Stability analysis of fractional nabla difference COVID-19 model

Exploring local and global stability of COVID-19 through numerical schemes

Respiratory sensitivity and pneumonia are possible outcomes of the coronavirus (COVID-19). Surface characteristics like temperature and sunshine affect how long the virus survives. This research article analyzes COVID-19 mathematical model behavior based on symptomatic and non-symptomatic individuals. In the reproductive model, the best result indicates the intensity of the epidemic. Our model remained stable at a certain point under controlled conditions after we evaluated a specific element. This approach is in place of traditional approaches such as Euler's and Runge-Kutta's. An unusual numerical approach known as the non-standard finite difference (NSFD) scheme is used in this article. This numerical approach gives us positivity. A dependable numerical analysis allowed us to evaluate different approaches and verify our theoretical results. Unlike the widely used Euler and RK4 approaches, we investigated the benefits of implementing NSFD schemes. By numerically simulating COVID-19 in a variety of scenarios, we demonstrated how our theoretical concepts work. The simulation findings support the usefulness of both approaches.

From pandemic to endemic: Divergence of COVID-19 positive-tests and hospitalization numbers from SARS-CoV-2 RNA levels in wastewater of Rochester, Minnesota

Traditionally, public health surveillance relied on individual-level data but recently wastewater-based epidemiology (WBE) for the detection of infectious diseases including COVID-19 became a valuable tool in the public health arsenal. Here, we use WBE to follow the course of the COVID-19 pandemic in Rochester, Minnesota (population 121,395 at the 2020 census), from February 2021 to December 2022. We monitored the impact of SARS-CoV-2 infections on public health by comparing three sets of data: quantitative measurements of viral RNA in wastewater as an unbiased reporter of virus level in the community, positive results of viral RNA or antigen tests from nasal swabs reflecting community reporting, and hospitalization data. From February 2021 to August 2022 viral RNA levels in wastewater were closely correlated with the oscillating course of COVID-19 case and hospitalization numbers. However, from September 2022 cases remained low and hospitalization numbers dropped, whereas viral RNA levels in wastewater continued to oscillate. The low reported cases may reflect virulence reduction combined with abated inclination to report, and the divergence of virus levels in wastewater from reported cases may reflect COVID-19 shifting from pandemic to endemic. WBE, which also detects asymptomatic infections, can provide an early warning of impending cases, and offers crucial insights during pandemic waves and in the transition to the endemic phase.

Stability analysis and optimal control of a fractional-order generalized SEIR model for the COVID-19 pandemic

In view of the spread of corona virus disease 2019 (COVID-19), this paper proposes a fractional-order generalized SEIR model. The non-negativity of the solution of the model is discussed. Based on the established threshold R 0 , the existence of the disease-free equilibrium and endemic equilibrium is analyzed. Then, sufficient conditions are established to ensure the local asymptotic stability of the equilibria. The parameters of the model are identified based on the statistical data of COVID-19 cases. Furthermore, the validity of the model for describing the COVID-19 outbreak is verified. Meanwhile, the accuracy of the relevant theoretical results are also verified. Considering the relevant strategies of COVID-19 prevention and control, the fractional optimal control problem (FOCP) is proposed. Numerical schemes for Riemann-Liouville (R-L) fractional-order adjoint system with transversal conditions is presented. Based on the relevant statistical data, the corresponding FOCP is numerically solved, and the control effect of the COVID-19 outbreak under the optimal control strategy is discussed.

Ulam-Hyers stability of tuberculosis and COVID-19 co-infection model under Atangana-Baleanu fractal-fractional operator

The intention of this work is to study a mathematical model for fractal-fractional tuberculosis and COVID-19 co-infection under the Atangana-Baleanu fractal-fractional operator. Firstly, we formulate the tuberculosis and COVID-19 co-infection model by considering the tuberculosis recovery individuals, the COVID-19 recovery individuals, and both disease recovery compartment in the proposed model. The fixed point approach is utilized to explore the existence and uniqueness of the solution in the suggested model. The stability analysis related to solve the Ulam-Hyers stability is also investigated. This paper is based on Lagrange's interpolation polynomial in the numerical scheme, which is validated through a specific case with a comparative numerical analysis for different values of the fractional and fractal orders.

