Dynamics of fractional order COVID-19 model with a case study of Saudi Arabia

The analysis of a new fractional model to the Zika virus infection with mutant

We present a new mathematical model to analyze the dynamics of the Zika virus (ZV) disease with the mutant under the real confirmed cases in Colombia. We give the formulation of the model initially in integer order derivative and then extend it to a fractional order system in the sense of the Mittag-Leffler kernel. We study the properties of the model in the Mittag-Leffler kernel and establish the result. The basic reproduction of the fractional system is computed. The equilibrium points of the Zika virus model are obtained and found that the endemic equilibria exist when the threshold is greater than unity. Further, we show that the model does not possess the backward bifurcation phenomenon. The numerical procedure to solve the problem using the Atangana-Baleanu derivative is shown using the newly established numerical scheme. We consider the real cases of the Zika virus in Colombia outbreak are considered and simulate the model using the nonlinear least square curve fit and computed the basic reproduction number R 0 = 0.4942 , whereas in previous work (Alzahrani et al., 2021) [1], the authors computed the basic reproduction number R 0 = 0.5447 . This is due to the fact that our work in the present paper provides better fitting to the data when using the fractional order model, and indeed the result regarding the data fitting using the fractional model is better than integer order model. We give a sensitivity analysis of the parameters involved in the basic reproduction number and show them graphically. The results obtained through the present numerical method converge to its equilibrium for the fractional order, indicating the proposed scheme's reliability.

Modeling the dynamics of COVID-19 with real data from Thailand

In recent years, COVID-19 has evolved into many variants, posing new challenges for disease control and prevention. The Omicron variant, in particular, has been found to be highly contagious. In this study, we constructed and analyzed a mathematical model of COVID-19 transmission that incorporates vaccination and three different compartments of the infected population: asymptomatic [Formula: see text], symptomatic [Formula: see text], and Omicron [Formula: see text]. The model is formulated in the Caputo sense, which allows for fractional derivatives that capture the memory effects of the disease dynamics. We proved the existence and uniqueness of the solution of the model, obtained the effective reproduction number, showed that the model exhibits both endemic and disease-free equilibrium points, and showed that backward bifurcation can occur. Furthermore, we documented the effects of asymptomatic infected individuals on the disease transmission. We validated the model using real data from Thailand and found that vaccination alone is insufficient to completely eradicate the disease. We also found that Thailand must monitor asymptomatic individuals through stringent testing to halt and subsequently eradicate the disease. Our study provides novel insights into the behavior and impact of the Omicron variant and suggests possible strategies to mitigate its spread.

An Efficient Analytical Approach to Investigate Fractional Caudrey–Dodd–Gibbon Equations with Non-Singular Kernel Derivatives

Fractional calculus is at this time an area where many models are still being developed, explored, and used in real-world applications in many branches of science and engineering where non-locality plays a key role. Although many wonderful discoveries have already been reported by researchers in important monographs and review articles, there is still a great deal of non-local phenomena that have not been studied and are only waiting to be explored. As a result, we can continually learn about new applications and aspects of fractional modelling. In this study, a precise and analytical method with non-singular kernel derivatives is used to solve the Caudrey–Dodd–Gibbon (CDG) model, a modification of the fifth-order KdV equation (fKdV). The fractional derivative is taken into account by the Caputo–Fabrizio (CF) derivative and the Atangana–Baleanu derivative in the Caputo sense (ABC). This model illustrates the propagation of magneto-acoustic, shallow-water, and gravity–capillary waves in a plasma medium. The dynamic behaviour of the acquired solutions has been represented in a number of two- and three-dimensional figures. A number of simulations are also performed to demonstrate how the resulting solutions physically behave with respect to fractional order. The significance of the current research is that new solutions are obtained by using a strong analytical approach. Utilizing a fractional derivative operator to solve equivalent models is another benefit of this approach. The results of the present work have similar aspects to the symmetry of partial differential equations.

