# Spreading of fake news, competence, and learning: kinetic modeling and
numerical approximation

Jonathan Franceschi Department of Mathematics and Computer Science, University
of Ferrara, Via Machiavelli 30, 44121 Ferrara, Italy
<EMAIL_ADDRESS>Lorenzo Pareschi Department of Mathematics
and Computer Science, University of Ferrara, Via Machiavelli 30, 44121
Ferrara, Italy (lorenzo.pareschi@unife.it)Center for Modeling, Computing and
Statistic (CMCS), University of Ferrara, Via Muratori 9, 44121 Ferrara, Italy

###### Abstract

The rise of social networks as the primary means of communication in almost
every country in the world has simultaneously triggered an increase in the
amount of fake news circulating online. This fact became particularly evident
during the 2016 U.S. political elections and even more so with the advent of
the COVID-19 pandemic. Several research studies have shown how the effects of
fake news dissemination can be mitigated by promoting greater competence
through lifelong learning and discussion communities, and generally rigorous
training in the scientific method and broad interdisciplinary education. The
urgent need for models that can describe the growing infodemic of fake news
has been highlighted by the current pandemic. The resulting slowdown in
vaccination campaigns due to misinformation and generally the inability of
individuals to discern the reliability of information is posing enormous risks
to the governments of many countries. In this research using the tools of
kinetic theory we describe the interaction between fake news spreading and
competence of individuals through multi-population models in which fake news
spreads analogously to an infectious disease with different impact depending
on the level of competence of individuals. The level of competence, in
particular, is subject to an evolutionary dynamic due to both social
interactions between agents and external learning dynamics. The results show
how the model is able to correctly describe the dynamics of diffusion of fake
news and the important role of competence in their containment.

Keywords: fake news, compartmental models, competence, learning dynamics,
interacting agents, socio-economic kinetic models, mean-field models

##  1 Introduction

With the rise of the Internet, connections among people has become easier than
ever; so has been for the availability of information and its accessibility.
As such, the Internet is also the source of unprecedented collective
phenomena, some of which, however, cast shadows on our contemporary society
[3]. Indeed, the dissemination of heavily biased, or worse, downright false
information, once relatively moderate in size, and limited possibly to the
class of hoaxes and scams, exploded in the last decade, creating the broader
category of  _fake news_. The urgent need for models that can describe the
increasing spread of fake news has been highlighted by the current COVID-19
pandemic. Governments in many countries have found themselves in enormous
difficulty because of the slowdown in vaccination campaigns due to the spread
of false information and the inability of individuals to discern the
authenticity of such information [12, 24].

Let us first briefly recall some of the main challenges in modeling fake news.
First of all, one of the priorities is to introduce a definition of fake news
with a consensus that is wide enough to make research works relatable. In this
direction, one of the most accepted (though not universally so) traits for
fake news to be labeled so is _purpose_ : fake news is intentionally false
news [40, 37, 2, 17]. The concept of purpose seems to be useful when
differentiating between theories that are focused on the content rather than
on the conveyor [40]. In this sense, a piece of information that is
accidentally false (e.g., by inaccuracy) is substantially different, both
semantically and stylistically, from a maliciously fabricated one.

Next, the challenge is to detect fake news. The majority of recent lines of
research in the direction of automatic detection moves toward the aid of _big
data_ and artificial intelligence tools [34, 11, 33, 39]. An alternative
strategy is instead _component-based_ and focuses on the analysis of the
multiple parts involved in the diffusion of the fake news, that is, both on
the side of the creator and on the side of the user, but also on the
linguistics and semantics of the actual content and on its style, and finally
on the social context of the information (see [40] and references therein).

Different lines are possible, though. Information theory, for instance, has
been used to model fake news: in [8], fake news are defined as time series
with an inherent bias, that is, its expectation is nonzero, involved in a
stochastic process of which the user tries to judge the likelihood of the
truth, together with noise.

Finally, epidemiological theory has been proving for long to be fertile ground
for modeling of fake news [13, 14], especially in the somewhat broader
category of rumor-spreading dynamics. The analogy between rumors and epidemics
has often proved fruitful: Daley and Kendall [13] took inspiration by the
classical works of Kermack and Mckendrick [20, 18] to propose a SIR-like model
involving ignorant, spreader and stifler agents who played the role of the
susceptible, infectious and recovered ones in [20]. Since the seminal paper
[13], rumor-spreading dynamics has taken ideas from epidemiological models to
improve their prediction accuracy.

Recently, networks theory delved in this direction, too [10, 7, 41, 38].
Substantial research has merged networks and epidemiology through for instance
classical compartmental models like the SIR (both epidemiological in [31] and
rumor-oriented [32]), the SIS [16] and the SIRS [22, 35]. Moreover, more
sophisticated epidemiological models have been developed, like the SEIZ model
[5] describing the evolution in time of the compartments of susceptible,
exposed, infectious and skeptic agents, which has been adapted to the analysis
of fake news dissemination (see, e.g., [25, 19]). In particular, the tendency
seems to be to define the skeptic agents like the ones who are aware of the
information but do not actively spread it [19]. In a symmetric fashion,
spreaders need not to believe a piece of information to be able to spread it
(this is especially useful when thinking that bots are often encountered in
social networks [36], both for legitimate and malicious purposes). This
descriptions are also sensible in terms of matching the model with data
available.

