Fractional Coronavirus Mathematical Model: Numerical and Theoretical Treatments

This paper deals with modeling of mathematical biological experiments using the iterative fractional integral equations following type(1)w(u)=h(u)+∫u0u(u−r)βΓ(β+1)K(r,w(w(r)))dr(1)where u0, u ∈ [a, b], w, h ∈ C([a, b] × [a, b]), K ∈ C([a, b] × [a, b]). We propose that the mathematical model (1) containing the iterative integral of fractional order that is the best method in the studying this field. We establish the existence and uniqueness solutions for fractional iterative integral equation by using the technique function h non-expansive mappings. Also, we show the results of the system of fractional iterative integral equation by using the technique of non-expansive operators.

We present a generalization of Darbo’s fixed point theorem in this article, and we use it to investigate the solvability of an infinite system of fractional order integral equations in ℓ p 1 ≤ p < ∞ space. The fundamental tool in the presentation of our proofs is the measure of noncompactness mnc approach. The suggested fixed point theory has the advantage of relaxing the constraint of the domain of compactness, which is necessary for several fixed point theorems. To illustrate the obtained result in the sequence space, a numerical example is provided. Also, we have applied it to an integral equation involving fractional integral by another function that is the generalization of many fixed point theorems and fractional integral equations.

This paper introduces a novel simulation approach that employs Caputo and Caputo-Fabrizio fractional derivative operators to explore the solution behavior of the fractional pollution model for a network of three lakes jointed by canals. Two input models are addressed by leveraging a purportedly innovative approximation techniques based on Gegenbauer wavelet polynomials (GWPs) and fractional Simpson's 1/3 rule (FSR). The spectral collocation method (SCM), leveraging the distinctive properties of GWPs are utilized to convert the model under consideration into a set of algebraic equations. The measurement of the residual error function (REF) confirms the precision and efficacy of the SCM. Additionally, for the second method, a numerical simulation of the resulting system of fractional integral equations (FIEs) is carried out using the FSR. Comparative analysis with the Runge-Kutta fourth order method (RK4M) highlights the efficacy of the techniques developed to simulate the solution behavior of such models, offering simple and efficient simulation tools.

Although differential transform method (DTM) is a highly efficient technique in the approximate analytical solutions of fractional differential equations, applicability of this method to the system of fractional integro-differential equations in higher dimensions has not been studied in detail in the literature. The major goal of this paper is to investigate the applicability of this method to the system of two-dimensional fractional integral equations, in particular to the two-dimensional fractional integro-Volterra equations. We deal with two different types of systems of fractional integral equations having some initial conditions. Computational results indicate that the results obtained by DTM are quite close to the exact solutions, which proves the power of DTM in the solutions of these sorts of systems of fractional integral equations.

This paper is concerned with the existence and uniqueness of positive solutions for a Volterra nonlinear fractional system of integral equations. Our analysis relies on a fixed point theorem of a sum operator. The conditions for the existence and uniqueness of a positive solution to the system are established. Moreover, an iterative scheme is constructed for approximating the solution. The case of quadratic system of fractional integral equations is also considered.

In recent decades, many researchers have pointed out that derivatives and integrals of the non-integer order are well suited for describing various real-world materials, for example, polymers. It has also been shown that fractional-order mathematical models are more effective than integer-order mathematical models. Thereby, given these considerations, the investigation of qualitative properties, in particular, Ulam-type stabilities of fractional differential equations, fractional integral equations, etc., has now become a highly attractive subject for mathematicians, as this represents an important field of study due to their extensive applications in various branches of aerodynamics, biology, chemistry, the electrodynamics of complex media, polymer science, physics, rheology, and so on. Meanwhile, the qualitative concepts called Ulam–Hyers–Mittag-Leffler (U-H-M-L) stability and Ulam–Hyers–Mittag-Leffler–Rassias (U-H-M-L-R) stability are well-suited for describing the characteristics of fractional Ulam-type stabilities. The Banach contraction principle is a fundamental tool in nonlinear analysis, with numerous applications in operational equations, fractal theory, optimization theory, and various other fields. In this study, we consider a nonlinear fractional Volterra integral equation (FrVIE). The nonlinear terms in the FrVIE contain multiple variable delays. We prove the U-H-M-L stability and U-H-M-L-R stability of the FrVIE on a finite interval. Throughout this article, new sufficient conditions are obtained via six new results with regard to the U-H-M-L stability or the U-H-M-L-R stability of the FrVIE. The proofs depend on Banach’s fixed-point theorem, as well as the Chebyshev and Bielecki norms. In the particular case of the FrVIE, an example is delivered to illustrate U-H-M-L stability.

