**1. Introduction**

In late December 2019, the emerging epidemic of Coronavirus, Sars-Cov-2 (Covid-19) first appeared in China. Then, it spread on a very large scale to most of the world's countries. For this reason the World Health Organization (WHO) has declared it a global pandemic. The main distinguishing characteristic of this pandemic is its very high contagion power and subsequently, which gives it a spectacular speed of propagation. Recently [1], it was associated with the Spanish flu of 1918 in this regard. For this reason, efforts to forecast, predict, and model the spread of Coronavirus (Covid-19) global pandemic are accumulating to understand its contamination mode [2–6]. The objective of these efforts is to explore ways for

predicting its evolution curve to be able to skirt the contamination, in the absence of an effective vaccine serving as an antidote against this disease.

In this context, in a recent study [7] on the spread of Covid-19 in 15 different countries, we have showed that the pandemic spread in a given population follows a sigmoidal law. Indeed, the curve of variation of the cumulative number of infected persons *I* is composed of 3 phases, a linear phase (slow growth) followed by second exponential phase (fast growth), then another stable phase (slow growth). Given that, in our laboratory, we are used to using the sigmoidal model to characterize the percolation phenomenon occurring in colloidal systems [8, 9].We suggest in the present work to apply the theory of percolation theory on the cumulative number of infected people in order to better understand the dynamics of the virus spread.

Percolation is a mathematical model which first introduced by S.R Braodblent and J. Hammersely in 1957 [10]. As a geometric phase transition, he described largescale connectivity from other links that are randomly established on small scales. Such connectivity brought the studied system from one state of disorder to another more ordered state. This model has been widely applied to explain phase transitions in a very wide variety of physical systems such as alloys, complex fluids, semiconductors, communication networks, etc. As examples of the application of the percolation model, we can cite the following transitions: Insulating conductive composite material (Insulator/conductor), Glass transition (Liquid/glass), Polymer gels (Liquid/gels), Quarks in nuclear matter (confinement/non confinement) [11].

Meanwhile, in our laboratory, we used to study several kinds of colloidal systems, in particular, the microemulsions water in oil, so-called reverse micelles. These are nano-droplets of water surrounded by monolayers of surfactant and dispersed in oily phases. In these systems, a percolation phenomenon has often been highlighted and widely studied. In fact, in reverse micelle systems, the percolation phenomenon was manifested by a spectacular increase in the transport properties (viscosity, dielectric constant electrical conductivity, etc.) of the micelles for a volume fraction, *ϕ<sup>d</sup>* or a well-determined temperature *Tp* [8, 9]. The critical micellar volume fraction and the temperature signaling the appearance of percolation phenomenon are called the percolation thresholds (*ϕ<sup>d</sup>*,*<sup>c</sup>*,*Tp*). This increase has been explained by the fusion of neighboring micelles, a fusion which in turn leads to the formation of a percolating network and the exchange of charge carriers (or matter). In the bibliography, the studies were focused on the viscosity, the electrical conductivity [12] and, of the dielectric constant of the reverse micelles around the percolation threshold. Theoretically, two models have been proposed [13] to justify the percolation-related experimental observations: a static model and a dynamic model. In the static model, the sudden change in the conduction of the system is caused by the formation of a bicontinuous phase of the organic solvent and water at the percolation threshold. Otherwise, micellar fusions are established between the neighboring micelles giving rise to micellar exchanges of the charge and the material. Furthermore, concerning the dynamic model, it is a model that attributes this behavior to the Brownian movement of droplets. Such movement induces random mutual collisions between the different micelles accompanied by exchanges of charge and matter.

#### **2. Electrical percolation versus Covid-19 pandemic percolation**

#### **2.1 Phenomenological description**

It seems imperative to mention that we are dealing here with two systems with several similarities in dynamics. Regarding the reverse micellar systems, these are indeed spherical particles of nanometric sizes in Brownian (random) motion. This

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

movement leads to permanent inter-micellar collisions giving rise to mutual exchanges of matter and electrical charges. Therefore, reverse micelles transport phenomena such as viscosity *η* and electrical conductivity *σ* increase exponentially. On the other side, they are infected people circulating randomly (carriers of the virus). This circulation leads to the virus spread in various ways (contamination of surfaces, the spread of droplets carrying the virus, handshaking with contaminated people...) which giving rise to cumulative number of infected people *Ic*. We report in **Figure 1** the schematic behavior of two systems. We also present in **Table 1** the detailed analogy between the two systems.

