A Novel Radial Basis and Sigmoid Neural Network Combination to Solve the Human Immunodeficiency Virus System in Cancer Patients

Abstract

The purpose of this work is to design a novel process based on the deep neural network (DNN) process to solve the dynamical human immunodeficiency virus (HIV-1) infection system in cancer patients (HIV-1-ISCP). The dual hidden layer neural network structure using the combination of a radial basis and sigmoid function with twenty and forty neurons is presented for the solution of the nonlinear HIV-1-ISCP. The mathematical form of the model is divided into three classes named cancer population cells (T), healthy cells (H), and infected HIV (I) cells. The validity of the designed novel scheme is proven through the comparison of the results. The optimization is performed using a competent scale conjugate gradient procedure, the correctness of the proposed numerical approach is observed through the reference results, and negligible values of the absolute error are around 10⁻³ to 10⁻⁴. The database numerical solutions are achieved from the Runge–Kutta numerical scheme, and are used further to reduce the mean square error by taking 72% of the data for training, while 14% of the data is taken for testing and substantiations. To authenticate the credibility of this novel procedure, graphical plots using different performances are derived.

1. Introduction

Many types of cancer have been found to develop often in patients with acquired immunodeficiency syndrome (AIDS) and most of these cancers are brought on by viruses [1]. Kaposi’s sarcoma (KPS) is a particularly common tumor seen in cancer patients, which develops from lymphocytes that have lymphatic or blood channels on their borders. Several bodily tissues, such as the lymph nodes and lungs, are also susceptible to cancer. Usually, however, they appear as tumors on the skin, like those found inside the mouth [2]. The AIDS-based outbreak is among the most common form of KPS and is found in the whole world, even in the United States. However, herpesviruses, not the human immunodeficiency virus (HIV), are the real perpetrators. HIV is a type of infection that causes AIDS, which is among the primary causes of KPS. Being positive for the virus or infected does not automatically mean that a person has AIDS. The virus can remain in the human body for a long time before causing severe forms of the illness. When a virus seriously compromises the immune system, some illnesses or additional medical conditions, such as malignancies, may manifest. This is the initial identified symptom of the illness, known as AIDS. When HIV compromises an individual’s immune system, those with the KPS herpesvirus become ill [3]. Moderate and increasing non-lymphomas Hodgkin’s (NLPH) with B-cell features are among the illnesses that define AIDS. It is frequent to observe the unique signs of NLPH and Hodgkin’s sickness, as the rate of significant central nerve NLPH is more than three thousand times greater, while the incidence of generalized NLH is over a hundred times greater in HIV-positive individuals [4]. The immunological observation concept was first put forth by Paul Ehrlich, who said that the human body’s immunological model should be regularly and continuously exploited to eradicate tumors [5]. The discovery of cancer antibodies and the ability of an immunoglobulin to inhibit the growth of tumors in rats by opposing those antigens support this theory [6].

Over the past 20 years, a number of mathematical systems have been designed to understand the dynamics of HIV-1 and its spreading process. Various investigations have shown there are a few ways in which HIV-1 propagates in real life, which can directly affect the T cells [7,8]. The majority of these systems indicate the spreading of these infections in the circulation mechanism [9]. HIV-1 propagation from one cell to another is estimated, which is more significant and effective for the free virus elements [10]. A process of HIV-1 transmission from one cell to another is considered significant for understanding the HIV dynamics based on the lymphatic system to obtain information about cellular infection and viral production. Culshaw et al. and Lou et al. [11,12] provided a framework using HIV-1 transmission in cell cultures from one cell to another cell.

The present study is related to the design of a novel computing process by applying the deep neural network (DNN) process in order to present the numerical solutions of the dynamical HIV-1 infection system in cancer patients (HIV-1-ISCP). The neural network structure of two hidden layers is presented by using the combination of a radial basis and sigmoid function with twenty and forty neurons in order to solve the nonlinear dynamics of the HIV-1-ISCP. This study is proposed for the first time, and is based on two different hidden layer structures (the radial basis and sigmoid function), different numbers of neurons (twenty and forty), and the optimization process of the scale conjugate gradient (SCG) to solve the nonlinear dynamics of the HIV-1-ISCP. The correctness of the solver is observed through the comparison with the literature results, which gives confidence to the authors to propose a reliable novel stochastic process based on the DNN using two different hidden layers and different numbers of neurons. The competency of the stochastic schemes was explored to solve various kinds of applications that arise in biological models, economic systems, robot models, and singular systems.