A memory effect model to predict COVID-19: analysis and simulation

On 19 September 2020, the Centers for Disease Control and Prevention (CDC) recommended that asymptomatic individuals, those who have close contact with infected person, be tested. Also, American society for biological clinical comments on testing of asymptomatic individuals. So, we proposed a new mathematical model for evaluating the population-level impact of contact rates (social-distancing) and the rate at which asymptomatic people are hospitalized (isolated) following testing due to close contact with documented infected people. The model is a deterministic system of nonlinear differential equations that is fitted and parameterized by least square curve fitting using COVID-19 pandemic data of Pakistan from 1 October 2020 to 30 April 2021. The fractional derivative is used to understand the biological process with crossover behavior and memory effect. The reproduction number and conditions for asymptotic stability are derived diligently. The most common non-integer Caputo derivative is used for deeper analysis and transmission dynamics of COVID-19 infection. The fractional-order Adams-Bashforth method is used for the solution of the model. In light of the dynamics of the COVID-19 outbreak in Pakistan, non-pharmaceutical interventions (NPIs) in terms of social distancing and isolation are being investigated. The reduction in the baseline value of contact rates and enhancement in hospitalization rate of symptomatic can lead the elimination of the pandemic.

Fractional order SEIQRD epidemic model of Covid-19: A case study of Italy

The fractional order SEIQRD compartmental model of COVID-19 is explored in this manuscript with six different categories in the Caputo approach. A few findings for the new model's existence and uniqueness criterion, as well as non-negativity and boundedness of the solution, have been established. When RCovid19<1 at infection-free equilibrium, we prove that the system is locally asymptotically stable. We also observed that RCovid 19<1, the system is globally asymptotically stable in the absence of disease. The main objective of this study is to investigate the COVID-19 transmission dynamics in Italy, in which the first case of Coronavirus infection 2019 (COVID-19) was identified on January 31st in 2020. We used the fractional order SEIQRD compartmental model in a fractional order framework to account for the uncertainty caused by the lack of information regarding the Coronavirus (COVID-19). The Routh-Hurwitz consistency criteria and La-Salle invariant principle are used to analyze the dynamics of the equilibrium. In addition, the fractional-order Taylor's approach is utilized to approximate the solution to the proposed model. The model's validity is demonstrated by comparing real-world data with simulation outcomes. This study considered the consequences of wearing face masks, and it was discovered that consistent use of face masks can help reduce the propagation of the COVID-19 disease.

Role of Vaccines in Controlling the Spread of COVID-19: A Fractional-Order Model

In this paper, we present a fractional-order mathematical model in the Caputo sense to investigate the significance of vaccines in controlling COVID-19. The Banach contraction mapping principle is used to prove the existence and uniqueness of the solution. Based on the magnitude of the basic reproduction number, we show that the model consists of two equilibrium solutions that are stable. The disease-free and endemic equilibrium points are locally stably when R0<1 and R0>1 respectively. We perform numerical simulations, with the significance of the vaccine clearly shown. The changes that occur due to the variation of the fractional order α are also shown. The model has been validated by fitting it to four months of real COVID-19 infection data in Thailand. Predictions for a longer period are provided by the model, which provides a good fit for the data.

Optimal control strategies to combat COVID-19 transmission: A mathematical model with incubation time delay

The coronavirus disease 2019, started spreading around December 2019, still persists in the population all across the globe. Though different countries have been able to cope with the disease to some extent and vaccination for the same has been developed, it cannot be ignored that the disease is still not on the verge of completely eradicating, which in turn creates a need for having deeper insights of the disease in order to understand it well and hence be able to work towards its eradication. Meanwhile, using mitigation strategies like non-pharmaceutical interventions can help in controlling the disease. In this work, our aim is to study the dynamics of COVID-19 using compartmental approach by applying various analytical methods. We obtain formula for important tools like R0 and establish the stability of disease-free equilibrium point for R0<1. Further, based on R0, we discuss the stability and existence of the endemic equilibrium point. We incorporate various control strategies possible and using optimal control theory, study their expected positive impacts on the spread of the disease. Later, using a biologically feasible set of parameters, we numerically analyse the model. We even study the trend of the outbreak in China, for over 120 days, where the active cases rise up to a peak and then the curve flattens.