From a qualitative viewpoint, we find that the evolution of the number of people infected over time followed the same curve as the evolution of the micelles' electrical conductivity as a function of temperature. These are two trays separated by an exponential progression. We have measured the electrical conductivity of water/ AOT/reverse micelles of composition *<sup>W</sup>*<sup>0</sup> <sup>¼</sup> ½ � *water* ½ � *AOT* <sup>¼</sup> 30, *<sup>ϕ</sup><sup>m</sup>* <sup>¼</sup> *VAOT Visooc*tan *<sup>e</sup>* ¼ 0*:*1, 0*:*15, 0*:*2, with an increase in temperature. Results are reported in **Figure 2** and compared with the evolution of the cumulative number of detected cases of Covid-19 in china since the first case.

#### **2.2 Theoretical background**

#### *2.2.1 Sigmoid Boltzmann equation*

Authors [13, 14] are used to using the sigmoid Boltzmann equation (SBE) to model the percolation process in reverse micelles. It is important to elucidate some notions on the SBE equation so that both the relevance of this equation and the definitions of both of its terms are understood.

**Figure 1.**

*Schematic representation of the analogy between reverse micelles and the virus spread.*


#### **Table 1.**

*Detailed similarities between reverse micelles and the pandemic spread.*

**Figure 2.**

*(a) Electrical conductivity of water/AOT/isooctane reverse micelles (W*<sup>0</sup> ¼ 30, *ϕ<sup>m</sup>* ¼ 0*:*1, 0*:*15, 0*:*2 *vs Temperature (b) Confirmed cases of infected persons with covid-19 since the first case in China.*

To describe the evolution of a quantity evolving as a function of a variable, the Sigmoid-Boltzmann equation (SBE) has the following form:

$$\mathcal{Y} = \frac{\mathcal{Y}\_i - \mathcal{Y}\_r}{1 + \exp\left(\pi - \pi\_0\right)/\Delta\mathfrak{x}} + \mathcal{Y}\_r \tag{1}$$

Rearranging Eq. (1),

$$\mathbf{y} = \mathbf{y}\_r \left[ \mathbf{1} + \left( \frac{\mathbf{y}\_i - \mathbf{y}\_r}{\mathbf{y}\_r} \right) \times \left\{ \mathbf{1} + \exp \left( \mathbf{x} - \mathbf{x}\_0 \right) / \Delta \mathbf{x} \right\}^{-1} \right] \tag{2}$$

where *y* is the measured magnitude of the system which depends on *x*, *yi* and *yr* are the left and right asymptotes of *y*, *x*<sup>0</sup> is the center (where *y* returns the mean of *yi* and *yr*), and Δ*x* is the constant interval of the independent variable that controls the rise profile or decrease from *yi* to *yr*(for a large Δ*x*, the rise is gradual while for small Δ*x*, the rise is rapid).

The equation therefore essentially deals with the switching of a variable from an initial state (state of *yi* ) to a final state (state of *yr*) through a transition *x*0.

#### *2.2.2 Application for electrical percolation of reverse micelles*

By applying this equation to the electrical conductivity of the inverse micelles, *σ*, evolving as a function of temperature, *T*, the equation, therefore, deals with the switching of the conductivity from an initial state, *σi*, to a final state, *σ <sup>f</sup>* , passing through the transition, *Tp*, (percolation temperature).

$$\sigma = \sigma\_f \left[ 1 + \left( \frac{\sigma\_i - \sigma\_f}{\sigma\_f} \right) \times \left\{ 1 + \exp \left( T - T\_p \right) / \Delta T \right\}^{-1} \right] \tag{3}$$

From an experimental point of view, it is often convenient to use the logarithmic scale to better highlight the variations. The equation therefore becomes:

$$\log \sigma = \log \sigma\_f \left[ \mathbf{1} + \left( \frac{\log \sigma\_i - \log \sigma\_f}{\log \sigma\_f} \right) \times \left\{ \mathbf{1} + \exp \left( T - T\_p \right) / \Delta T \right\}^{-1} \right] \tag{4}$$

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

## *2.2.3 Application of SBE to the cumulative number of infected persons Ic with Covid-19*

In a recent work, we modeled the evolution of the cumulative number of infected people *Ic* in 15 countries with the sigmoid Boltzmann equation (SBE), evolving as a function of time, *t*. The SBE equation deals with the switching of numbers, *Ic*, from an initial state, *Ii*, to a final state, *I*max, through the transition, *tp*. At this point, we can consider that the transition, *tp*, corresponds to what is called the pandemic peak.

$$I\_c = I\_{c,\max} \left[ \mathbf{1} + \left( \frac{I\_i - I\_{c,\max}}{I\_{\max}} \right) \times \left\{ \mathbf{1} + \exp \left( t - t\_p \right) / \Delta t \right\}^{-1} \right] \tag{5}$$