A computational method for a class of systems of nonlinear variable-order fractional quadratic integral equations

The main aim of this paper is to use the operational matrices of fractional integration of Haar wavelets to find the numerical solution for a nonlinear system of two-dimensional fractional partial Volterra integral equations. To do this, first we present the operational matrices of fractional integration of Haar wavelets. Then we apply these matrices to solve systems of two-dimensional fractional partial Volterra integral equations (2DFPVIE). Also, we present the error analysis and convergence as well. At the end, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

Fractional differential equations and fractional integral equations have gained considerable importance and attention due to their applications in many engineering and scientific disciplines. Gronwall-Bellman inequalities are important tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solutions of Fractional differential equations and fractional integral equations. In this paper, we discuss a class of integral inequalities with pth power, which includes a nonconstant term outside the integrals. Using the definitions and properties of modified Riemann-Liouville fractional derivative and Riemann-Liouville fractional integral, the upper bounds of the unknown function is estimated explicitly. The derived result can be applied in the study of qualitative properties of solutions of fractional integral equations. Introduction Fractional differential equations and fractional integral equations have gained considerable importance and attention due to their applications in many engineering and scientific disciplines. Gronwall-Bellman inequalities [1, 2] are important tools in the study of existence, uniqueness, boundedness, stability, invariant manifolds and other qualitative properties of solutions of fractional differential equations and fractional integral equations. In 2011, Abdeldaim et al. [3] studied a new iterated integral inequality with pth power ds d d g u h s u s u s f u t u s t ] ] ) ( ) ( [ ) ( ) ( )[ ( ) ( ) ( 0 0 0 0              . (1) In 2014, El-Owaidy, Abdeldaim, and El-Deeb [4] discussed a new nonlinear integral inequality with a nonconstant term outside the integrals ds s u s h ds s u s g t f t u t

Background: The number of asymptomatic infected patients with coronavirus disease 2019 (COVID-2019) is rampaging around the world but limited information aimed on risk factors of asymptomatic infections. The purpose of this study is to investigate the risk factors of symptoms onset and clinical features in asymptomatic COVID-19 infected patients.Methods: A retrospective study was performed in 70 asymptomatic COVID-2019 infected patients confirmed by nucleic acid tests in Hunan province, China between 28 January 2020 and 18 February, 2020. The epidemiological, clinical features and laboratory data were reviewed and analyzed. Presence or absence at the onset of symptoms was taken as the outcome. A Cox regression model was performed to evaluate the potential predictors of the onset of symptoms.Results: The study included 36 males and 34 females with a mean age of 33.24±20.40 years (range, 0.5-84 years). There were 22 asymptomatic carriers developed symptoms during hospitalization isolated observation, and diagnosed as confirmed cases, while 48 cases remained asymptomatic throughout the course of disease. Of 70 asymptomatic patients, 14 (14/70, 20%) had underlying diseases, 3 (3/70, 4.3%) had drinking history, and 11 (11/70, 15.7%) had smoking history. 22 patients developed symptoms onset of fever (4/22, 18.2%), cough (13/22, 59.1%), chest discomfort (2/22, 9.1%), fatigue (1/22, 4.5%), pharyngalgia (1/22, 4.5%) during hospitalization; only one (1/22, 4.5%) patient developed signs of both cough and pharyngalgia. Abnormalities on chest CT were detected among 35 of the 69 patients (50.7%) after admission, except for one pregnant woman had not been examined. 4 (4/70, 5.7%) and 8 (8/70, 11.4%) cases showed leucopenia and lymphopenia. With the effective antiviral treatment, all the 70 asymptomatic infections had been discharged, none cases developed severe pneumonia, admission to intensive care unit, or died. The mean time from nucleic acid positive to negative was 13.2±6.84 days. Cox regression analysis showed that smoking history (P=0.028, hazard ratio=4.49, 95% CI 1.18-17.08) and existence of pulmonary disease (P=0.038, hazard ratio=7.09, 95% CI 1.12-44.90) were risk factors of the onset of symptoms in asymptomatic carries.Conclusion: The initially asymptomatic patients can develop mild symptoms and have a good prognosis. History of smoking and pulmonary disease was prone to illness onset in asymptomatic patients, and it is necessary to be highly vigilant to those patients.

