# Comparison of the Evolution of the COVID-19 Disease between Romania and Italy
## 2 Time-Dependent SIR Model
In the time dependent SIR model (4) the corresponding systems of ordinary differential equations are non-linear and therefore it is impossible to obtain the exact solution to a particular initial condition problem. Therefore we use the Mathematica 10.4. command NDSolve, which gives results in terms of interpolating function objects. These interpolating functions are representing the numerical solution to a particular initial condition problem as an approximate function. For the time integration the Runge-Kutta 4 th order method was used, while for the interpolation of the solution function the Hermite function was used. We used the following Mathematica code:  
\[ \begin{array}{ll}\text{In}[1]:=R0[t_{-}]:=&\text{Piecewise}[\{[R0,\ 0\leq t \leq N1\},\{-(R0-R1/N2(t-N1)\ +R0,N1<t\\ &\leq N1+N2\},\{R1,N1+N2<t\\ &\leq N1+N2+N3\},\{(R2-R1/N4(t-(N1+N2+N3+N4))+R2,N1+N2\\ &\qquad+N3<t\leq N1+N2+N3+N4\},\{R2,N1+N2+N3+N4<t\\ &\leq N1+N2+N3+15N4\})]]\\ \end{array} \]  
\[ \begin{array}{ll}\text{In}[2]:=\text{sol1}=&\text{NDSolve}[\{v^{\prime}[t]==v[ t]*(R0[t]*u[t]-1),u^{\prime}[t]=\\ &=-R0[t]\ u[t]\ v[t],w^{\prime}[t]==v[t],v[0]==1,u[0]==1-r1,w[0]\\ &==0\},\{v,u,w\},\{t,0,N1+N2+N3+15N4\}]\\ \end{array} \]  
Note that the only difference of \(R_{0}=R_{0}(t)\) in Figures 5 and 7 are the time intervals (the levels of higher \(R_{0H}=4.0\) and lower \(R_{0L}=0.5\) is the same). The time intervals in Figure 4 are as follows: for \(t\in[0,3.5]\) we have \(R_{0}=4.0\), for \(t\in[3.5,5.5]\) we have the descending line, for \(t\in[5.5,10]\) we have \(R_{1}=0.5\), for \(t\in[10,15]\) we have the ascending line and for \(t\in[15,25]\) we have again \(R_{2}=4.0\). The time intervals in Figure 6 are as follows: for \(t\in[0,4]\) we have \(R_{0}=4.0\), for \(t\in[4,6]\) we have the descending line, for \(t\in[6,10]\) we have \(R_{1}=0.5\), for \(t\in[10,15]\) we have the ascending line and for \(t\in[15,25]\) we have again \(R_{2}=4.0\). \(\bullet\) For Italy, the parameters of the model (4) are adjusted to real data are found to be  
\[r1=2.319\cdot 10^{-7};\ N1=1.183;\ N2=3.630;\ N3=0.726;\]  
\[N4=1.6;R0=4.28;R1=0.4;\frac{1}{\gamma}=13\ days.\]  
The parameter \(R2\) was varied in the range of [1.2,3]. Parameter value \(N2+N3=\Delta t\) was varied in the range of [4.3,6.3]. Note that in the dimensionless model (4) this corresponds to a range from almost two months to more than three months. For Italy, the time is in units \(\tau\times 13\) days, for Romania the time is in units \(\tau\times 14\) days. \(\bullet\) For Romania, the parameters of the model (4) adjusted to real data are found to be:  
\[r1=1.5\cdot 10^{-7};\ N1=1.42;\ N2=2.60;\ N3=1.33;\]  
\[N4=1.6;R0=4.80;R1=0.83;\frac{1}{\gamma}=14\ days.\]  
The parameter \(R2\) was varied in the range of [1.5,2.5]. Parameter \(N2+N3=\Delta t\) was varied in the range of [2,8]. Note that in the dimensionless model (4) this corresponds to a range from one to almost four months. Figure 7: The evolution of the infected fraction \(v=\frac{l}{N}\), as follows from the double transition of \(R_{0}=R_{0}(t)\) defined in Figure 6. In this case the 2 nd wave is much smaller than the first one. Figure 11: Italy: first wave of magnitude 1.79%. One time unit corresponds to 13 days. The timescale starts on 22 February 2020. Figure 12: Italy: estimated second wave of magnitude 7.99%. One time unit corresponds to 13 days. The timescale starts on 22 February 2020. Figure 10: Relation between the 1 st and 2 nd wave in Italy, if \(R_{2}=1.6\). The overall picture. One time unit corresponds to 13 days. The timescale starts on 15 February 2020. The estimated time of formation of the 2 nd peak is 7.8 months after the beginning (in our model 22 February 2020) of the epidemic. According to [16] our model predicts the 2 nd peak about the 15 th of October 2020, if \(R_{2}=1.6\). Assuming \(R_{2}=1.6\), the extent of the 2 nd wave is 44.5-times larger than the extent of the first wave:  
\[\frac{v_{2}^{*}}{v_{1}^{*}}=44.5\]  
The 1 st peak is at 1.79%, while the 2 nd peak is at 7.99%, if \(R_{2}=1.6\).