# Determination of Number of Infected Cells and Concentration of Viral Particles in Plasma during HIV-1 Infections Using Shehu Transformation

Authors: M. Higazy; Sudhanshu Aggarwal; Y. S. Hamed
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, the authors determine the number of infected cells and concentration of infected (viral) particles in plasma during HIV-1 (human immunodeficiency virus type one) infections using Shehu transformation. For this, the authors first defined some useful properties of Shehu transformation with proof and then applied Shehu transformation on the mathematical representation of the HIV-1 infection model. The mathematical model of HIV-1 infections contains a system of two simultaneous ordinary linear differential equations with initial conditions. Results depict that Shehu transformation is very effective integral transformation for determining the number of infected cells and concentration of viral particles in plasma during HIV-1 infections.


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## Body

## 1. Introduction

Phylogenetically, two types of HIV are recognized, HIV-1 and HIV-2. There is evidence [1–3] which supports the hypothesis that HIV-1 was first transmitted from chimpanzees to man and HIV-2 transferred from sooty mangabey monkey to humans. Today, there are millions of people infected by HIV and many more each day have their lives affected by it.Researchers [1] have now concluded that HIV is not an airborne virus like that influenza virus. HIV is not transmitted by insects such as fleas, and it is not transmitted by mosquitoes. HIV is not transmitted by hugging, touching, and shaking hands with infected persons. HIV is transmitted in humans through four main body fluids: semen, blood, breast milk, and vaginal secretions. Other unusual case of HIV transmission has been documented, but upon investigation, all have involved some exchange of one of these infected body fluids.HIV is a virus. Viruses are microscopic biological agents that are technically not considered to be living, as they do not meet the scientific requirements for what constitutes life. They are only able to replicate and reproduce themselves through the use of other cells. Specially, they infect themselves to a host cell and deliver their genetic material to the cell. They then hijack the cells’ mechanisms for reproduction and use the cell to replicate many versions of themselves. So, it is very essential to determine the number of infected cells and concentration of viral particles in plasma during HIV-1 infections.Integral transformation is one of the linchpins of modern science and engineering which is essential for finding the exact solutions of complex problems which appears in engineering, business, natural sciences, optical science, computers, and modern mathematics. Nowadays, Integral transformations play a predominant role in mathematics, medical science, chemical engineering, radar and signal processing, physics, fluid mechanics, and theory of elasticity [4, 5].Many scholars [6, 7] used different integral transformations for obtaining the solutions of complex problems of mathematics, biology, chemistry, mechanics, physics, etc. Recently, Aggarwal with others [8–10] comparatively discussed many new integral transformations and determined the solutions of system of simultaneous ordinary linear differential equations by applying these new integral transformations on them. Maitama and Zhao [11] solved constant differential equations by defining Shehu transformation. In year 2019, Aggarwal and others [12–14] used Shehu transformation and determined the solutions of many problems such as growth and decay problems, famous Abel’s problem of mechanics, and linear first kind Volterra integral equations. Aggarwal and Singh [15] defined Shehu transformation of probability integral which frequently appears in statistics. Many complex problems of medicine and electrical engineering such as problem of determining total red blood cells in a hospital patient undergoing surgery and problem of two-loop electrical circuit can be modeled using differential equation and their system [16, 17]. Perelson et al. [18] discussed the HIV-1 model with the help of system of differential equations. Mathematical analysis of HIV-1 dynamics in vivo was given by Perelson and Nelson [19]. Perelson [20] gave the modeling of viral and immune system. Nelson and Perelson [21] used delay differential equation and analyzed the HIV-1 infection model. Stafford et al. [22] discussed the plasma virus concentration during primary HIV infection. Rong et al. [23] gave the mathematical modeling of HIV-1 infection and drug therapy. Aggarwal et al. [24] applied Laplace transform for determining the solutions of population growth and decay problems.To determine the number of infected cells and concentration of viral particles in plasma during HIV-1 infections by applying Shehu transformation on the mathematical model of HIV-1 infections is the main aim of this paper.


## 2. Definition of Shehu Transformation

The Shehu transformation of the functionωt,t≥0 is given by [11](1)Sωt=∫0∞ωte−ε/σtdt=Tε,σ,ε>0,σ>0.


## 3. Some Important Properties of Shehu Transform

The authors present important characteristics of Shehu transformation in this part of the paper.


