The human immunodeficiency viruses are two species of Lentivirus that infect humans. Over time, they cause acquired immunodeficiency syndrome, a condition in which progressive immune system failure allows life-threatening opportunistic infections and cancers to thrive. Human immunodeficiency virus infection came from a type of chimpanzee in Central Africa. Studies show that immunodeficiency viruses may have jumped from chimpanzees to humans as far back as the late 1800s. Over decades, human immunodeficiency viruses slowly spread across Africa and later into other parts of the world. The Susceptible-Infected-Recovered (SIR) models are significant in studying disease dynamics. In this paper, we have studied the effect of irresponsible immigrants on HIV/AIDS dynamics by formulating and considering different methods. Euler, Runge Kutta, and a Non-standard finite difference (NSFD) method are developed for the same problem. Numerical experiments are performed at disease-free and endemic equilibria points at different time step sizes ‘ℎ’. The results reveal that, unlike Euler and Runge Kutta, which fail for large time step sizes, the proposed Non-standard finite difference (NSFD) method gives a convergence solution for any time step size. Our proposed numerical method is bounded, dynamically consistent, and preserves the positivity of the continuous solution, which are essential requirements when modeling a prevalent disease.

Acquired Immunodeficiency Syndrome (AIDS) is caused by the Human Immunodeficiency Virus (HIV). HIV/AIDS is the most destructive disease faced by humanity. There are serious consequences for the community, economy, and public health. People infected with Human Immunodeficiency Virus (HIV) may harbor the virus for many years without clinical signs of disease. Eventually, it destroys the body’s immune system, increases the risk of certain diseases, damages body organs such as the brain, kidneys, and heart, and results in death. Human Immunodeficiency Virus (HIV) deteriorates the blood deformation of some commonly used immune systems, which protect the body from disease. HIV/AIDS currently kills approximately 2 million people worldwide each year. Sexual relations with an infected person and exchanging infected blood normally cause Human Immunodeficiency Virus (HIV) transmission. The infected mother also transfers Human Immunodeficiency Virus (HIV) to her newborn. People with Acquired Immunodeficiency Syndrome (AIDS) are susceptible to many disease infections that do not normally cause disease in healthy people [^{+} T-Cells depending on the viral load via Caputo Fabrizio derivative [

In this work, we consider the HIV/AIDS model presented by [

With initial conditions

The system

and non-negative initial conditions are given as,

The model modified into total population

For the sake of simplicity, the above system can be re-written as

where

The classical solution of the deterministic model

The goal of the current section is to show the unique existence of the solution by the well-known Banach fixed theorem stated as.

and we choose the space of continuous functions

The application of the contraction mapping principle seeks the following two conditions,

for the operators

In this subsection, we show the explicit estimates for

Here,

The sufficiently small initial conditions can be considered. For arbitrary initial needs, one may choose a larger ball radius. Still, the permissible restriction on the radius can be obtained after the second condition on the contractility of the operator.

For the contraction condition, we consider the following two elements from

From

The Lipschitz constants

Condition

To understand the optimal behavior of the solution vector

Let us have a comparison of the two functions appearing on the RHS of the above inequalities

The functions have a specific intersection point

The above important consideration leads to the important result of an optimal solution.

This section is dedicated to discussing the steady states of the system

For the disease-free steady states, there are no infective and full-blown Acquired Immunodeficiency Syndrome (AIDS) patients. Hence

From this, we obtain the reproduction number

The disease persists at the endemic equilibrium state, and E*= gives the equilibrium point

We note here that

The Euler method for the studied model can be obtained as follows

In this section, we make RK- 4 scheme for the studied model. Considering the system of

Non-Standard finite-difference (NSFD) scheme for the studied model can be constructed as

Use

Similarly, we obtain

Jacobian matrix for the system

Clearly two Eigen value i.e.,

Since finding the Eigenvalue of the above matrix is quite tricky, the spectral radius (most considerable Eigen value) was calculated using MATLAB, which is depicted in

In this portion, we discuss our findings. The following are values of the parameters used in our model.

In

A comparison of the proposed method and the Runge Kutta technique describing the proportions of immunodeficiency for different values of

In

The effect of irresponsible immigrants on human immunodeficiency has been studied in this article. The existence of the solutions incorporating the contraction mapping principle and self-mapping is discussed. Euler, Runge Kutta, and Non-standard Finite Difference (NSFD) methods are formulated to solve the studied model. Numerical experiments are performed at disease-free equilibrium (DFE), and endemic equilibrium (EE) points at different time-step sizes. The obtained results are analyzed and compared. We concluded that the Euler and Runge Kutta methods fail to converge at large time step sizes, while the proposed method gives results that combine to actual steady states for any time step size. Moreover, the Non-standard Finite Difference (NSFD) method is bounded, dynamically consistent, and preserves the positivity of the solution, which are essential requirements when modeling a prevalent disease.

Thanks to our families and colleagues who supported us morally.

^{+}T-cells depending on the viral load via Caputo Fabrizio derivative