The rest of the paper is summarized as follows: Materials and Methods are presented in Section 2. The results of the HIV-1-ISCP are presented in Section 3. Section 4 presents the conclusions and future research directions.

2. Materials and Methods

The mathematical form of the HIV-1-ISCP model is divided into three classes named cancer population cells (T), healthy cells (H), and infected HIV (I) cells, as shown in Equation (1), while the used parameters in Model (1) are shown in

Table 1. Description of the parameters for the nonlinear dynamical HIV-1-ISCP model.

Parameter	Details
$T(x)$	Cancer cells
$I(x)$	HIV infected cells
$H(x)$	Healthy cells
${\gamma }_1$	Rate of unrestrained proliferation cells of cancer
${\gamma }_2$	Healthy cells based on inherent progression
${\eta }_1$	Rate of immune killing using the tumor cells
${\eta }_2$	Infection rate coefficient
$\mu$	Active carrying capacity of the system
$\sigma$	Rate of losing cells of immune
${\rho }_1$	Total immune system killing infection effects
$k_1,\hspace{0.17em}{k}_2,\hspace{0.17em}{k}_3$	Initial conditions
${\lambda }_H$	Infection force
${\mu }_R$	Natural death rate per capita
$v$	Time

[13,14]:

$$\left\{\begin{array}{l}{\displaystyle \frac{dT(x)}{dx}}=\left[-{\gamma }_1\left({\displaystyle \frac{I(x)+H(x)+T(x)}{\mu }}-1\right)-{\eta }_{1}H(x)\right]T(x),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}T(0)=k_1,\\ {\displaystyle \frac{dH(x)}{dx}}=\left[-{\gamma }_2\left({\displaystyle \frac{I(x)+H(x)+T(x)}{\mu }}-1\right)-\sigma {\eta }_{1}T(x)-{\eta }_{2}I(x)\right]H(x),\hspace{1em}H(0)=k_2,\\ {\displaystyle \frac{dI(x)}{dx}}={\eta }_{2}I(x)H(x)-{\rho }_{1}I(x),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}I(0)=k_3,\end{array}\right.$$

(1)

2.1. DNN Procedure

The process of the DNN is categorized into multiple layers (input, output, hidden). The mathematical DNN process using the activation radial basis and sigmoid functions is implemented by taking twenty and forty neurons to solve the nonlinear dynamics based on the HIV-1-ISCP. The first hidden layer is based on the radial basis, while the sigmoid function is used in the second hidden layer, which is presented mathematically as:

$$\left[\begin{array}{l}{y}_1\\ y_2\\ y_3\\ .\\ .\\ .\\ y_{20}\end{array}\right]={\delta }_1\left(\left[\begin{array}{l}{w}_{1,1}\\ w_{1,2}\\ w_{1,3}\\ .\\ .\\ .\\ w_{1,20}\end{array}\right][x]+\left[\begin{array}{l}{b}_{1,1}\\ b_{1,2}\\ b_{1,3}\\ \begin{array}{c}.\\ \begin{array}{l}.\\ .\end{array}\\ b_{1,20}\end{array}\end{array}\right]\right),$$

(2)

$$\left[\begin{array}{l}{z}_1\\ z_2\\ z_3\\ .\\ .\\ .\\ z_{40}\end{array}\right]={\delta }_2\left(\left[\begin{array}{ccccccc}{\psi }_{1,1}& {\psi }_{2,1}& {\psi }_{3,1}& {\psi }_{4,1}& .& .& {\psi }_{20,1}\\ {\psi }_{1,2}& {\psi }_{2,2}& {\psi }_{3,2}& {\psi }_{4,1}& .& .& {\psi }_{20,2}\\ {\psi }_{1,3}& {\psi }_{2,3}& {\psi }_{3,3}& {\psi }_{4,1}& .& .& {\psi }_{20,3}\\ .& .& .& .& .& .& .\\ .& .& .& .& .& .& .\\ .& .& .& .& .& .& .\\ {\psi }_{1,40}& {\psi }_{2,40}& {\psi }_{3,40}& {\psi }_{4,40}& .& .& {\psi }_{20,40}\end{array}\right]\left[\begin{array}{l}{y}_1\\ y_2\\ y_3\\ \begin{array}{c}.\\ \begin{array}{l}.\\ .\end{array}\\ y_{40}\end{array}\end{array}\right]+\left[\begin{array}{l}{b}_{2,1}\\ b_{2,2}\\ b_{2,3}\\ \begin{array}{c}.\\ .\\ \begin{array}{c}.\\ b_{2,40}\end{array}\end{array}\end{array}\right]\right),$$

