Mathematical Modeling of COVID-19 with Vaccination Using Fractional Derivative: A Case Study

Vaccination against any infectious disease is considered to be an effective way of controlling it. This paper studies a fractional order model with vaccine efficacy and waning immunity. We present the model's dynamics under vaccine efficacy, the impact of immunization, and the waning of the vaccine on coronavirus infection disease. We analyze the model under their equilibrium points. The model under the equilibrium points is discussed and proven that it is locally asymptotically stable if (Formula presented.) is lesser than unity. We present the backward bifurcation hypothesis of the model and show that there is a parameter (Formula presented.) that causes the backward bifurcation in the imperfect vaccine model. We show certain assumptions when (Formula presented.) for the imperfect vaccine case;the model is both stable globally asymptotically at the disease-free ((Formula presented.)) and endemic cases ((Formula presented.)). By using infected cases from the recent wave throughout Pakistan, we shall estimate the model parameters and calculate the numerical value of the basic reproductive number (Formula presented.). We present the comprehensive graphical results for the realistic parameter values and show many useful suggestions regarding the elimination of the infection from society. The vaccination efficacy that provides an important role in disease elimination is discussed in detail. © 2023 by the authors.

A stochastically perturbed co-infection epidemic model for COVID-19 and hepatitis B virus.

A new co-infection model for the transmission dynamics of two virus hepatitis B (HBV) and coronavirus (COVID-19) is formulated to study the effect of white noise intensities. First, we present the model equilibria and basic reproduction number. The local stability of the equilibria points is proved. Moreover, the proposed stochastic model has been investigated for a non-negative solution and positively invariant region. With the help of Lyapunov function, analysis was performed and conditions for extinction and persistence of the disease based on the stochastic co-infection model were derived. Particularly, we discuss the dynamics of the stochastic model around the disease-free state. Similarly, we obtain the conditions that fluctuate at the disease endemic state holds if min ( R H s , R C s , R HC s ) > 1 . Based on extinction as well as persistence some conditions are established in form of expression containing white noise intensities as well as model parameters. The numerical results have also been used to illustrate our analytical results.

Mathematical analysis of new variant Omicron model driven by Lévy noise and with variable-order fractional derivatives

In this paper, we study a variable-order fractional mathematical model driven by Lévy noise describing the new variant of COVID-19 (Omicron virus). Based on our analysis and discussion under a new set of sufficient conditions, we prove the existence and uniqueness of the related solution. Moreover, we discuss the stability analysis of the corresponding Omicron virus model by employing Ulam–Hyers and Ulam–Hyers–Rassias stabilities in Banach spaces. Finally, we present some numerical results and comparative studies to show clearly the importance of our results and its effects on behaviors of the new variant model.

A Mathematical Model of Vaccinations Using New Fractional Order Derivative.

Purpose: This paper studies a simple SVIR (susceptible, vaccinated, infected, recovered) type of model to investigate the coronavirus's dynamics in Saudi Arabia with the recent cases of the coronavirus. Our purpose is to investigate coronavirus cases in Saudi Arabia and to predict the early eliminations as well as future case predictions. The impact of vaccinations on COVID-19 is also analyzed. Methods: We consider the recently introduced fractional derivative known as the generalized Hattaf fractional derivative to extend our COVID-19 model. To obtain the fitted and estimated values of the parameters, we consider the nonlinear least square fitting method. We present the numerical scheme using the newly introduced fractional operator for the graphical solution of the generalized fractional differential equation in the sense of the Hattaf fractional derivative. Mathematical as well as numerical aspects of the model are investigated. Results: The local stability of the model at disease-free equilibrium is shown. Further, we consider real cases from Saudi Arabia since 1 May−4 August 2022, to parameterize the model and obtain the basic reproduction number R0v≈2.92. Further, we find the equilibrium point of the endemic state and observe the possibility of the backward bifurcation for the model and present their results. We present the global stability of the model at the endemic case, which we found to be globally asymptotically stable when R0v>1. Conclusion: The simulation results using the recently introduced scheme are obtained and discussed in detail. We present graphical results with different fractional orders and found that when the order is decreased, the number of cases decreases. The sensitive parameters indicate that future infected cases decrease faster if face masks, social distancing, vaccination, etc., are effective.

A mathematical model of coronavirus transmission by using the heuristic computing neural networks.