### 3.1. Linearity Property of Shehu Transformation

If Shehu transform of functionsω1t and ω2t is T1ε,σ and T2ε,σ, respectively, then Shehu transform of lω1t+mω2t is given by lT1ε,σ+mT2ε,σ, where l,m are arbitrary constants.Proof.
Using (1), we obtain(2)Sωt=∫0∞ωte−ε/σtdt,⟹Slω1t+mω2t=∫0∞lω1t+mω2te−ε/σtdt⟹Slω1t+mω2t=l∫0∞ω1te−ε/σtdt+m∫0∞ω2te−ε/σtdt⟹Slω1t+mω2t=lSω1t+mSω2t⟹Slω1t+mω2t=lT1ε,σ+mT2ε,σ,where l,m are arbitrary constants.


### 3.2. Scaling Property of Shehu Transformation

IfSωt=Tε,σ, then Sωkt=1/kTε/k,σ.Proof.
Using (1), we obtain(3)Sωt=∫0∞ωte−ε/σtdt⟹Sωkt=∫0∞ωkte−ε/σtdt.
Substitutingkt=p⟹kdt=dp in (3), we have(4)Sωkt=1k∫0∞ωpe−ε/σp/kdp⟹Sωkt=1k∫0∞e−ε/kp/σωpdp⟹Sωkt=1kTεk,σ.


### 3.3. Translation Property of Shehu Transformation

IfSωt=Tε,σ, then Sektωt=Tε−kσ,σ.Proof.
Using (1), we obtain(5)Sωt=∫0∞ωte−ε/σtdt⟹Sektωt=∫0∞ektωte−ε/σtdt⟹Sektωt=∫0∞ωte−ε/σ−ktdt=∫0∞ωte−ε−kσ/σtdt=Tε−kσ,σ.


## 3.1. Linearity Property of Shehu Transformation

If Shehu transform of functionsω1t and ω2t is T1ε,σ and T2ε,σ, respectively, then Shehu transform of lω1t+mω2t is given by lT1ε,σ+mT2ε,σ, where l,m are arbitrary constants.Proof.
Using (1), we obtain(2)Sωt=∫0∞ωte−ε/σtdt,⟹Slω1t+mω2t=∫0∞lω1t+mω2te−ε/σtdt⟹Slω1t+mω2t=l∫0∞ω1te−ε/σtdt+m∫0∞ω2te−ε/σtdt⟹Slω1t+mω2t=lSω1t+mSω2t⟹Slω1t+mω2t=lT1ε,σ+mT2ε,σ,where l,m are arbitrary constants.


## 3.2. Scaling Property of Shehu Transformation

IfSωt=Tε,σ, then Sωkt=1/kTε/k,σ.Proof.
Using (1), we obtain(3)Sωt=∫0∞ωte−ε/σtdt⟹Sωkt=∫0∞ωkte−ε/σtdt.
Substitutingkt=p⟹kdt=dp in (3), we have(4)Sωkt=1k∫0∞ωpe−ε/σp/kdp⟹Sωkt=1k∫0∞e−ε/kp/σωpdp⟹Sωkt=1kTεk,σ.


## 3.3. Translation Property of Shehu Transformation

IfSωt=Tε,σ, then Sektωt=Tε−kσ,σ.Proof.
Using (1), we obtain(5)Sωt=∫0∞ωte−ε/σtdt⟹Sektωt=∫0∞ektωte−ε/σtdt⟹Sektωt=∫0∞ωte−ε/σ−ktdt=∫0∞ωte−ε−kσ/σtdt=Tε−kσ,σ.