In this paper we follow this pathway: borrowing ideas from kinetic theory [15,
27], we combine a classical compartmental approach inspired by epidemiology
[18, 20] with a kinetic description of the effects of competence [29, 28]. We
refer also to the recent work [6] concerning evolutionary models for
knowledge. In fact, the first wave of initiatives addressing fake news focused
on news production by trying to limit citizen exposure to fake news. This can
be done by fact-checking, labeling stories as fake, and eliminating them
before they spread. Unfortunately, this strategy has already been proven not
to work, it is indeed unrealistic to expect that only high quality, reliable
information will survive. As a result, governments, international
organizations, and social media companies have turned their attention to
digital news consumers, and particularly children and young adults. From
national campaigns in several countries to the OECD, there is a wave of action
to develop new curricula, online learning tools, and resources that foster the
ability to “spot fake news” [26].

It is therefore of paramount importance to build models capable of describing
the interplay between the dissemination of fake news and the creation of
competence among the population. To this end, the approach we have followed in
this paper falls within the recent socio-economic modeling described by
kinetic equations (see [27] for a recent monograph on the subject). More
precisely, we adapted the competence model introduced in [29, 28] to a
compartmental model describing fake news dissemination. Such a model allows
not only to introduce competence as a static feature of the dynamics but as an
evolutionary component both taking into account learning by interactions
between agents and possible interventions aimed at educating individuals in
the ability to identify fake news. Furthermore, in our modeling approach
agents may have memory of fake news and as such be permanently immune to it
once it has been detected, or fake news may not have any inherent
peculiarities that would make it memorable enough for the population to
immunize themselves against it in the future. The approach can be easily
adapted to other compartmental models present in the literature, like the ones
previously discussed [5, 25, 32].

The rest of the manuscript is organized as follows. In Section 2 we introduce
the structured kinetic model describing the spread of fake news in presence of
different competence levels among individuals. The main properties of the
resulting kinetic models are also analyzed. Next, Section 3 is devoted to
study the Fokker-Planck approximation of the kinetic model and to derive the
corresponding stationary states in terms of competence. Several numerical
results are then presented in Section 4 that illustrate the theoretical
findings and the capability of the model to describe transition effects in the
spread of fake news due to the interaction between epidemiological and
competence parameters. Some concluding remarks are reported in the last
Section together with details on the theoretical results and the numerical
methods in two separate appendices.

##  2 Fake news spreading in a socially structured population

In this section, we introduce a structured model for the dissemination of fake
news in presence of different levels of skills among individuals in detecting
the actual veracity of information, by combining a compartmental model in
epidemiology and rumor-spreading analysis [18, 14] with the kinetic model of
competence evolution proposed in [29].

We consider a population of individuals divided into four classes. The
oblivious ones, still not aware of the news; the reflecting ones, who are
aware of the news and are evaluating how to act; the spreader ones, who
actively disseminate the news and the silent ones, who have recognized the
fake news and do not contribute to its spread. Terminology, when describing
this compartmental models, is not fully established; however, the dominant
one, inspired by epidemiology, refers to the definitions provided by Daley
[14] of a population composed of ignorant, spreader and stifler individuals.
The class of reflecting agents can be referred to as a group that has a time-
delay before taking a decision and enter an active compartment [5, 25].

Notation, i.e., the choice of letters to represent the compartments, is even
more scattered and somewhat confusing. In Table 1 for readers’ convenience we
have summarized some of the different possible choices of letters and
terminology found in literature. Given the widespread use of epidemiological
models compared to fake news models, in order to make the analogies easier to
understand, we chose to align with notations conventionally used in
epidemiology. Therefore, in the rest of the paper we will describe the
population in terms of susceptible agents (S), who are the oblivious ones;
exposed agents (E), who are in the time-delay compartment after exposure and
before shifting into an active class; infectious agents (I), who are the
spreader ones and finally removed agents (R) who are aware of the news but not
actively engaging in its spread.

Note that this subdivision of the population does not take into account actual
beliefs of agents about the truth of the news, so that removed agents, for
instance, need not be actually skeptic, nor the spreaders need to actually
believe the news. To simplify the mathematical treatment, as in the original
works by Daley and Kendall [13, 14], we ignored the possible ‘active’ effects
of the population of removed individuals by interacting with other
compartments and producing immunization among susceptible (the role of skeptic
individuals in [5, 25]) and remission among spreaders (the role of stiflers in
[32]). Of course, the model easily generalizes to include these additional
dynamics.

The main novelty in our approach is to consider an additional structure on the
population based on the concept of competence of the agents, here understood
as the ability to assess and evaluate information.

SEIR (this paper) | DK [13, 14] | ISR [32] | SEIZR [5] | SEIZ [25]
---|---|---|---|---
Category name
Susceptible | Ignorant | Ignorant | Susceptible | Susceptible
Exposed | - | - | Idea incubator | Exposed
Infectious | Spreader | Spreader | Idea adopter | Infectious
Removed | Stifler | Stifler | Skeptic/Recovered | Skeptic
Variable notation
S | X | I | S | S
E | - | - | E | E
I | Y | S | I | I
R | Z | R | Z/R | Z
Table 1: Different compartments and notations for some of the models found in
literature.

Let us suppose that agents in the system are completely characterized by their
competence $x\in X\subseteq\mathbb{R}^{+}$, measured in a suitable unit. We
denote by $f_{S}=f_{S}(x,t)$, $f_{E}=f_{E}(x,t)$, $f_{I}=f_{I}(x,t)$,
$f_{R}=f_{R}(x,t)$, the competence distribution at time $t>0$ of susceptible,
exposed, infectious and removed individuals, respectively. Aside from natality
or mortality concerns (i.e., the social network is a closed system—nobody
enters or leaves it during the diffusion of the fake news, which is a common
assumption, based on the average lifespan of fake news) we therefore have:

$\int_{X}\bigl{(}f_{S}(x,t)+f_{E}(x,t)+f_{I}(x,t)+f_{R}(x,t)\bigr{)}\,\mathrm{d}x=1,\quad
t>0,$

which implies that we will refer to

$\displaystyle S(t)$ $\displaystyle=\int_{X}f_{S}(x,t)\,\mathrm{d}x,$
$\displaystyle E(t)$ $\displaystyle=\int_{X}f_{E}(x,t)\,\mathrm{d}x,$
$\displaystyle I(t)$ $\displaystyle=\int_{X}f_{I}(x,t)\,\mathrm{d}x,$
$\displaystyle R(t)$ $\displaystyle=\int_{X}f_{R}(x,t)\,\mathrm{d}x$

as the fractions of the population that are susceptible, exposed, infected, or
recovered respectively. We also denote the relative mean competences as

$\displaystyle m_{S}(t)$ $\displaystyle=\int_{X}xf_{S}(x,t)\,\mathrm{d}x,$
$\displaystyle m_{E}(t)$ $\displaystyle=\int_{X}xf_{E}(x,t)\,\mathrm{d}x,$
$\displaystyle m_{I}(t)$ $\displaystyle=\int_{X}xf_{I}(x,t)\,\mathrm{d}x,$
$\displaystyle m_{R}(t)$ $\displaystyle=\int_{X}xf_{R}(x,t)\,\mathrm{d}x.$

###  2.1 A SEIR model describing fake news dynamics

The fake news dynamics proceeds as follows: a susceptible agent gets to know
it by a spreader. At this point, the now-exposed agent evaluates the piece of
information—the reflecting, or delay, stage—and decide whether to share it
with other individuals (and turning into a spreader themselves) or to keep
silent, removing themselves by the dissemination process.

Figure 1: SEIR diagram with transition rates.

When the dynamic is independent from the knowledge of individuals, the model
can be expressed by the following system of ODEs

$\left\\{\begin{aligned} \frac{\mathrm{d}S}{\mathrm{d}t}&=-\beta
SI+(1-\alpha)\gamma I\\\ \frac{\mathrm{d}E}{\mathrm{d}t}&=\beta SI-\delta E\\\
\frac{\mathrm{d}I}{\mathrm{d}t}&=(1-\eta)\delta E-\gamma I,\\\
\frac{\mathrm{d}R}{\mathrm{d}t}&=\eta\delta E+\alpha\gamma
I,\end{aligned}\right.$ (1)

with $S+E+I+R=1$ and where $\beta$ is the contact rate between the class of
the susceptible and the class of infectious, $\delta$ is the rate at which
agents make their decision about spreading the news or not, $1-\eta$ is the
portion of agents who become infectious and $\gamma$ is the rate at which
spreaders remove themselves from the compartment, due, e.g., to loss of
interest in sharing the news or forgetfulness. Finally, $\alpha$ is related to
the specificity of the fake news and the probability of individulas to
remember it. A probability of $0$ means that the fake news has not any
inherent peculiarity (e.g., in terms of content, structure, style, …) that can
make it memorable enough for the population to ‘immunize’ against it in the
future, while a probability of $1$ allows for the agents to have the full
ability to not fall for that fake news a second time. The various parameters
have been summarized in Table 2. The diagram of the SEIR model (1) is shown in
Figure 1.

Parameter | Definition
---|---
$\beta$ | contact rate between susceptible and infected individuals
$1/\delta$ | average decision time on whether or not to spread fake news
$\eta$ | probability of deciding not to spread fake news
$1/\gamma$ | average duration of a fake news
$\alpha$ | probability of remembering fake news
Table 2: Parameters definition in the SEIR model (1).

It is straightforward to notice that when $\alpha$ and $\eta$ are zero, system
(1) specializes in a classic SEIS epidemiological model. This is consistent
with treating the dissemination of non-specific fake news in a population as
the spread of a disease with multiple strains, for which a durable
immunization is never attained. In this case system (1) has two equilibrium
states: a disease-free equilibrium state $(1,0,0)$ and an endemic equilibrium
state $\tilde{P}=(\tilde{S},\tilde{E},\tilde{I})$ where

$\tilde{S}=\frac{1}{R_{0}},\qquad\tilde{E}=\frac{\gamma}{\gamma+\delta}\biggl{(}1-\frac{1}{R_{0}}\biggr{)},\qquad\tilde{I}=\frac{\delta}{\gamma+\delta}\biggl{(}1-\frac{1}{R_{0}}\biggr{)},$
(2)

and $R_{0}=\beta/\gamma$ is the basic reproduction number. It is known [21]
that if $R_{0}>1$ the endemic equilibrium state $\tilde{P}$ of system (1) is
globally asymptotically stable.

If instead $\alpha>0$ or $\eta>0$, there also is the possibility to
permanently immunize against fake news with those traits; moreover, both
infectious and exposed agents eventually vanish, leaving only the susceptible
and removed compartments populated. In the case of maximum specificity of the
fake news, i.e., $\alpha=1$, the stationary equilibrium state has the form

$S(t)\to S^{\infty},\quad E(t)\to 0,\quad I(t)\to 0,\quad R(t)\to
R^{\infty}=1-S^{\infty},$ (3)

where $S^{\infty}$ is solution of the nonlinear equation

$\log\frac{S_{0}}{S^{\infty}}=\frac{\beta}{\gamma}(1-\eta)(1-S^{\infty}),$ (4)

in which $S_{0}$ is the initial datum $S(t=0)$.

We refer to [5, 25, 32] for the inclusion of additional interaction dynamics,
taking into account counter-information effects due to the removed population
interacting against susceptible and infectious, and the relative analysis of
the resulting equilibrium states.