(3)

$$\left[\begin{array}{c}T(x)\\ H(x)\\ I(x)\end{array}\right]=\left(\left[\begin{array}{ccccccc}{\omega }_{1,1}& {\omega }_{2,1}& {\omega }_{3,1}& .& .& .& {\omega }_{40,1}\\ {\omega }_{1,2}& {\omega }_{2,2}& {\omega }_{3,2}& .& .& .& {\omega }_{40,2}\\ {\omega }_{1,3}& {\omega }_{2,3}& {\omega }_{3,3}& .& .& .& {\omega }_{40,3}\end{array}\right]\left[\begin{array}{l}{z}_1\\ z_2\\ z_3\\ .\\ .\\ .\\ z_{40}\end{array}\right]+\left[\begin{array}{c}{b}_{3,1}\\ b_{3,2}\\ b_{3,3}\end{array}\right]\right),$$

(4)

where ${\delta }_1$ and ${\delta }_2$ denote the activation functions using the radial basis and sigmoid neural networks; b represents the bias; w, $\psi$, and ω are the weight vectors for the 1, 2, and output layers; y and z signify the 1st and 2nd layers; and T(x), H(x), and I(x) are the outputs.

2.2. Radial Basis and Sigmoid Activation Functions

The process of the radial basis neural network in the first hidden layer is applied as an activation function, while the sigmoid function is presented in the second hidden layer. The first hidden layer based on the radial basis neural network is generally implemented as an activation function. This function defines the nonlinearity of the network, which allows the recognition of complicated designs. The radial basis with a fixed shape parameter 1 is one of the best choices, which shortens the model and decreases the overfitting risk. The radial basis presents a universal approximation, which shows the calculation of a continuous function with random correctness. This function is considered continuous, computationally able to be calculated, and smooth, which is helpful for modeling the complicated association. The sigmoid function based on the log-sigmoid is applied to the inputs 0 and 1. This function is differentiable, which permits the training process using the optimization of the gradient performances. The neural network performances are used in the general calculations to approximate any sort of continuous function based on the random accuracy. The radial basis and sigmoid functions are mathematically shown as follows:

$${\delta }_1=\exp (-m^2) \mathrm{and} {\delta }_2={\displaystyle \frac{1}{1+\exp (-m)}}, \mathrm{where} m={\displaystyle \sum _{j=1}^s\left(w_j{z}_j\right)}+b,$$

(5)

where s represents the number of neurons. Figure 1 shows the DNN process, which starts from the input single layer. This is crossed to the radial basis (first hidden layer) by taking 20 neurons, then it is passed to the sigmoid function (second hidden layer) by taking 40 neurons, and then output layers are generated.

In order to solve the mathematical HIV-1-ISCP model, Figure 2 illustrates a novel DNN procedure that combines the radial basis and log-sigmoid activation functions with twenty and forty neurons in the respective hidden layers. The number of epochs is selected as 1200 for cases 1 to 3, while the optimization performance is provided by applying the structure of the SCG, which is used in the neural network training. The SCG is an iterative procedure applied to lessen the cost function by regulating the network weights. It presents an improved version of the conjugate gradient traditional solver, which has a scaling factor to control the step size throughout the entire optimization procedure. When compared to alternative techniques to store information in the Hessian matrix, it is more effective in terms of memory usage. By using the suggested number of cycles and achieving a small change in the cost function numbers, the convergence requirements for this technique are verified. The last phase of Figure 2 indicates that the radial basis and sigmoid functions are implemented in the hidden layers 1 and 2, respectively. In recent decades, the SCG process has been applied in many submissions, like image restoration and simulation of accelerating charged polymers.

Table 2. Parameter settings of the DNN process for solving the nonlinear HIV-1-ISCP model.

