# Fractional-Order SIR Epidemic Model for Transmission Prediction of COVID-19 Disease
## 4 Numerical Simulations
### Transmission Modeling for Italy and Spain  
In this section, we implement the fractional-order SIR model in order to predict COVID-19 transmission for Italy and Spain.  
In order to model the spread of COVID-19, we have model parameters, as follows \(\{\alpha_{1},\alpha_{2},\alpha_{3},\gamma,\beta,N\}\). Accounting that the order \(\alpha_{3}\) does not affect model output \(I\), for simplicity, we assume that \(\alpha_{3}=1\). Parameter \(N\) is arbitrary given for each country. Therefore, using square lost function, we have to estimate four parameters \(\{\alpha_{1},\alpha_{2},\gamma,\beta\}=\arg\min\sum_{k=0}^{K}(\hat{I}_{k}-l_{ k})^{2}\), where \(\hat{I}_{k}\) and \(l_{k}\) denotes modeled and actual number of infected individuals, respectively. In this paper, optimal model parameters are obtained the use of the Monte Carlo method based on the genetic algorithm.  
The starting point of simulation regarding the Italy epidemic is 21 February, when there were 19 active cases and 0 removed, so \(I_{0}=19\), \(R_{0}=0\), \(N=60\),461,828 and \(S_{0}=60\),461,726. Using the above-presented optimization method for \(K=150\), we obtain parameters, as follows \(\beta=3.3108\), \(\gamma=0.1333\), \(\alpha_{1}=0.4091\) and \(\alpha_{2}=0.9964\). Figure 4 presents the time plots of modeled and actual disease transmission.  
As we can see in Figure 4, the fractional-order SIR model is effective in modeling the COVID-19 transmission in Italy. The Root Mean Square Error, defined as \(RMSE_{n}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(\hat{I}_{k}-l_{k})^{2}}\) with \(n\) denoting prediction range, \(l_{k}\) and \(\hat{I}_{k}\) being the actual and modeled active cases, respectively, is as \(RMSE_{150}=3.283\times 10^{3}\). Note that, for the first 30 days, the modeled transmission is higher than the actual one. This can be a result of relatively low daily tests when compared to the rest of the period. In contrast, for \(k>120\), the actual transmission is significantly higher than the modeled one. It may be the result of gradually easing the restrictions (from May 16, \(k=85\)).  
The starting point of simulation regarding the Spain epidemic is 27 February, when there were 31 active cases and 0 removed, so \(I_{0}=31\), \(R_{0}=0\), \(N=46\),757,240 and \(S_{0}=46\),757,209. Using the optimization method for \(K=80\), we obtain parameters, as follows \(\beta=3.76\), \(\gamma=0.2708,\alpha_{1}=0.5374\) and \(\alpha_{2}=0.5983\). Figure 5 presents time plots of modeled and actual disease transmission.  
Figure 3: Simulation results for various \(\alpha_{3}\) and \(\alpha_{1}=\alpha_{2}=1\).  
Figure 4: Modeled vs. actual disease transmission in Italy.  
Figure 5: Modeled vs. actual disease transmission in Spain.  
The fractional-order SIR model is quite effective in modeling the COVID-19 transmission in Spain, as we can see in Figure 5. However, for \(k>85\), the actual transmission is significantly higher than the modeled one. Again, it may result from easing the restrictions (from 2 May, \(k=65\)). Furthermore, a resurgence is occurring in Spain, and the model is not capable of modeling this process. It is confirmed by RMSE, which, for \(n=80\), is equal to \(RMSE_{80}=7.506\times 10^{3}\), but for \(n=150\) is as \(RMSE_{150}=3.083\times 10^{4}\). In this case, using time-varying model parameters may improve the fractional-order SIR model performances. However, using more complex models may be necessary in this case.