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# Quasilinearization-Lagrangian method to solve the HIV infection model of CD4 $$^+$$ + T cells

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## Abstract

In this paper, the quasilinearization and Lagrangian methods are used for solving a model of the HIV infection of CD4$$^+$$T cells. This approach is based on the Lagrangian method by using the collocation points of transformed Hermite polynomials. The quasilinearization method is used for converting the non-linear problem to a sequence of linear equations and the Hermite Lagrangian method is applied for solving linear equations at each iteration. In the end, the obtained results have been compared with some other well-known results and show that the present method is efficient.

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... Gandomani [31] applied the Müntz-Legendre Polynomials technique to solve the HIV model numerically. Parand et al. [32] employed the Quasi-linearization Lagrangian method (QLM) for finding the numerical solution of the HIV model. Attaullah et al. [33] employed the Galerkin time discretization approach for the HIV disease transmission dynamics with a non-linear supply rate. ...
... The primary purpose of such mathematical models is to get insights into disease mechanisms and how to mitigate the disease. Herein, we considered a mathematical model proposed by Parand et al. [32] consists of three firstorder nonlinear differential equations that characterize the dynamics of healthy/infected T-cells and free virus particles. The population of healthy T-cells represented by T(t), infected T-cells by I(t), and virus particles by V(t) which is described as follow: Table 1 summarizes the initial conditions of the state variables and a detailed description of each parameter involved in the model. ...
... Values [32,35] ...
Article
In developing nations, the human immunodeficiency virus (HIV) infection, which can lead to acquired immunodeficiency syndrome (AIDS), has become a serious infectious disease. It destroys millions of people and costs incredible amounts of money to treat and control epidemics. In this research, we implemented a Legendre wavelet collocation scheme for the model of HIV infection and compared the new findings to previous findings in the literature. The findings demonstrate the precision and practicality of the suggested approach for approximating the solutions of HIV model. Additionally, establish an autonomous non-linear model for the transmission dynamics of healthy CD4⁺ T-cells, infected CD4⁺ T-cells and free particles HIV with a cure rate. Through increased human immunity, the cure rate contributes to a reduction in infected cells and viruses. Using the Routh-Hurwitz criterion, we determine the basic reproductive number and assess the stability of the disease-free equilibrium and unique endemic equilibrium of the model. Furthermore, numerical simulations of the novel model are presented using the suggested approach to demonstrate the efficiency of the key findings.
... The research at hand, pertains to the infection spread dynamics of human immunodeficiency virus (HIV) of CD4 + T-cells model [21]. The HIV infection model depends upon three components; i.e., CD4 + T cells infected by the HIV, elements of free HIV in the blood and susceptible cells. ...
... Rate of virus free CD4 + T cells equations as follows [21]: ...
... Many numerical as well as analytical schemes have been exploited for the solution of infection dynamics for HIV mathematical models [18,21,24,45]. The determinist solvers/procedures have their own specific measures and advantages as well as drawbacks and limitations. ...
Article
In the investigations presented here, an efficient computing approach is applied to solve Human Immunodeficiency Virus (HIV) infection spread. This approach involves CD4+ T-cells by feed-forward artificial neural networks (FF-ANNs) trained with particle swarm optimization (PSO) and interior point method (IPM), i.e., FF-ANN-PSO-IPM. In the proposed solver FF-ANN-PSO-IPM, the FF-ANN models of differential equations are used to develop the fitness functions for an infection model of T-cells. The training of networks through minimization problem are proficiently conducted by integrated heuristic capability of PSO-IPM. The reliability, stability and exactness of the proposed FF-ANN-PSO-IPM are established through comparison with outcomes of standard numerical procedure with Adams method for both single and multiple autonomous trials with precision of order 4 to 8 decimal places of accuracy. The statistical measures are effectively used to validate the outcomes of the proposed FF-ANN-PSO-IPM.
... The HIV model was also solved using the Legendre wavelets [19] and the Exponential collocation method [22] in 2016. Parand et al. [12] proposed a numerical scheme known as the quasilinearization-lagrangian method and also the shifted Lagrangian Jacobi collocation scheme [13] in 2017. Mirzaee and Nasrin [10] studied the HIV model using the orthonormal Bernstein collocation method in 2018. ...
... Accepted manuscript to appear in IJB Table 5 compares the present results with QL [12] in the time interval [1.2,3] in step size of 0.2 for H(t) , I(t) and V(t). The results obtained for N = 50 using QL are easily obtained by the present method for N = 15. ...
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In this research work, we study the Human Immunodeficiency Virus (HIV) infection on helper T cells governed by a mathematical model consisting of a system of three first-order nonlinear differential equations. The objective of the analysis is to present an approximate mathematical solution to the model that gives the count of the numbers of uninfected and infected helper T cells and the number of free virus particles present at a given instant of time. The system of nonlinear ODEs is converted into a system of nonlinear algebraic equations using spectral collocation method with three different basis functions such as Chebyshev, Legendre and Jacobi polynomials. Some factors such as the production of helper T cells and infection of these cells play a vital role in infected and uninfected cell counts. Detailed error analysis is done to compare our results with the existing methods. It is shown that the spectral collocation method is a very reliable, efficient and robust method of solution compared to many other solution procedures available in the literature. All these results are presented in the form of tables and figures.
