# Comparison of the Evolution of the COVID-19 Disease between Romania and Italy
## 2 Time-Dependent SIR Model
Figure 13 shows the relation between fraction \(\frac{v_{2}^{*}}{v_{1}^{*}}\) and \(R_{2}\) in range of \(R_{2}\in[1.5,2.1]\). Figure 14 shows that \(T_{2}\) tends to 5 months, if \(R_{2}\) exceeds 3 (i.e., the 2 nd peak could appear in August). But, if \(R_{2}\in[1.6,2.0]\) the 2 nd peak might appear between September and November. Varying R2 and N2 in model (4) we generated the results for extent \(v_{2}^{*}\) and time \(T_{2}\). For fitting to linear regression line we used the Mathematica command LinearModelFit while for fitting to exponential dependence \(f(x)=a+be^{-cx}\) the command NonlinearModelFit of Mathematica 10.4 was used. \[N4=1.6;R0=4.28;R1=0.4;\frac{1}{\gamma}=13\ days.\]0 Italy: fitted exponential dependence \(T_{2}(R_{2})\approx 4.77+242.35e^{-2.81\cdot R_{2}}\). The fraction \(\frac{v_{2}^{*}}{v_{1}^{*}}\) depends linearly on \(R_{2}\). The (linear) correlation (Pearson) coefficient is found to be 0.999865. The functorial dependence is found to be \(\frac{v_{2}^{*}}{v_{1}^{*}}=-97.31036+89.0063\cdot R_{2}\). In Table 1 the estimates, standard errors, \(t\)-Statistics and the corresponding \(p\)-value for coefficients \(a\) and \(b\) in \(y=a+bx\) are presented. The \(p\)-value stands for the null-hypothesis that the coefficients \(a,b\) are equal to zero (\(y\leftrightarrow\frac{v_{2}^{*}}{v_{1}^{*}}\) and \(x\leftrightarrow R_{2}\)). The fitted line and the simulated data are shown in \[N4=1.6;R0=4.28;R1=0.4;\frac{1}{\gamma}=13\ days.\]1  
Next, we fitted \(T_{2}\) in months vs. \(R_{2}\) as follows \(T_{2}(R_{2})=a+be^{-c\cdot R_{2}}\). We found \(a=4.7714696\), \(b=242.350437\) and \(c=2.80811033\). The fitted line and the simulated data are shown in \[N4=1.6;R0=4.28;R1=0.4;\frac{1}{\gamma}=13\ days.\]2  
We fitted the extent fraction \(\frac{v_{2}^{*}}{v_{1}^{*}}\) in relation to \(\Delta t\) and found \(\frac{v_{2}^{*}}{v_{1}^{*}}=44.8798615-0.09911836\cdot\Delta t\). In Table 2 the estimates, standard errors, \(t\)-Statistics and the corresponding \(p\)-value for coefficients \(a\) and \(b\) in \(y=a+bx\) are presented. The \(p\)-value stands for the null-hypothesis that the coefficients \(a,b\) are equal to zero (\(y\leftrightarrow\frac{v_{2}^{*}}{v_{1}^{*}}\) and \(x\leftrightarrow R_{2}\)). The fitted line and the simulated data are shown in \[N4=1.6;R0=4.28;R1=0.4;\frac{1}{\gamma}=13\ days.\]3 The correlation coefficient was found to be \(-0.943522\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Estimate** & **Standard Error** & \(t\)**-Statistic** & \(p\)**-Value** \\ \hline
1 & \(-97.3136\) & \(4.16505\) & \(-23.3643\) & \(2.67331\times 10^{-6}\) \\ \hline x & \(89.0063\) & \(2.06599\) & \(43.0817\) & \(1.27156\times 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistics table for coefficients. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Estimate** & **Standard Error** & \(t\)**-Statistic** & \(p\)**-Value** \\ \hline
1 & \(44.8799\) & \(0.0871619\) & \(514.903\) & \(5.24398\times 10^{-13}\) \\ \hline x & \(-0.0991184\) & \(0.015565\) & \(-6.36801\) & \(0.00141214\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistics table for coefficients. Finally, we fitted the date of the second peak vs. \(\Delta t\) in terms of \(\tau\). The correlation was found to be \(T_{2}=3.6552199083+0.8801832505\cdot\Delta t\). The functorial dependence was found to be 0.999797 with the statistical table for the estimates, standard errors, \(t\)-Statistics and the corresponding \(p\)-value for coefficients \(a\) and \(b\) in \(y=a+bx\) as stated in Table 3. The \(p\)-value stands for the null-hypothesis that the coefficient s \(a\), \(b\) are equal to zero (\(y\leftrightarrow T_{2}\) and \(x\leftrightarrow\Delta t\)). The fitted line and the simulated data are shown in Figure 16.