• **China**

The number of infected cases reaches maximum of around 82011.46. The rise to this maximum lasts approximately 28 days. The transition point to a stable state is marked around day 18 (pandemic peak) since the first case detected.

#### • **Tunisia**

The number of infected cases reaches maximum of around 1142.57. The rise to this maximum lasts for approximately 58 days. The transition point to a stable state is marked around day 32 (approximately 1 month) (pandemic peak) since the first case detected.

#### • **Italy**

The number of infected cases reaches maximum of around 227262.40. The rise to this maximum lasts for approximately 87.58 days. The transition point to a stable state is marked around day 63 (approximately 2 months) (pandemic peak) since the first case detected.

#### • **Germany**

The number of infected cases reaches maximum of around 227262.40. The rise to this maximum lasts approximately 93.302 days. The transition point to a stable state is marked at day 71 (approximately 2.5 months) (pandemic peak) since the first case detected.

#### • **France**

The number of infected cases reaches maximum of around 154264.94. The rise to this maximum lasts approximately 98.873 days. The transition point to a stable state is marked around day 75 (pandemic peak) since the first case detected.

#### **3.2 The contamination frequency** *Ac*

By plotting the Arrhenius curve *Ln I*ð Þ versus <sup>1</sup> *<sup>t</sup>* (**Figure 5**), we have calculated the contamination frequency, *Ac*, from the intercept of its linear portion. The results for the 5 studied countries are listed in **Table 3**. The low values observed for China and

**Figure 5.** *Arrhenius plot of the number of cumulative infected persons Iversus days.*

Tunisia reflect a reduction in the contamination frequency in these two countries. This can be attributed to the anticipation of the authorities of these two countries for reducing the mobility of people by imposing general confinement very early. However, the huge values observed for the 3 European countries reflect a very high contamination frequency. This can be explained by a delay in taking decisions relating to the restriction of people's movement.

#### **3.3 The dynamic behavior of virus spread during the critical period**

The determination of the critical exponents (Eqs. (17) and (18)) before (*s*) and after (*ts*) the percolation point informed us about the percolation model which dominated the spread of the virus in each country. The results reported in **Table 3** show values have largely deviated from theoretical values. This represents a very strong tendency toward the dynamic model for all countries. This reveals that the chains of covid-19 virus transmission are randomly distributed in space and the contamination is dispersed in all regions. However, this general trend does not prevent us from observing deviations toward the static model, especially before the pandemic peak (*s* value), for Tunisia and China.


**Table 3.**

*The critical exponents and contamination frequency for the 5 studied country.*

*Propagation Analysis of the Coronavirus Pandemic on the Light of the Percolation Theory DOI: http://dx.doi.org/10.5772/intechopen.97772*

#### **3.4 Speed of the pandemic spread**

In the first section of this article, we stated that the differential *<sup>∂</sup><sup>I</sup> <sup>∂</sup><sup>t</sup>* estimation of the variation rate in the number of people infected *I* as a function of the change in the number of days *t* since the first case is, therefore, the speed of the pandemic spread *Vp* (Eq. (19)). For the 5 studied countries, we measured this differential. The results were reported in **Figure 6**. As we can see, the speeds of the virus spread in the different countries gradually increase over time until they reach a maximum value *Vp* max , then they start to decrease. It is crucial to note that the dates corresponding to the maximum speeds reached coincide with the dates recorded as pandemic peaks *tp*( *<sup>t</sup> tp* ¼ 1) that have been determined by the Boltzmann sigmoid equation (SBE).

Furthermore, we can also notice that the speeds achieved take on considerable values except for Tunisia (*Vp* max ≈55 cases per day), a country which Covid-19 did not greatly affect.