Qualitative analysis of a fractional-order two-strain epidemic model with vaccination and general non-monotonic incidence rate

In this paper, a fractional-order two-strain epidemic model with vaccination and general non-monotonic incidence rate is analyzed. The studied problem is formulated using susceptible, infectious and recovered compartmental model. A Caputo fractional operator is incorporated in each compartment to describe the memory effect related to an epidemic evolution. First, the global existence, positivity and boundedness of solutions of the proposed model are proved. The basic reproduction numbers associated with studied problem are calculated. Four steady states are given, namely the disease-free equilibrium, the strain 1 endemic equilibrium, the strain 2 endemic equilibrium, and the endemic equilibrium associated with both strains. By considering appropriate Lyapunov functions, the global stability of the equilibrium points is proven according to the model parameters. Our modeling approach using a generalized non-monotonic incidence functions encloses a variety of fractional-order epidemic models existing in the literature. Finally, the theoretical findings are illustrated using numerical simulations.

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.

Dynamic stability and optimal control of SIS q I q RS epidemic network

We develop a complex network-based SISq Iq RS model, calculate the thresholdR 0 of infectious disease transmission and analyze the stability of the model. In the model, three control measures including isolation and vaccination are considered, where the isolation is structured in isolation of susceptible nodes and the isolation of infected nodes. We regard these three kinds of controls as time-varying variables, and obtain the existence and the solution of the optimal control by using the optimal control theory. With regard to the stability of the model, sensitivity analysis of the parameters and optimal control, we carry out numerical simulations. From the simulation results, it is obvious that when the three kinds of controls exist simultaneously, the scale and cost of the disease are minimal. Finally, we fit the real data of COVID-19 to the numerical solution of the model.

A case study of 2019-nCoV in Russia using integer and fractional order derivatives

In this article, we define a mathematical model to analyze the outbreaks of the most deadly disease of the decade named 2019-nCoV by using integer and fractional order derivatives. For the case study, the real data of Russia is taken to perform novel parameter estimation by using the Trust Region Reflective (TRR) algorithm. First, we define an integer order model and then generalize it by using fractional derivatives. A novel optimal control problem is derived to see the impact of possible preventive measures against the spread of 2019-nCoV. We implement the forward-backward sweep method to numerically solve our proposed model and control problem. A number of graphs have been plotted to see the impact of the proposed control practically. The Russian data-based parameter estimation along with the proposal of a mathematical model in the sense of Caputo fractional derivative that contains the memory term in the system are the main novel features of this study.

Threshold of Stochastic SIRS Epidemic Model from Infectious to Susceptible Class with Saturated Incidence Rate Using Spectral Method

Stochastic SIRS models play a key role in formulating and analyzing the transmission of infectious diseases. These models reflect the environmental changes of the diseases and their biological mechanisms. Therefore, it is very important to study the uniqueness and existence of the global positive solution to investigate the asymptotic properties of the model. In this article, we investigate the dynamics of the stochastic SIRS epidemic model with a saturated incidence rate. The effects of both deterministic and stochastic distribution from infectious to susceptible are analyzed. Our findings show that the occurrence of symmetry breaking as a function of the stochastic noise has a significant advantage over the deterministic one to prevent the spread of the infectious diseases. The larger stochastic noise will guarantee the control of epidemic diseases with symmetric Brownian motion. Periodic outbreaks and re-infection may occur due to the existence of feedback memory. It is shown that the endemic equilibrium is stable under some suitable initial conditions, taking advantage of the symmetry of the large amount of contact structure. A numerical method based on Legendre polynomials that converts the given stochastic SIRS model into a nonlinear algebraic system is used for the approximate solution. Finally, some numerical experiments are performed to verify the theoretical results and clearly show the sharpness of the obtained conditions and thresholds.

Modelling the effect of non-pharmaceutical interventions on COVID-19 transmission from mobility maps

The world has faced the COVID-19 pandemic for over two years now, and it is time to revisit the lessons learned from lockdown measures for theoretical and practical epidemiological improvements. The interlink between these measures and the resulting change in mobility (a predictor of the disease transmission contact rate) is uncertain. We thus propose a new method for assessing the efficacy of various non-pharmaceutical interventions (NPI) and examine the aptness of incorporating mobility data for epidemiological modelling. Facebook mobility maps for the United Arab Emirates are used as input datasets from the first infection in the country to mid-Oct 2020. Dataset was limited to the pre-vaccination period as this paper focuses on assessing the different NPIs at an early epidemic stage when no vaccines are available and NPIs are the only way to reduce the reproduction number (R0). We developed a travel network density parameter βt to provide an estimate of NPI impact on mobility patterns. Given the infection-fatality ratio and time lag (onset-to-death), a Bayesian probabilistic model is adapted to calculate the change in epidemic development with βt. Results showed that the change in βt clearly impacted R0. The three lockdowns strongly affected the growth of transmission rate and collectively reduced R0 by 78% before the restrictions were eased. The model forecasted daily infections and deaths by 2% and 3% fractional errors. It also projected what-if scenarios for different implementation protocols of each NPI. The developed model can be applied to identify the most efficient NPIs for confronting new COVID-19 waves and the spread of variants, as well as for future pandemics.

