Dynamics of a fractional order mathematical model for COVID-19 epidemic

"Does a Respiratory Virus Have an Ecological Niche, and If So, Can It Be Mapped?" Yes and Yes

Although the utility of Ecological Niche Models (ENM) and Species Distribution Models (SDM) has been demonstrated in many ecological applications, their suitability for modelling epidemics or pandemics, such as SARS-Cov-2, has been questioned. In this paper, contrary to this viewpoint, we show that ENMs and SDMs can be created that can describe the evolution of pandemics, both in space and time. As an illustrative use case, we create models for predicting confirmed cases of COVID-19, viewed as our target "species", in Mexico through 2020 and 2021, showing that the models are predictive in both space and time. In order to achieve this, we extend a recently developed Bayesian framework for niche modelling, to include: (i) dynamic, non-equilibrium "species" distributions; (ii) a wider set of habitat variables, including behavioural, socio-economic and socio-demographic variables, as well as standard climatic variables; (iii) distinct models and associated niches for different species characteristics, showing how the niche, as deduced through presence-absence data, can differ from that deduced from abundance data. We show that the niche associated with those places with the highest abundance of cases has been highly conserved throughout the pandemic, while the inferred niche associated with presence of cases has been changing. Finally, we show how causal chains can be inferred and confounding identified by showing that behavioural and social factors are much more predictive than climate and that, further, the latter is confounded by the former.

Dynamics of a fractional order mathematical model for COVID-19 epidemic transmission

To achieve the aim of immediately halting spread of COVID-19 it is essential to know the dynamic behavior of the virus of intensive level of replication. Simply analyzing experimental data to learn about this disease consumes a lot of effort and cost. Mathematical models may be able to assist in this regard. Through integrating the mathematical frameworks with the accessible disease data it will be useful and outlay to comprehend the primary components involved in the spreading of COVID-19. There are so many techniques to formulate the impact of disease on the population mathematically, including deterministic modeling, stochastic modeling or fractional order modeling etc. Fractional derivative modeling is one of the essential techniques for analyzing real-world issues and making accurate assessments of situations. In this paper, a fractional order epidemic model that represents the transmission of COVID-19 using seven compartments of population susceptible, exposed, infective, recovered, the quarantine population, recovered-exposed, and dead population is provided. The fractional order derivative is considered in the Caputo sense. In order to determine the epidemic forecast and persistence, we calculate the reproduction number R 0 . Applying fixed point theory, the existence and uniqueness of the solutions of fractional order derivative have been studied . Moreover, we implement the generalized Adams-Bashforth-Moulton method to get an approximate solution of the fractional-order COVID-19 model. Finally, numerical result and an outstanding graphic simulation are presented.

Analysis of a Fractional-Order COVID-19 Epidemic Model with Lockdown

The outbreak of the coronavirus disease (COVID-19) has caused a lot of disruptions around the world. In an attempt to control the spread of the disease among the population, several measures such as lockdown, and mask mandates, amongst others, were implemented by many governments in their countries. To understand the effectiveness of these measures in controlling the disease, several mathematical models have been proposed in the literature. In this paper, we study a mathematical model of the coronavirus disease with lockdown by employing the Caputo fractional-order derivative. We establish the existence and uniqueness of the solution to the model. We also study the local and global stability of the disease-free equilibrium and endemic equilibrium solutions. By using the residual power series method, we obtain a fractional power series approximation of the analytic solution. Finally, to show the accuracy of the theoretical results, we provide some numerical and graphical results.

Modeling the stochastic within-host dynamics SARS-CoV-2 infection with discrete delay

In this paper, a new mathematical model that describes the dynamics of the within-host COVID-19 epidemic is formulated. We show the stochastic dynamics of Target-Latent-Infected-Virus free within the human body with discrete delay and noise. Positivity and uniqueness of the solutions are established. Our study shows the extinction and persistence of the disease inside the human body through the stability analysis of the disease-free equilibrium [Formula: see text] and the endemic equilibrium [Formula: see text], respectively. Moreover, we show the impact of delay tactics and noise on the extinction of the disease. The most interesting result is even if the deterministic system is inevitably pandemic at a specific point, extinction will become possible in the stochastic version of our model.

Stability Analysis of an Extended SEIR COVID-19 Fractional Model with Vaccination Efficiency

This work is aimed at presenting a new numerical scheme for COVID-19 epidemic model based on Atangana-Baleanu fractional order derivative in Caputo sense (ABC) to investigate the vaccine efficiency. Our construction of the model is based on the classical SEIR, four compartmental models with an additional compartment V of vaccinated people extending it SEIRV model, for the transmission as well as an effort to cure this infectious disease. The point of disease-free equilibrium is calculated, and the stability analysis of the equilibrium point using the reproduction number is performed. The endemic equilibrium's existence and uniqueness are investigated. For the solution of the nonlinear system presented in the model at different fractional orders, a new numerical scheme based on modified Simpson's 1/3 method is developed. Convergence and stability of the numerical scheme are thoroughly analyzed. We attempted to develop an epidemiological model presenting the COVID-19 dynamics in Italy. The proposed model's dynamics are graphically interpreted to observe the effect of vaccination by altering the vaccination rate.

