A stochastic computational scheme for the computer epidemic virus with delay effects

: This work aims to provide the numerical performances of the computer epidemic virus model with the time delay effects using the stochastic Levenberg-Marquardt backpropagation neural networks (LMBP-NNs). The computer epidemic virus model with the time delay effects is categorized into four dynamics, the uninfected S ( x ) computers, the latently infected L ( x ) computers, the breaking-out B ( x ) computers, and the antivirus PC’s aptitude R ( x ). The LMBP-NNs approach has been used to numerically simulate three cases of the computer virus epidemic system with delay effects. The stochastic framework for the computer epidemic virus system with the time delay effects is provided using the selection of data with 11%, 13%, and 76% for testing, training, and verification together with 15 neurons. The proposed and data-based Adam technique

These investigations aim to provide the numerical performances of the computer epidemic virus model with the time delay effects using the stochastic Levenberg-Marquardt backpropagation neural networks (LMBP-NNs). The stochastic LMBP-NNs approach has never been used to illustrate the dynamics of a computer epidemic virus system with delay effects. On the other hand, stochastic solvers have been designated to solve various complex, nonlinear, and stiff dynamical systems. To mention some of the applications are food chain systems [26], the coronavirus dynamical model [27], HIV infection system [28], singular thermal explosion theory model [29], eye surgery system [30], differential model of the smoking model [31], and singular model [32]. The novel study features are signified as: • A nonlinear computer epidemic virus model with the time delay effects is presented numerically.
• Three cases of the computer virus epidemic system with the delay effects have been numerically stimulated using the LMBP-NNs scheme. • The exactness of the LMBP-NNs solver is performed based on the overlapping of the proposed and data-based reference Adam method. • The consistency of the LMBP-NNs solver is authenticated by using the absolute error (AE) performances of the computer virus epidemic system with the delay effects. • The state transitions (STs) measures, regression actions, correlation performances, error histograms (EHs), and mean square error (MSE) measures are provided using the LMBP-NNs solver for the computer epidemic virus system with the delay effects. The other paper parts are organized as follows: Section 2 shows the computer epidemic virus model. The designed network structure is shown in Section 3. Section 4 narrates the results simulations. Conclusions are discussed in the final Section.

Mathematical model
This section performs the mathematical structure of the computer epidemic virus system with the delay effects. The system is categorized into four dynamics, the uninfected S(x) computers, the latently infected L(x) computers, the breaking-out B(x) computers and the antivirus PC's aptitude R(x). The mathematical structure of the computer virus epidemic system with the delay effects along with the initial conditions (ICs) is given as [33]: The parameter based on the computer epidemic virus dynamical model with the time delay effects is tabulated in Table 1, while the flow illustrations are provided in Figure 1.

Methodology: LMBP-NNs scheme
In this section, the structure of the LMBP-NNs scheme is presented to the computer dynamical epidemic virus system with the delay effects by using the necessary performances of the procedure along with its implementation. Figure 2 shows the optimization performances of the multi-layer stochastic process based on the LMBP-NNs technique. The stochastic framework of the computer epidemic virus model with the time delay effects is provided using data selection with 11%, 13% and 76% for testing, training and verification. Fifteen numbers of neurons have been used in this study to solve the delay differential model.

AIMS Mathematics
Volume 8, Issue 1, 148-163.  The setting of the parameters based on the LMBP-NNs procedure is specified to the computer dynamical epidemic virus system with the delay effects provided in Table 2. The slight alteration and variation can conclude the poor performances, i.e., untimely convergence. Consequently, these options will be carefully included after substantial numerical trial and expertise. The LMB algorithms implementation process and necessary additional theoretical details are shown in [34,35].

