Galerkin time discretization scheme for the transmission dynamics of HIV infection with non-linear supply rate

Attaullah; Ramzi Drissi; Wajaree Weera; Attaullah; Ramzi Drissi; Wajaree Weera

doi:10.3934/math.2022630

AIMS Mathematics

2022, Volume 7, Issue 6: 11292-11310. doi: 10.3934/math.2022630

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Galerkin time discretization scheme for the transmission dynamics of HIV infection with non-linear supply rate

1.
Department of Mathematics and Statistics, Bacha Khan University, Charsadda 24420, Pakistan
2.
Department of Insurance, College of Islamic Economics and Finance, Umm Al-Qura University, Makkah, 24243, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand

Received: 17 December 2021 Revised: 19 March 2022 Accepted: 28 March 2022 Published: 11 April 2022
MSC : 34A12, 34K28

The present work implements the continuous Galerkin-Petrov method (cGP(2)-method) to compute an approximate solution of the model for HIV infection of $ \text{CD4}^{+} $ T-cells. We discuss and analyse the influence of different clinical parameters on the model. The work also depicts graphically that how the level of $ \text{CD4}^{+} $ T-cells varies with respect to the emerging parameters in the model. Simultaneously, the model is solved using the fourth-order Runge Kutta (RK4) method. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with those obtained through the RK4 method. A numerical comparison between the results of the cGP (2) method and the RK4 method reveals that the proposed technique is a promising tool for the approximate solution of non-linear systems of differential equations. The present study highlights the accuracy and efficiency of the proposed schemes as in comparison to the other traditional schemes, for example, the Laplace adomian decomposition method (LADM), variational iteration method (VIM), homotopy analysis method (HAM), homotopy perturbation method (HAPM), etc. In this study, two different versions of the HIV model are considered. In the first one, the supply of new $ \text{CD4}^{+} $ T-cells from the thymus is constant, while in the second, we consider the production of these cells as a monotonically decreasing function of viral load. The experiments show that the lateral model provides more reasonable predictions than the former model.
- HIV infection,
- CD4$ ^{+} $ T-cells,
- continuous Galerkin-Petrov method,
- non-linear supply rate
Citation: Attaullah, Ramzi Drissi, Wajaree Weera. Galerkin time discretization scheme for the transmission dynamics of HIV infection with non-linear supply rate[J]. AIMS Mathematics, 2022, 7(6): 11292-11310. doi: 10.3934/math.2022630

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Abstract

The present work implements the continuous Galerkin-Petrov method (cGP(2)-method) to compute an approximate solution of the model for HIV infection of $ \text{CD4}^{+} $ T-cells. We discuss and analyse the influence of different clinical parameters on the model. The work also depicts graphically that how the level of $ \text{CD4}^{+} $ T-cells varies with respect to the emerging parameters in the model. Simultaneously, the model is solved using the fourth-order Runge Kutta (RK4) method. Finally, the validity and reliability of the proposed scheme are verified by comparing the numerical and graphical results with those obtained through the RK4 method. A numerical comparison between the results of the cGP (2) method and the RK4 method reveals that the proposed technique is a promising tool for the approximate solution of non-linear systems of differential equations. The present study highlights the accuracy and efficiency of the proposed schemes as in comparison to the other traditional schemes, for example, the Laplace adomian decomposition method (LADM), variational iteration method (VIM), homotopy analysis method (HAM), homotopy perturbation method (HAPM), etc. In this study, two different versions of the HIV model are considered. In the first one, the supply of new $ \text{CD4}^{+} $ T-cells from the thymus is constant, while in the second, we consider the production of these cells as a monotonically decreasing function of viral load. The experiments show that the lateral model provides more reasonable predictions than the former model.

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