Modeling and analysis of the dynamics of novel coronavirus (COVID-19) with Caputo fractional derivative

Dynamics of infectious disease mathematical model through unsupervised stochastic neural network paradigm

The viruses has spread globally and have been impacted lives of people socially and economically, which causes immense suffering throughout the world. Thousands of people died and millions of illnesses were brought, by the outbreak worldwide. In order to control the coronavirus pandemic, mathematical modeling proved to be an invaluable tool for analyzing and determining the potential and severity of the illness. This work proposed and assessed a deterministic six-compartment model with a novel stochastic neural network. The significance of the proposed model was demonstrated by numerical simulation in which the results are agreed with sensitivity analysis. Furthermore, the efficacy of stochastic neural network has been proven with the help of numerical simulations. Some investigations have been conducted through graphs and tables that how the vaccination process is helpful to minimize stress in society. The numerical simulations also focused on preventing the community-wide spread of the disease. The lowest residual errors have been achieved by our proposed stochastic neural network and compared with numerical solvers to assess the accuracy and robustness of the proposed approach.

Dynamic analysis of deterministic and stochastic SEIR models incorporating the Ornstein-Uhlenbeck process

In this paper, we introduce a Susceptible-Exposed-Infected-Recovered (SEIR) epidemic model and analyze it in both deterministic and stochastic contexts, incorporating the Ornstein-Uhlenbeck process. The model incorporates a nonlinear incidence rate and a saturated treatment response. We establish the basic properties of solutions and conduct a comprehensive stability analysis of the system's equilibria to assess its epidemiological relevance. Our results demonstrate that the disease will be eradicated from the population when R0<1, while the disease will persist when R0>1. Furthermore, we explore various bifurcation phenomena, including transcritical, backward, saddle-node, and Hopf, and discuss their epidemiological implications. For the stochastic model, we demonstrate the existence of a unique global positive solution. We also identify sufficient conditions for the disease extinction and persistence. Additionally, by developing a suitable Lyapunov function, we establish the existence of a stationary distribution. Several numerical simulations are conducted to validate the theoretical findings of the deterministic and stochastic models. The results provide a comprehensive demonstration of the disease dynamics in constant as well as noisy environments, highlighting the implications of our study.

Modelling epidemiological dynamics with pseudo-recovery via fractional-order derivative operator and optimal control measures

In this study, a new deterministic mathematical model based on fractional-order derivative operator that describes the pseudo-recovery dynamics of an epidemiological process is developed. Fractional-order derivative of Caputo type is used to examine the effect of memory in the spread process of infectious diseases with pseudo-recovery. The well-posedness of the model is qualitatively investigated through Banach fixed point theory technique. The spread of the disease in the population is measured by analysing the basic reproduction of the model with respect to its parameters through the sensitivity analysis. Consequently, the analysis is extended to the fractional optimal control model where time-dependent preventive strategy and treatment measure are characterized by Pontryagin's maximum principle. The resulting Caputo fractional-order optimality system is simulated to understand how both preventive and treatment controls affect the pseudo-recovery dynamics of infectious diseases in the presence of memory. Graphical illustrations are shown to corroborate the qualitative results, and to demonstrate the importance of memory effects in infectious disease modelling. It is shown that time-dependent preventive strategy and treatment measure in the presence of memory engenders significant reduction in the spread of the disease when compared with memoryless situation.

New solutions of time-fractional cancer tumor models using modified He-Laplace algorithm

Cancer develops through cells when mutations build up in different genes that control cell proliferation. To treat these abnormal cells and minimize their growth, various cancer tumor samples have been modeled and analyzed in literature. The current study is focused on the investigation of more generalized cancer tumor model in fractional environment, where net killing rate is taken into account in different domains. Three types of killing rates are considered in the current study including time and position dependent killing rates, and concentration of cells based killing rate. A hybrid mechanism is proposed in which different homotopies are used with perturbation technique and Laplace transform. This leads to a convenient algorithm to tackle all types of fractional derivatives efficiently. The convergence and error bounds of the proposed scheme are computed theoretically by proving related theorems. In the next phase, convergence and validity is analyzed numerically by calculating residual errors round the fractional domain. It is observed that computed errors are very less in the entire fractional domain. Moreover, comparative analysis of Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu (AB) fractional derivatives is also performed graphically to discern the effect of different fractional approaches on the solution profile. Analysis asserts the reliability of proposed methodology in the matter of intricate fractional tumor models, and hence can be used to other complex physical phenomena.

A new investigation on fractionalized modeling of human liver

This study focuses on improving the accuracy of assessing liver damage and early detection for improved treatment strategies. In this study, we examine the human liver using a modified Atangana-Baleanu fractional derivative based on the mathematical model to understand and predict the behavior of the human liver. The iteration method and fixed-point theory are used to investigate the presence of a unique solution in the new model. Furthermore, the homotopy analysis transform method, whose convergence is also examined, implements the mathematical model. Finally, numerical testing is performed to demonstrate the findings better. According to real clinical data comparison, the new fractional model outperforms the classical integer-order model with coherent temporal derivatives.