###  2.2 The interplay with competence and learning

In the following, we combine the evolution of the densities according to the
SEIR model (1) with the competence dynamics proposed in [29]. We refer to the
degree of competence that an individual can gain or loose in a single
interaction from the background as $z\in\mathbb{R}^{+}$; in what follows we
denote by $C(z)$ the bounded-mean distribution of $z$, satisfying

$\int_{\mathbb{R}^{+}}C(z)\,\mathrm{d}z=1,\quad\int_{\mathbb{R}^{+}}zC(z)\,\mathrm{d}z=m_{B}.$

Assuming a susceptible agent has a competence level $x$ and interacts with
another one belonging to the various compartments in the population and having
a competence level $x_{\ast}$, their levels after the interaction will be
given by

$\displaystyle\left\\{\begin{aligned}
x^{\prime}&=(1-\lambda_{S}(x))x+\lambda_{CJ}(x)x_{\ast}+\lambda_{BS}(x)z+\kappa_{SJ}x\\\
x_{\ast}^{\prime}&=(1-\lambda_{J}(x_{\ast}))x_{\ast}+\lambda_{CS}(x_{\ast})x+\lambda_{BJ}(x_{\ast})z+\tilde{\kappa}_{SJ}x_{\ast},\end{aligned}\right.$
$\displaystyle J\in\\{S,E,I,R\\}$ (5)

where $\lambda_{S}(\cdot)$ and $\lambda_{BS}(\cdot)$ quantify the amount of
competence lost by susceptible individuals by the natural process of
forgetfulness and the amount gained by susceptible individuals from the
background, respectively. $\lambda_{CJ}$, instead, models the competence
gained through the interaction with members of the class $J$, with
$J\in\\{S,E,I,R\\}$; a possible choice for $\lambda_{CJ}(x)$ is
$\lambda_{CJ}(x)=\lambda_{CJ}\chi(x\geq\bar{x})$, where $\chi(\cdot)$ is the
characteristic function and $\bar{x}\in X$ a minimum level of competence
required to the agents for increasing their own skills by interactions.
Finally, $\kappa_{SJ}$ and $\tilde{\kappa}_{SJ}$ are independent and
identically distributed zero-mean random variables with the same variance
$\sigma(t)$ to consider the non-deterministic nature of the competence
acquisition process.

The binary interactions involving the exposed agents can be similarly defined

$\displaystyle\left\\{\begin{aligned}
x^{\prime}&=(1-\lambda_{E}(x))x+\lambda_{CJ}(x)x_{\ast}+\lambda_{BE}(x)z+\kappa_{EJ}x\\\
x_{\ast}^{\prime}&=(1-\lambda_{J}(x_{\ast}))x_{\ast}+\lambda_{CE}(x_{\ast})x+\lambda_{BJ}(x_{\ast})z+\tilde{\kappa}_{EJ}x_{\ast},\end{aligned}\right.$
$\displaystyle J\in\\{S,E,I,R\\}$ (6)

the same holds for the interactions concerning the infectious fraction of the
population

$\displaystyle\left\\{\begin{aligned}
x^{\prime}&=(1-\lambda_{I}(x))x+\lambda_{CJ}(x)x_{\ast}+\lambda_{BI}(x)z+\kappa_{IJ}x\\\
x_{\ast}^{\prime}&=(1-\lambda_{J}(x_{\ast}))x_{\ast}+\lambda_{CI}(x_{\ast})x+\lambda_{BJ}(x_{\ast})z+\tilde{\kappa}_{IJ}x_{\ast},\end{aligned}\right.$
$\displaystyle J\in\\{S,E,I,R\\}$ (7)

and finally we have the interactions regarding the removed agents

$\displaystyle\left\\{\begin{aligned}
x^{\prime}&=(1-\lambda_{R}(x))x+\lambda_{CJ}(x)x_{\ast}+\lambda_{BR}(x)z+\kappa_{RJ}x\\\
x_{\ast}^{\prime}&=(1-\lambda_{J}(x_{\ast}))x_{\ast}+\lambda_{CR}(x_{\ast})x+\lambda_{BJ}(x_{\ast})z+\tilde{\kappa}_{RJ}x_{\ast},\end{aligned}\right.$
$\displaystyle J\in\\{S,E,I,R\\}.$ (8)

It is reasonable to assume that both the processes of gain and loss of
competence from the interaction with other agents or with the background in
(5)–(8) are bounded by zero. Therefore we suppose that if
$J,H\in\\{S,E,I,R\\}$, and if
$\lambda_{J}\in[\lambda_{J}^{-},\lambda_{J}^{+}]$, with $\lambda_{J}^{-}>0$
and $\lambda_{J}^{+}<1$, and $\lambda_{CJ}(x),\lambda_{BJ}(x)\in[0,1]$ then
$\kappa_{HJ}$ may, for example, be uniformly distributed in
$[-1+\lambda_{J}^{+},1-\lambda_{J}^{+}]$.

In order to combine the compartmental model SEIR with the evolution of the
competence levels given by equations (5)–(8) we introduce the interaction
operator $Q_{HJ}(\cdot,\cdot)$ following the standard Boltzmann-type theory
[27]. As earlier, we will denote with $J$ a suitable compartment of the
population, i.e., $H,J\in\\{S,E,I,R\\}$, and we will use the brackets
$\langle\cdot\rangle$ to indicate the expectation with respect to the random
variable $\kappa_{HJ}$. Thus, if $\psi(x)$ is an observable function, then the
action of $Q_{HJ}(f_{H},f_{J})(x,t)$ on $\psi(x)$ is given by

$\int_{\mathbb{R}^{+}}Q_{SJ}(f_{S},f_{J})\psi(x)\,\mathrm{d}x=\left\langle\int_{\mathbb{R}^{2}_{+}}f_{S}(x,t)f_{J}(x_{\ast},t)\bigl{(}\psi(x^{\prime})-\psi(x)\bigr{)}\,\mathrm{d}x_{\ast}\mathrm{d}x\right\rangle,$
(9)

with $x^{\prime}$ defined by (5)

