A Collocation Technique via Pell-Lucas Polynomials to Solve Fractional Differential Equation Model for HIV/AIDS with Treatment Compartment

Abstract

In this study, a numerical method based on the Pell-Lucas polynomials (PLPs) is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment. The HIV/AIDS mathematical model with a treatment compartment is divided into five classes, namely, susceptible patients (S), HIV-positive individuals (I), individuals with full-blown AIDS but not receiving ARV treatment (A), individuals being treated (T), and individuals who have changed their sexual habits sufficiently (R). According to the method, by utilizing the PLPs and the collocation points, we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations. Also, the error analysis is presented for the Pell-Lucas approximation method. The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people. To demonstrate the usefulness of this method, the applications are made on the numerical example with the help of MATLAB. In addition, four cases of the fractional order derivative (p = 1, p = 0.95, p = 0.9, p = 0.85) are examined in the range [0, 200]. Owing to applications, we figured out that the outcomes have quite decent errors. Also, we understand that the errors decrease when the value of N increases. The figures in this study are created in MATLAB. The outcomes indicate that the presented method is reasonably sufficient and correct.