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Quasilinearization-Lagrangian method to solve the HIV infection model of CD4 $$^+$$ + T cells
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.
... 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] ...
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. ...
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. ...
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. ...
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 ...
In this paper, we use Kansa method for solving the system of differential equations in the area of biology. One of the challenges in Kansa method is picking out an optimum value for Shape parameter in Radial Basis Function to achieve the best result of the method because there are not any available analytical approaches for obtaining optimum Shape parameter. For this reason, we design a genetic algorithm to detect a close optimum Shape parameter. The experimental results show that this strategy is efficient in the systems of differential models in biology such as HIV and Influenza. Furthermore, we prove that using Pseudo-Combination formula for crossover in genetic strategy leads to convergence in the nearly best selection of Shape parameter.
https://authors.elsevier.com/a/1giCM1M3Yj9Eb7
The intension of the present work is to present the stochastic numerical approach for solving human immunodeficiency virus (HIV) infection model of cluster of differentiation 4 of T-cells, i.e., CD4+ T cells. A reliable integrated intelligent computing framework using layered structure of neural network with different neurons and their optimization with efficacy of global search by genetic algorithms supported with rapid local search methodology of active-set method, i.e., hybrid of GA-ASM, is used for solving the HIV infection model of CD4+ T cells. A comparison between the present results for different neurons-based models and the numerical values of the Runge–Kutta method reveals that the present intelligent computing techniques is trustworthy, convergent and robust. Statistics-based observation on different performance indices further demonstrates the applicability, effectiveness and convergence of the present schemes.
In this paper, a hybrid method based on the collocation and Newton-Kantorovich methods is used for solving the nonlinear singular Thomas-Fermi equation. At first, by using the Newton-Kantorovich method, the nonlinear problem is converted to a sequence of linear differential equations, and then, the fractional order of rational Legendre functions are introduced and used for solving linear differential equations at each iteration based on the collocation method. Moreover, the boundary conditions of the problem by using Ritz method without domain truncation method are satisfied. In the end, the obtained results compare with other published in the literature to show the performance of the method, and the amounts of residual error are very small, which indicates the convergence of the method.
In this article, we introduce a fractional order of rational Bessel functions
collocation method (FRBC) for solving the Thomas-Fermi equation.
The problem is defined in the semi-infinite domain and has a singularity at
$x = 0$ and its boundary condition occurs at infinity. We solve the problem
on the semi-infinite domain without any domain truncation or transformation
of the domain of the problem to a finite domain. This approach at first,
obtains a sequence of linear differential equations by using the
quasilinearization method (QLM), then at each iteration the equation is
solves by FRBC method. To illustrate the reliability of this work,
we compare the numerical results of the present method with some well-known
results, to show that the new method is accurate, efficient and applicable.
The In this paper, a new Iteration Algorithm is examined to provide an approximate solution of a model for HIV infection of CD4+ T-Cells. This method allows the solution of governing differential equation calculated in the form of an infinite series, with components that can be easily calculated. The reliability and the efficiency of proposed approach is demonstrated in different time intervals by numerical example. All computations have been carried out by computer code written in Mathematica 9.0
The study used the Quasilinearization method to solve Volterra's model for popu-lation growth of a species within a closed system is proposed. This model is a nonlinear integro-differential where the integral term represents the effect of toxin. First we convert this model to a nonlinear ordinary differential equation, then approximate the solution of this equation by treating the nonlinear terms as a perturbation about the linear ones. Finally we compare this method with the other methods and come to the conclusion that the Quasilinearization method gives excellent results.
We pioneered the application of the quasilinearization method (QLM) to resonance calculations. The quartic anharmonic oscillator (kx 2 /2) + λx 4 with a negative coupling constant λ was chosen as the simplest example of the resonant potential. The QLM has been suggested recently for solving the bound state Schrödinger equation after conversion into Riccati form. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries. Comparison of our approximate analytic expressions for the resonance energies and wavefunctions obtained in the first QLM iteration with the exact numerical solutions demonstrate their high accuracy in the wide range of the negative coupling constant. The results enable accurate analytic estimates of the effects of the coupling constant variation on the positions and widths of the resonances.
In this paper we propose a collocation method for solving some well-known classes of Lane–Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.
