# Using Epidemiological Models to Predict the Spread of Information on Twitter
## 1 Introduction  
Social media have become a great tool for the dissemination of information, especially if we consider the dissemination of fake news. They are easy to use and free, and for this reason, in some cases, they are the main source of information. Therefore, it is important to understand not only how news spreads on social media but also the period of time when a news story is popular or the moment when the peak of maximum popularity of the news item occurs. For example, for a company that produces technological devices, it can be advantageous to predict when there will be maximum interest in a topic on social networks in order to launch a new product at the right time, maximizing earnings. Similarly, with respect to fake news, these factors are also fundamental, since predicting the peak of diffusion of fake news allows us to apply countermeasures to block such content (see more details in [1]). Fake news can be crucial for the outcome of some political decisions or social events, such as the American presidential elections of 2016 or Brexit, as shown in [2]; therefore, a good way of analyzing fake news is required. Several models are available to study the spread of information, including suitable epidemiological models based on ordinary differential equations [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13], models based on partial differential equations [14; 15; 16; 17; 18] and models based on stochastic differential equations [19; 20; 21]. In this work, in particular, we consider models based on ordinary differential equation systems that are also used in mathematical epidemiology to study the spread of an epidemic (for more epidemiological models and details of models used in mathematical epidemiology, see [22]). Research activity on epidemiological models is widespread and does not only concern the construction of models capable of characterizing the actual spread of epidemics but also the derivation of numerical methods that are able to preserve the fundamental features of the related exact solution, such as the equilibrium point properties, are important for locating and analyzing peaks; positivity [23; 24; 25; 26; 27]; the oscillation frequency (if any) [28; 29]; and inherent conservation laws [30; 31; 32]. In mathematical epidemiology, there are different kinds of models based on ordinary differential equation systems, including:  
* \"Ode systems-based models\" that divide the population into different classes named according to the state in which various individuals find themselves, such as susceptible, infected, dead, recovered, etc. Each equation of the system describes the evolution of these classes;
* Stochastic models that use stochastic differential equation systems;
* Models with delay that use delayed differential equations in order to consider the incubation period of a virus. In this paper, we consider models of the first class by treating news as a virus and studying its evolution through ordinary differential equations systems. Consequently, to archive our goal of predicting the trend of news and its peak, we have to find both the right model and the best possible set of parameters for it in order to obtain the most realistic result that fits the real data that we collected from social media site \"Twitter\" using the tools shown in the next section. Moreover, we also want to understand if there are some common points that characterize the diffusion of different kinds of news. One relevant aspect to take into consideration, as underlined in [2], is that a key role is played by the grade of access to the Internet of the population. Countries with the highest grade are more vulnerable to the spread of fake news. However, this cannot be the only reason. There are also other important factors, such as the person (or the group of persons) who shares the news first or the period of time when the news is popular, which could have a considerable impact on the grade of the news diffusion, as we show in the section dedicated to numerical experiments. The paper is organized as follows.