The impact of vaccination on the modeling of COVID-19 dynamics: a fractional order model

The coronavirus disease 2019 (COVID-19) is a recent outbreak of respiratory infections that have affected millions of humans all around the world. Initially, the major intervention strategies used to combat the infection were the basic public health measure, nevertheless, vaccination is an effective strategy and has been used to control the incidence of many infectious diseases. Currently, few safe and effective vaccines have been approved to control the inadvertent transmission of COVID-19. In this paper, the modeling approach is adopted to investigate the impact of currently available anti-COVID vaccines on the dynamics of COVID-19. A new fractional-order epidemic model by incorporating the vaccination class is presented. The fractional derivative is considered in the well-known Caputo sense. Initially, the proposed vaccine model for the dynamics of COVID-19 is developed via integer-order differential equations and then the Caputo-type derivative is applied to extend the model to a fractional case. By applying the least square method, the model is fitted to the reported cases in Pakistan and some of the parameters involved in the models are estimated from the actual data. The threshold quantity ( R 0 ) is computed by the Next-generation method. A detailed analysis of the fractional model, such as positivity of model solution, equilibrium points, and stabilities on both disease-free and endemic states are discussed comprehensively. An efficient iterative method is utilized for the numerical solution of the proposed model and the model is then simulated in the light of vaccination. The impact of important influential parameters on the pandemic dynamics is shown graphically. Moreover, the impact of different intervention scenarios on the disease incidence is depicted and it is found that the reduction in the effective contact rate (up to 30%) and enhancement in vaccination rate (up to 50%) to the current baseline values significantly reduced the disease new infected cases.

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 whenR 0 < 1 at the disease-free case. ForR 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 isR 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.

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 numberR 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 versusR 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.

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.

Investigation of fractal-fractional HIV infection by evaluating the drug therapy effect in the Atangana-Baleanu sense

In this paper, we apply the fractal-fractional derivative in the Atangana-Baleanu sense to a model of the human immunodeficiency virus infection of CD$ 4^{+} $ T-cells in the presence of a reverse transcriptase inhibitor, which occurs before the infected cell begins producing the virus. The existence and uniqueness results obtained by applying Banach-type and Leray-Schauder-type fixed-point theorems for the solution of the suggested model are established. Stability analysis in the context of Ulam's stability and its various types are investigated in order to ensure that a close exact solution exists. Additionally, the equilibrium points and their stability are analyzed by using the basic reproduction number. Three numerical algorithms are provided to illustrate the approximate solutions by using the Newton polynomial approach, the Adam-Bashforth method and the predictor-corrector technique, and a comparison between them is presented. Furthermore, we present the results of numerical simulations in the form of graphical figures corresponding to different fractal dimensions and fractional orders between zero and one. We analyze the behavior of the considered model for the provided values of input factors. As a result, the behavior of the system was predicted for various fractal dimensions and fractional orders, which revealed that slight changes in the fractal dimensions and fractional orders had no impact on the function's behavior in general but only occur in the numerical simulations.

A simple model to estimate the transmissibility of the Beta, Delta, and Omicron variants of SARS-COV-2 in South Africa

The COVID-19 pandemic caused multiple waves of mortality in South Africa, where three genetic variants of SARS-COV-2 and their ancestral strain dominated consecutively. State-of-the-art mathematical modeling approach was used to estimate the time-varying transmissibility of SARS-COV-2 and the relative transmissibility of Beta, Delta, and Omicron variants. The transmissibility of the three variants were about 73%, 87%, and 276% higher than their preceding variants. To the best of our knowledge, our model is the first simple model that can simulate multiple mortality waves and three variants' replacements in South Africa. The transmissibility of the Omicron variant is substantially higher than that of previous variants.