## 4. Shehu Transformation of Derivatives

IfSωt=Tε,σ, then(a)
Sω′t=ε/σTε,σ−ω0(b)
Sω″t=ε2/σ2Tε,σ−ε/σω0−ω′0(c)
Sωρt=ερ/σρTε,σ−∑k=0ρ−1ε/σρ−k+1ωk0Proof.
(a)
Using (1), we obtain(6)Sωt=∫0∞ωte−ε/σtdt⟹Sω′t=∫0∞ω′te−ε/σtdt=ωte−ε/σt0∞−∫0∞−εσωte−ε/σtdt=limt⟶∞ωte−ε/σt−ω0+εσ∫0∞ωte−ε/σtdt=0−ω0+εσSωt⟹Sω′t=εσTε,σ−ω0.(b)
We have(7)Sω′t=εσTε,σ−ω0=εσSωt−ω0⟹Sω″t=εσSω′t−ω′0=εσεσSωt−ω0−ω′0⟹Sω″t=ε2σ2Sωt−εσω0−ω′0⟹Sω″t=ε2σ2Tε,σ−εσω0−ω′0.(c)
We have(8)Sω″t=ε2σ2Tε,σ−εσω0−ω′0=ε2σ2Sωt−εσω0−ω′0⟹Sω‴t=ε2σ2Sω′t−εσω′0−ω″0=ε2σ2εσSωt−ω0−εσω′0−ω″0⟹Sω‴t=ε3σ3Sωt−ε2σ2ω0−εσω′0−ω″0=ε3σ3Tε,σ−ε2σ2ω0−εσω′0−ω″0.
In general, we have(9)Sωρt=ερσρSωt−ερ−1σρ−1ω0−ερ−2σρ−2ω′0−⋯⋯⋯−ωρ−10⟹Sωρt=ερσρTε,σ−ερ−1σρ−1ω0−ερ−2σρ−2ω′0−⋯⋯⋯−ωρ−10⟹Sωρt=ερσρTε,σ−∑k=0ρ−1εσρ−k+1ωk0.


## 5. Inverse Shehu Transformation

The inverse Shehu transformation ofTε,σ designated by S−1Tε,σ is another function ωt having the property that Sωt=Tε,σ.


## 6. Linearity Property of Inverse Shehu Transformation

IfS−1T1ε,σ=ω1t and S−1T2ε,σ=ω2t, then(10)S−1lT1ε,σ+mT2ε,σ=lS−1T1ε,σ+mS−1T2ε,σ⟹S−1lT1ε,σ+mT2ε,σ=lω1t+mω2t,where l,m are arbitrary constants.


## 7. Relationship between Shehu and Laplace Transformations

IfSωt=Tε,σ and Lωt=Uε where Lωt=∫0∞e−εtωtdt=Uε is the Laplace transformation of ωt, then Tε,σ=Uε/σ.


## 8. Determination of the Total Number of Infected Cells and Concentration of Viral Particles in Plasma during HIV-1 Infections Using Shehu Transformation