A nonstandard finite difference scheme for a time-fractional model of Zika virus transmission

In this work, we investigate the transmission dynamics of the Zika virus, considering both a compartmental model involving humans and mosquitoes and an extended model that introduces a non-human primate (monkey) as a second reservoir host. The novelty of our approach lies in the later generalization of the model using a fractional time derivative. The significance of this study is underscored by its contribution to understanding the complex dynamics of Zika virus transmission. Unlike previous studies, we incorporate a non-human primate reservoir host into the model, providing a more comprehensive representation of the disease spread. Our results reveal the importance of utilizing a nonstandard finite difference (NSFD) scheme to simulate the disease's dynamics accurately. This NSFD scheme ensures the positivity of the solution and captures the correct asymptotic behavior, addressing a crucial limitation of standard solvers like the Runge-Kutta Fehlberg method (ode45). The numerical simulations vividly demonstrate the advantages of our approach, particularly in terms of positivity preservation, offering a more reliable depiction of Zika virus transmission dynamics. From these findings, we draw the conclusion that considering a non-human primate reservoir host and employing an NSFD scheme significantly enhances the accuracy and reliability of modeling Zika virus transmission. Researchers and policymakers can use these insights to develop more effective strategies for disease control and prevention.

Mathematical assessment of monkeypox disease with the impact of vaccination using a fractional epidemiological modeling approach

This present paper aims to examine various epidemiological aspects of the monkeypox viral infection using a fractional-order mathematical model. Initially, the model is formulated using integer-order nonlinear differential equations. The imperfect vaccination is considered for human population in the model formulation. The proposed model is then reformulated using a fractional order derivative with power law to gain a deeper understanding of disease dynamics. The values of the model parameters are determined from the cumulative reported monkeypox cases in the United States during the period from May 10th to October 10th, 2022. Besides this, some of the demographic parameters are evaluated from the population of the literature. We establish sufficient conditions to ensure the existence and uniqueness of the model's solution in the fractional case. Furthermore, the stability of the endemic equilibrium of the fractional monkeypox model is presented. The Lyapunov function approach is used to demonstrate the global stability of the model equilibria. Moreover, the fractional order model is numerically solved using an efficient numerical technique known as the fractional Adams-Bashforth-Moulton method. The numerical simulations are conducted using estimated parameters, considering various values of the fractional order of the Caputo derivative. The finding of this study reveals the impact of various model parameters and fractional order values on the dynamics and control of monkeypox.

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.

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.

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.

Isolation in the control of epidemic

Among many epidemic prevention measures, isolation is an important method to control the spread of infectious disease. Scholars rarely study the impact of isolation on disease dissemination from a quantitative perspective. In this paper, we introduce an isolation ratio and establish the corresponding model. The basic reproductive number and its biological explanation are given. The stability conditions of the disease-free and endemic equilibria are obtained by analyzing its distribution of characteristic values. It is shown that the isolation ratio has an important influence on the basic reproductive number and the stability conditions. Taking the COVID-19 in Wuhan as an example, isolating more than 68% of the population can control the spread of the epidemic. This method can provide precise epidemic prevention strategies for government departments. Numerical simulations verify the effectiveness of the results.

Numerical simulation of a Caputo fractional epidemic model for the novel coronavirus with the impact of environmental transmission☆

The coronavirus infectious disease (COVID-19) is a novel respiratory disease reported in 2019 in China. The infection is very destructive to human lives and caused millions of deaths. Various approaches have been made recently to understand the complex dynamics of COVID-19. The mathematical modeling approach is one of the considerable tools to study the disease spreading pattern. In this article, we develop a fractional order epidemic model for COVID-19 in the sense of Caputo operator. The model is based on the effective contacts among the population and environmental impact to analyze the disease dynamics. The fractional models are comparatively better in understanding the disease outbreak and providing deeper insights into the infectious disease dynamics. We first consider the classical integer model studied in recent literature and then we generalize it by introducing the Caputo fractional derivative. Furthermore, we explore some fundamental mathematical analysis of the fractional model, including the basic reproductive number R 0 and equilibria stability utilizing the Routh-Hurwitz and the Lyapunov function approaches. Besides theoretical analysis, we also focused on the numerical solution. To simulate the model, we use the well-known generalized Adams–Bashforth Moulton Scheme. Finally, the influence of some of the model essential parameters on the dynamics of the disease is demonstrated graphically.

Modeling nosocomial infection of COVID-19 transmission dynamics

COVID-19 epidemic has posed an unprecedented threat to global public health. The disease has alarmed the healthcare system with the harm of nosocomial infection. Nosocomial spread of COVID-19 has been discovered and reported globally in different healthcare facilities. Asymptomatic patients and super-spreaders are sough to be among of the source of these infections. Thus, this study contributes to the subject by formulating a S E I H R mathematical model to gain the insight into nosocomial infection for COVID-19 transmission dynamics. The role of personal protective equipment θ is studied in the proposed model. Benefiting the next generation matrix method,R 0 was computed. Routh-Hurwitz criterion and stable Metzler matrix theory revealed that COVID-19-free equilibrium point is locally and globally asymptotically stable wheneverR 0 < 1 . Lyapunov function depicted that the endemic equilibrium point is globally asymptotically stable whenR 0 > 1 . Further, the dynamics behavior ofR 0 was explored when varying θ . In the absence of θ , the value ofR 0 was 8.4584 which implies the expansion of the disease. When θ is introduced in the model,R 0 was 0.4229, indicating the decrease of the disease in the community. Numerical solutions were simulated by using Runge-Kutta fourth-order method. Global sensitivity analysis is performed to present the most significant parameter. The numerical results illustrated mathematically that personal protective equipment can minimizes nosocomial infections of COVID-19.

A study on fractional HBV model through singular and non-singular derivatives

The current study's aim is to evaluate the dynamics of a Hepatitis B virus (HBV) model with the class of asymptomatic carriers using two different numerical algorithms and various values of the fractional-order parameter. We considered the model with two different fractional-order derivatives, namely the Caputo derivative and Atangana-Baleanu derivative in the Caputo sense (ABC). The considered derivatives are the most widely used fractional operators in modeling. We present some mathematical analysis of the fractional ABC model. The fixed-point theory is used to determine the existence and uniqueness of the solutions to the considered fractional model. For numerical results, we show a generalized Adams-Bashforth-Moulton (ABM) method for Caputo derivative and an Adams type predictor-corrector (PC) algorithm for Atangana-Baleanu derivatives. Finally, the models are numerically solved using computational techniques and obtained results graphically illustrated with a wide range of fractional-order values. We compare the numerical results for Caputo and ABC derivatives graphically. In addition, a new variable-order fractional network of the HBV model is proposed. Considering the fact that most communities interact with each other, and the rate of disease spread is affected by this factor, the proposed network can provide more accurate insight for the modeling of the disease.

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.

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.

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.

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

The novel coronavirus disease or COVID-19 is still posing an alarming situation around the globe. The whole world is facing the second wave of this novel pandemic. Recently, the researchers are focused to study the complex dynamics and possible control of this global infection. Mathematical modeling is a useful tool and gains much interest in this regard. In this paper, a fractional-order transmission model is considered to study its dynamical behavior using the real cases reported in Saudia Arabia. The classical Caputo type derivative of fractional order is used in order to formulate the model. The transmission of the infection through the environment is taken into consideration. The documented data since March 02, 2020 up to July 31, 2020 are considered for estimation of parameters of system. We have the estimated basic reproduction number ( R 0 ) for the data is 1.2937 . The Banach fixed point analysis has been used for the existence and uniqueness of the solution. The stability analysis at infection free equilibrium and at the endemic state are presented in details via a nonlinear Lyapunov function in conjunction with LaSalle Invariance Principle. An efficient numerical scheme of Adams-Molten type is implemented for the iterative solution of the model, which plays an important role in determining the impact of control measures and also sensitive parameters that can reduce the infection in the general public and thereby reduce the spread of pandemic as shown graphically. We present some graphical results for the model and the effect of the important sensitive parameters for possible infection minimization in the population.

An SIR epidemic model for COVID-19 spread with fuzzy parameter: the case of Indonesia

The aim of this research is to construct an SIR model for COVID-19 with fuzzy parameters. The SIR model is constructed by considering the factors of vaccination, treatment, obedience in implementing health protocols, and the corona virus-load. Parameters of the infection rate, recovery rate, and death rate due to COVID-19 are constructed as a fuzzy number, and their membership functions are used in the model as fuzzy parameters. The model analysis uses the generation matrix method to obtain the basic reproduction number and the stability of the model's equilibrium points. Simulation results show that differences in corona virus-loads will also cause differences in the transmission of COVID-19. Likewise, the factors of vaccination and obedience in implementing health protocols have the same effect in slowing or stopping the transmission of COVID-19 in Indonesia.

A time-delay COVID-19 propagation model considering supply chain transmission and hierarchical quarantine rate

In this manuscript, we investigate a novel Susceptible-Exposed-Infected-Quarantined-Recovered (SEIQR) COVID-19 propagation model with two delays, and we also consider supply chain transmission and hierarchical quarantine rate in this model. Firstly, we analyze the existence of an equilibrium, including a virus-free equilibrium and a virus-existence equilibrium. Then local stability and the occurrence of Hopf bifurcation have been researched by thinking of time delay as the bifurcation parameter. Besides, we calculate direction and stability of the Hopf bifurcation. Finally, we carry out some numerical simulations to prove the validity of theoretical results.