In this study, the nonlinear mathematical model of COVID-19 is investigated by stochastic solver using the scaled conjugate gradient neural networks (SCGNNs). The nonlinear mathematical model of COVID-19 is represented by coupled system of ordinary differential equations and is studied for three different cases of initial conditions with suitable parametric values. This model is studied subject to seven class of human population N(t) and individuals are categorized as: susceptible S(t), exposed E(t), quarantined Q(t), asymptotically diseased IA (t), symptomatic diseased IS (t) and finally the persons removed from COVID-19 and are denoted by R(t). The stochastic numerical computing SCGNNs approach will be used to examine the numerical performance of nonlinear mathematical model of COVID-19. The stochastic SCGNNs approach is based on three factors by using procedure of verification, sample statistics, testing and training. For this purpose, large portion of data is considered, i.e., 70%, 16%, 14% for training, testing and validation, respectively. The efficiency, reliability and authenticity of stochastic numerical SCGNNs approach are analysed graphically in terms of error histograms, mean square error, correlation, regression and finally further endorsed by graphical illustrations for absolute errors in the range of 10-05 to 10-07 for each scenario of the system model.

Application of piecewise fractional differential equation to COVID-19 infection dynamics.

We proposed a new mathematical model to study the COVID-19 infection in piecewise fractional differential equations. The model was initially designed using the classical differential equations and later we extend it to the fractional case. We consider the infected cases generated at health care and formulate the model first in integer order. We extend the model into Caputo fractional differential equation and study its background mathematical results. We show that the fractional model is locally asymptotically stable when R 0 < 1 at the disease-free case. For R 0 ≤ 1 , we show the global asymptotical stability of the model. We consider the infected cases in Saudi Arabia and determine the parameters of the model. We show that for the real cases, the basic reproduction is R 0 ≈ 1 . 7372 . We further extend the Caputo model into piecewise stochastic fractional differential equations and discuss the procedure for its numerical simulation. Numerical simulations for the Caputo case and piecewise models are shown in detail.

A Chat Recommender System for COVID-19 Support based in Textual Sentence Embeddings

With the emergence of the COVID-19 pandemic, the demand for health services has exponentially increased, which caused the saturation of hospital beds and a high death toll. Motivated by the need to provide more agility in patients' attendance and unburden the health services, this work proposes a solution for automatic attendance via a Recommender System that uses sentence embeddings of text messages to train an LSTM classifier. This classifier can provide recommendations of a course of action for patients, instructing them to stay at home or seek medical support. Our numerical results validate the proposed solution and corroborate its reasonable accuracy rate. © 2021 ACM.

Dynamical analysis of coronavirus disease with crowding effect, and vaccination: a study of third strain.

Countries affected by the coronavirus epidemic have reported many infected cases and deaths based on world health statistics. The crowding factor, which we named "crowding effects," plays a significant role in spreading the diseases. However, the introduction of vaccines marks a turning point in the rate of spread of coronavirus infections. Modeling both effects is vastly essential as it directly impacts the overall population of the studied region. To determine the peak of the infection curve by considering the third strain, we develop a mathematical model (susceptible-infected-vaccinated-recovered) with reported cases from August 01, 2021, till August 29, 2021. The nonlinear incidence rate with the inclusion of both effects is the best approach to analyze the dynamics. The model's positivity, boundedness, existence, uniqueness, and stability (local and global) are addressed with the help of a reproduction number. In addition, the strength number and second derivative Lyapunov analysis are examined, and the model was found to be asymptotically stable. The suggested parameters efficiently control the active cases of the third strain in Pakistan. It was shown that a systematic vaccination program regulates the infection rate. However, the crowding effect reduces the impact of vaccination. The present results show that the model can be applied to other countries' data to predict the infection rate.

Mathematical modeling and analysis of COVID-19: A study of new variant Omicron.

We construct a new mathematical model to better understand the novel coronavirus (omicron variant). We briefly present the modeling of COVID-19 with the omicron variant and present their mathematical results. We study that the Omicron model is locally asymptotically stable if the basic reproduction number R 0 < 1 , while for R 0 ≤ 1 , the model at the disease-free equilibrium is globally asymptotically stable. We extend the model to the second-order differential equations to study the possible occurrence of the layers(waves). We then extend the model to a fractional stochastic version and studied its numerical results. The real data for the period ranging from November 1, 2021, to January 23, 2022, from South Africa are considered to obtain the realistic values of the model parameters. The basic reproduction number for the suggested data is found to be approximate R 0 ≈ 2 . 1107 which is very close to the actual basic reproduction in South Africa. We perform the global sensitivity analysis using the PRCC method to investigate the most influential parameters that increase or decrease R 0 . We use the new numerical scheme recently reported for the solution of piecewise fractional differential equations to present the numerical simulation of the model. Some graphical results for the model with sensitive parameters are given which indicate that the infection in the population can be minimized by following the recommendations of the world health organizations (WHO), such as social distances, using facemasks, washing hands, avoiding gathering, etc.