Assessing the potential impact of COVID-19 Omicron variant: Insight through a fractional piecewise model

We consider a new mathematical model for the COVID-19 disease with Omicron variant mutation. We formulate in details the modeling of the problem with omicron variant in classical differential equations. We use the definition of the Atangana-Baleanu derivative and obtain the extended fractional version of the omicron model. We study mathematical results for the fractional model and show the local asymptotical stability of the model for infection-free case ifR 0 < 1 . We show the global asymptotically stable of the model for the disease free case whenR 0 ≤ 1 . We show the existence and uniqueness of solution of the fractional model. We further extend the fractional order model into piecewise differential equation system and give a numerical algorithm for their numerical simulation. We consider the real cases of COVID-19 in South Africa of the third wave March 2021-Sep 2021 and estimate the model parameters and getR 0 ≈ 1 . 4004 . The real parameters values are used to show the graphical results for the fractional and piecewise model.

Fractional-order backstepping strategy for fractional-order model of COVID-19 outbreak

The coronavirus disease (COVID-19) pandemic has impacted many nations around the world. Recently, new variant of this virus has been identified that have a much higher rate of transmission. Although vaccine production and distribution are currently underway, non-pharmacological interventions are still being implemented as an important and fundamental strategy to control the spread of the virus in countries around the world. To realize and forecast the transmission dynamics of this disease, mathematical models can be very effective. Various mathematical modeling methods have been proposed to investigate the transmission patterns of this new infection. In this paper, we utilized the fractional-order dynamics of COVID-19. The goal is to control the prevalence of the disease using non-pharmacological interventions. In this paper, a novel fractional-order backstepping sliding mode control (FOBSMC) is proposed for non-pharmacological decisions. Recently, new variant of this virus have been identified that have a much higher rate of transmission, so finally the effectiveness of the proposed controller in the presence of new variant of COVID-19 is investigated.

Stochastic differential equation model of Covid-19: Case study of Pakistan

In recent years, the world lived a horrible nightmare named the Covid-19 pandemic. It's changing lifestyle caused the closure of critical establishments like schools, sports halls, companies, etc., as well as causing damage and menace for humanity. Mathematical models are helpful tools for modeling and analysing the infectious disease transmission This article presents a stochastic model of the Covid-19 epidemic for a population with five compartments. We give a numerical analysis of the proposed stochastic model. Also, we compare it with results of the corresponding deterministic model.

A mathematical model for SARS-CoV-2 in variable-order fractional derivative

A new coronavirus mathematical with hospitalization is considered with the consideration of the real cases from March 06, 2021 till the end of April 30, 2021. The essential mathematical results for the model are presented. We show the model stability whenR 0 < 1 in the absence of infection. We show that the system is stable locally asymptotically whenR 0 < 1 at infection free state. We also show that the system is globally asymptotically stable in the disease absence whenR 0 < 1 . Data have been used to fit accurately to the model and found the estimated basic reproduction number to beR 0 = 1.2036 . Some graphical results for the effective parameters are drawn for the disease elimination. In addition, a variable-order model is introduced, and so as to handle the outbreak effectively and efficiently, a genetic algorithm is used to produce high-quality control. Numerical simulations clearly show that decision-makers may develop helpful and practical strategies to manage future waves by implementing optimum policies.

A mathematical COVID-19 model considering asymptomatic and symptomatic classes with waning immunity

The spread of COVID-19 to more than 200 countries has shocked the public. Therefore, understanding the dynamics of transmission is very important. In this paper, the COVID-19 mathematical model has been formulated, analyzed, and validated using incident data from West Java Province, Indonesia. The model made considers the asymptomatic and symptomatic compartments and decreased immunity. The model is formulated in the form of a system of differential equations, where the population is divided into seven compartments, namely Susceptible Population (S0), Exposed Population (E), Asymptomatic Infection Population (IA), Symptomatic Infection Population (YS), Recovered Population (Z), Susceptible Populations previously infected (Z0), and Quarantine population (Q). The results show that there has been an outbreak of COVID-19 in West Java Province, Indonesia. This can be seen from the basic reproduction number of this model, which is 3.180126127 (R0>1). Also, the numerical simulation results show that waning immunity can increase the occurrence of outbreaks; and a period of isolation can slow down the process of spreading COVID-19. So if a strict social distancing policy is enforced like a quarantine, the outbreak will lessen.