$\int_{\mathbb{R}^{+}}Q_{EJ}(f_{E},f_{J})\psi(x)\,\mathrm{d}x=\left\langle\int_{\mathbb{R}^{2}_{+}}f_{E}(x,t)f_{J}(x_{\ast},t)\bigl{(}\psi(x^{\prime})-\psi(x)\bigr{)}\,\mathrm{d}x_{\ast}\mathrm{d}x\right\rangle,$
(10)

with $x^{\prime}$ defined by (6)

$\int_{\mathbb{R}^{+}}Q_{IJ}(f_{I},f_{J})\psi(x)\,\mathrm{d}x=\left\langle\int_{\mathbb{R}^{2}_{+}}f_{I}(x,t)f_{J}(x_{\ast},t)\bigl{(}\psi(x^{\prime})-\psi(x)\bigr{)}\,\mathrm{d}x_{\ast}\mathrm{d}x\right\rangle,$
(11)

with $x^{\prime}$ defined by (7),

$\int_{\mathbb{R}^{+}}Q_{RJ}(f_{R},f_{J})\psi(x)\,\mathrm{d}x=\left\langle\int_{\mathbb{R}^{2}_{+}}f_{R}(x,t)f_{J}(x_{\ast},t)\bigl{(}\psi(x^{\prime})-\psi(x)\bigr{)}\,\mathrm{d}x_{\ast}\mathrm{d}x\right\rangle,$
(12)

with $x^{\prime}$ defined by (8). All the above operators preserve the total
number of agents as the unique interaction invariant, corresponding to
$\psi(\cdot)\equiv 1$.

The system then reads:

$\left\\{\begin{aligned} \frac{\partial f_{S}(x,t)}{\partial
t}&=-K(x,t)f_{S}(x,t)+(1-\alpha(x))\gamma(x)f_{I}(x,t)+\sum_{\mathclap{J\in\\{S,E,I,R\\}}}Q_{SJ}(f_{S},f_{J})(x,t),\\\
\frac{\partial f_{E}(x,t)}{\partial
t}&=K(x,t)f_{S}(x,t)-\delta(x)f_{E}(x,t)+\sum_{\mathclap{J\in\\{S,E,I,R\\}}}Q_{EJ}(f_{E},f_{J})(x,t),\\\
\frac{\partial f_{I}(x,t)}{\partial
t}&=\delta(x)(1-\eta(x))f_{E}(x,t)-\gamma(x)f_{I}(x,t)+\sum_{\mathclap{J\in\\{S,E,I,R\\}}}Q_{IJ}(f_{I},f_{J})(x,t),\\\
\frac{\partial f_{R}(x,t)}{\partial
t}&=\delta(x)\eta(x)f_{E}(x,t)+\alpha(x)\gamma(x)f_{I}(x,t)+\sum_{\mathclap{J\in\\{S,E,I,R\\}}}Q_{RJ}(f_{R},f_{J})(x,t),\end{aligned}\right.$
(13)

where the function

$K(x,t)=\int_{\mathbb{R}^{+}}\beta(x,x_{\ast})f_{I}(x_{\ast},t)\,\mathrm{d}x_{\ast}$

is responsible for the contagion, $\beta(x,x_{\ast})$ being the contact rate
between agents with competence levels $x$ and $x_{\ast}$. In the above
formulation we also assumed $\beta$, $\gamma$, $\delta$, $\eta$ and $\alpha$
functions of $x$. Note that, clearly, the most important parameters influenced
by individuals’ competence are $\beta(x,x_{\ast})$, since individuals have the
highest rates of contact with people belonging to the same social class, and
thus with a similar level of competence, $\delta(x)$ as individuals with
greater competence invest more time in checking the authenticity of
information, and $\eta(x)$, which characterizes individuals’ decision to
spread fake news. On the other hand, the values of $\gamma$ and $\alpha$ we
may assume to be less influenced by the level of expertise of individuals.

###  2.3 Properties of the kinetic SEIR model with competence

In this section we analyze some of the properties of the Boltzmann system
(13). First let us consider the reproducing ratio in presence of knowledge.

By integrating system (13) against $x$, and considering only the compartments
of individuals which may disseminate the fake news we have

$\left\\{\begin{aligned}
\frac{\mathrm{d}E(t)}{\mathrm{d}t}&=\int_{X}K(x,t)f_{S}(x,t)\,dx-\int_{X}\delta(x)f_{E}(x,t)\,dx,\\\
\frac{\mathrm{d}I(t)}{\mathrm{d}t}&=\int_{X}\delta(x)(1-\eta(x))f_{E}(x,t)\,dx-\int_{X}\gamma(x)f_{I}(x,t)\,dx.\\\
\end{aligned}\right.$ (14)

In the above derivation we used the fact that the Boltzmann interaction terms
describing knowledge evolution among agents preserve the total number of
individuals and therefore vanish. Following the analysis in [4], and omitting
the details for brevity, we obtain a reproduction number generalizing the
classical one

$R_{0}(t)=\frac{\int_{X}K(x,t)f_{S}(x,t)\,dx}{\int_{X}\gamma(x)f_{I}(x,t)\,dx}.$
(15)

Next, following [15], we can prove uniqueness of the solution of (13) in the
simplified case of constant parameters: $\beta(x,x_{\ast})=\beta>0$,
$\gamma(x)=\gamma>0$, $\delta(x)=\delta>0$, $\eta(x)=\eta\in[0,1]$,
$\alpha(x)=\alpha\in[0,1]$. In this case, exploiting the fact that the
interaction operator $Q(\cdot,\cdot)$ has a natural connection with the
Fourier transform by choosing its kernel $\mathrm{e}^{-ix\xi}$ as test
function, we can analyze the system (13) with the Fourier transforms of the
densities as unknowns.