Mathematical modeling and analysis of the SARS-Cov-2 disease with reinfection.

The COVID-19 infection which is still infecting many individuals around the world and at the same time the recovered individuals after the recovery are infecting again. This reinfection of the individuals after the recovery may lead the disease to worse in the population with so many challenges to the health sectors. We study in the present work by formulating a mathematical model for SARS-CoV-2 with reinfection. We first briefly discuss the formulation of the model with the assumptions of reinfection, and then study the related qualitative properties of the model. We show that the reinfection model is stable locally asymptotically when R0<1. For R0≤1, we show that the model is globally asymptotically stable. Further, we consider the available data of coronavirus from Pakistan to estimate the parameters involved in the model. We show that the proposed model shows good fitting to the infected data. We compute the basic reproduction number with the estimated and fitted parameters numerical value is R0≈1.4962. Further, we simulate the model using realistic parameters and present the graphical results. We show that the infection can be minimized if the realistic parameters (that are sensitive to the basic reproduction number) are taken into account. Also, we observe the model prediction for the total infected cases in the future fifth layer of COVID-19 in Pakistan that may begin in the second week of February 2022.

Are smart contracts too smart for Supply Chain 4.0? A blockchain framework to mitigate challenges

Purpose: Smart contracts are self-executing computer programmes that have the potential to be used in several applications instead of traditional written contracts. With the recent rise of smart systems (e.g. Internet of things) and digital platforms (e.g. blockchain), smart contracts are gaining high interest in both business and academia. In this work, a framework for smart contracts was proposed with using reputation as the system currency, and conducts currency mining through fulfilling the physical commitments that are agreed upon. Design/methodology/approach: A game theory model is developed to represent the proposed system, and then a system dynamics simulator is used to check the response of the blockchain with different sizes. Findings: The numerical results showed that the proposed system could identify the takeover attacks and protect the blockchain from being controlled by an outsider. Another important finding is that careful setting of the maximum currency amount can improve the scalability of the blockchain and prevent the currency inflation. Research limitations/implications: This work is proposed as a conceptual framework for supply chain 4.0. Future work will be dedicated to implement and experiment the proposed framework for other characteristics that may be encountered in the context of supply chain 4.0, such as different suppliers' tiers, different customer typologies and smart logistics applications, which may reveal other challenges and provide additional interesting insights. Practical implications: By using the proposed framework, smart contracts and blockchains can be implemented to handle many issues in the context of operations and supply chain 4.0, especially in times of turbulence such as the COVID-19 global pandemic crisis. Originality/value: This work emphasizes that smart contracts are not too smart to be applied in the context of supply chain 4.0. The proposed framework of smart contracts is expected to serve supply chain 4.0 by automating the knowledge work and enabling scenario planning through the game theory model. It will also improve online transparency and order processing in real-time through secured multitier connectivity. This can be applied in global supply chain functions backed with digitization, notably during the time of the pandemic, in which e-commerce and online shopping have changed the rules of the game. © 2021, Emerald Publishing Limited.

Mathematical modeling and optimal control of the COVID-19 dynamics.

We are considering a new COVID-19 model with an optimal control analysis when vaccination is present. Firstly, we formulate the vaccine-free model and present the associated mathematical results involved. Stability results for R 0 < 1 are shown. In addition, we frame the model with the vaccination class. We look at the mathematical results with the details of the vaccine model. Additionally, we are considering setting controls to minimize infection spread and control. We consider four different controls, such as prevention, vaccination control, rapid screening of people in the exposed category, and people who are identified as infected without screening. Using the suggested controls, we develop an optimal control model and derive mathematical results from it. In addition, the mathematical model with control and without control is resolved by the forward-backward Runge-Kutta method and presents the results graphically. The results obtained through optimal control suggest that controls can be useful for minimizing infected individuals and improving population health.

Numerical Investigations through ANNs for Solving COVID-19 Model.

