# Using Epidemiological Models to Predict the Spread of Information on Twitter
## 2 Epidemiology-Based Information Spread Models
 \tag{1}\]  
The dynamics of interactions of the three classes of the population are represented by the following graph (Figure 1):  
As we previously anticipated, there are many other models that can be used to describe news spread. An example is the IESZ model, which, contrary to the ISR model, was developed to study the spread of information, although it is an ODE system-based model, like the previously discussed model. It divides the population into four classes as follows:  
* I(t), i.e., ignorant class. As in the ISR model, this group represents the class of people that can see the news;
* E(t), i.e., exposed class. The class of people who have been exposed to the news but have not yet shared it;
* S(t), i.e., spreader class. The class of people who spread the news;
* Z(t), i.e., skeptic class. The class of people who saw the news but choose to ignore it. Figure 1: Graph of the ISR model. As before, to formulate the model, we have to introduce some parameters, taking into account that a \"contact\" between two individuals occurs when one of them reads a tweet or a comment of the other. Therefore, we introduce:  
* \(\omega\): the contact rate between a member of the ignorant class and a member of the spreader class;
* \(b\): the contact rate between a member of the ignorant class and a member of the skeptic class;
* \(p\): the probability of transition from the ignorants class to the spreader class after a meeting between a member of the former class with a member of the latter class;
* \(l\): the probability of transition from the ignorants class to the skeptic class;
* \(\rho\): the contact rate between a member of the exposed class and a member of the spreader class;
* \(\epsilon\): the transition rate from the exposed class to the spreader class;
* \(N\): population size, which is the sum of the sizes of each class. The following graph (Figure 2) summarizes all the parameters and the possible transitions from one class to another. The model is obtained as follows [10]:  
\[ \begin{cases}I^{\prime}(t)=-\frac{\omega tS}{N}-b\frac{IZ}{N}\\ E^{\prime}(t)=(1-p)\omega\frac{tS}{N}+(1-l)b\frac{IZ}{N}-\rho E\frac{S}{N}- \epsilon E\\ S^{\prime}(t)=\frac{\rho\omega IS}{N}+\rho E\frac{S}{N}+\epsilon E\\ Z^{\prime}(t)=lbI\frac{Z}{N}\end{cases}  \tag{2}\]  
As we can see, there is not a direct transition from the class of ignorants to the class of exposed individuals. In fact, a member of the class of ignorants can see the news in the form of a tweet; then, he can encounter a member of the class of spreaders and become exposed with a probability of \((1-p)\) if he does not share the news, or he can become a spreader himself if he chooses to share the news, with a probability of \(p\). On the contrary, if he reads the news but does not show any particular reaction, he transitions to the skeptic class, with a probability of \(l\), while if he believes in the news, even after a period of time but he does not share it, he passes to the exposed class, with a probability of \((1-l)\). Figure 2: Graph of the IESZ model. An application of these two models can be seen in Figure 3, where, on the left, we report the plot of the solution of model (1) in correspondence with example parameters:  
\[\beta=0.2;\quad\gamma=0.3;\quad\kappa=1.2 \tag{3}\]  
while on the right, we present the solution of model (2) in correspondence with example parameters:  
\[\omega=4.3;\quad b=8;\quad p=0.7;\quad l=0.8;\quad\rho=1.38\times 10^{-6};\quad \epsilon=0.03; \tag{4}\]  
We observe that the two models are very different. Not only the dynamics differ among the different populations, but the results that we obtain if we use the two models to predict the evolution of a news item also differ. Indeed, the ISR Model foresees the transition from the class of ignorants to that of spreaders, then to that of recovered, so the spreaders function (S(t)) is, at first, an increasing function; then, it reaches a maximum inside the interval under consideration, and finally, it decreases.