# Stiffness Analysis to Predict the Spread Out of Fake Information
## 2 Stiffness Analysis of the SIR Model for the Diffusion of Fake Information  
As aforementioned, the SIR model was first introduced in 1927 by [PERSON] and [PERSON] [31], even if seminal contributions on compartmental models are also advisable in [30; 32; 33; 34]. It represents an extremely simple mathematical model for describing the transmission of an infectious disease. This model describes the mutual interactions of three populations of individuals: the population S of susceptible people, i.e., the healthy individuals who can contract the disease; the population \(I\) of the infected, i.e., individuals who have contracted the disease and are able to transmit it; and the population \(R\) of the recovered, that is, individuals who are healed. In our model for the spread of fake information, above populations are described as follows:  
* \(S(t)\): potentially authoring the spreading of fake news;
* \(I(t)\): the wide variety of authors highly active in posting fake information;
* \(R(t)\): authors who are inactive to the spreading of fake news. The model is based on the continuous interaction between susceptible and infected individuals along time. The corresponding system of ODEs is then given by:  
\[\left\{ \begin{aligned} \frac{dS(t)}{dt}&=-\beta S(t)I(t), \\ \frac{dI(t)}{dt}&=\beta S(t)I(t)-\alpha I(t),\\ \frac{dR(t)}{dt}&=\alpha I(t),\end{aligned} \right. \tag{1}\]  
where \(\alpha\) is the rate of recovery and \(\beta\) the contact rate. Specifically, since the purpose of our method is to compare the impact of fake news in different countries, the parameters \(\beta\) and \(\alpha\), are related to two important indices, commonly used to describe the social, economic and cultural performance of our society. In particular,  
\[\beta=\frac{i}{10},\quad\alpha=\frac{h}{100},\]  
where \(i\) is the _internet penetration index_ of the country and \(h\) is the human development index of the same country. These values are commonly provided in the annual report of United Nations Development Programme [45] (also see the projects [46; 47]). In general, the value of \(\alpha\) is smaller than that of \(\beta\) because it is easier to spread a lie than reaffirming the truth. Table 1 shows, for instance, the values of \(\alpha\) and \(\beta\) for selected countries, i.e., Australia, Brazil, France, India, Italy, Mexico, Mozambique and the USA, referring to the year 2019. **Remark 1**. : _It is crucial to highlight the different power of the denial of a fake news and that of the fake news itself [48]. Fake information circulates online much faster than true one and is more prone to be shared by users who encounter it; on the other hand, the re-assessment of the truth is nowhere viral and reaches far fewer people than those who have read or spread fake information. Due to this intrinsic characteristic of true news compared to false news, the former require a much greater commitment on the part of individual users than that related to the spread of fake news. Above all, the spread out of fake information does not necessarily require a particularly strong human presence: often fake news are circulated by bots created specifically by someone or shared by fake accounts, so they do not correspond to real people. Real people, on the other hand, are necessary and represent the only option to restore the truth. This motivates the choice of the recovery rate a linked to human development index per country and the contact rate \(\beta\) to the spread of the internet in the same country._  
In our analysis, it is relevant to consider a linearization around the initial value vector  
\[\left[ \begin{array}{c}S_{0}\\ I_{0}\\ R_{0}\end{array} \right]=\left[ \begin{array}{c}S(0)\\ I(0)\\ R(0)\end{array} \right],\]  
leading to  
\[\left\{ \begin{aligned} \frac{dS(t)}{dt}&=\beta S_{0}I_{0}- \beta I_{0}S(t)-\beta S_{0}I(t)+\text{high order terms},\\ \frac{dI(t)}{dt}&=-\beta S_{0}I_{0}+\beta I_{0}S(t)+(\beta S_{0}- \alpha)I(t)+\text{high order terms},\\ \frac{dR(t)}{dt}&=\alpha I(t),\end{aligned} \right.