The authors present Shehu transformation for determining the number of infected cells and concentration of viral particles in plasma during HIV-1 infections in this part of the paper.Perelson et al. [18] gave the mathematical model of HIV-1 infections as(11)dωdt=αφω0−βωdφdt=−γφ,with(12)ω0=ω0,φ0=φ0,where ω=ωt=number of  infected cells at any timeω0=number of potentially infected cells at the begining of therapyφ=φt=concentration of viral particles in plasmaφ0=initial concentration of viral particles in plasmaα=rate of infectionsβ=rate of loss of virus producing cellsγ=rate constant of  viral particle clearance.The matrix representation of the above system is(13)dLdt=ML+NtwithL0=R,Where dL/dt=dω/dtdφ/dt,M=−βαω00−γ,L=ωφ,N=00,L0=ω0φ0, and R=ω0φ0.Taking Shehu transformation of system (11) and using initial condition (12), we have(14)εσ+βSω−αω0Sφ=ω0εσ+γSφ=φ0.The primitives of (14) are determined by(15)Sω=ω0−αω0φ0ε/σ+γε/σ+β−αω00ε/σ+γ=ω0σε+σγ+φ0ασε+σγε+βσ=ω0σε+βσ+ω0φ0ασ2ε+σγε+βσ=ω0σε+βσ+ω0φ0αβ−γσε+γσ−σε+βσ,(16)Sφ=ε/σ+βω00φ0ε/σ+β−αω00ε/σ+γ=φ01+βσσ1+σγ1+βσ=φ0σε+γσ.Taking inverse Shehu transformation of equations (15) and (16), we have(17)ω=S−1ω0σε+βσ+ω0φ0αβ−γσε+γσ−σε+βσ=ω0S−1σε+βσ+ω0φ0αβ−γS−1σε+γσ−σε+βσ=ω0S−1σε+βσ+ω0φ0αβ−γS−1σε+γσ−S−1σε+βσ=ω0e−βt+ω0φ0αβ−γe−γt−e−βt,(18)φ=S−1φ0σε+γσ=φ0S−1σε+γσ=φ0e−γt.Equations (17) and (18) give the required number of infected cells ω and concentration φ of viral particles in plasma during HIV-1 infections.For explaining the importance of equations (17) and (18), we consider different combinations of the values of the factors, namely, the number of potentially infected cells at the beginning of therapy ω0, rate of loss of virus producing cells β, initial concentration of viral particles in plasma φ0, rate of infection α, and rate constant of viral particle clearance γ, and determine the corresponding values of ω and φ for different values of time t. These values are represented in the Tables 1–7.Table 1
Concentration of viral particles in plasmaφ for different values of time t and rate constant of viral particle clearance γ with initial concentration of viral particles in plasma φ0=15copies/mL (see Figure 1).
Time,t (days)φ0=15copies/mLγ=2.08day−1γ=2.09day−1γ=2.10day−1γ=2.11day−1γ=2.12day−1015151515150.1012.1812.1712.1612.1512.130.209.909.889.869.849.820.308.048.017.997.967.940.406.536.506.486.456.420.505.305.285.255.225.200.604.314.284.254.234.200.703.503.473.453.423.400.802.842.822.802.772.750.902.312.292.272.252.231.001.871.861.841.821.80Table 2
Concentration of viral particles in plasmaφ for different values of time t and initial concentration of viral particles in plasma φ0 with rate constant of viral particle clearance γ=2.08day−1 (see Figure 2).
Time,t (days)γ=2.08day−1φ0=15copies/mLφ0=16copies/mLφ0=17copies/mLφ0=18copies/mLφ0=19copies/mL015161718190.1012.1813.0013.8114.6215.430.209.9010.5511.2111.8712.530.308.048.579.119.6410.180.406.536.967.407.838.270.505.305.666.016.366.720.604.314.594.885.175.450.703.503.733.964.204.430.802.843.033.223.413.600.902.312.462.612.772.921.001.872.002.122.252.37Table 3
The number of infected cellsω for different values of time t and rate of loss of virus producing cells β with ω0=500,φ0=15copies/mL,α=0.10day−1,andγ=2.08day−1 (see Figure 3).
Time,t (days)ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1β=0.65day−1β=0.66day−1β=0.67day−1β=0.68day−1β=0.69day−105005005005005001469.3465.43461.59457.78454.012271.02266.42261.9257.47253.113144.73140.93137.23133.63130.13475.9673.2470.6268.1065.66539.7137.9136.1934.5532.99620.7419.618.5217.5116.55710.8310.139.488.878.3085.655.244.854.494.1692.952.712.482.282.09101.541.401.271.151.05Table 4
The number of infected cellsω for different values of time t and number of potentially infected cells at the beginning of therapy ω0 with β=0.65day−1,φ0=15copies/mL,α=0.10day−1,andγ=2.08day−1 (see Figure 4).
Time,t (days)β=0.65day−1,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1ω0=500ω0=502ω0=504ω0=506ω0=50805005025045065081469.30471.18473.05474.93476.812271.02272.10273.18274.27275.353144.73145.31145.89146.47147.05475.9676.2776.5776.8877.18539.7139.8740.0240.1840.34620.7420.8220.920.9821.07710.8310.8710.9110.9611.0085.655.675.705.725.7492.952.962.972.993.00101.541.551.551.561.56Table 5
The number of infected cellsω for different values of time t and initial concentration of viral particles in plasma φ0 with β=0.65day−1,ω0=500,α=0.10day−1,andγ=2.08day−1 (see Figure 5).
