An integral equation approach for long COVID-19

Introduction

A Volterra-Fredholm Integral Equation Model of the second kind was created for determining the variables associated with recovery R(t) of long COVID symptoms. It was deduced that long COVID symptoms were dependent on many independent variables. Each of these variables was included in a multiple regression analysis to verify each variable's significance and the significant variables were added to the Volterra-Fredholm Integral LC Model. The LC (Long COVID) model was modified from the COVID-19 models previously built by one of the authors [2,3] . The model also includes the infection rate 𝛾.

Integral Methods 2.1 Volterra-Fredholm Integral Equation

The integral equation model was based on the Volterra Fredholm Integral equation of the second kind. We approximated the number of individuals with Long COVID symptoms (%), using a double integral with a weekly singular integrand. The constant, 𝑅 0 , was the initial recovery rate that was available from [4] . The constant, 𝛾, was found from [4] , was the rate of recovery from Long COVID. Using available data, we created a logistic 4P model represented by R(T, t) function. The coefficient of determination of the logistic 4P model was 0.78. Within this model, gender, race, vaccination status, death and age impact factor, and heart disease were included as significant factors impacting recovery from Long COVID. In the Volterra Fredholm Integral model 𝑅 0 = 0.17, α = 0.0019, β = 0.17712, 𝛿 = 0.4936, 𝜆 = 0.1221, ɛ = 0.0442, 𝛾 = 0.11, t was the number of months since contracting COVID, and R(t) was the percent of recovered individuals after contracting COVID 19. Here T was the number of months since contracting COVID, R(T) was the % number of recovered individuals. From known data it was deduced that 6%-8% adults who have contacted COVID are suffering from Long COVID [5] . R(t) was given by: 𝑅(0.5, 𝑡) = 10.47599+

𝑑𝑡 𝑑𝑇

The Recovery Rate was given by: We chose to look at the Hispanic population (% of all adults in the United States) as multi-generation families living in the same household. This might have contributed more to the spread of COVID in Hispanic populations compared to other demographics. [9] Variable P-Value Vaccination with Booster 0.0968<0.1 Race 0.0560<0.1 The multiple regression equation was given by: Y =-43.5006 + 1.0125δ -0.1311β + 0.8026ɛ+ 0.02153λ Y is the percentage of all adults currently experiencing Long COVID. δ is gender (Male). β is the vaccination status with the booster. ɛ is heart disease. λ is race (Hispanic). As the percentage of adults with the factors, male, Hispanic, and heart disease are increased, the number of all adults currently experiencing Long COVID is increased as well. These factors are directly proportional to each other. However, as the number of people vaccinated with the booster is increased, the number of all adults currently experiencing Long COVID decreased

Multiple Linear Regression

Gaussian Quadrature Method

We used the Gaussian Quadrature Method to approximate the Volterra Fredholm integral equation. The Gaussian Quadrature method is a numerical approximation method for approximating integrals that are challenging to compute through analytical methods of integration. The original integral is converted to an integral defined in the closed interval [-1, 1], as the basis functions for this method are based on Legendre Polynomials. The formulation is based on orthogonal polynomials, with the common domain of integration as indicated above from [-1, 1], so the formulation is exact for polynomials of degree 2n -1 or less. This exact rule is known as the Gauss-Legendre Quadrature rule. The Quadrature rule will only be an accurate approximation to the integral above if f (x) is wellapproximated by a polynomial of degree 2n-1 or less on [-1, 1]. The Gauss-Legendre Quadrature rule is not typically used for integrable functions with endpoint singularities, as this does not apply to our integral we could use this method of approximation. The following formulation summarizes the process ∫ 𝑓(𝑥)𝑑𝑥 = Table 4 shows the approximation of the number of months after contracting Long COVID at 5 nodes. It is evident from the table that 5 nodes were sufficient in the accuracy of results.

Neural Network Model

3.6

The combined result by using the nodes from Table 5 The final result of the Neural Network Model is given by the following

Currently Experiencing Long COVID (Y) = 6.26-0.83 N1.1 + 0.81 N2.1 0.08 N3.1 For the Neural Network Model, we imputed specific values for each variable and obtained the following results in Tables 6 and6.1.

Assumptions

One of the drawbacks of our model was that we only considered the Hispanic population. The model that shows the data of currently experiencing long COVID by ever experienced long COVID, plotted into a logistic 4P model only had a significance of 0.78. We extrapolated our model results taken from a data set that had recorded data from Jan 4 th -Jan 16 th of 2023. When initially determining what factors had an impact on Long COVID, Age was determined to not be significant. The Death/Age Impact factor was created due to the idea that many individuals of old age passed away from Long COVID. This factor was created by dividing the percentage of deaths (based on 8 different age ranges) by the midpoint of age distribution, resulting in 8 different values. [10]