Stability and numerical analysis via non-standard finite difference scheme of a nonlinear classical and fractional order model

In this paper, we develop a new mathematical model for an in-depth understanding of COVID-19 (Omicron variant). The mathematical study of an omicron variant of the corona virus is discussed. In this new Omicron model, we used idea of dividing infected compartment further into more classes i.e asymptomatic, symptomatic and Omicron infected compartment. Model is asymptotically locally stable whenever R0<1 and when R0≤1 at disease free equilibrium the system is globally asymptotically stable. Local stability is investigated with Jacobian matrix and with Lyapunov function global stability is analyzed. Moreover basic reduction number is calculated through next generation matrix and numerical analysis will be used to verify the model with real data. We consider also the this model under fractional order derivative. We use Grunwald-Letnikov concept to establish a numerical scheme. We use nonstandard finite difference (NSFD) scheme to simulate the results. Graphical presentations are given corresponding to classical and fractional order derivative. According to our graphical results for the model with numerical parameters, the population's risk of infection can be reduced by adhering to the WHO's suggestions, which include keeping social distances, wearing facemasks, washing one's hands, avoiding crowds, etc.

Interval estimation for nabla fractional order linear time-invariant systems

In this paper, we provide a framework to achieve interval estimation for nabla Caputo fractional order linear time-invariant (LTI) systems in the presence of bounded model uncertainties. Interval observers based on fractional order positive systems theory are designed by possessing desired stable and positive error dynamics. Specifically, the basic concepts and conditions for guaranteeing stability and positivity of the considered systems are derived systematically by finding the system responses. Using the developed criteria and the structure of Luenberger-type observers, a classic interval observer is designed directly which further extends the system classes of interval estimation. Besides, due to the possible absence of a gain matrix which ensures positivity requirement, a more general interval observer design scheme is given by exploiting the coordinate transformation technique. Finally, some simulated cases including fault detection and fractional order circuits related scenarios are developed to validate the usefulness and practicality of the framework.

Numerical analysis of Atangana-Baleanu fractional model to understand the propagation of a novel corona virus pandemic

In this manuscript, we formulated a new nonlinear SEIQR fractional order pandemic model for the Corona virus disease (COVID-19) with Atangana-Baleanu derivative. Two main equilibrium points F0∗,F1∗ of the proposed model are stated. Threshold parameter R0 for the model using next generation technique is computed to investigate the future dynamics of the disease. The existence and uniqueness of solution is proved using a fixed point theorem. For the numerical solution of fractional model, we implemented a newly proposed Toufik-Atangana numerical scheme to validate the importance of arbitrary order derivative ρ and our obtained theoretical results. It is worth mentioning that fractional order derivative provides much deeper information about the complex dynamics of Corona model. Results obtained through the proposed scheme are dynamically consistent and good in agreement with the analytical results. To draw our conclusions, we explore a complete quantitative analysis of the given model for different quarantine levels. It is claimed through numerical simulations that pandemic could be eradicated faster if a human community selfishly adopts mandatory quarantine measures at various coverage levels with proper awareness. Finally, we have executed the joint variability of all classes to understand the effectiveness of quarantine policy on human population.

A fractal fractional order vaccination model of COVID-19 pandemic using Adam’s moulton analysis

The pandemic caused by coronaviruses (SARS-COV-2) is a zoonotic disease targeting the respiratory tract of active humans. Few mild symptoms of fever and tiredness get cured without any medicinal aid, whereas some severe symptoms of dry cough with breathing illness led to perceived risk of secondary transmission. This paper studies the effectiveness of vaccination in Covid-19​ pandemic disease by modelling three compartments susceptible, vaccinated and infected (SVI) of Atangana Baleanu of Caputo (ABC) type derivatives in non-integer order. The disease dynamics is analysed and its stability is performed. Numerical approximation is derived using Adam’s Moulton method and simulated to forecast the results for controllability of pandemic spread.