Time,t (days)β=0.65day−1,ω0=500,α=0.10day−1,γ=2.08day−1φ0=15copies/mLφ0=16copies/mLφ0=17copies/mLφ0=18copies/mLφ0=19copies/mL05005005005005001469.3483.19497.07510.96524.842271.02280.00288.98297.97306.953144.73149.64154.55159.45164.36475.9678.5581.1483.7386.32539.7141.0642.4243.7745.13620.7421.4422.1522.8623.57710.8311.2011.5611.9312.3085.655.846.046.236.4292.953.053.153.253.35101.541.591.651.701.75Table 6
The number of infected cellsω for different values of time t and rate of infection α with φ0=15copies/mL,β=0.65day−1,ω0=500,andγ=2.08day−1 (see Figure 6).
Time,t (days)φ0=15copies/mL,β=0.65day−1,ω0=500,γ=2.08day−1α=0.06day−1α=0.07day−1α=0.08day−1α=0.09day−1α=0.10day−105005005005005001385.99406.82427.64448.47469.302217.12230.59244.07257.54271.023115.30122.65130.01137.37144.73460.4364.3268.2072.0875.96531.5833.6135.6437.6839.71616.4917.5518.6119.6720.7478.619.169.7210.2710.8384.494.785.075.365.6592.352.502.652.802.95101.221.301.381.461.54Table 7
The number of infected cellsω for different values of time t and rate constant of viral particle clearance γ with φ0=15copies/mL,β=0.65day−1,ω0=500,andγ=2.08day−1 (see Figure 7).
Time,t (days)φ0=15copies/mL,β=0.65day−1,ω0=500,γ=2.08day−1γ=2.08day−1γ=2.09day−1γ=2.10day−1γ=2.11day−1γ=2.12day−105005005005005001469.30468.50467.71466.92466.132271.02270.24269.47268.71267.963144.73144.25143.78143.31142.84475.9675.7075.4475.1874.93539.7139.5739.4339.2939.16620.7420.6620.5920.5220.45710.8310.7910.7510.7110.6785.655.635.615.595.5792.952.942.932.922.91101.541.531.531.521.52From Table1, it can be concluded that as time t increases from 0 to 1 day, the concentration of viral particles in plasma φ decreases for all the five combinations, namely, φ0=15copies/mL,γ=2.08day−1φ0=15copies/mL,γ=2.09day−1φ0=15copies/mL,γ=2.10day−1φ0=15copies/mL,γ=2.11day−1φ0=15copies/mL,γ=2.12day−1. From this table, it is also clear that the concentration of viral particles in plasma φ decreases as rate constant of viral particle clearance γ increases from 2.08day−1 to 2.12day−1. Figure 1 supports the results of Table 1.Figure 1
Concentration of viral particles in plasmaφ for different values of time t and rate constant of viral particle clearance γ with initial concentration of viral particles in plasma φ0=15copies/mL.Table2 shows that as time t increases from 0 to 1 day, the concentration of viral particles in plasma φ decreases for all the five combinations, namely, φ0=15copies/mL,γ=2.08day−1φ0=16copies/mL,γ=2.08day−1φ0=17copies/mL,γ=2.08day−1φ0=18copies/mL,γ=2.08day−1φ0=19copies/mL,γ=2.08day−1. From this table, it is also clear that the concentration of viral particles in plasma φ increases as initial concentration of viral particles in plasma φ0 increases from 15copies/mL to 19copies/mL. The graph plotted in Figure 2 supports the results of Table 2.Figure 2
Concentration of viral particles in plasmaφ for different values of time t and initial concentration of viral particles in plasma φ0 with rate constant of viral particle clearance γ=2.08day−1.From Table3, it can be concluded that as therapy time t increases from 0 to 10 days, the number of infected cells ω decreases for all the five combinations, namely, ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.66day−1ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.67day−1ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.68day−1ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.69day−1. From this table, it is also clear that the number of infected cells ω decreases as rate of loss of virus producing cells β increases from 0.65day−1 to 0.69day−1. The graph of Figure 3 supports the results of Table 3.Figure 3
The number of infected cellsω for different values of time t and rate of loss of virus producing cells β with ω0=500,φ0=15copies/mL,α=0.10day−1,andγ=2.08day−1.From Table4, it can be concluded that as therapy time t increases from 0 to 10 days, the number of infected cells ω decreases for all the five combinations, namely, ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1ω0=502,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1ω0=504,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1ω0=506,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1ω0=508,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1. From this table, it is also clear that the number of infected cells ω increases as number of potentially infected cells at the beginning of therapy ω0 increases from 500 to 508. Figure 4 supports the results of Table 4.Figure 4
