# Comparison of the Evolution of the COVID-19 Disease between Romania and Italy
## 2 Time-Dependent SIR Model  
In this section, the classical SIR model and the time-dependent SIR model are briefly described [13]. In the classical SIR model:  
\[\frac{dS}{d\tau}=-\beta IS,\ \frac{dI}{d\tau}=\beta IS-\gamma I,\ \frac{dR}{d\tau}= \gamma I, \tag{1}\]  
Figure 1: The comparative evolution of the number of cases of diseases per day of evolution of the COVID-19 infection: (**a**) Romania, (**b**) Italy. where \(S(\tau)\), \(I(\tau)\) and \(R(\tau)\) denotes the number of susceptible persons, infected persons and persons recovered at time \(\tau\), respectively, the population of size \(N\) is closed, i.e., \(N=S(\tau)+I(\tau)+R(\tau)\). The constants \(\beta N\) and \(\gamma\) stand for: \(\beta N\) denotes the (average) number of susceptible persons an infected person comes into contact with (and is possible to spreads the disease), while \(\frac{1}{\gamma}\) denotes the average time an infected person takes to recover (i.e., the time of being infectious). Therefore, the quotient \(R_{0}=\frac{\beta N}{\gamma}\) is constant and represents the number of new infections caused by an infected person before he/she recovers. In this paper \(R_{0}\) is considered as time-dependent: \(R_{0}=R_{0}(t)\); changing after restrictions are announced. Rewriting Equation (1) in a non-dimensional form, we define:  
\[u=\frac{S}{N},\;v=\frac{I}{N},\;w=\frac{R}{N},\;t=\gamma\tau \tag{2}\]  
to obtain  
\[\frac{du}{dt}=-R_{0}uv,\;\frac{dv}{dt}=v(R_{0}u-1),\;\frac{dw}{dt}=v. \tag{3}\]  
From the dynamical point of view [14; 15], the \((u,v)-\) plane is invariant, \(v=0\) is the line of singular points, and \((u,v)=(0,1)\) is unstable if \(R_{0}>1\), while all other singular points \(v\in[0,1)\) are stable. Note that in Model (3) all variables are less than (or equal to) one (since representing the percentage of the total population). Note that the exact formula characterizing \(R_{0}\) is \(\frac{\beta S_{0}}{\gamma}\) where \(S_{0}\) is the starting number of susceptible persons. However, if the population \(N\) is quite big and there are only few initially infected, the density \(S_{0}/N\) is near \(1\), thus the approximation \(\frac{\beta S_{0}}{\gamma}\approx\frac{\beta N}{\gamma}\) is possible. We assume (see Figure 2) the following time dependence for \(R_{0}=R_{0}(t)\):  
* After introduction of the restrictions/interventions, at time \(t_{1}\), the value of \(R_{0}\) decreases to a certain (lower constant) value \(R_{1}\) (i.e., \(R_{0}>R_{1}\)),
* While the restrictions are in force this value remains constant (for time \(t\in[t_{1},t_{2}]\)),
* After the end of the restrictions/interventions, after time \(t_{2}\), the value of \(R_{0}\) increases to a certain (higher) value \(R_{2}\) (i.e., \(R_{2}>R_{1}\)),
* The total duration of restrictions is denoted by \(\Delta t=t_{2}-t_{1}\),
* The transitions \(R_{0}\to R_{1}\) and \(R_{1}\to R_{2}\) appear in a time comparable to \(\frac{1}{\gamma}\), which is much less than \(\Delta t=t_{2}-t_{1}\). Figure 2: Time dependent \(R_{0}\), as following from the assumptions described above. The restrictions/interventions start at time \(t_{1}\) and end at time \(t_{2}\). During the restrictions it is assumed that the value of \(R_{0}\) decreases, after the restrictions have ended it is assumed that the value of \(R_{0}\) increases again, but to a slightly lower value than before the restrictions were introduced. This type of time dependent \(R_{0}\) in a SIR-model causes the evolution of the infected-fraction \(v=v(t)\), as shown in Figure 3. The maximum of the 1 st wave is \(v_{1}^{*}\) and appears at time \(T_{1}\). The maximum of the 2 nd wave is \(v_{2}^{*}\) and appears at time \(T_{2}\). By \(R_{3}\) a third value for \(R_{0}\) is denoted (see Figure 4), as one considers again some restrictions to avoid the in-avoidable 2 nd peak. Note also that the travels were one of the first things that were blocked by the restrictions, so closed population assumption is legitimate under the regime \(R_{1}\).