In this paper, the model of HIV infection of CD4 ⁺ T cells is considered as a system of fractional differential equations. Then, a numerical method by using collocation method based on the Müntz-Legendre polynomials to approximate solution of the model is presented. The application of the proposed numerical method causes fractional differential equations system to convert into the algebraic equations system. The new system can be solved by one of the existing methods. Finally, we compare the result of this numerical method with the result of the methods have already been presented in the literature.
In this paper, the nonlinear Hunter–Saxton equation, which is a famous partial differential equation, is solved by using a hybrid numerical method based on the quasilinearization method and the bivariate generalized fractional order of the Chebyshev functions (B-GFCF) collocation method. First, using the quasilinearization method, the equation is converted into a sequence of linear partial differential equations (LPD), and then these LPDs are solved using the B-GFCF collocation method. A very good approximation of solutions is obtained, and comparisons show that the obtained results are more accurate than the results of other researchers.
In this paper, the nonlinear singular Thomas–Fermi differential equation for neutral atoms is solved using the fractional order of rational Chebyshev orthogonal functions (FRCs) of the first kind, , on a semi-infinite domain, where is an arbitrary numerical parameter. First, using the quasilinearization method, the equation be converted into a sequence of linear ordinary differential equations (LDEs), and then these LDEs are solved using the FRCs collocation method. Using 300 collocation points, we have obtained a very good approximation solution and the value of the initial slope , highly accurate to 37 decimal places.
In this paper, a new algorithm based on the fractional order of rational Euler functions (FRE) is introduced to study the Thomas-Fermi (TF) model which is a nonlinear singular ordinary differential equation on a semi-infinite interval. This problem, using the quasilinearization method (QLM), converts to the sequence of linear ordinary differential equations to obtain the solution. For the first time, the rational Euler (RE) and the FRE have been made based on Euler polynomials. In addition, the equation will be solved on a semi-infinite domain without truncating it to a finite domain by taking FRE as basic functions for the collocation method. This method reduces the solution of this problem to the solution of a system of algebraic equations. We demonstrated that the new proposed algorithm is efficient for obtaining the value of \( y'(0)\) , \( y(x)\) and \( y'(x)\) . Comparison with some numerical and analytical solutions shows that the present solution is highly accurate.
In this paper, the Legendre wavelet method for solving a model for HIV infection of \(\hbox {CD}4^{+}\,\hbox {T}\)-cells is studied. The properties of Legendre wavelets and its operational matrices are first presented and then are used to convert into algebraic equations. Also the convergence and error analysis for the proposed technique have been discussed. Illustrative examples have been given to demonstrate the validity and applicability of the technique. The efficiency of the proposed method has been compared with other traditional methods and it is observed that the Legendre wavelet method is more convenient than the other methods in terms of applicability, efficiency, accuracy, error, and computational effort.
In this paper we improve the quasilinearization method by barycentric Lagrange interpolation because of its numerical stability and computation speed to achieve a stable semi analytical solution. Then we applied the improved method for solving the Fin problem which is a nonlinear equation that occurs in the heat transferring. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The modified QLM is iterative but not perturbative and gives stable semi analytical solutions to nonlinear problems without depending on the existence of a smallness parameter. Comparison with some numerical solutions shows that the present solution is applicable.
In this paper, an exponential method is presented for the approximate solutions of the HIV infection model of CD4+T. The method is based on exponential polynomials and collocation points. This model problem corresponds to a system of nonlinear ordinary differential equations. Matrix relations are constructed for the exponential functions. By aid of these matrix relations and the collocation points, the proposed technique transforms the model problem into a system of nonlinear algebraic equations. By solving the system of the algebraic equations, the unknown coefficients are computed and thus the approximate solutions are obtained. The applications of the method for the considered problem are given and the comparisons are made with the other methods.
In this paper, the Laplace Adomian Decomposition Method is implemented to give an approximate solution of nonlinear ordinary differential equation systems, such as a model for HIV infection of CD4+T cells. The technique is described and illustrated with numerical example. Some plots are presented to show the reliability and simplicity of the methods.
In this paper we propose a Lagrangian method for solving Lane–Emden equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on a Modified generalized Laguerre functions Lagrangian method. The method reduces the solution of this problem to the solution of a system of algebraic equations. We also present the comparison of this work with some well-known results and show that the present solution is acceptable.
In this article, a variational iteration method (VIM) is performed to give approximate and analytical solutions of nonlinear ordinary differential equation systems such as a model for HIV infection of CD4+ T cells. A modified VIM (MVIM), based on the use of Padé approximants is proposed. Some plots are presented to show the reliability and simplicity of the methods.
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