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.
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 function is given by [11]
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 functions and is and , respectively, then Shehu transform of is given by , where are arbitrary constants.
Proof. Using (1), we obtainwhere are arbitrary constants.
3.2. Scaling Property of Shehu Transformation
If , then
Proof. Using (1), we obtainSubstituting in (3), we have
3.3. Translation Property of Shehu Transformation
If , then
Proof. Using (1), we obtain
4. Shehu Transformation of Derivatives
If , then(a)(b)(c)
Proof. (a)Using (1), we obtain(b)We have(c)We haveIn general, we have
5. Inverse Shehu Transformation
The inverse Shehu transformation of designated by is another function having the property that
6. Linearity Property of Inverse Shehu Transformation
If and , thenwhere are arbitrary constants.
7. Relationship between Shehu and Laplace Transformations
If and where is the Laplace transformation of , then
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 aswithwhere
The matrix representation of the above system isWhere , and .
Taking Shehu transformation of system (11) and using initial condition (12), we have
The primitives of (14) are determined by
Taking inverse Shehu transformation of equations (15) and (16), we have
Equations (17) and (18) give the required number of infected cells and concentration 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 rate of loss of virus producing cells , initial concentration of viral particles in plasma , rate of infection and rate constant of viral particle clearance , and determine the corresponding values of and for different values of time . These values are represented in the Tables 1–7.
From Table 1, it can be concluded that as time increases from 0 to 1 day, the concentration of viral particles in plasma decreases for all the five combinations, namely, . From this table, it is also clear that the concentration of viral particles in plasma decreases as rate constant of viral particle clearance increases from to . Figure 1 supports the results of Table 1.
Table 2 shows that as time increases from 0 to 1 day, the concentration of viral particles in plasma decreases for all the five combinations, namely, . From this table, it is also clear that the concentration of viral particles in plasma increases as initial concentration of viral particles in plasma increases from to . The graph plotted in Figure 2 supports the results of Table 2.
From Table 3, it can be concluded that as therapy time increases from 0 to 10 days, the number of infected cells decreases for all the five combinations, namely, . From this table, it is also clear that the number of infected cells decreases as rate of loss of virus producing cells increases from to . The graph of Figure 3 supports the results of Table 3.
From Table 4, it can be concluded that as therapy time increases from 0 to 10 days, the number of infected cells decreases for all the five combinations, namely, . From this table, it is also clear that the number of infected cells increases as number of potentially infected cells at the beginning of therapy increases from to . Figure 4 supports the results of Table 4.
From Table 5, it can be concluded that as therapy time increases from 0 to 10 days, the number of infected cells decreases for all the five combinations, namely, . From this table, it is also clear that the number of infected cells increases as initial concentration of viral particles in plasma increases from to . The graph plotted in Figure 5 supports the results of Table 5.
Table 6 shows that as therapy time increases from 0 to 10 days, the number of infected cells decreases for all the five combinations, namely, . From this table, it is also clear that the number of infected cells increases as rate of infection increases from to . From the graph plotted in Figure 6, it is also clear that the number of infected cells increases as rate of infection increases from to .
Table 7 shows that as therapy time increases from 0 to 10 days, the number of infected cells decreases for all the five combinations, namely, . From this table, it is also clear that the number of infected cells decreases as rate constant of viral particle clearance increases from to . Figure 7 supports the results of Table 7.
Shehu and inverse Shehu transformations of frequently used functions are presented in Tables 8 and 9, respectively.
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.
Data Availability
The datasets used to support the findings of this paper are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by Taif University Researchers Supporting Project (TURSP-2020/155), Taif University, Taif, Saudi Arabia.