Index	Settings
Dataset	Runge–Kutta
Hidden layer 1	Radial basis
Hidden layer 2	Sigmoid function
Neurons in the first hidden layer	20
Neurons in the second hidden layer	40
Hidden layers	2
Minimum gradient	10−10
Samples assortment	Random
Maximum verification failures	Inf
Performance goal	0
Training time	Inf
Training data	0.72
Testing data	0.14
Authentication data	0.14
Maximum epochs for training	1200
Training	SCG
Stoppage standards	Default

shows the parameters based on the DNN using two hidden layers for solving the nonlinear HIV-1-ISCP model.

In the DNN, including the process of two hidden layers with different activation radial basis and log-sigmoid functions along with different neurons can provide several benefits. Some of them are as follows:

Increased complexity: Both hidden layers can acquire more multifaceted associations in the inputs and outputs.

Improved generalization: The combination of both activation radial basis and log-sigmoid functions provides the support of the network in order to improve the hidden data.

Enhanced feature extraction: The activation radial basis and log-sigmoid functions have been used in this research, which shows that the radial basis is helpful for feature extraction, whereas the log-sigmoid is appropriate for probabilistic outcomes.

Apprehending nonlinear relations: The mixture of both activation functions can efficiently capture the nonlinear relation in the inputs and outputs.

Reduced overfitting: Different neurons in both hidden layers are used to avoid the problem of overfitting through decreasing the system’s capacity to learn the training data.

The software Mathematica v13.3 was used for the dataset of the HIV-1-ISCP model, while the training of the dataset was performed using the MATLAB software (R2019a).

3. Results and Discussion

In this section, the numerical solutions based on three different cases of the nonlinear HIV-1-ISCP model are presented. The validity of the designed DNN based on the structure of two different hidden layers is also discussed using the comparison with the literature results.

Case 1: The updated form of System (1) by taking the suitable values in Model (1) as follows [13]:

$$\left\{\begin{array}{l}{\displaystyle \frac{dT(x)}{dx}}=\left[-0.1795\left({\displaystyle \frac{I(x)+H(x)+T(x)}{1500}}-1\right)-0.001H(x)\right]T(x),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}T(0)=0.1,\\ {\displaystyle \frac{dH(x)}{dx}}=\left[-0.002\left({\displaystyle \frac{I(x)+H(x)+T(x)}{1500}}-1\right)-0.00001T(x)-0.002I(x)\right]H(x),\hspace{1em}H(0)=0.2,\\ {\displaystyle \frac{dI(x)}{dx}}=0.002I(x)H(x)-0.03I(x),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}I(0)=0.3.\end{array}\right.$$

(6)

Table 3. The comparison of the results for the first case of the HIV-1-ISCP [13].

$\mathit{x}$	$\mathit{T}(\mathit{x})$	$\widehat{\mathit{T}}(\mathit{x})$	$\mathit{H}(\mathit{x})$	$\widehat{\mathit{H}}(\mathit{x})$	$\mathit{I}(\mathit{x})$	$\widehat{\mathit{I}}(\mathit{x})$
0	0.10000016943	0.10868775969	0.20000008321	0.19777037341	0.29999785078	0.29666379600
0.1	0.10181070891	0.10054160977	0.20059785133	0.19910492158	0.29113385624	0.30027524875
0.2	0.10365435168	0.10475029814	0.20119756273	0.19752226400	0.28253192998	0.29733258735
0.3	0.10553010512	0.10906943671	0.20179884254	0.19730820783	0.27418555979	0.29725420585
0.4	0.10743964687	0.10964018943	0.20240186724	0.19709838780	0.26609013390	0.29603841816
0.5	0.10938490222	0.11115338137	0.20300721939	0.19706652181	0.25822556442	0.29569799059
0.6	0.11136543800	0.111435116972	0.20361440339	0.19605622218	0.25059632840	0.29477258299
0.7	0.11338115446	0.113757188816	0.20422330888	0.19604788037	0.24319377896	0.29440618652
0.8	0.11543283457	0.116365231945	0.20483395732	0.19585288523	0.23601153161	0.29264639065
0.9	0.11752328540	0.117294283045	0.20544696957	0.19487912232	0.22903667420	0.29127034342
1	0.11965051111	0.112513456419	0.20606160205	0.19659756280	0.22227046540	0.29487454521

,

Table 4. The comparison of the results for the second case of the HIV-1-ISCP [13].