Are the upper bounds for new SARS-CoV-2 infections in Germany useful?

At the end of 2019, an outbreak of a new coronavirus, called SARS–CoV–2, was reported in China and later in other parts of the world. First infection reported in Germany by the end of January 2020 and on March 16th, 2020 the federal government announced a partial lockdown in order to mitigate the spread. Since the dynamics of new infections started to slow down, German states started to relax the confinement measures as to May 6th, 2020. As a fall back option, a limit of 50 new infections per 100,000 inhabitants within seven days was introduced for each district in Germany. If a district exceeds this limit, measures to control the spread of the virus should be taken. Based on a multi–patch SEAIRD–type model, we will simulate the effect of choosing a specific upper limit for new infections. We investigate, whether the politically motivated bound is low enough to detect new outbreaks at an early stage. Subsequently, we introduce an optimal control problem to tackle the multi–criteria problem of finding a bound for new infections that is low enough to avoid new outbreaks, which might lead to an overload of the health care system, but is large enough to curb the expected economic losses.

A Comprehensive Mathematical Model for SARS-CoV-2 in Caputo Derivative

In the present work, we study the COVID-19 infection through a new mathematical model using the Caputo derivative. The model has all the possible interactions that are responsible for the spread of disease in the community. We first formulate the model in classical differential equations and then extend it into fractional differential equations using the definition of the Caputo derivative. We explore in detail the stability results for the model of the disease-free case when R0<1. We show that the model is stable locally when R0<1. We give the result that the model is globally asymptotically stable whenever R0≤1. Further, to estimate the model parameters, we consider the real data of the fourth wave from Pakistan and provide a reasonable fitting to the data. We estimate the basic reproduction number for the proposed data to be R0=1.0779. Moreover, using the real parameters, we present the numerical solution by first giving a reliable scheme that can numerically handle the solution of the model. In our simulation, we give the graphical results for some sensitive parameters that have a large impact on disease elimination. Our results show that taking into consideration all the possible interactions can describe COVID-19 infection.

Mathematical assessment of constant and time-dependent control measures on the dynamics of the novel coronavirus: An application of optimal control theory

The coronavirus infectious disease (COVID-19) is a novel respiratory disease reported in 2019 in China. The COVID-19 is one of the deadliest pandemics in history due to its high mortality rate in a short period. Many approaches have been adopted for disease minimization and eradication. In this paper, we studied the impact of various constant and time-dependent variable control measures coupled with vaccination on the dynamics of COVID-19. The optimal control theory is used to optimize the model and set an effective control intervention for the infection. Initially, we formulate the mathematical epidemic model for the COVID-19 without variable controls. The model basic mathematical assessment is presented. The nonlinear least-square procedure is utilized to parameterize the model from actual cases reported in Pakistan. A well-known technique based on statistical tools known as the Latin-hypercube sampling approach (LHS) coupled with the partial rank correlation coefficient (PRCC) is applied to present the model global sensitivity analysis. Based on global sensitivity analysis, the COVID-19 vaccine model is reformulated to obtain a control problem by introducing three time dependent control variables for isolation, vaccine efficacy and treatment enhancement represented byu 1 ( t ) ,u 2 ( t ) andu 3 ( t ) , respectively. The necessary optimality conditions of the control problem are derived via the optimal control theory. Finally, the simulation results are depicted with and without variable controls using the well-known Runge-Kutta numerical scheme. The simulation results revealed that time-dependent control measures play a vital role in disease eradication.

Pandemic fatigue impact on COVID-19 spread: A mathematical modelling answer to the Italian scenario

The COVID-19 outbreak has generated, in addition to the dramatic sanitary consequences, severe psychological repercussions for the populations affected by the pandemic. Simultaneously, these consequences can have related effects on the spread of the virus. Pandemic fatigue occurs when stress rises beyond a threshold, leading a person to feel demotivated to follow recommended behaviours to protect themselves and others. In the present paper, we introduce a new susceptible-infected-quarantined-recovered-dead (SIQRD) model in terms of a system of ordinary differential equations (ODE). The model considers the countermeasures taken by sanitary authorities and the effect of pandemic fatigue. The latter can be mitigated by fear of the disease's consequences modelled with the death rate in mind. The mathematical well-posedness of the model is proved. We show the numerical results to be consistent with the transmission dynamics data characterising the epidemic of the COVID-19 outbreak in Italy in 2020. We provide a measure of the possible pandemic fatigue impact. The model can be used to evaluate the public health interventions and prevent with specific actions the possible damages resulting from the social phenomenon of relaxation concerning the observance of the preventive rules imposed.

