# Using Epidemiological Models to Predict the Spread of Information on Twitter
## 2 Epidemiology-Based Information Spread Models  
As anticipated, there are different ways to study the spread of information on social networks, among which we considered the epidemiological ones. From a mathematical point of view, there is an analogy between describing the evolution of an epidemic and the process of dissemination of information. An epidemic spreads by affecting the majority of the population, (as occurred, for example, with the COVID-19 pandemic in 2020); likewise, news can be compared to a contagious virus spread by all those who share it on social networks. We consider epidemiological models based on \(ODE_{S}\) systems that study epidemics by dividing population in different classes. Such models comprise different types: from the easiest one, the SI (susceptible-infected) model, that takes into consideration only susceptible individuals who can be infected and infected individuals (see [22] for more details), to more complex models that involve dead, recovered, hospitalized and quarantined individuals, etc. With the aim of studying information diffusion on the web or on social media, we chose classes of individuals taking into account the available data in order to validate the model using real data. Hence, we focused on models that were developed to study epidemics but that can be easily adapted to describe the dissemination of information or models that have some characteristics in common with epidemic models but that were developed explicitly to study information spread, i.e., the ISR (ignorants spreaders recovered) model and the IESZ (ignorants exposed spreaders skeptics) model ([10; 11; 12; 33]). Before introducing these two models, a brief explanation of the terminology used in this paper is required. In epidemiology, there is a lot of models with similar names, such as the SIR (susceptible infected recovered) model introduced in 1927 by [PERSON] and [PERSON] in their work called _\"A contribution to the mathematical theory of epidemics\"_. However, here, the notation is different. In the case of the SIR model, the population is divided into three different classes, which are described as follows:  
* S(t), i.e., _susceptible_ class: the class of people who can be infected;
* I(t), i.e., _infectious_ class: the class of people who are infected;
* R(t), i.e., _recovered_ class: the class of people who have recovered from disease. In our case, although we can use the word \"susceptible\" to refer people who can see a tweet containing news, the word \"infectious\" to refer to people who saw the news and decided to share it, and the word \"recovered\" to refer to people who recovered because they lost their interest in the news or because they do not believe in that news anymore. It is more appropriate to apply some changes to the notation by considering the ISR model, which refers to three different classes of the population as follows:  
* I(t), i.e, _ignorant_ class. This group plays the role the susceptible class, because its members are all users of the social network who can see the news, but they have not seen it yet, so they ignore it;
* S(t), i.e, _spreader_ class. This group plays the role of the infectious class because its members are the users that spread the news, exactly as infected individuals can spread a disease during an epidemic;
* R(t), i.e, _recovered_ class. This group plays the role of the recovered class, such as in the SIR model of epidemics; they are individuals who do not spread the news anymore. Using the previous notation, we introduce the following parameters:
* \(\beta\): the news spreading rate;
* \(\gamma\): the recovery rate;
* \(k\): the average number of connections among individuals;
* \(N\): the population size. The model is obtained as follows [11]:  
\[ \begin{cases}I^{\prime}(t)=-\frac{\beta k(t)S(t)}{N}\\ S^{\prime}(t)=\frac{\beta k(t)S(t)}{N}-\frac{\gamma kS(t)S(t)}{N}\\ R^{\prime}(t)=\frac{\gamma kS(t)S(t)}{N}\\ \end{cases}