Unravelling the dynamics of the COVID-19 pandemic with the effect of vaccination, vertical transmission and hospitalization

The coronavirus disease 2019 (COVID-19) is caused by a newly emerged virus known as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), transmitted through air droplets from an infected person. However, other transmission routes are reported, such as vertical transmission. Here, we propose an epidemic model that considers the combined effect of vertical transmission, vaccination and hospitalization to investigate the dynamics of the virus's dissemination. Rigorous mathematical analysis of the model reveals that two equilibria exist: the disease-free equilibrium, which is locally asymptotically stable when the basic reproduction number ( R 0 ) is less than 1 (unstable otherwise), and an endemic equilibrium, which is globally asymptotically stable when R 0 > 1 under certain conditions, implying the plausibility of the disease to spread and cause large outbreaks in a community. Moreover, we fit the model using the Saudi Arabia cases scenario, which designates the incidence cases from the in-depth surveillance data as well as displays the epidemic trends in Saudi Arabia. Through Caputo fractional-order, simulation results are provided to show dynamics behaviour on the model parameters. Together with the non-integer order variant, the proposed model is considered to explain various dynamics features of the disease. Further numerical simulations are carried out using an efficient numerical technique to offer additional insight into the model's dynamics and investigate the combined effect of vaccination, vertical transmission, and hospitalization. In addition, a sensitivity analysis is conducted on the model parameters against the R 0 and infection attack rate to pinpoint the most crucial parameters that should be emphasized in controlling the pandemic effectively. Finally, the findings suggest that adequate vaccination coupled with basic non-pharmaceutical interventions are crucial in mitigating disease incidences and deaths.

Modeling epidemic flow with fluid dynamics

In this paper, a new mathematical model based on partial differential equations is proposed to study the spatial spread of infectious diseases. The model incorporates fluid dynamics theory and represents the epidemic spread as a fluid motion generated through the interaction between the susceptible and infected hosts. At the macroscopic level, the spread of the infection is modeled as an inviscid flow described by the Euler equation. Nontrivial numerical methods from computational fluid dynamics (CFD) are applied to investigate the model. In particular, a fifth-order weighted essentially non-oscillatory (WENO) scheme is employed for the spatial discretization. As an application, this mathematical and computational framework is used in a simulation study for the COVID-19 outbreak in Wuhan, China. The simulation results match the reported data for the cumulative cases with high accuracy and generate new insight into the complex spatial dynamics of COVID-19.

A curative and preventive treatment fractional model for plant disease in Atangana–Baleanu derivative through Lagrange interpolation

The growth of the world populations number leads to increasing food needs. However, plant diseases can decrease the production and quality of agricultural harvests. Mathematical models are widely used to model and interpret plant diseases, showing viruses’ transmission dynamics and effects. In this paper, we investigate the dynamics of the treatments of plant diseases via the Atangana–Baleanu derivative in the sense of Caputo (ABC). We study the existence and uniqueness of solutions of curative and preventive treatment fractional model for plant disease. By using Lagrange interpolation, we give numerical simulations and investigate the results at various fractional orders under specific parameters. The results show that the increase of the roguing rate for the most infected plant or the decrease of the rate of planting in the infected area will reduce the plant disease transmissions. For balancing the plant production, the decision-makers can plant in other areas in which there are no infected cases.