Ranking non-pharmaceutical interventions against Covid-19 global pandemic using global sensitivity analysis-Effect on number of deaths

In this study, we use Global Sensitivity Analysis (GSA) to rank four non-pharmaceutical interventions (NPIs) in a deterministic compartmental model that might control Covid-19 related deaths in the United States. The NPIs are social distancing, isolation of infected individuals, identifying asymptomatically infected individuals through testing, and the use of face masks. The model uses a fear-based behavioral model that leads unmasked susceptible individuals to wear masks. The model parameters are estimated from the reported deaths for the United States of America from March 1, 2020 to November 26, 2020. Two GSA tools, the Sobol' sesntivity indices and Partial Rank Correlation Coefficient are used to obtain the rankings of the input parameters at different stages of the disease propagation. We found that social distancing and outward mask efficiency alone decreases the output uncertainty by 25-45%. Sobol' second order indices show that the combined effect of social distancing with increased mask usage and identifying and isolating asymptomatically infected individuals decreases uncertainty an additional 10%.

A dynamical study of SARS-COV-2: A study of third wave

The coronavirus still an epidemic in most countries of the world and put the people in danger with so many infected cases and death. Considering the third wave of corona virus infection and to determine the peak of the infection curve, we suggest a new mathematical model with reported cases from March 06, 2021, till April 30, 2021. The model provides an accurate fitting to the suggested data, and the basic reproduction number calculated to be R 0 = 1 . 2044 . We study the stability of the model and show that the model is locally as well as globally asymptotically stable when R 0 < 1 , for the disease free case. The parameters that are sensitive to the basic reproduction number, their effect on the model variables are shown graphically. We can observe that the suggested parameters can decrease efficiently the infection cases of the third wave in Pakistan. Further, our model suggests that the infection peak is to be May 06, 2021. The present results determine that the model can be useful in order to predict other countries data.

Estimation of the doubling time and reproduction number for COVID-19

In this paper, we have calculated the basic reproduction number (R₀) and doubling time (T_d) for the novel Coronavirus disease 2019 (COVID-19). The calculation is performed for March 2020 from the data provided by worldometer. We have investigated the data for Germany and Bangladesh. The calculation of R₀ is performed based on SIR model. The parameter T_d is estimated based on the new cases of each day. Since T_d and R₀ in use to judge the lockdowns and other measures to prevent spreading of the virus, we have provided simple approximation of both parameters.

On fractional approaches to the dynamics of a SARS-CoV-2 infection model including singular and non-singular kernels

Covid-19 (2019-nCoV) disease has been spreading in China since late 2019 and has spread to various countries around the world. With the spread of the disease around the world, much attention has been paid to epidemiological knowledge. This knowledge plays a key role in understanding the pattern of disease transmission and how to prevent a larger population from contracting it. In the meantime, one should not overlook the significant role that mathematical descriptions play in epidemiology. In this paper, using some known definitions of fractional derivatives, which is a relatively new definition in differential calculus, and then by employing them in a mathematical framework, the effects of these tools in a better description of the epidemic of a SARS-CoV-2 infection is investigated. To solve these problems, efficient numerical methods have been used which can provide a very good approximation of the solution of the problem. In addition, numerical simulations related to each method will be provided in solving these models. The results obtained in each case indicate that the used approximate methods have been able to provide a good description of the problem situation.