The current investigations of the COVID-19 spreading model are presented through the artificial neuron networks (ANNs) with training of the Levenberg-Marquardt backpropagation (LMB), i.e., ANNs-LMB. The ANNs-LMB scheme is used in different variations of the sample data for training, validation, and testing with 80%, 10%, and 10%, respectively. The approximate numerical solutions of the COVID-19 spreading model have been calculated using the ANNs-LMB and compared viably using the reference dataset based on the Runge-Kutta scheme. The obtained performance of the solution dynamics of the COVID-19 spreading model are presented based on the ANNs-LMB to minimize the values of fitness on mean square error (M.S.E), along with error histograms, regression, and correlation analysis.

Investigation of fractal-fractional order model of COVID-19 in Pakistan under Atangana-Baleanu Caputo (ABC) derivative.

This manuscript addressing the dynamics of fractal-fractional type modified SEIR model under Atangana-Baleanu Caputo (ABC) derivative of fractional order y and fractal dimension p for the available data in Pakistan. The proposed model has been investigated for qualitative analysis by applying the theory of non-linear functional analysis along with fixed point theory. The fractional Adams-bashforth iterative techniques have been applied for the numerical solution of the said model. The Ulam-Hyers (UH) stability techniques have been derived for the stability of the considered model. The simulation of all compartments has been drawn against the available data of covid-19 in Pakistan. The whole study of this manuscript illustrates that control of the effective transmission rate is necessary for stoping the transmission of the outbreak. This means that everyone in the society must change their behavior towards self-protection by keeping most of the precautionary measures sufficient for controlling covid-19.

SARS-CoV-2 and Rohingya Refugee Camp, Bangladesh: Uncertainty and How the Government Took Over the Situation.

Background: Bangladesh hosts more than 800,000 Rohingya refugees from Myanmar. The low health immunity, lifestyle, access to good healthcare services, and social-security cause this population to be at risk of far more direct effects of COVID-19 than the host population. Therefore, evidence-based forecasting of the COVID-19 burden is vital in this regard. In this study, we aimed to forecast the COVID-19 obligation among the Rohingya refugees of Bangladesh to keep up with the disease outbreak's pace, health needs, and disaster preparedness. Methodology and Findings: To estimate the possible consequences of COVID-19 in the Rohingya camps of Bangladesh, we used a modified Susceptible-Exposed-Infectious-Recovered (SEIR) transmission model. All of the values of different parameters used in this model were from the Bangladesh Government's database and the relevant emerging literature. We addressed two different scenarios, i.e., the best-fitting model and the good-fitting model with unique consequences of COVID-19. Our best fitting model suggests that there will be reasonable control over the transmission of the COVID-19 disease. At the end of December 2020, there will be only 169 confirmed COVID-19 cases in the Rohingya refugee camps. The average basic reproduction number (R0) has been estimated to be 0.7563. Conclusions: Our analysis suggests that, due to the extensive precautions from the Bangladesh government and other humanitarian organizations, the coronavirus disease will be under control if the maintenance continues like this. However, detailed and pragmatic preparedness should be adopted for the worst scenario.

Non-standard finite difference scheme and analysis of smoking model with reversion class.

Smokers are at more risk to COVID-19 as the entertainment of smoking because their fingers are in touch with lips regularly during smoking that increases the probability of transmission of virus from hand to mouth. On other hand the smokers may have lung disease (or reduced lung capacity) which would greatly increase risk of serious illness especially COVID-19. For this esteem, in this research work, we first formulate a mathematical model contains the reversion class. Then, using different techniques for finding the local and global stability of the presented model related to equilibrium points that are free smoking and positive smoking equilibrium points. As the model consisting on the nonlinear equations, so we use the non-standard finite difference (NSFD) scheme, ODE45 and RK4 methods to find the numerical results. Finally, we show the graphs numerically through MATLAB.

Mathematical modeling for novel coronavirus (COVID-19) and control.

In the present investigations, we construct a new mathematical for the transmission dynamics of corona virus (COVID-19) using the cases reported in Kingdom of Saudi Arabia for March 02 till July 31, 2020. We investigate the parameters values of the model using the least square curve fitting and the basic reproduction number is suggested for the given data is ℛ0 ≈ 1.2937. The stability results of the model are shown when the basic reproduction number is ℛ0 < 1. The model is locally asymptotically stable when ℛ0 < 1. Further, we show some important parameters that are more sensitive to the basic reproduction number ℛ0 using the PRCC method. The sensitive parameters that act as a control parameters that can reduce and control the infection in the population are shown graphically. The suggested control parameters can reduce dramatically the infection in the Kingdom of Saudi Arabia if the proper attention is paid to the suggested controls.