Methodology: MWNN-GA-ASA scheme
This section presents the solutions to the dynamical form of the mathematical model. These types of mathematical dynamical models have been reported in various studies [36][37][38]. The numerical results of the computer epidemic virus of the time delay system using the LMBP-NNs scheme are shown in this section. The mathematical form of each case is shown as: The solutions to the computer epidemic virus system with the delay effects using the LMBP-NNs scheme are provided for three variations based on the ICs. The output, hidden, and input layer construction is presented in Figure 3.  Figure 4 based on the computer virus epidemic system with the delay effects. These gradient operator performances have been provided as 6.9708×10 -8 , 7.3422×10 -8 , and 9.9456×10 -8 . These graphical plot representations indicate the convergence of the LMBP-NNs scheme of the computer epidemic virus system. Figure 5 validates the fitting cure design to perform the numerical simulations of the computer virus epidemic system with the delay effects. The graphical curve plots compare the results for each case of the model. The error plots through the training, verification, and testing performances have been indicated in the computer virus epidemic model with the delay effects using the LMBP-NNs procedure. The EHs illustrations and the regression measures have been presented in Figure 5 using the computer virus epidemic system using the LMBP-NNs procedure. The EHs values have been presented as 1.43×10 -6 , 2.68×10 -7 and 2.08×10 -6 for each model variation using the LMBP-NNs procedure. The regression plot illustrations are reported in Figure 6 to indicate the correlation measures. It is observed that the correlation measures are reported as 1 for individual cases of the model. The testing, training, and authentication plots label the exactness of the LMBP-NNs procedure to indicate the numerical simulations of the model using the LMBP-NNs procedure. The convergence plots obtained through the MSE measures using the testing, validation, training, epochs, and complexity are provided in Table 3 for the computer epidemic virus model using the LMBP-NNs procedure.    The performance outcomes and absolute error (AE) values from three validation, testing, and training cases utilizing the LMBP-NNs approach are shown in Figures 7 and 8. These plots indicate the correctness of the LMBP-NNs scheme for the computer epidemic virus system with the time delay effects. Figure 7 shows the comparison values through the achieved and reference results for solving the computer virus epidemic system with the delay effects. The matching of the obtained, and reference results provides the correctness of LMBP-NNs scheme for the computer epidemic virus model. The AE measures for the LMBP-NNs stochastic approach using three different cases of the model are derived in Figure 8. The computer epidemic virus model with the time delay effects is categorized into four dynamics, the uninfected S(x) computers, the latently infected L(x) computers, the breaking-out B(x) computers, and the antivirus PC's aptitude R(x). The AE for the uninfected PC's S(x) is derived as 10 -5 to 10 -6 , 10 -5 to 10 -7 , and 10 -5 to 10 -8 for individual cases of the computer epidemic virus model. The AE performances for the latently infected L(x) computers are derived as 10 -5 to 10 -6 , 10 -6 to 10 -7 , and 10 -6 to 10 -8 for cases 1-3. The AE values for the breaking-out PC's B(x) are performed as 10 -5 to 10 -6 , 10 -6 to 10 -7 , and 10 -5 to 10 -7 . Similarly, the AE measures for the antivirus PC's ability R(x) are performed as 10 -5 to 10 -8 , 10 -6 to 10 -8 and 10 -5 to 10 -7 for individual cases of the computer epidemic virus model. These precise and accurate measures indicate the correctness of the LMBP-NNs scheme for the computer epidemic virus model with the time delay effects.

Conclusions
This study aims to solve the computer epidemic virus model with the time delay effects using the Levenberg-Marquardt backpropagation neural networks. The mathematical form of the model is categorized into four dynamics, the uninfected S(x) computers, the latently infected L(x) computers, the breaking-out B(x) computers, and the antivirus PC's aptitude R(x). Finally, a few concluding remarks of the current study are presented as follows: • A nonlinear computer epidemic virus model with the time delay effects has been numerically solved. • The computer epidemic virus model's delay factors complicate the dynamic system. To numerically formulate the computer epidemic virus model with delay effects, the stochastic approach based on Levenberg-Marquardt backpropagation neural networks is one of the appropriate options. • The stochastic framework for the computer virus epidemic model with the delay effects has been provided using data selection with 11%, 13%, and 76% for testing, training, and verification. • Fifteen hidden neurons have been used to solve the computer virus epidemic model with the delay effects. • Stochastic LMBP-NNs procedure's exactness has been performed by overlapping the proposed and data-based reference Adam method. • The AE values are provided in suitable measures, which are performed as 10 -4 to 10 -7 for each category of the computer virus epidemic system with the delay effects.