Mathematical modeling and analysis of the novel Coronavirus using Atangana-Baleanu derivative

The novel Coronavirus infection disease is becoming more complex for the humans society by giving death and infected cases throughout the world. Due to this infection, many countries of the world suffers from great economic loss. The researchers around the world are very active to make a plan and policy for its early eradication. The government officials have taken full action for the eradication of this virus using different possible control strategies. It is the first priority of the researchers to develop safe vaccine against this deadly disease to minimize the infection. Different approaches have been made in this regards for its elimination. In this study, we formulate a mathematical epidemic model to analyze the dynamical behavior and transmission patterns of this new pandemic. We consider the environmental viral concentration in the model to better study the disease incidence in a community. Initially, the model is constructed with the derivative of integer-order. The classical epidemic model is then reconstructed with the fractional order operator in the form of Atangana-Baleanu derivative with the nonsingular and nonlocal kernel in order to analyze the dynamics of Coronavirus infection in a better way. A well-known estimation approach is used to estimate model parameters from the COVID-19 cases reported in Saudi Arabia from March 1 till August 20, 2020. After the procedure of parameters estimation, we explore some basic mathematical analysis of the fractional model. The stability results are provided for the disease free case using fractional stability concepts. Further, the uniqueness and existence results will be shown using the Picard-Lendelof approach. Moreover, an efficient numerical scheme has been proposed to obtain the solution of the model numerically. Finally, using the real fitted parameters, we depict many simulation results in order to demonstrate the importance of various model parameters and the memory index on disease dynamics and possible eradication.

Application of reinforcement learning for effective vaccination strategies of coronavirus disease 2019 (COVID-19)

Since December 2019, the new coronavirus has raged in China and subsequently all over the world. From the first days, researchers have tried to discover vaccines to combat the epidemic. Several vaccines are now available as a result of the contributions of those researchers. As a matter of fact, the available vaccines should be used in effective and efficient manners to put the pandemic to an end. Hence, a major problem now is how to efficiently distribute these available vaccines among various components of the population. Using mathematical modeling and reinforcement learning control approaches, the present article aims to address this issue. To this end, a deterministic Susceptible-Exposed-Infectious-Recovered-type model with additional vaccine components is proposed. The proposed mathematical model can be used to simulate the consequences of vaccination policies. Then, the suppression of the outbreak is taken to account. The main objective is to reduce the effects of Covid-19 and its domino effects which stem from its spreading and progression. Therefore, to reach optimal policies, reinforcement learning optimal control is implemented, and four different optimal strategies are extracted. Demonstrating the efficacy of the proposed methods, finally, numerical simulations are presented.

The influence of awareness campaigns on the spread of an infectious disease: a qualitative analysis of a fractional epidemic model

Mass-media coverage is one of the most widely used government strategies on influencing public opinion in times of crisis. Awareness campaigns are highly influential tools to expand healthy behavior practices among individuals during epidemics and pandemics. Mathematical modeling has become an important tool in analyzing the effects of media awareness on the spread of infectious diseases. In this paper, a fractional-order epidemic model incorporating media coverage is presented and analyzed. The problem is formulated using susceptible, infectious and recovered compartmental model. A long-term memory effect modeled by a Caputo fractional derivative is included in each compartment to describe the evolution related to the individuals’ experiences. The well-posedness of the model is investigated in terms of global existence, positivity, and boundedness of solutions. Moreover, the disease-free equilibrium and the endemic equilibrium points are given alongside their local stabilities. By constructing suitable Lyapunov functions, the global stability of the disease-free and endemic equilibria is proven according to the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0. Finally, numerical simulations are performed to support our analytical findings. It was found out that the long-term memory has no effect on the stability of the equilibrium points. However, for increased values of the fractional derivative order parameter, each solution reaches its equilibrium state more rapidly. Furthermore, it was observed that an increase of the media awareness parameter, decreases the magnitude of infected individuals, and consequently, the height of the epidemic peak.

A Mathematical Model of COVID-19 Pandemic: A Case Study of Bangkok, Thailand

In this study, we propose a new mathematical model and analyze it to understand the transmission dynamics of the COVID-19 pandemic in Bangkok, Thailand. It is divided into seven compartmental classes, namely, susceptible (S), exposed (E), symptomatically infected (I s ), asymptomatically infected (I a ), quarantined (Q), recovered (R), and death (D), respectively. The next-generation matrix approach was used to compute the basic reproduction number denoted as R cvd19 of the proposed model. The results show that the disease-free equilibrium is globally asymptotically stable if R cvd19 < 1. On the other hand, the global asymptotic stability of the endemic equilibrium occurs if R cvd19 > 1. The mathematical analysis of the model is supported using numerical simulations. Moreover, the model's analysis and numerical results prove that the consistent use of face masks would go on a long way in reducing the COVID-19 pandemic.

A robust study on 2019-nCOV outbreaks through non-singular derivative

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.