Modeling and simulation of the novel coronavirus in Caputo derivative.

The Coronavirus disease or COVID-19 is an infectious disease caused by a newly discovered coronavirus. The COVID-19 pandemic is an inciting panic for human health and economy as there is no vaccine or effective treatment so far. Different mathematical modeling approaches have been suggested to analyze the transmission patterns of this novel infection. this paper, we investigate the dynamics of COVID-19 using the classical Caputo fractional derivative. Initially, we formulate the mathematical model and then explore some the basic and necessary analysis including the stability results of the model for the case when R 0 < 1 . Despite the basic analysis, we consider the real cases of coronavirus in China from January 11, 2020 to April 9, 2020 and estimated the basic reproduction number as R 0 ≈ 4.95 . The present findings show that the reported data is accurately fit the proposed model and consequently, we obtain more realistic and suitable parameters. Finally, the fractional model is solved numerically using a numerical approach and depicts many graphical results for the fractional order of Caputo operator. Furthermore, some key parameters and their impact on the disease dynamics are shown graphically.

The dynamics of COVID-19 with quarantined and isolation.

In the present paper, we formulate a new mathematical model for the dynamics of COVID-19 with quarantine and isolation. Initially, we provide a brief discussion on the model formulation and provide relevant mathematical results. Then, we consider the fractal-fractional derivative in Atangana-Baleanu sense, and we also generalize the model. The generalized model is used to obtain its stability results. We show that the model is locally asymptotically stable if R 0 < 1 . Further, we consider the real cases reported in China since January 11 till April 9, 2020. The reported cases have been used for obtaining the real parameters and the basic reproduction number for the given period, R 0 ≈ 6.6361 . The data of reported cases versus model for classical and fractal-factional order are presented. We show that the fractal-fractional order model provides the best fitting to the reported cases. The fractional mathematical model is solved by a novel numerical technique based on Newton approach, which is useful and reliable. A brief discussion on the graphical results using the novel numerical procedures are shown. Some key parameters that show significance in the disease elimination from the society are explored.

Modeling the impact of non-pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study.

Coronavirus disease (COVID-19) is the biggest public health challenge the world is facing in recent days. Since there is no effective vaccine and treatment for this virus, therefore, the only way to mitigate this infection is the implementation of non-pharmaceutical interventions such as social-distancing, community lockdown, quarantine, hospitalization or self-isolation and contact-tracing. In this paper, we develop a mathematical model to explore the transmission dynamics and possible control of the COVID-19 pandemic in Pakistan, one of the Asian countries with a high burden of disease with more than 200,000 confirmed infected cases so far. Initially, a mathematical model without optimal control is formulated and some of the basic necessary analysis of the model, including stability results of the disease-free equilibrium is presented. It is found that the model is stable around the disease-free equilibrium both locally and globally when the basic reproduction number is less than unity. Despite the basic analysis of the model, we further consider the confirmed infected COVID-19 cases documented in Pakistan from March 1, till May 28, 2020 and estimate the model parameters using the least square fitting tools from statistics and probability theory. The results show that the model output is in good agreement with the reported COVID-19 infected cases. The approximate value of the basic reproductive number based on the estimated parameters is R 0 ≈ 1.87 . The effect of low (or mild), moderate, and comparatively strict control interventions like social-distancing, quarantine rate, (or contact-tracing of suspected people) and hospitalization (or self-isolation) of testing positive COVID-19 cases are shown graphically. It is observed that the most effective strategy to minimize the disease burden is the implementation of maintaining a strict social-distancing and contact-tracing to quarantine the exposed people. Furthermore, we carried out the global sensitivity analysis of the most crucial parameter known as the basic reproduction number using the Latin Hypercube Sampling (LHS) and the partial rank correlation coefficient (PRCC) techniques. The proposed model is then reformulated by adding the time-dependent control variables u 1(t) for quarantine and u 2(t) for the hospitalization interventions and present the necessary optimality conditions using the optimal control theory and Pontryagin's maximum principle. Finally, the impact of constant and optimal control interventions on infected individuals is compared graphically.