... The concept of the Lyapunov function is used to figure out a new set of conditions that keep the steady states stable. The dynamical behavior of various infectious diseases are described using the idea of mathematical modeling (see [17][18][19][20][21][22][23][24][25][26][27][28][29][30] for detail information). Attaullah et al. [31] established a mathematical model for the dynamics of Human Immunodeficiency Virus (HIV) infection. ...
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Mathematical modelling has been extensively used to measure intervention strategies for the control of contagious conditions. Alignment between different models is pivotal for furnishing strong substantiation for policymakers because the differences in model features can impact their prognostications. Mathematical modelling has been widely used in order to better understand the transmission, treatment, and prevention of infectious diseases. Herein, we study the dynamics of a human immunodeficiency virus (HIV) infection model with four variables: S (t), I (t), C (t), and A (t) the susceptible individuals; HIV infected individuals (with no clinical symptoms of AIDS); HIV infected individuals (under ART with a viral load remaining low), and HIV infected individuals (with two different incidence functions (bilinear and saturated incidence functions). A novel numerical scheme called the continuous Galerkin-Petrov method is implemented for the solution of the model. The influence of different clinical parameters on the dynamical behavior of S (t), I (t), C (t) and A (t) is described and analyzed. All the results are depicted graphically. On the other hand, we explore the time-dependent movement of nanofluid in porous media on an extending sheet under the influence of thermal radiation, heat flux, hall impact, variable heat source, and nanomaterial. The flow is considered to be 2D, boundary layer, viscous, incompressible, laminar, and unsteady. Sufficient transformations turn governing connected PDEs into ODEs, which are solved using the proposed scheme. To justify the envisaged problem, a comparison of the current work with previous literature is presented.
... Table 2 shows some approaches applied to this model. [44] Homotopy Perturbation method(HPM) 2007 Alomari et al. [24] Homotopy Analysis method(HAM) 2011 Merdan et al. [45] Variational Iteration Method(VIM) 2011 Ongun [48] Laplace Adomian Decomposition Method(LADM) 2011 Dogun [17] Multistep Laplace Adomian Decomposition Method(MLADM) 2012 Khan et al. [40] Iterative Homotopy Perturbation Transform Method(IHPTM) 2012 Yüzbaşi [73] Bessel Collocation Method(BCM) 2012 Atangana et al. [5] Homotopy Decomposition Method(HDM) 2014 Chen [11] Padé-Adomian Decomposition Method(PADM) 2015 Venkatesh et al. [68] Legendre Wavelets Method(LWM) 2016 Kajani et al. [23] Müntz-Legendre Method(MLM) 2016 El-Baghdady et al. [18] Legendre Collocation Method(LCM) 2017 Parand et al. [52] Shifted Lagrangian Jacobi Method (SLG) 2018 Parand et al. [51] Quasilinearization-Lagrangian Method (QLM) 2018 Parand et al. [53] Pseudospectral Legendre Method (PLM) 2018 Parand et al. [52] Shifted Boubaker Lagrangian Method (SBLM) 2018 Parand et al. [54] Shifted Chebyshev Polynomial Method (SCP) 2019 Umar et al. [66] Genetic Algorithm Active Set Method (GA-ASM) 2020 Oluwaseun et al. [47] Block Method (BM) 2021 Thirumalai et al. [65] Spectral Collocation Method (SCM) 2021 Umar et al. [67] Neuro Swarm Intelligent Computing (NSIC) 2021 Hassani et al. [29] Generalized Shifted Jacobi Polynomials (GSJP) 2022 ...
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The authors examine a model for the interaction of HIV with CD4[sup +] T cells that considers four populations: uninfected T cells, latently infected T cells, actively infected T cells, and free virus. Using this model they show that many of the puzzling quantitative features of HIV infection can be explained simply. They also consider effects of AZT on viral growth and T-cell population dynamics. The model exhibits two steady states, an uninfected state in which no virus is present and an endemically infected state, in which virus and infected T cells are present. They show that if N, the number of infections virions produced per actively infected T cell, is less a critical value, N[sub crit], then the uninfected state is the only steady state in the nonnegative orthant, and this state is stable. For N > N[sub crit], the uninfected state is unstable, and the endemically infected state can be either stable, or unstable and surrounded by a stable limit cycle. Using numerical bifurcation techniques they map out the parameter regimes of these various behaviors. Oscillatory behavior seems to lie outside the region of biologically realistic parameter value. When the endemically infected state is stable, it is characterized by a reduced number of T cells compared with the uninfected state. Thus T-cell depletion occurs through the establishment of a new steady state. The dynamics of the establishment of this new steady state are examined both numerically and via the quasi-steady-state approximation. They develop approximations for the dynamics at early times in which the free virus rapidly binds to T cells, during an intermediate time scale in which the virus grows exponentially, and a third time scale on which viral growth slows and the endemically infected steady state is approached. Using the quasi-steady-state approximation the model can be simplified to two ordinary differential equations that summarize much of the dynamical behavior. 65 refs., 14 figs., 2 tabs.
Spectral Methods for Differential Problems. Institute of Numerical Analysis
• C I Gheorghiu