Mathematical model for the novel coronavirus (2019-nCOV) with clinical data using fractional operator

Coronavirus infection (COVID-19) is a considerably dangerous disease with a high demise rate around the world. There is no known vaccination or medicine until our time because the unknown aspects of the virus are more significant than our theoretical and experimental knowledge. One of the most effective strategies for comprehending and controlling the spread of this epidemic is to model it using a powerful mathematical model. However, mathematical modeling with a fractional operator can provide explanations for the disease's possibility and severity. Accordingly, basic information will be provided to identify the kind of measure and intrusion that will be required to control the disease's progress. In this study, we propose using a fractional-order SEIARPQ model with the Caputo sense to model the coronavirus (COVID-19) pandemic, which has never been done before in the literature. The stability analysis, existence, uniqueness theorems, and numerical solutions of such a model are displayed. All results were numerically simulated using MATLAB programming. The current study supports the applicability and influence of fractional operators on real-world problems.

The fractional-order discrete COVID-19 pandemic model: stability and chaos

This paper presents and investigates a new fractional discrete COVID-19 model which involves three variables: the new daily cases, additional severe cases and deaths. Here, we analyze the stability of the equilibrium point at different values of the fractional order. Using maximum Lyapunov exponents, phase attractors, bifurcation diagrams, the 0-1 test and approximation entropy (ApEn), it is shown that the dynamic behaviors of the model change from stable to chaotic behavior by varying the fractional orders. Besides showing that the fractional discrete model fits the real data of the pandemic, the simulation findings also show that the numbers of new daily cases, additional severe cases and deaths exhibit chaotic behavior without any effective attempts to curb the epidemic.

The effect of the Caputo fractional difference operator on a new discrete COVID-19 model

This study aims to generalize the discrete integer-order SEIR model to obtain the novel discrete fractional-order SEIR model of COVID-19 and study its dynamic characteristics. Here, we determine the equilibrium points of the model and discuss the stability analysis of these points in detail. Then, the non-linear dynamic behaviors of the suggested discrete fractional model for commensurate and incommensurate fractional orders are investigated through several numerical techniques, including maximum Lyapunov exponents, phase attractors, bifurcation diagrams and C 0 algorithm. Finally, we fitted the model with actual data to verify the accuracy of our mathematical study of the stability of the fractional discrete COVID-19 model.

Dynamical Behavior of a Fractional Order Model for Within-Host SARS-CoV-2

The prime objective of the current study is to propose a novel mathematical framework under the fractional-order derivative, which describes the complex within-host behavior of SARS-CoV-2 by taking into account the effects of memory and carrier. To do this, we formulate a mathematical model of SARS-CoV-2 under the Caputo fractional-order derivative. We derived the conditions for the existence of equilibria of the model and computed the basic reproduction number R0. We used mathematical analysis to establish the proposed model’s local and global stability results. Some numerical resolutions of our theoretical results are presented. The main result of this study is that as the fractional derivative order increases, the approach of the solution to the equilibrium points becomes faster. It is also observed that the value of R0 increases as the value of β and πv increases.

Mathematical modeling and stability analysis of the time-delayed SAIM model for COVID-19 vaccination and media coverage

Since the COVID-19 outbreak began in early 2020, it has spread rapidly and threatened public health worldwide. Vaccination is an effective way to control the epidemic. In this paper, we model a SAIM equation. Our model involves vaccination and the time delay for people to change their willingness to be vaccinated, which is influenced by media coverage. Second, we theoretically analyze the existence and stability of the equilibria of our model. Then, we study the existence of Hopf bifurcation related to the two equilibria and obtain the normal form near the Hopf bifurcating critical point. Third, numerical simulations based two groups of values for model parameters are carried out to verify our theoretical analysis and assess features such as stable equilibria and periodic solutions. To ensure the appropriateness of model parameters, we conduct a mathematical analysis of official data. Next, we study the effect of the media influence rate and attenuation rate of media coverage on vaccination and epidemic control. The analysis results are consistent with real-world conditions. Finally, we present conclusions and suggestions related to the impact of media coverage on vaccination and epidemic control.

