# Solution for the Mathematical Modeling and Future Prediction of the COVID-19 Pandemic Dynamics
## 4 Prediction of Future Pandemic Evolution
### Simulation Results  
In Figure 5, the black curve provides a visual representation of the evolution of the number of new cases \"experimentally determined\" in relation to time. The other five coloured curves represent responses of the proposed model, based on continuous transfer functions by applying at its inputs the same values as the \"experimental ones\" (for which the real number of cases was obtained). In this case, different variations of the number of poles and zeros for obtaining the best fit were chosen, to analyze an extended evaluation of the best solution. As available below, the estimated curves follow the black curve (the real data curve) closely. The curves represent the evolution of the case numbers in 120 days. The following interval between \([121,180]\) days represents the prediction offered through the model of the continuous transfer vector (which contains as elements the transfer functions \(\text{H}_{1}(\text{s})\), \(\text{H}_{2}(\text{s})\), \(\text{H}_{3}(\text{s})\), \(\text{H}_{4}(\text{s})\) and \(\text{H}_{5}(\text{s})\)). The model performance is highlighted using the cumulated relative error in percentages defined by:  
\[\text{crep}=100\cdot\frac{\sum_{i=1}^{\text{N}}\left|y_{\text{sim,k}}-y_{\text{ rk}}\right|}{\sum_{i=1}^{\text{N}}\left|y_{\text{r,k}}\right|}\left[\%\right] \tag{34}\]  
where: crep--the cumulative relative error (%), N--the number of the pairs of the considered data samples, \(y_{\text{r,k}}\)--the real value (the \"experimental\" value) of the considered output signal associated with the sample k and \(y_{\text{sim,k}}\)--the simulated value of the output signal associated with the sample k, obtained through the simulation of one of the proposed mathematical models. Consequently, the fit between the two responses is given by:  
\[\text{fit}=100\%-\text{crep} \tag{35}\]  
where the crep value is computed using the data corresponding to the considered responses. The degrees of the polynomials of the numerators and of the denominators of the five transfer functions are centralized in Table 3 (these polynomials depend on the complex variable \"s\" associated with the Laplace transform). In Figure 6, the best solution using Estimation_3 from Table 3 is depicted with a fit of 97.23%. The difference in obtaining a 100% fit is given by the inconsistency of data gathering due to the not optimal feedback provided through the national official platforms on weekends affecting the validation model as seen in the interval between day 135 and day  
Figure 5: The representation of transfer functions estimation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**The Curve** & **No. of Zeros for Each** & **No. of Poles for Each** \\  & **Transfer Function** & **Transfer Function** \\ \hline Estimation\_1 & [1 1 1 1 1] & [4 4 3 3 3] \\ \hline Estimation\_2 & [1 1 1 2 2] & [8 8 7 7 7] \\ \hline Estimation\_3 & [1 2 1 1 2] & [4 4 3 3 3] \\ \hline Estimation\_4 & [1 1 2 1 1] & [6 6 6 6 6] \\ \hline Estimation\_5 & [2 2 2 2 2] & [6 5 5 4 4] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The centralizer of the parameters associated with the continuous transfer functions. 160. During this time interval, the graphic shows a few oscillations, but the estimated model does not, so this could be a natural outcome due to the inconsistency mentioned before. The five coloured responses presented in Figure 7 are the responses of the proposed ARX models (with different parameters) in comparison with the \"experimental response\". The simulations of the five proposed ARX models are made in the same conditions as the simulation presented in Figure 5.  
In Figure 7, the errors are visible due to consistent distances (visible on several time domains) between the real curve and the estimated one. The ARX model estimation shows a slightly less performance than in the case of the continuous transfer function estimation which was detailed above.