Knowing that at least one person is infected therefore *Ii* ¼ 1 and taking into account that the maximum number of infected people, *I*max, takes huge values, so *I*max≻≻≻*Ii*, the Eq. (5) becomes:

$$I\_c = I\_{c,\max} \left[ \mathbf{1} - \left\{ \mathbf{1} + \exp \left( t - t\_p \right) / \Delta t \right\}^{-1} \right] \tag{6}$$

On a logarithmic scale, the sigmoid equation becomes:

$$\log I\_{\varepsilon} = \log I\_{\varepsilon, \max} \left[ \mathbf{1} + \left( \frac{\log I\_{i} - \log I\_{\varepsilon, \max}}{\log I\_{\varepsilon, \max}} \right) \times \left\{ \mathbf{1} + \exp \left( t - t\_{p} \right) / \Delta t \right\}^{-1} \right] \tag{7}$$

Considering that at least one person is infected, so *Ii* ¼ 1, the Eq. (7) becomes:

$$\log I\_{\varepsilon} = \log I\_{c,\max} \left[ \mathbf{1} - \left\{ \mathbf{1} + \exp \left( t - t\_p \right) / \Delta t \right\}^{-1} \right] \tag{8}$$

For the spread of the pandemic in a given population, the Boltzmann equation (SBE) allows us to derive important parameters describing the spread of the virus. The most important parameters are the time interval, Δ*t*, the percolation time, *tp*, and the maximum number of infected persons, *Ic*, max :

• The pandemic percolation time *tp*

This is the transition point corresponding to the pandemic percolation threshold. At this time, the number of infected people rises drastically until the day with the maximum propagation speed. After that time the speed of virus spread decreases and the number of infected cases per day begins to decrease. It represents the pandemic peak

• Maximum number of infected persons *Ic*, max

It is a cumulative number of infected people signaling the stabilization of the epidemic crisis in a population. The real maximum number of people infected is in the vicinity of this number.

• The time constant Δ*t*

It is called the time constant. It characterizes the rise of the exponential part (gradually or abrupt). Generally, the epidemic state stabilizes when the number of infected cases *N* reaches almost the maximum number of infected cases *I* ¼ 0*:*99*Ic*, max which corresponds to maximum time, *t*max of pandemic spread:

Considering Eq. (6), the *t*max value can be calculated from the following equation:

$$t\_{\text{max}} = \text{2.19} \Delta t + t\_p \tag{9}$$

So, we can estimate the time necessary for the stabilization of the epidemic state of each country *t*max from Eq. (9). From an epidemiological point of view, the Δ*t* value must necessarily depend on the effectiveness of the precautionary and preventive measures taken by the authorities of each country.

#### *2.2.4 The characteristic contamination frequency*

For reverse micelles systems, the energy required to make two micelles appear and activate the exchange of charges between them is called the activation energy of percolation. This energy constitutes a barrier to be crossed for this energy constitutes a barrier to be crossed to put two micelles in contact and trigger the exchange of charges. It takes the form of the famous empirical law of Arrhenius [13]:

$$\sigma = A \exp\left(-E\_p / RT\right) \tag{10}$$

Where *σ* is the electrical conductivity of the micelles. *R* is the universal constant of ideal gases. *T* is the temperature. *A* is a pre-exponential factor having the same unit as *σ*. Typically, the refinement of Arrhenius law was effected by the Eyring transition state theory [15, 16], where the pre-exponential factor *A* is shown as the entropic factor of the transition state. In this theory the transition kinetic constant *k* is correlated with temperature by law analogous to that of Arrhenius:

$$k = A \exp\left(-\Delta H/RT\right) \tag{11}$$

$$k = \frac{K\_B T}{h} \exp\left(-\Delta S/R\right) \exp\left(-\Delta H/RT\right) \tag{12}$$

Where *k* is kinetic constant, Δ*S* and Δ*H* are the transition state entropy and enthalpy, *KB* is the Boltzmann constant and *h* is the Planck constant. In another report [16], the pre-exponential factor *A* was also correlated to motion, rate, disorder, speed … of particles. Tian Hao [6] considers that the movement of infected individuals and individuals exposed to contamination is analogous to that of the movement of particles in various systems such as granular systems, colloidal systems… He suggests that the equations of conductivity and viscosity are also applicable in the description of the viruses spread. In his work, he asserts that Eyring Rate Process Theory and Free Volume Concept [6, 17] are applicable on the movement of individuals carrying viruses which means that the more the infected and exposed individuals have the more free volume to move, the higher the probability of transmission.