New classifications of monotonicity investigation for discrete operators with Mittag-Leffler kernel

This paper deals with studying monotonicity analysis for discrete fractional operators with Mittag-Leffler in kernel. The $ \nu- $monotonicity definitions, namely $ \nu- $(strictly) increasing and $ \nu- $(strictly) decreasing, are presented as well. By examining the basic properties of the proposed discrete fractional operators together with $ \nu- $monotonicity definitions, we find that the investigated discrete fractional operators will be $ \nu^2- $(strictly) increasing or $ \nu^2- $(strictly) decreasing in certain domains of the time scale $ \mathbb{N}_a: = \{a, a+1, \dots\} $. Finally, the correctness of developed theories is verified by deriving mean value theorem in discrete fractional calculus.

A class of delay SIQR-V models considering quarantine and vaccination: Validation based on the COVID-19 perspective

To contain the novel SARS-CoV-2 (COVID-19) spreading worldwide, governments generally adopt two measures: quarantining the infected people and vaccinating the susceptible people. To investigate the disease latency's influence on the transmission characteristics of the system, we establish a new SIQR-V (susceptible-infective-quarantined-recovered-vaccinated) dynamic model that focus on the effectiveness of quarantine and vaccination measures in the scale-free network. We use theoretical analysis and numerical simulation to explore the evolution trend of different nodes and factors influencing the system stability. The study shows that both the complexity of the network and latency delay can affect the evolution trend of the infected nodes in the system. Still, only latency delay can destroy the stability of the system. In addition, through the parameter sensitivity analysis of the basic reproduction number, we find that the effect of the vaccination parameter α on the basic reproduction number R 0 is more significant than that of transmission rate β and quarantine parameter σ . It shows that vaccination is one of the most effective public policies to prevent infectious diseases' spread. Finally, we calculate the basic reproduction numbers that are greater than one for Germany and Pakistan under COVID-19 and validate the model's effectiveness based on the disease data of COVID-19 in Germany. The results show that the changing trend of the infected population in Germany based on the SIQR-V model is roughly the same as that reflected by the actual epidemic data in Germany. Therefore, providing suggestions and guidance for treating infectious diseases based on this model can effectively reduce the harm caused by the outbreak of contagious diseases.

Stochastic covid-19 model with fractional global and classical piecewise derivative

Several methodologies have been advocated in the last decades with the aim to better understand behaviours displayed by some real-world problems. Among which, stochastics modelling and fractional modelling, fuzzy and others. These methodologies have been suggested to threat specific problems; however, It have been noticed that some problems exhibit different patterns as time passes by. Randomness and nonlocality can be combined to depict complex real-world behaviours. It has been observed that, covid-19 virus spread does not follow a single pattern; sometimes we obtained stochastic behaviours, another nonlocal behaviour and others. In this paper, we shall consider a covid-19 model with fractional stochastic behaviours. More precisely a covid-19 model, where the model considers nonlocalities and randomness is suggested. Then a comprehensive analysis of the model is conducted. Numerical simulations and illustrations are done to show the efficiency of the model.

Analysis of a COVID-19 compartmental model: a mathematical and computational approach

In this note, we consider a compartmental epidemic mathematical model given by a system of differential equations. We provide a complete toolkit for performing both a symbolic and numerical analysis of the spreading of COVID-19. By using the free and open-source programming language Python and the mathematical software SageMath, we contribute for the reproducibility of the mathematical analysis of the stability of the equilibrium points of epidemic models and their fitting to real data. The mathematical tools and codes can be adapted to a wide range of mathematical epidemic models.

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.

Fractional dynamic system simulating the growth of microbe

There are different approaches that indicate the dynamic of the growth of microbe. In this research, we simulate the growth by utilizing the concept of fractional calculus. We investigate a fractional system of integro-differential equations, which covers the subtleties of the diffusion between infected and asymptomatic cases. The suggested system is applicable to distinguish the presentation of growth level of the infection and to approve if its mechanism is positively active. An optimal solution under simulation mapping assets is considered. The estimated numerical solution is indicated by employing the fractional Tutte polynomials. Our methodology is based on the Atangana-Baleanu calculus (ABC). We assess the recommended system by utilizing real data.