Indeed, given a function $f(x)\in L_{1}(\mathbb{R}^{+})$, its Fourier
transform is defined as

$\hat{f}(\xi)=\int_{\mathbb{R}}\mathrm{e}^{-ix\xi}f(x)\,\mathrm{d}x.$

The system (13) becomes

$\left\\{\begin{aligned} \frac{\partial\hat{f}_{S}(\xi,t)}{\partial t}&=-\beta
I(t)\hat{f}_{S}(\xi,t)+(1-\alpha)\gamma\hat{f}_{I}(\xi,t)+\sum_{\mathclap{J\in\\{S,E,I,R\\}}}\hat{Q}_{SJ}(\hat{f}_{S},\hat{f}_{J})(\xi,t),\\\
\frac{\partial\hat{f}_{E}(\xi,t)}{\partial t}&=\beta
I(t)\hat{f}_{S}(\xi,t)-\delta\hat{f}_{E}(\xi,t)+\sum_{\mathclap{J\in\\{S,E,I,R\\}}}\hat{Q}_{EJ}(\hat{f}_{E},\hat{f}_{J})(\xi,t),\\\
\frac{\partial\hat{f}_{I}(\xi,t)}{\partial
t}&=\delta(1-\eta)\hat{f}_{E}(\xi,t)-\gamma\hat{f}_{I}(\xi,t)+\sum_{\mathclap{J\in\\{S,E,I,R\\}}}\hat{Q}_{IJ}(\hat{f}_{I},\hat{f}_{J})(\xi,t),\\\
\frac{\partial\hat{f}_{R}(\xi,t)}{\partial
t}&=\delta\eta\hat{f}_{E}(\xi,t)+\alpha\gamma\hat{f}_{I}(\xi,t)+\sum_{\mathclap{J\in\\{S,E,I,R\\}}}\hat{Q}_{RJ}(\hat{f}_{R},\hat{f}_{J})(\xi,t),\end{aligned}\right.$
(16)

where the operators $\hat{Q}_{HJ}(\hat{f}_{H},\hat{f}_{J})$ are defined in
terms of the Fourier transforms of their arguments for $J\in\\{S,E,I,R\\}$, so
that

$\hat{Q}_{HJ}(\hat{f}_{H},\hat{f}_{J})=\langle\hat{f}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)\rangle\hat{f}_{J}(\lambda_{CJ}\xi,t)-\hat{f}_{H}(\xi,t)J(t),$

where $A_{HJ}$, with $H,J\in\\{S,E,I,R\\}$ is defined as

$A_{HJ}=1-\lambda_{H}+\kappa_{H}.$ (17)

We suppose that the parameters satisfy the condition

$\nu=\max_{H,J\in\\{S,E,I,R\\}}[\lambda_{CJ}^{2}+\langle
A_{JH}^{2}\rangle]<1,$ (18)

which will prove useful in the proof.

As in [15] we recall a class of metrics which is of natural use in bilinear
Boltzmann equations [27]. Let $f$ and $g$ be probability densities. Then, for
$s>0$ we define

$d_{s}(f,g)=\sup_{\xi\in\mathbb{R}}\frac{\lvert\hat{f}(\xi)-\hat{g}(\xi)\rvert}{\lvert\xi\rvert^{s}},$
(19)

which is finite whenever $f$ and $g$ have equal moments up to the integer part
of $s$ or to $s-1$ if $s$ is an integer.

We have the following result.

######  Theorem 1.

Let $J\in\\{S,E,I,R\\}$, and let $f_{J}(x,t)$ and $g_{J}(x,t)$ be two
solutions of the system (13) with initial values $f_{J}(x,0)$ and $g_{J}(x,0)$
such that $d_{2}(f_{J}(x,0),g_{J}(x,0)$ is finite. Then, condition (18)
implies that the Fourier based distance $d_{2}(f_{J}(x,t),g_{J}(x,t)$ decays
exponentially (in time) to zero, so that

$\sum_{\mathclap{J\in\\{S,E,I,R\\}}}d_{2}(f_{J}(x,t),g_{J}(x,t)\leq\sum_{\mathclap{J\in\\{S,E,I,R\\}}}d_{2}(f_{J}(x,0),g_{J}(x,0)\mathrm{e}^{-(1-\nu)t}.$

For the details of the proof we refer to Appendix A.

##  3 Mean-field approximation

A highly useful tool to obtain information analytically on the large-time
behavior of Boltzmann-type models are scaling techniques; in particular the
so-called quasi-invariant limit [27], which allows to derive the corresponding
mean-field description of the kinetic model (13).

Indeed, let us consider the case in which the interactions between agents
produce small variations of the competence. We scale the quantities involved
in the binary interactions (5)-(8) accordingly

$\displaystyle\lambda_{CJ}\to\varepsilon\lambda_{CJ},$
$\displaystyle\lambda_{BJ}\to\varepsilon\lambda_{BJ},$
$\displaystyle\lambda_{J}\to\varepsilon\lambda_{J},$
$\displaystyle\sigma\to\varepsilon\sigma,$ (20)

where $J\in\\{S,E,I,R\\}$ and the functions involved in the dissemination of
the fake news, as well

$\displaystyle\beta(x,x_{\ast})\to\varepsilon\beta(x,x_{\ast}),$
$\displaystyle\gamma(x)\to\varepsilon\gamma(x),$
$\displaystyle\delta(x)\to\varepsilon\delta(x),$
$\displaystyle\eta(x)\to\varepsilon\eta(x).$ (21)

We denote by $Q^{\varepsilon}_{HJ}(\cdot,\cdot)$ the scaled interaction terms.
Omitting the dependence on time on mean values and re-scaling time as $t\to
t/\varepsilon$, we obtain up to $\mathcal{O}(\varepsilon)$

$\displaystyle\frac{1}{\varepsilon}\int_{\mathbb{R}^{+}}Q_{SJ}^{\varepsilon}(f_{S},f_{J})\psi(x)\,\mathrm{d}x$
$\displaystyle\approx\int_{\mathbb{R}^{+}}\biggl{[}-\psi(x)^{\prime}(\lambda_{S}xJ-\lambda_{CJ}m_{J}-\lambda_{BS}m_{B}J)+\frac{\sigma}{2}\psi(x)^{\prime\prime}x^{2}J\biggr{]}f_{S}(x,t)\,\mathrm{d}x$
$\displaystyle\frac{1}{\varepsilon}\int_{\mathbb{R}^{+}}Q_{EJ}^{\varepsilon}(f_{E},f_{J})\psi(x)\,\mathrm{d}x$
$\displaystyle\approx\int_{\mathbb{R}^{+}}\biggl{[}-\psi(x)^{\prime}(\lambda_{E}xJ-\lambda_{CJ}m_{J}-\lambda_{BE}m_{B}J)+\frac{\sigma}{2}\psi(x)^{\prime\prime}x^{2}J\biggr{]}f_{E}(x,t)\,\mathrm{d}x$
$\displaystyle\frac{1}{\varepsilon}\int_{\mathbb{R}^{+}}Q_{IJ}^{\varepsilon}(f_{I},f_{J})\psi(x)\,\mathrm{d}x$
$\displaystyle\approx\int_{\mathbb{R}^{+}}\biggl{[}-\psi(x)^{\prime}(\lambda_{I}xJ-\lambda_{CJ}m_{J}-\lambda_{BI}m_{B}J)+\frac{\sigma}{2}\psi(x)^{\prime\prime}x^{2}J\biggr{]}f_{I}(x,t)\,\mathrm{d}x$
$\displaystyle\frac{1}{\varepsilon}\int_{\mathbb{R}^{+}}Q_{RJ}^{\varepsilon}(f_{R},f_{J})\psi(x)\,\mathrm{d}x$
$\displaystyle\approx\int_{\mathbb{R}^{+}}\biggl{[}-\psi(x)^{\prime}(\lambda_{R}xJ-\lambda_{CJ}m_{J}-\lambda_{BR}m_{B}J)+\frac{\sigma}{2}\psi(x)^{\prime\prime}x^{2}J\biggr{]}f_{R}(x,t)\,\mathrm{d}x,$

where we used a Taylor expansion for small values of $\varepsilon$ of

$\psi(x^{\prime})=\psi(x)+(x^{\prime}-x)\psi^{\prime}(x)+\frac{(x^{\prime}-x)^{2}}{2}\psi^{\prime\prime}(x)+\mathcal{O}(\varepsilon^{2})$

in (9)-(12) and the scaled interaction rules (5)-(8).

###  3.1 Stationary solutions of Fokker-Planck SEIR models

Let us impose that $\varepsilon\to 0$, following [27] from the computations of
the previous section we formally obtain the Fokker-Planck system

$\displaystyle\frac{\partial f_{S}(x,t)}{\partial t}$
$\displaystyle=-K(x,t)f_{S}(x,t)+(1-\alpha(x))\gamma(x)f_{I}(x,t)+\frac{\partial}{\partial
x}[(x\lambda_{S}-\overline{m}(t)-\lambda_{BS}m_{B})f_{S}(x,t)]$
$\displaystyle\phantom{=}{}+\frac{\sigma}{2}\frac{\partial^{2}}{\partial
x^{2}}(x^{2}f_{S}(x,t))$ (22) $\displaystyle\frac{\partial
f_{E}(x,t)}{\partial t}$
$\displaystyle=K(x,t)f_{S}(x,t)-\delta(x)f_{E}(x,t)+\frac{\partial}{\partial
x}[(x\lambda_{E}-\overline{m}(t)-\lambda_{BE}m_{B})f_{E}(x,t)]$
$\displaystyle\phantom{=}{}+\frac{\sigma}{2}\frac{\partial^{2}}{\partial
x^{2}}(x^{2}f_{E}(x,t))$ (23) $\displaystyle\frac{\partial
f_{I}(x,t)}{\partial t}$
$\displaystyle=\delta(x)(1-\eta(x))f_{E}(x,t)-\gamma(x)f_{I}(x,t)+\frac{\partial}{\partial
x}[(x\lambda_{I}-\overline{m}(t)-\lambda_{BI}m_{B})f_{I}(x,t)]$
$\displaystyle\phantom{=}{}+\frac{\sigma}{2}\frac{\partial^{2}}{\partial
x^{2}}(x^{2}f_{I}(x,t))$ (24) $\displaystyle\frac{\partial
f_{R}(x,t)}{\partial t}$
$\displaystyle=\delta(x)\eta(x)f_{E}(x,t)+\alpha(x)\gamma(x)f_{I}(x,t)+\frac{\partial}{\partial
x}[(x\lambda_{R}-\overline{m}(t)-\lambda_{BR}m_{B})f_{R}(x,t)]$
$\displaystyle\phantom{=}{}+\frac{\sigma}{2}\frac{\partial^{2}}{\partial
x^{2}}(x^{2}f_{R}(x,t))$ (25)

where now

$\overline{m}(t)=\lambda_{CS}m_{S}(t)+\lambda_{CE}m_{E}(t)+\lambda_{CI}m_{I}(t)+\lambda_{CR}m_{R}(t).$

We can consider the mean values system associated to (22)–(25) in the case of
constant epidemiological parameters