So, for the virus spread, Arrhenius law can be applied to the cumulative number of infected cases *I* as follows:

$$I = A\_c \exp\left(-t\_C/t\right) \tag{13}$$

Having the dimension of time, *tC* is equivalent to the Arrhenius temperature *Ea R* [16], *Ac* is a pre-exponential factor considered as the contamination frequency factor and *t* is the number of days. The contamination frequency *Ac* can be determined from the slope of the linear portion of the curve:

$$
\ln I = \ln A\_C - \frac{t\_C}{t} \tag{14}
$$

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

#### *2.2.5 Scaling laws and critical exponents*

We have mentioned before that the rationalization of the experimental results resulted in two theoretical approaches discussing the state of a system during the percolation phenomenon. A static model which suggests that at the time of percolation, the charge carriers merge with their neighbors, thus establishing a network of connected particles exchanging charges. Besides, a dynamic model whose particles carrying the charges are in permanent collision, merge randomly at the percolation threshold, and exchange the charges. The scheme depicted in **Figure 3** simplifies the difference between the two percolation models.

Theoretically, scale laws have been established to identify the predominant model for a given system. These laws are explained by two equations, which are applied only around the percolation point. For electrical percolation in reverse micelles systems, induced by temperature variation, these equations take the following forms:

$$
\sigma = \beta \left( T\_p - T \right)^{-s}, T \prec T\_p \tag{15}
$$

$$
\sigma = \chi \left( \mathbf{T} - T\_p \right)^{t\_\iota}, \mathbf{T} \succeq T\_p \tag{16}
$$

Where *β* and *γ* are free parameters. *S* and *ts* are called the critical exponents. These two laws are valid near to the temperature of the percolation, *Tp*. Critical exponent values exhibit the difference between static percolation and dynamic percolation [13]. In fact, in the case of static percolation, the predicted theoretical values of critical exponents were found to be *ts* ¼ 1*:*6 and *s* ¼ 0*:*7. Divergence of these values was used to be the proof of dynamic percolation.

In the case of the propagation of Covid-19, these two equations apply to the vicinity of the peak pandemic as follows:

$$I = \beta \left(\mathbf{t}\_p - \mathbf{t}\right)^{-\prime}, \mathbf{t} \preccurlyeq \mathbf{t}\_p \tag{17}$$

$$I = \mathbf{y}\left(\mathbf{t} - \mathbf{t}\_p\right)^{t\_l}, \mathbf{t} \succ t\_p \tag{18}$$

From an epidemiological point of view, the static percolation model corresponds to the formation of geolocated chains of transmission of the virus [17]. However, the dynamic model reflects randomly distributed chains of transmissions in space.

**Figure 3.** *Schematic representation of the two percolation models.*

#### *2.2.6 Speed of the pandemic spread*

Typically, to detect the percolation threshold (temperature, volume fraction … ), a differential method is used [9, 13]. The method consists of determining the maximum of the curve *∂σ=∂T* versus the temperature (volume fraction...). This curve measures the speed of the variation of electrical conductivity as temperature varies.

Similarly, we can consider that the differential, *∂Ic=∂t*, measures the speed of propagation of the contamination *Vp* (and therefore of the virus) as a function of time. Thus, the maximum speed corresponds to the maximum curve *∂Ic=∂t* versus time (Days).

$$V\_p = \frac{\partial I\_c}{\partial t} \tag{19}$$

#### **3. Application to the cases of China, Tunisia, France, Italy and Germany**

Using data provided by Johns Hopkins university resource center [19] on the number of infected cases in 5 countries (China, France, Germany, Italy and Tunisia) during 6 months of the virus spread since the day of the first case detection of until June 30.

#### **3.1 The cumulative number of people infected** *Ic*

We plotted in **Figure 4** the variation of the cumulative number of infected people in each country versus the number of days since the first detected case. The existence of 3 stages can be discerned: a first phase, where the number of cases varies gradually over time. A second phase in which the number of cases unexpectedly rises very quickly and a third phase in which the rate of increase in the number of cases is stabilized overtime. We have therefore adjusted all these curves with the

#### **Figure 4.**

*Variation of the cumulative number of confirmed case Ic versus days since first case t. The red line ( ) represents the sigmoid Boltzmann equation fit (SBE).*

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


**Table 2.**

*Fit parameters of number of infected cases Ic (cumulative) with the SBE equation, for the different countries studied.*

sigmoid Boltzmann equation (SBE) (Eq. (6)). The results are listed in **Table 2**. The fit results can therefore be interpreted with respect to the evolution of the epidemic situation as follows:
