# Soft Computing Paradigms to Find the Numerical Solutions of a Nonlinear Influenza Disease Model
## 3 Simulations and Results
The routine for the fitness function as given in Equations (4)-(9) is developed in the Matlab software package while the optimization is conducted as per procedure of pseudocode in Table 1 using the 'GA' and 'fmincon' routines of the optimization toolbox. The proposed solutions of the nonlinear influenza disease model are specified in the best weight vector form shown in the Equations (12)-(15) for each nonlinear influenza disease model. The graphical illustrations of these weight vectors are shown in Figure 2.  
\[\hat{S}(u)=\frac{0.6622}{1+e^{-(-10.636u-1.1152)}}-\frac{1.3369}{1+e^{-(-3.6682 u-7.9483)}}-\ldots+\frac{0.3752}{1+e^{-(-0.096u-4.1269)}}, \tag{12}\] \[\hat{I}(u)=\frac{-14.9676}{1+e^{-(0.0720u-16.7745)}}-\frac{4.1637} {1+e^{-(-15.649u-16.512)}}-\ldots-\frac{15.4637}{1+e^{-(-19.7976u-1.1137)}},\] (13) \[\hat{R}(u)=\frac{3.2898}{1+e^{-(-1.6034u-7.6633)}}-\frac{1.6034} {1+e^{-(-8.138u-11.8697)}}-\ldots-\frac{6.3094}{1+e^{-(-1.468u-10.9883)}},\] (14) \[C(u)=\frac{-0.8800}{1+e^{-(0.4167u-11.3515)}}-\frac{0.8127}{1+e^{ -(-0.8127u-11.351)}}-\ldots-\frac{5.8006}{1+e^{-(-16.500u-4.6960)}}. \tag{15}\]  
The proposed outputs are obtained using the above systems ((12)-(15)) as provided in Appendix C for the 0-1 range with a 0.1 step size to present the solutions for each influenza disease category. The plots of the best weight vectors are derived in the Figure 2a-d. The mean and best results, i.e., the outcomes of the proposed methodology based on the average and best fitted weights of ANNs, compared with the reference solutions, i.e., the outcomes of the systems calculated with the Adams numerical technique, to get the nonlinear influenza disease models are also plotted in Figure 2e-h. The solutions obtained by the ANN-GA-ASM coincided with the [PERSON] solutions to present the solutions for each category of influenza disease. These results are consistent with the perfection and precision of the proposed ANN-GA-ASM. Figure 3 shows the AE values to provide solutions for each influenza category. One may observe that the AE best values for the susceptible (\(S\)), infectious (\(I\)), recovered (\(R\)) and cross-immune (\(C\)) groups based on the influenza disease model lie in the ranges of \(10^{-2}\)-\(10^{-4}\), \(10^{-3}\)-\(10^{-6}\), \(10^{-2}\)-\(10^{-4}\) and \(10^{-2}\)-\(10^{-4}\), respectively. Meanwhile, the AE mean values for the above categories of susceptible, infectious, recovered and cross-immune lie around \(10^{-1}\)-\(10^{-3}\), \(10^{-2}\)-\(10^{-4}\), \(10^{-1}\)-\(10^{-3}\) and \(10^{-1}\)-\(10^{-2}\), respectively. Figure 4 represents the performances based on the EVAF, MAD and TIC operators to solve each category of the influenza disease nonlinear model. It is designated that the best EVAF, MAD and TIC presentations of the susceptible category lie around \(10^{-2}\)-\(10^{-4}\), \(10^{-2}\)-\(10^{-3}\) and \(10^{-6}\)-\(10^{-8}\) for EVAF, MAD and TIC, respectively. The infectious best category lies in the range of \(10^{-2}\)-\(10^{-4}\), \(10^{-3}\)-\(10^{-4}\) and \(10^{-7}\)-\(10^{-8}\) for EVAF, MAD and TIC, respectively. The recovered best category lies in the range of \(10^{-3}\)-\(10^{-4}\), \(10^{-2}\)-\(10^{-3}\) and \(10^{-6}\)-\(10^{-8}\) for EVAF, MAD and TIC, respectively. The cross-immune best category lies in the range of \(10^{-2}\)-\(10^{-4}\), \(10^{-2}\)-\(10^{-3}\) and \(10^{-7}\)-\(10^{-8}\) for EVAF, MAD and TIC, respectively. These indications make it possible to affirm that the designed ANN-GA-ASM is precise and accurate. Figure 2: Best set of weight vectors and comparison of proposed and reference solutions obtained by using Adams and optimization algorithm to solve the influenza nonlinear disease model. Figure 3: AE values based on best and mean solutions for each category of the influenza disease nonlinear model.