To assess the speed of propagation between the studied countries, we also have adjusted our findings by estimating the root mean square speed of propagation *VPRMS* for each country:

$$V\_{PRMS} = \sqrt{\frac{\sum\_{i} V\_{pi}^{2}}{n}} \tag{20}$$

Where *Vpi* the spread speed for day *i* and *n* is the number of days since the first case. In **Figure 7**, we have depicted the variation of the root mean square speed of Covid-19 in the studied 5 countries. As reported, the highest *VPRMS* was recorded for Italy exceeding the speed of *VPRMS* ¼ 2000 cases/day. However, the lowest one was recorded for Tunisia with a value that does not exceed *VPRMS* ¼ 19 cases/day.

**Figure 6.** *Speed of the pandemic spread Vp versus <sup>t</sup> tp for the 5 studied countries.*

**Figure 7.** *The root mean square speed of propagation VPRMS since the first case until 30 June, for the 5 studied country.*

### **4. Conclusions**

We analyzed epidemic data on the cumulative number of cases infected with covid-19 in 5 countries affected differently by the pandemic. The data describe the state of the virus spread during more than 6 months of its appearance (appeared at the beginning in China (December 2019)). This analysis was consisted of the application of the percolation theory on the evolution of the cumulative number of people infected *Ic* in each country. Based on an analogy between the spread of Covid-19 in a population and the electrical percolation of reverse micelles, we introduced the covid-19 pandemic percolation. First, adjusting the data using the Boltzmann sigmoid equation (SBE) made it possible to derive important parameters related to the spread of the pandemic: The first represents the cumulative number of people infected *Ic*, max , from which the epidemic state in each country begins to stabilize. The second is the pandemic peak time *tp* (threshold of pandemic percolation), this is a critical day corresponding to the transformation of the epidemic situation from serious to stable state. The third is a time interval Δ*t* that we have called the time constant. This constant represents the momentum of exponential pandemic progression in each country. It allowed us to estimate the time necessary *t*max for the stabilization of the epidemic state in each country. Comparing the value of this constant in each country makes it possible to assess the effectiveness of the preventive measures taken by the authorities.

By applying the Arrhenius law on the cumulative number of cases in each country, we introduced a characteristic contamination factor this rate measures the frequency of interpersonal contact in each country. This permits us to compare contamination frequency following the traffic restriction measures taken by each country (border closure, general confinement, geo-located confinement, etc.). The application of scale laws at the vicinity of the percolation threshold *tp* has shown a total predominance of the dynamic percolation model of contamination. This predominance has been interpreted by the existence of transmission chains of the virus randomly distributed in each country (and not local chains).

Finally, the determination of the propagation speed *Vp* of covid-19 in each country showed that the maximum propagation speed was recorded during the *Propagation Analysis of the Coronavirus Pandemic on the Light of the Percolation Theory DOI: http://dx.doi.org/10.5772/intechopen.97772*

pandemic peak *tp* (previously determined by SBE equation). Moreover, the calculation of the quadratic mean root square speed of propagation *VPRMS* allowed us to compare the evolution of the speed in each country.

Finally, all of the results show that Tunisia and China have implemented the most effective strategies to combat the first wave of Covid-19. Indeed, Tunisia's authorities have opted for general containment (lockdown) throughout the country since the second week after the first confirmed case of Covid-19 was discovered. According to our findings, this strategy significantly reduced the frequency of contact between individuals carrying the virus, as well as the speed of covid-19 propagation over the next three months. As a result, the total number of cases infected with covid-19 remained very low. This clearly demonstrates the effectiveness of this strategy in combating the pandemic. In addition, the second country which showed low propagation indicators (contact frequency, speed of propagation, total number of infected people) is China. This is due to China's adoption of a strategy known as targeted containment. That is, the confinement was limited to the city of Wuhan (the site of the virus's appearance). Such a strategy has clearly demonstrated its effectiveness by significantly slowing propagation. However, the three European countries (Italy, France, and Germany) experienced relatively high propagation rates as well as a high number of covid-19 cases. These outcomes are the result of the authorities' failure to implement general containment procedures on the right time.

This further proves that the most effective strategy to bypass the spread of covid-19 is either general containment or targeted containment.
