# Soft Computing Paradigms to Find the Numerical Solutions of a Nonlinear Influenza Disease Model
## 3 Simulations and Results  
The relative presentations of the results obtained using the Adams method are used to certify or analyze the accuracy of the ANN-GA-ASM. Moreover, the statistical performance is testified for the precision, reliability and accuracy of the proposed structure. The updated form of the nonlinear model of influenza illness using the appropriate parameters as defined in reported study [13]:  
\[\left\{ \begin{array}{ll}S^{\prime}(u)=0.02-(50I(u)+0.02)S(u)+0.5C(u),&S(0)=0.8,\\ I^{\prime}(u)=50(S(t)+0.05C(u))I(u)-73.02I(u),&I(0)=0.1,\\ R^{\prime}(u)=73I(u)+47.5C(u)I(u)-1.02R(u),&R(0)=0.04,\\ C^{\prime}(u)=R(u)-(0.52+50I(u))C(u),&C(0)=0.06,\end{array} \right. \tag{10}\]  
The fitness function for the influenza disease system (10) becomes:  
\[ \begin{array}{l}E=\frac{1}{N}\sum\limits_{i=1}^{N}\left(\begin{array}{c} \big{[}\hat{S}_{i}^{\prime}-0.02+50\hat{S}_{i}\hat{I}_{i}+0.02\hat{S}_{i}-0.5 \hat{C}_{i}\big{]}^{2}\\ +\big{[}\hat{I}_{i}^{\prime}-50\hat{S}_{i}\hat{I}_{i}-2.5\hat{C}_{i}\hat{I}_{ i}+73.02I_{i}\big{]}^{2}\\ +\big{[}\hat{R}_{i}^{\prime}-73\hat{I}_{i}-47.5C_{i}\hat{I}_{i}+1.02\hat{R}_{ i}\big{]}^{2}\\ +\big{[}\hat{C}_{i}^{\prime}-\hat{R}_{i}+0.52\hat{C}_{i}-50\hat{I}_{i}\hat{C}_{ i}\big{]}^{2}\end{array}\right)\\ +\frac{1}{4}\Big{[}\big{(}\hat{S}_{0}-0.8\big{)}^{2}+\big{(}\hat{I}_{0}-0.1 \big{)}^{2}+\big{(}\hat{R}_{0}-0.04\big{)}^{2}+\big{(}\hat{C}_{0}-0.06\big{)}^{ 2}\Big{]}.\end{array} \right.\end{array} \tag{11}\] \begin{table}
\begin{tabular}{l} \hline \hline
**GA process starts** \\ **Inputs:** Select the chromosomes with the same numbers as decision variable of \\ ANN models for solving influenza disease system: \\ **\(W=[\mathbf{a},\mathbf{w},\mathbf{b}]\)** \\ **Population:** The chromosomes set is given as: \\ **\(W=[\mathbf{W}_{S},\mathbf{W}_{I},W_{R},W_{C}]\)**, for \(\mathbf{W}_{S}=[\mathbf{p}_{S},\mathbf{\omega}_{S},\mathbf{q}_{S}]\), \(\mathbf{W}_{I}=[\mathbf{p}_{I},\mathbf{\omega}_{I},\mathbf{q}_{I}]\), \\ **\(W_{R}=[\mathbf{p}_{R},\mathbf{\omega}_{R},\mathbf{q}_{R}]\)** and \(\mathbf{W}_{R}=[\mathbf{p}_{R},\mathbf{\omega}_{R},\mathbf{q}_{R}]\)** \\ **Output:** Global vectors are \(\mathbf{W}_{\text{B-GA}}\) \\ **Initialization:** For variables selection, modify the weight vector. \\ **Fitness calculation:** Determine the fitness \(E\) of each member of the population (P) using Equations (4)-(9) \\ \(\bullet\)**Stopping standards:** Terminate if \([E=10^{-21}]\), \([\text{TolCon}=\text{TolFun}=10^{-21}]\). \\ [StallLimit = 140], \([\text{PopSize = 210}]\) and \([\text{Generations = 120}]\) are achieved. \\ Move to **storage** \\ **Ranking:** Rank precise weight vector in the designated population based on appropriateness on fitness E, i.e., better the fitness more be the rank of the individual. \\ **Storage:** Save \(\mathbf{W}_{\text{B-GA}}\), iterations, FIT, function counts and time. \\ **End of GA** \\ \hline
**ASM** \\ **Inputs:**\(W_{\text{B-GA}}\). \\ **Output:**\(W_{\text{GA-ASM}}\) is the best GA-ASM weight vectors. \\ **Initialize:**\(W_{\text{B-GA}}\), iterations, assignments and other values. \\ **Termination:** Stop if \([E=10^{-20}]\), \([\text{Iterations = 1200}]\), \([\text{TolX = TolCon = 10^{-20}}]\) \\ \([\text{TolFun}=10^{-21}]\) and \([\text{MaxFunEvals = 279,000}]\) meets. \\ **Evaluation of FIT:** Calculate \(W\) and \(E\) using networks 4–9. \\ **Adjusts:** Adjust ’fmincon’ for ASM, calculate \(E\) using 4–9 systems. \\ **Accumulate:** Transmute \(W_{\text{GA-ASM}}\), function counts, time iterations and FIT \\ **ASM End** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Pseudocode of optimization through ANN-GA-ASM to solve the influenza disease nonlinear system. Performance by optimization is demonstrated for the nonlinear influenza disease model using the proposed ANN-GA-ASM for 30 independent runs, i.e., sufficiently large multiple autonomous executions, having 30 numbers of hidden neurons in ANN models in set (3).