerical plots corresponding to various fractional orders between zero and one are presented to describe the dynamical behavior of the fractional model under consideration.

Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel

In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated series.

Modeling of fractional-order COVID-19 epidemic model with quarantine and social distancing.

Different countries of the world are facing a serious pandemic of corona virus disease (COVID-19). One of the most typical treatments for COVID-19 is social distancing, which includes lockdown; it will help to decrease the number of contacts for undiagnosed individuals. The main aim of this article is to construct and evaluate a fractional-order COVID-19 epidemic model with quarantine and social distancing. Laplace homotopy analysis method is used for a system of fractional differential equation (FDEs) with Caputo and Atangana-Baleanu-Caputo (ABC) fractional derivative. By applying the ABC and Caputo derivative, the numerical solution for fractional-order COVID-19 epidemic model is achieved. The uniqueness and existence of the solution is checked by Picard-Lindelof's method. The proposed fractional model is demonstrated by numerical simulation which is useful for the government to control the spread of disease in a practical way.