$\mathit{x}$	$\mathit{T}(\mathit{x})$	$\widehat{\mathit{T}}(\mathit{x})$	$\mathit{H}(\mathit{x})$	$\widehat{\mathit{H}}(\mathit{x})$	$\mathit{I}(\mathit{x})$	$\widehat{\mathit{I}}(\mathit{x})$
0	0.2000017596	0.20672928236	0.3000004514	0.296868317750	0.39999350745	0.396895963115
0.1	0.2036203843	0.20274680488	0.3008950644	0.300453914746	0.38818162865	0.398755989912
0.2	0.2073055561	0.20889790587	0.3017927952	0.295910556060	0.37671481077	0.396283569317
0.3	0.2110571379	0.21285846167	0.3026933066	0.296958616589	0.36558757916	0.398265169235
0.4	0.2148764204	0.21182550649	0.3035966306	0.296177205247	0.35478949532	0.392529695129
0.5	0.2187649056	0.21786449577	0.3045028295	0.294783608812	0.34431028450	0.393938035846
0.6	0.2227239001	0.22232468833	0.3054119255	0.293705720489	0.33414021356	0.393138719155
0.7	0.2267548696	0.22959061858	0.3063239578	0.292720504741	0.32427001165	0.393024504442
0.8	0.2308587164	0.22967777474	0.3072388593	0.293132906098	0.31469109403	0.393240135109
0.9	0.2350371310	0.237549332324	0.30815669210	0.293286405664	0.30539499855	0.387324826792
1	0.2392908428	0.229888828815	0.30907735224	0.292377815516	0.29637375030	0.390382212614

and

Table 5. The comparison of the results for the third case of the HIV-1-ISCP [13].

$\mathit{x}$	$\mathit{T}(\mathit{x})$	$\widehat{\mathit{T}}(\mathit{x})$	$\mathit{H}(\mathit{x})$	$\widehat{\mathit{H}}(\mathit{x})$	$\mathit{I}(\mathit{x})$	$\widehat{\mathit{I}}(\mathit{x})$
0	0.3000007094	0.304066489582	0.4000001497	0.398591491498	0.4999983571	0.496011729745
0.1	0.3054288289	0.301837431541	0.4011911130	0.396402639620	0.4852308158	0.489933006182
0.2	0.3109548473	0.320782076340	0.4023858733	0.394418744477	0.4708992905	0.493583836983
0.3	0.3165810809	0.320448887260	0.4035844771	0.396575641205	0.4569926818	0.492421288975
0.4	0.3223083941	0.323445853659	0.4047868367	0.396774634204	0.4434960538	0.492704489205
0.5	0.3281401774	0.331885125197	0.4059931721	0.395747127372	0.4303983361	0.493284746323
0.6	0.3340769185	0.328464097378	0.4072032912	0.395481359892	0.4176872355	0.492734841347
0.7	0.3401217296	0.335523769646	0.4084173831	0.395444373642	0.4053510683	0.494042069547
0.8	0.3462761785	0.346878349148	0.4096353977	0.392797123047	0.3933786779	0.492964820021
0.9	0.3525416759	0.366959035239	0.4108572420	0.396150276041	0.3817601399	0.490824781744
1	0.3589201975	0.343832788774	0.4120829043	0.395519994541	0.3704852824	0.492857310554

shows the comparison of the results for each case of the nonlinear HIV-1-ISCP using the DNN process.

Case 2: The updated form of System (1) by taking the suitable values in Model (1) as follows [13]:

$$\left\{\begin{array}{l}{\displaystyle \frac{dT(x)}{dx}}=\left[-0.1795\left({\displaystyle \frac{I(x)+H(x)+T(x)}{1500}}-1\right)-0.001H(x)\right]T(x),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}T(0)=0.2,\\ {\displaystyle \frac{dH(x)}{dx}}=\left[-0.002\left({\displaystyle \frac{I(x)+H(x)+T(x)}{1500}}-1\right)-0.00001T(x)-0.002I(x)\right]H(x),\hspace{1em}H(0)=0.3,\\ {\displaystyle \frac{dI(x)}{dx}}=0.002I(x)H(x)-0.03I(x),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}I(0)=0.4.\end{array}\right.$$

(7)

