# Shifted Lagrangian Jacobi collocation scheme for numerical solution of a model of HIV infection

## Abstract

In this paper, a system of nonlinear ordinary differential equations (NODEs), namely the equation model of the human immunodeficiency virus (HIV) infection of $$CD4^+ T$$ cells, is studied. Our approach is implemented by using the Shifted-Lagrangian Jacobi (SLJ) polynomials formed by Shifted-Jacobi-Gauss-Radau (SJ-GR) points. In a new insight, by applying Quasilinearization method (QLM) the system of NODE’s is simplified and changed into a system of Linear ordinary differential equations (LODE’s) and instead of working on a system of NODE’s, all processes and works are done on a system of LODE’s. Therefore, unlike the most of the current studies working on nonlinear algebraic equations, the problem is reduced to a system of linear algebraic equations. Then, to solve the problem and find the unknown approximation coefficients, a system of $$Ax=b$$ has been solved. At the end, the accuracy and reliability of this method are shown and comparisons with the other current work’s results are made.

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## Appendix

### Appendix

In this Appendix we are going to apply QLM to equations in (1). In first equation in (1), we are coping with

\begin{aligned} \frac{dT}{dx}= & {} c_1-c_2T(x)+c_3T(x)\left( 1-\frac{T(x)+I(x)}{T_{Max}}\right) -c_4V(x)T(x),\\ \frac{dT}{dx}= & {} c_1-c_2T(x)+c_3T(x) -c_3\frac{T^2(x)}{T_{Max}}-c_3\frac{T(x)I(x)}{T_{Max}}-c_4V(x)T(x). \end{aligned}

Applying the QLM, we will have

\begin{aligned} \frac{dT_{i+1}(x)}{dx}= & {} \underbrace{c_1-c_2T_{i}(x)+c_3T_{i}(x) -c_3(x)\frac{T_{i}^2(x)}{T_{Max}}-c_3\frac{T_{i}(x)I_{i}(x)}{T_{Max}}-c_4V_{i}(x)T_{i}(x)}_{A_1(x)}\nonumber \\&+\big (T_{i+1(x)}-T_{i}(x)\big ).\left( -c_2+c_3-c_3\frac{2T_{i}(x)}{T_{Max}}-c_3\frac{I_{i}(x)}{T_{Max}}-c_4V_{i}(x)\right) .\nonumber \\ \end{aligned}
(25)

All the i-th step forms are known. So then we can show these forms as

\begin{aligned} \frac{dT_{i+1}(x)}{dx}= & {} A_1(x)+\big (T_{i+1}(x)-T_{i}(x)\big ).\left( -c_2+c_3-c3\frac{2T_{i}(x)}{T_{Max}}-c_3\frac{I_{i}(x)}{T_{Max}}-c_4V_{i}(x)\right) ,\\ \frac{dT_{i+1}(x)}{dx}= & {} A_1(x)+T_{i+1}(x)\left( -c_2+c_3-c_3\frac{2T_{i}(x)}{T_{Max}}-c_3\frac{I_{i}(x)}{T_{Max}}-c_4V_{i}(x)\right) \\&\quad -\underbrace{T_{i}(x).\left( -c_2+c_3-c3\frac{2T_{i}(x)}{T_{Max}}-c_3\frac{I_{i}(x)}{T_{Max}}-c_4V_{i}(x)\right) }_{A_2(x)},\\ \frac{dT_{i+1}(x)}{dx}= & {} A_1(x)+T_{i+1}(x)\underbrace{\left( -c_2+c_3-c3\frac{2T_{i}(x)}{T_{Max}}-c_3\frac{I_{i}(x)}{T_{Max}}-c_4V_{i}(x)\right) }_{A_3(x)}+A_2(x),\\ \frac{dT_{i+1}(x)}{dx}= & {} A_3(x)T_{i+1}(x)+\underbrace{A_1(x)+A_2(x)}_{A_4(x)},\\ \frac{dT_{i+1}(x)}{dx}= & {} A_3(x)T_{i+1}(x)+A_4(x). \end{aligned}

Thus, at the end

\begin{aligned} A_3T_{i+1}(x)+A_4(x)-\frac{dT_{i+1}(x)}{dx}=0. \end{aligned}
(26)

Similarly, for the second and third equations in (1), we do the same

\begin{aligned} \frac{dI(x)}{dx}= & {} c_4V(x)T(x)-c_5I(x), \nonumber \\ \frac{dI_{i+1}(x)}{dx}= & {} \underbrace{ c_4V_{i}(x)T_{i}(x)-c_5I_{i}(x)}_{B_1(x)}+\big (I_{i+1}(x)-I_{i}(x)\big )(-c5), \nonumber \\ \frac{dI_{i+1}(x)}{dx}= & {} B_1(x)+(I_{i+1}(x)-I_{i}(x))(-c5), \nonumber \\ \frac{dI_{i+1}(x)}{dx}= & {} B_1(x)+I_{i+1}(x)(-c5)-\underbrace{I_{i}(x)(-c5)}_{B_2(x)}, \nonumber \\ \frac{dI_{i+1}(x)}{dx}= & {} B_1(x)+I_{i+1}(x)\underbrace{(-c5)}_{B_3(x)}+B_2(x), \nonumber \\ \frac{dI_{i+1}(x)}{dx}= & {} \underbrace{B_1(x)+B_2(x)}_{B_4(x)}+B_3(x)I_{i+1}(x), \nonumber \\ \frac{dI_{i+1}(x)}{dx}= & {} B_3(x)I_{i+1}(x)+B_4(x). \end{aligned}
\begin{aligned} B_3(x)I_{i+1}(x)+B_4(x)-\frac{dI_{i+1}(x)}{dx}=0. \end{aligned}
(27)

For the third equation,

\begin{aligned} \frac{dV}{dx}= & {} Nc_5I(x)-c_6V(x), \\ \frac{dV_{i+1}(x)}{dx}= & {} \underbrace{Nc_5I_{i}(x)-c_6V_{i}(x)}_{C_1(x)}+(V_{i+1}(x)-V_{i}(x))(-c_6), \\ \frac{dV_{i+1}(x)}{dx}= & {} C_1(x)+(V_{i+1}(x)-V_{i}(x))(-c_6), \\ \frac{dV_{i+1}(x)}{dx}= & {} C_1(x)+V_{i+1}(x)(-c_6)-\underbrace{V_{i}(x)(-c_6)}_{C_2(x)}, \\ \frac{dV_{i+1}(x)}{dx}= & {} C_1(x)+V_{i+1}(x)\underbrace{(-c_6)}_{C_3(x)}+C_2(x), \\ \frac{dV_{i+1}(x)}{dx}= & {} \underbrace{C_1(x)+C_2(x)}_{C_4(x)}+C_3(x)V_{i+1}(x), \\ \frac{dV_{i+1}(x)}{dx}= & {} C_3(x)V_{i+1}(x)+C_4(x). \end{aligned}
\begin{aligned} C_3(x)V_{i+1}(x)+C_4(x)-\frac{dV_{i+1}(x)}{dx}=0. \end{aligned}
(28)

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Parand, K., Latifi, S. & Moayeri, M.M. Shifted Lagrangian Jacobi collocation scheme for numerical solution of a model of HIV infection. SeMA 75, 379–398 (2018). https://doi.org/10.1007/s40324-017-0138-9

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• DOI: https://doi.org/10.1007/s40324-017-0138-9

### Keywords

• HIV $$CD4^+$$Tcells
• Shifted-Jacobi polynomials