The number of infected cellsω for different values of time t and number of potentially infected cells at the beginning of therapy ω0 with β=0.65day−1,φ0=15copies/mL,α=0.10day−1,andγ=2.08day−1.From Table5, it can be concluded that as therapy time t increases from 0 to 10 days, the number of infected cells ω decreases for all the five combinations, namely, ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1ω0=500,φ0=16copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1ω0=500,φ0=17copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1ω0=500,φ0=18copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1ω0=500,φ0=19copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1. From this table, it is also clear that the number of infected cells ω increases as initial concentration of viral particles in plasma φ0 increases from 15copies/mL to 19copies/mL. The graph plotted in Figure 5 supports the results of Table 5.Figure 5
The number of infected cellsω for different values of time t and initial concentration of viral particles in plasma φ0 with β=0.65day−1,ω0=500,α=0.10day−1,andγ=2.08day−1.Table6 shows that as therapy time t increases from 0 to 10 days, the number of infected cells ω decreases for all the five combinations, namely, ω0=500,φ0=15copies/mL,α=0.06day−1,γ=2.08day−1,β=0.65day−1ω0=500,φ0=15copies/mL,α=0.07day−1,γ=2.08day−1,β=0.65day−1ω0=500,φ0=15copies/mL,α=0.08day−1,γ=2.08day−1,β=0.65day−1ω0=500,φ0=15copies/mL,α=0.09day−1,γ=2.08day−1,β=0.65day−1ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1. From this table, it is also clear that the number of infected cells ω increases as rate of infection α increases from 0.06day−1 to 0.10day−1. From the graph plotted in Figure 6, it is also clear that the number of infected cells ω increases as rate of infection α increases from 0.06day−1 to 0.10day−1.Figure 6
The number of infected cellsω for different values of time t and rate of infection α with φ0=15copies/mL,β=0.65day−1,ω0=500,andγ=2.08day−1.Table7 shows that as therapy time t increases from 0 to 10 days, the number of infected cells ω decreases for all the five combinations, namely, ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.08day−1,β=0.65day−1ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.09day−1,β=0.65day−1ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.10day−1,β=0.65day−1ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.11day−1,β=0.65day−1ω0=500,φ0=15copies/mL,α=0.10day−1,γ=2.12day−1,β=0.65day−1. From this table, it is also clear that the number of infected cells ω decreases as rate constant of viral particle clearance γ increases from 2.08day−1 to 2.12day−1. Figure 7 supports the results of Table 7.Figure 7
The number of infected cellsω for different values of time t and rate constant of viral particle clearance γ with φ0=15copies/mL,β=0.65day−1,ω0=500,andα=0.10day−1.Shehu and inverse Shehu transformations of frequently used functions are presented in Tables8 and 9, respectively.Table 8
Shehu transformation of frequently used functions [12–15].
S.N.ωtSωt=Tε,σ11σ/ε2tσ/ε23t22σ/ε34tρ,ρ∈Nρ!σ/ερ+15tρ,ρ>−1Γρ+1σ/ερ+16eltσ/ε−lσ7sinltlσ2/ε2+l2σ28cosltσε/ε2+l2σ29sinhltlσ2/ε2−l2σ210coshltσε/ε2−l2σ211J0ltσ/ε2+l2σ2Table 9
Inverse Shehu transformation of frequently used functions [12–14].
S.N.Tε,σωt=S−1Tε,σ1σ/ε12σ/ε2t3σ/ε3t2/2!4σ/ερ+1,ρ∈Ntρ/ρ!5σ/ερ+1,ρ>−1tρ/Γρ+16σ/ε−lσelt7σ2/ε2+l2σ2sinlt/l8σε/ε2+l2σ2coslt9σ2/ε2−l2σ2sinhlt/l10σε/ε2−l2σ2coshlt11σ/ε2+l2σ2J0lt


## 9. Conclusions

In the present paper, the authors fruitfully determined the total number of infected cells and concentration of viral particles in plasma during HIV-1 infections using Shehu transformation. Results depict that Shehu transformation is very effective integral transformation for determining the total number of infected cells and concentration of viral particles in plasma during HIV-1 infections. The proposed method is better than the other methods because it provides the exact solution of critical problems in a very short time and without doing large calculation work. This study is very helpful in medical field and provides the solution of critical problems of determining total number of infected cells and concentration of viral particles in plasma during HIV-1 infections at the time of operation. The solutions of complex problems of science, medicine, space, pharmacokinetics, economics, business, and engineering such as the problem of residential segregation, arms race models, the problem related to drug distribution in the blood, problems of heat and mass transfer, radioactive decay problem, and the problem of tumor growth can be determined by applying Shehu transformation on them in future.



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*Source: 6624794-2020-12-07.xml*