Case 3: The updated form of System (1) by taking the suitable values in Model (1) as follows [13]:

$$\left\{\begin{array}{l}{\displaystyle \frac{dT(x)}{dx}}=\left[-0.1795\left({\displaystyle \frac{I(x)+H(x)+T(x)}{1500}}-1\right)-0.001H(x)\right]T(x),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}T(0)=0.3,\\ {\displaystyle \frac{dH(x)}{dx}}=\left[-0.002\left({\displaystyle \frac{I(x)+H(x)+T(x)}{1500}}-1\right)-0.00001T(x)-0.002I(x)\right]H(x),\hspace{1em}H(0)=0.4,\\ {\displaystyle \frac{dI(x)}{dx}}=0.002I(x)H(x)-0.03I(x),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}I(0)=0.5.\end{array}\right.$$

(8)

The reference solutions are calculated by using the single layer structure, which are provided in reference [13]. The numerical solutions of the first, second, and third cases of the HIV-1-ISCP are presented in the above Table 3, Table 4 and Table 5. In these tables, T(x), H(x), and I(x) are the reference solutions, while $\widehat{T}(x)$, $\widehat{H}(x)$, and $\widehat{I}(x)$ are the proposed solutions that have been calculated by using the novel stochastic scheme. The overlapping of the proposed and literature results is performed, which gives confidence to the authors that the designed DNN based on two hidden layers is consistent, reliable, and trustworthy, and can be implemented to solve the nonlinear HIV-1-ISCP model. The novel proposed DNN procedure is presented by applying the structure of transfer functions based on the radial basis and sigmoid, along with the SCG optimization, for deriving the solutions of the nonlinear HIV-1-ISCP model. The optimal training performances and the gradient measures are illustrated in Figure 3 by using the proposed stochastic procedure for solving the nonlinear HIV-1-ISCP model. The optimal form of the training based on the MSE is used to involve the large-scale frameworks of deep learning, complex optimization, and immense datasets. By keeping the structure of double hidden layers, the number of epochs was selected as 1200, which shows the training process in order to examine the better iteration performance based on the learning schemes over the entire training statistics. Figure 3 indicates the MSE performances, which are observed as 3.5965 × 10⁻⁶, 1.241 × 10⁻⁵, and 2.8844 × 10⁻⁵ at epochs 78, 32, and 23. The second phase of Figure 3 shows the representations based on the gradient, Mu, and validation checks. The gradient represents the function based on the optimization process, which is reported as 3.544 × 10⁻⁶, 2.3106 × 10⁻⁵, and 2.2911 × 10⁻⁵ for cases 1, 2, and 3, respectively. These representations signify the steepest propensity by refining a function, which is normally executed in the process of optimization in order to achieve the minimum and maximum function values. The value of the factor Mu indicates the step size applied to the system’s parameter throughout the learning phase. Authentication checks give a basis by which the characteristics of the network are justified. In order to perform the solutions of the nonlinear HIV-1-ISCP model, Figure 4 details the fitness (Fit) outcomes utilizing the test, target, and trained data error and outputs. Typically, the Fit function offers the best approach to measure specific results according to the goals that are expected. Figure 5 presents the values of the error histogram (EH) in order to present the nonlinear HIV-1-ISCP model by applying the DNN, radial basis, and sigmoid transfer functions along with the process of SCG. The EH is implemented to determine the accuracy of the system in order to evaluate the intended outputs. The EH performances for cases 1, 2, and 3 are provided as 4 × 10⁻⁴, 6.72 × 10⁻⁴, and 2.48 × 10⁻⁴. Regression is represented in Figure 6 and is calculated around 1 for every variant, which exhibits the perfect form of the modeling. Regression data are used to determine the best-fitting line, which represents the connection between each of the variables. MSE results are displayed in

Table 6. Different performances using the DNN for the nonlinear HIV-1-ISCP model.

Case	MSE			Gradient	Performance	Epochs	Time
	Training	Authorization	Testing
I	2.9456 × 10−6	3.5965 × 10−6	5.5755 × 10−6	3.54 × 10−6	2.72 × 10−6	78	01
2	9.8158 × 10−6	1.2410 × 10−5	1.2984 × 10−5	2.31 × 10−5	6.14 × 10−6	32	01
3	2.2233 × 10−5	2.8844 × 10−5	4.5467 × 10−5	2.29 × 10−5	1.75 × 10−5	23	04

together with complexity levels and cycle details.

