# Comparison of the Evolution of the COVID-19 Disease between Romania and Italy
## 2 Time-Dependent SIR Model
It is known that even in the restricted period a large number of people was traveling from Romania to Germany, for agriculture. The traveling between countries with the same risk-factor was possible after the end of the 1 st wave. Even the traveling from a country with higher risk factor to a country with lower risk factor was possible, but migrations were quite limited at this time and a quarantine was mandatory. We assumed the traveling persons were healthy and tested for COVID-19 and they return in the period close to some (low) multiple of \(\frac{1}{\gamma}\). This justifies the assumption of closed population also under the regime \(R_{2}\). Figure 4: Double transition \(R_{0}=R_{0}(t)\) with the parameter values \(R_{2}=R_{0}=4.0\) and \(R_{1}=0.5\). The start of restrictions is at time \(t_{1}=3.5\), the end of the restrictions is at time \(t_{2}=10\). Figure 3: First and second wave of the evolution, as derived from the assumption \(R_{0}=R_{0}(t)\) and the susceptible–infected–recovered (SIR) model; Equation (3). The start/end time of the restrictions: \(t_{1}\) and \(t_{2}\) are followed by two peaks at time \(T_{1}\) and \(T_{2}\), respectively. The extension of the successive waves are \(v_{1}^{*}\) and \(v_{2}^{*}\) respectively. First we adjusted the value of \(R_{0}\) at the beginning from the data [16; 17] by simply taking logarithms of \(I=I(t)\) and adjusting the data \((t,\log I)\) for the linear regression line. For Italy we found \(R_{0}\in[3.72,5.01]\), while for Romania we found \(R_{0}\in[3.56,4.8]\). The lower value corresponds to value \(\frac{1}{\gamma}=11\) days, while the higher value corresponds to \(\frac{1}{\gamma}=14\) days [11]. Next, we read from the graph \(I=I(t)\)[16; 17] the date of the beginning of the restrictions in Italy and Romania (i.e., we read \(t_{1}\)). Then we adjust the value of \(R_{1}\) for Italy and Romania so that the maximum value of the first wave \(v_{1}^{*}\) appears at real time \(T_{1}\), which in turn was read from the graph [16; 17] according to the real data. For the transition-time of \(R_{0}=R_{0}(t)\) from \(R_{0}\) to \(R_{1}\) and from \(R_{1}\) to \(R_{2}\) we assumed a value comparable to \(\frac{1}{\gamma}\), but still in accordance to the actual measures taken by authorities. The value of \(R_{1}\) and \(\gamma\) was determined to adjust the evolution of \(v=v(t)\) the to the real data. The values for \(R_{1}\) were checked for values \(R_{1}\in[0.4,\ 0.9]\), according to [18]. The crucial part of the model is the variation of the value for \(R_{3}\) (see also [19]). Additionally, we vary the value of \(t_{2}\) to get its effect on \(T_{2}\) and on the value of the extent fraction \(\frac{v_{2}^{*}}{v_{1}^{*}}\). In the paper we answer the following questions:  
* (Q1) How do the publicly announced interventions/restrictions affect parameter \(R_{1}\)? * (Q2) What is the time delay, \(\Delta T=T_{2}-T_{1}\), in the evolution of the spread of the disease for both countries? * (Q3) How does parameter \(R_{2}\) affect the fraction \(\frac{v_{2}^{*}}{v_{1}^{*}}\)? * (Q4) What is the effect of the parameter \(R_{2}\) on the time \(T_{2}\)? * (Q5) How does the parameter \(\Delta t=t_{2}-t_{1}\) (the duration of the restrictions) affect the fraction \(\frac{v_{2}^{*}}{v_{1}^{*}}\)? * (Q6) How does the parameter \(\Delta t\) affect the time \(T_{2}\)? * (Q7) What is the effect of possible re-restrictions on the extent and datum of the second peak? To answer these questions, we look at the system (3) with the real data output and vary the values of \(R_{1}\), \(R_{2}\), \(\Delta t\) for Italy and for Romania. For \(\frac{1}{\gamma}\) we used/assumed the value of 11-14 days, as reported in news. To obtain numerical solutions to Equations (3) we used Mathematica 10.4. In the sequel model (3) with \(R_{0}=R_{0}(t)\) as shown in Figure 3.  
\[\frac{du}{dt}=-R_{0}(t)uv,\ \frac{dv}{dt}=v(R_{0}(t)u-1),\ \frac{dw}{dt}=v. \tag{4}\]  
were adjusted to real data for Italy [16] and Romania [17].