A fractional-order model for CoViD-19 dynamics with reinfection and the importance of quarantine

Coronavirus disease 2019 (CoViD-19) is an infectious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Among many symptoms, cough, fever and tiredness are the most common. People over 60 years old and with associated comorbidities are most likely to develop a worsening health condition. This paper proposes a non-integer order model to describe the dynamics of CoViD-19 in a standard population. The model incorporates the reinfection rate in the individuals recovered from the disease. Numerical simulations are performed for different values of the order of the fractional derivative and of reinfection rate. The results are discussed from a biological point of view.

Multi-Model Selection and Analysis for COVID-19

In the face of an increasing number of COVID-19 infections, one of the most crucial and challenging problems is to pick out the most reasonable and reliable models. Based on the COVID-19 data of four typical cities/provinces in China, integer-order and fractional SIR, SEIR, SEIR-Q, SEIR-QD, and SEIR-AHQ models are systematically analyzed by the AICc, BIC, RMSE, and R means. Through extensive simulation and comprehensive comparison, we show that the fractional models perform much better than the corresponding integer-order models in representing the epidemiological information contained in the real data. It is further revealed that the inflection point plays a vital role in the prediction. Moreover, the basic reproduction numbers R0 of all models are highly dependent on the contact rate.

Control of COVID-19 dynamics through a fractional-order model

We investigate, through a fractional mathematical model, the effects of physical distance on the SARS-CoV-2 virus transmission. Two controls are considered in our model for eradication of the spread of COVID-19: media education, through campaigns explaining the importance of social distancing, use of face masks, etc., towards all population, while the second one is quarantine social isolation of the exposed individuals. A general fractional order optimal control problem, and associated optimality conditions of Pontryagin type, are discussed, with the goal to minimize the number of susceptible and infected while maximizing the number of recovered. The extremals are then numerically obtained.

Dynamics of SEIR epidemic model by optimal auxiliary functions method

The aim of the present work is to establish an approximate analytical solution for the nonlinear Susceptible, Exposed, Infected, Recovered (SEIR) model applied to novel coronavirus COVID-19. The mathematical model depending of five nonlinear differential equations, is studied and approximate solutions are obtained using Optimal Auxiliary Functions Method (OAFM). Our technique ensures a fast convergence of the solutions after only one iteration. The nonstandard part of OAFM is described by the presence of so-called auxiliary functions and of the optimal convergence-control parameters. We have a great freedom to select the auxiliary functions and the number of optimal convergence-control parameters which are optimally determined. Our approach is independent of the presence of small or large parameters in the governing equations or in the initial/boundary conditions, is effective, simple and very efficient.

A fractional order SITR mathematical model for forecasting of transmission of COVID-19 of India with lockdown effect

In this paper, we consider a mathematical model to explain, understanding, and to forecast the outbreaks of COVID-19 in India. The model has four components leading to a system of fractional order differential equations incorporating the refuge concept to study the lockdown effect in controlling COVID-19 spread in India. We investigate the model using the concept of Caputo fractional-order derivative. The goal of this model is to estimate the number of total infected, active cases, deaths, as well as recoveries from COVID-19 to control or minimize the above issues in India. The existence, uniqueness, non-negativity, and boundedness of the solutions are established. In addition, the local and global asymptotic stability of the equilibrium points of the fractional-order system and the basic reproduction number are studied for understanding and prediction of the transmission of COVID-19 in India. The next step is to carry out sensitivity analysis to find out which parameter is the most dominant to affect the disease's endemicity. The results reveal that the parameters η , μ and ρ are the most dominant sensitivity indices towards the basic reproductive number. A numerical illustration is presented via computer simulations using MATLAB to show a realistic point of view.

A fractional-order SIRD model with time-dependent memory indexes for encompassing the multi-fractional characteristics of the COVID-19

COVID-19 is a novel coronavirus affecting all the world since December last year. Up to date, the spread of the outbreak continues to complicate our lives, and therefore, several research efforts from many scientific areas are proposed. Among them, mathematical models are an excellent way to understand and predict the epidemic outbreaks evolution to some extent. Due to the COVID-19 may be modeled as a non-Markovian process that follows power-law scaling features, we present a fractional-order SIRD (Susceptible-Infected-Recovered-Dead) model based on the Caputo derivative for incorporating the memory effects (long and short) in the outbreak progress. Additionally, we analyze the experimental time series of 23 countries using fractal formalism. Like previous works, we identify that the COVID-19 evolution shows various power-law exponents (no just a single one) and share some universality among geographical regions. Hence, we incorporate numerous memory indexes in the proposed model, i.e., distinct fractional-orders defined by a time-dependent function that permits us to set specific memory contributions during the evolution. This allows controlling the memory effects of more early states, e.g., before and after a quarantine decree, which could be less relevant than the contribution of more recent ones on the current state of the SIRD system. We also prove our model with Italy's real data from the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University.