Figure 7 presents the comparative performances for the categories T(x), H(x), and I(x) based on the nonlinear HIV-1-ISCP model. This accurate overlapping of each class of the nonlinear HIV-1-ISCP model designates the accuracy of the designed DNN process using the process of two layers.

Table 7. AE values for solving the HIV-1-ISCP model.

x	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
T(x)	9 × 10−3	1 × 10−3	1 × 10−3	4 × 10−3	2 × 10−3	2 × 10−3	8 × 10−5	4 × 10−4	9 × 10−4	2 × 10−4	7 × 10−3
	7 × 10−3	9 × 10−4	2 × 10−3	2 × 10−3	3 × 10−3	9 × 10−4	4 × 10−4	3 × 10−3	1 × 10−3	3 × 10−3	9 × 10−3
	4 × 10−3	4 × 10−3	1 × 10−2	4 × 10−3	1 × 10−3	4 × 10−3	6 × 10−3	5 × 10−3	7 × 10−4	1 × 10−2	1 × 10−2
H(x)	2 × 10−3	3 × 10−4	1 × 10−3	9 × 10−4	5 × 10−4	7 × 10−5	3 × 10−4	2 × 10−4	6 × 10−4	2 × 10−4	3 × 10−3
	3 × 10−3	1 × 10−3	2 × 10−3	3 × 10−4	2 × 10−4	7 × 10−4	9 × 10−4	1 × 10−3	3 × 10−4	1 × 10−3	1 × 10−3
	1 × 10−3	2 × 10−3	3 × 10−3	2 × 10−4	2 × 10−3	2 × 10−3	3 × 10−3	4 × 10−3	2 × 10−3	7 × 10−3	8 × 10−3
I(x)	3 × 10−3	1 × 10−3	9 × 10−4	7 × 10−5	4 × 10−4	1 × 10−4	1 × 10−4	6 × 10−4	3 × 10−4	8 × 10−4	4 × 10−3
	3 × 10−3	5 × 10−5	1 × 10−3	2 × 10−3	3 × 10−3	1 × 10−4	2 × 10−4	1 × 10−3	3 × 10−3	2 × 10−3	2 × 10−3
	4 × 10−3	9 × 10−3	3 × 10−3	3 × 10−3	1 × 10−3	7 × 10−4	2 × 10−3	4 × 10−3	5 × 10−3	4 × 10−3	8 × 10−3

shows the values of AE for the classes T(x), H(x), and I(x) of the nonlinear HIV-1-ISCP model. The input values are selected as 0 and 1 with a 0.01 step size. The negligible performances based on AE for the classes T(x), H(x), and I(x) of the HIV-1-ISCP model indicate the correctness of the proposed scheme.

4. Conclusions

A novel computational design based on the deep neural network process to solve the dynamical HIV-1 infection system in cancer patients is presented in this paper. The neural network structure with two hidden layers using the combination of a radial basis and sigmoid function with twenty and forty neurons is presented for the solutions of the nonlinear HIV-1-ISCP. The mathematical model was categorized into three nonlinear dynamics named cancer population cells (T), healthy cells (H), and infected HIV (I) cells. Some of the concluding remarks of this study are presented as follows:

• The numerical representations of the dynamical HIV-1 infection system in cancer patients were proposed successfully by applying the proposed computational framework.
• A novel deep layer design was provided using the radial basis and sigmoid activation functions in the first and second hidden layers.
• The neural network is updated by taking twenty neurons in the first hidden layer and forty neurons in the second layer to solve the dynamical HIV-1 infection system in cancer patients.
• An optimization based on the competent scale conjugate gradient scheme was performed to present the solutions of the dynamical HIV-1 infection system in cancer patients.
• The perfection of the proposed numerical scheme was observed through the overlapping with the reference results, the Runge–Kutta scheme, and the negligible values of the absolute error, which were found to range from 10⁻³ to 10⁻⁴.
• The database solutions are used to lessen the MSE by taking 72%of the data for training, while 14%of the data was chosen for both authentication and testing.
• To credibility of the novel procedure is authenticated through the graphical plots based on different performances.

In future, the proposed double layer structure could be used to solve numerous nonlinear, fluid dynamics, food chain, epidemic system, and human balancing models [15,16].