Coronavirus Disease-19 Testing Strategies for Patients and Health Care Workers to Improve Workplace Safety.

Our aim in this paper is to develop a Legendre-Jacobi collocation approach for a nonlinear system of two-point boundary value problems with derivative orders at most two on the interval (0,T). The scheme is constructed based on the reduction of the system considered to its equivalent system of Volterra-Fredholm integral equations. The spectral rate of convergence for the proposed method is established in both L2- and $ L^{\infty } $- norms. The resulting spectral method is capable of achieving spectral accuracy for problems with smooth solutions and a reasonable order of convergence for non-smooth solutions. Moreover, the scheme is easy to implement numerically. The applicability of the method is demonstrated on a variety of problems of varying complexity. To the best of our knowledge, the spectral solution of such a nonlinear system of fractional differential equations and its associated nonlinear system of Volterra-Fredholm integral equations has not yet been studied in literature in detail. This gap in the literature is filled by the present paper.

Nonlinear Fractional Volterra integral equations (FVIEs) of the first kind present challenges due to their intricate nature, combining fractional calculus and integral equations. In this research paper, we introduce a novel method for solving such equations using Leibniz integral rules. Our study focuses on a thorough analysis and application of the proposed algorithm to solve fractional Volterra integral equations. By using Leibniz integral rules, we offer a fresh perspective on handling these equations, shedding light on their fundamental properties and behaviours. As a result of this study, we anticipate contributing distinctively to the broader development of analytical tools and techniques. By bridging the gap between fractional calculus and integral equations, our approach not only offers a valuable computational methodology but also paves the way for new insights into the application domains in which such equations arise.

Fractional integrals, fractional derivatives, fractional integral equations, and fractional differential equations are numerically solved by Sinc methods. Sinc methods are able to deal with singularities of the weakly singular integral equations of Riemann-Liouville and Caputo type. The convergence of the numerical method is numerically examined and shows exponential behavior. Different examples are used to demonstrate the effective derivation of numerical solutions for different types of fractional differential and integral equations, linear and non-linear ones. Equations of mixed ordinary and fractional derivatives, integro-differential equations are solved using Sinc methods. We demonstrate that the numerical calculation needed in fractional calculus can be gained with high accuracy using Sinc methods.

Carotid artery disease is a major cause of ischemic stroke, the risk of which is directly related to the severity of stenosis and presence of symptoms.1,2 Stroke is the third leading cause of death in the United States, with approximately three quarters of a million strokes per year. Stroke is the leading cause of functional impairment, with more than 20% of survivors requiring institutional care and up to one third having a permanent disability.3 More worrisome, however, is the fact that as the population ages, the number of patients having strokes appears to be increasing.4 The pathophysiology of stroke may be broadly classified as hemorrhagic, embolic, or ischemic. The majority of strokes are caused by embolic events due to atheroemboli from the carotid artery, the ascending aorta, and arch vessels or cardiac thromboembolism from the left atrium or ventricle. It is estimated that carotid artery stenosis is responsible for 15% to 20% of all strokes.5 As percutaneous treatment options expand, there is uncertainty about appropriate therapy for carotid disease. This document will focus on 3 current controversies: (1) carotid artery revascularization in asymptomatic patients, (2) carotid artery stenting (CAS) in patients who are considered to be at increased surgical risk for carotid endarterectomy (CEA), and (3) the current role for CAS in patients at average surgical risk. ### Prevalence and Natural History The prevalence of asymptomatic extracranial carotid stenosis (≥50%) in persons >65 years of age is estimated to be between 5% and 10%, whereas ≤1% of patients are estimated to have a severe narrowing (>80%).6 In asymptomatic patients with ≥50% carotid artery stenoses, the annual risk of stroke is between 1% and 4.3%.2,7 Long-term (10- to 15-year) cohort studies in asymptomatic patients with moderate to severe carotid stenosis demonstrate an ipsilateral stroke rate between 0.9% and 1.1% per …