$\displaystyle\frac{\mathrm{d}m_{S}(t)}{\mathrm{d}t}$ $\displaystyle=-\beta
I(t)m_{S}(t)+(1-\alpha)\gamma m_{I}(t)+\lambda S(t)(m(t)-m_{B})/2-\lambda
m_{S}(t)$ (26) $\displaystyle\frac{\mathrm{d}m_{E}(t)}{\mathrm{d}t}$
$\displaystyle=\beta I(t)m_{S}(t)-\delta m_{E}(t)+\lambda
E(t)(m(t)-m_{B})/2-\lambda m_{E}(t)$ (27)
$\displaystyle\frac{\mathrm{d}m_{I}(t)}{\mathrm{d}t}$
$\displaystyle=\delta(1-\eta)m_{E}(t)-\gamma m_{I}(t)+\lambda
I(t)(m(t)-m_{B})/2-\lambda m_{I}(t)$ (28)
$\displaystyle\frac{\mathrm{d}m_{R}(t)}{\mathrm{d}t}$
$\displaystyle=\delta\eta m_{E}(t)+\alpha\gamma m_{I}(t)+\lambda
R(t)(m(t)-m_{B})/2-\lambda m_{R}(t),$ (29)

with

$m(t)=m_{S}(t)+m_{E}(t)+m_{I}(t)+m_{R}(t).$

In the case $\alpha>0$ or $\eta>0$, we know that $E(t)\to 0$, $I(t)\to 0$,
$S(t)\to S^{\infty}$ and $R(t)\to R^{\infty}=1-S^{\infty}$ due to mass
conservation, so that $m_{E}(t)\to 0$ and $m_{I}(t)\to 0$ as well. Thus,
adding all the equations together leads us to

$\displaystyle\frac{\lambda}{2}(m_{S}^{\infty}+m_{R}^{\infty})+\frac{\lambda}{2}m_{B}$
$\displaystyle=\lambda(m_{S}^{\infty}+m_{R}^{\infty}),$ (30)

i.e., $m_{S}^{\infty}+m_{R}^{\infty}=m_{B}$. At this point, adding together
equations (22) to (25) gives us

$0=\lambda\frac{\partial}{\partial
x}(x-m_{B})\sum_{\mathclap{J\in\\{S,R\\}}}f_{J}^{\infty}(x)+\frac{\sigma}{2}\frac{\partial^{2}}{\partial
x^{2}}\biggl{(}x^{2}\sum_{\mathclap{\quad
J\in\\{S,R\\}}}f_{J}^{\infty}(x)\biggr{)},$

which has as solution an inverse Gamma density

$f^{\infty}(x)=f_{S}^{\infty}(x)+f_{R}^{\infty}(x)=\frac{k^{\mu}}{\Gamma(\mu)}\frac{\mathrm{e}^{-k/x}}{x^{1+\mu}},$
(31) $\displaystyle\mu=1+\frac{2\lambda}{\sigma},$ $\displaystyle\quad
k=(\mu-1)m_{B}.$

It is straightforward to see that the scaled Gamma densities

$f_{S}^{\infty}(x)=S^{\infty}\frac{k^{\mu}}{\Gamma(\mu)}\frac{\mathrm{e}^{-k/x}}{x^{1+\mu}}\quad
f_{R}^{\infty}(x)=(1-S^{\infty})\frac{k^{\mu}}{\Gamma(\mu)}\frac{\mathrm{e}^{-k/x}}{x^{1+\mu}}$

are solutions of the system (22)–(25).

If, instead, $\alpha=\eta=0$, we find again the same solution as (31), but in
this case $J\to\tilde{J}$, where $\tilde{J}$ are defined as in (2).

In Figure 2 we report two examples of the stationary solutions where we chose
the competence variable $z$ to be uniformly distributed in $[0,1]$: in the
first case (left) we considered $\alpha=\eta=0$, while in the second case
(right) we set $\alpha=0.2$ and $\eta=0.1$.

Figure 2: Exact solutions for competence distributions at the end of
epidemic(31) for $\lambda=0.1$, $\mu=5$, $m_{B}=0.5$ (solid), $m_{B}=3$ (dash-
dotted), $\tilde{S}=0.5$, $\tilde{E}=0.1$, $\tilde{I}=0.4$ (left) and
$S^{\infty}=0.6$, $R^{\infty}=0.4$ (right). Left: case $\alpha=\eta=0$; right:
case $\alpha=0.2$, $\eta=0.1$.

##  4 Numerical examples

In this section we present some numerical tests to show the characteristics of
the model in describing the dynamics of fake news dissemination in a
population with a competence-based structure.

To begin with, we validate the Fokker-Planck model obtained as the _quasi-
invariant limit_ of the Boltzmann equation: we will do so through a Monte
Carlo method for the competence distribution (see [27], Chapter 5 for more
details). Next, we approximate the Fokker-Planck systems (22)–(25) by
generalizing the structure-preserving numerical scheme [30] to explore the
interplay between competence and disseminating dynamics in the more realistic
case of epidemiological parameters dependent on the competence level (see
Appendix B). Lastly, we investigate how the fake news’ diffusion would impact
differently on different classes of the population defined in terms of their
capabilities of interacting with information.

###  4.1 Test 1: Numerical quasi-invariant limit

In this test we show that the mean-field Fokker-Planck system (22)–(25)
obtained under the quasi-invariant scaling (20) and (21) is a good
approximation of the Boltzmann models (13) when $\varepsilon\ll 1$. We do so
by using a Monte Carlo method with $N=10^{4}$ particles, starting with a
uniform distribution of competence $f_{0}(x)=\frac{1}{2}\chi(x\in[0,2])$,
where $\chi(\cdot)$ is the indicator function, and performing various
iterations until the stationary state was reached; next, the distributions
were averaged over the next 500 iterations. We considered constant competence-
related parameters $\lambda_{CJ}=\lambda_{BJ}$ and
$\lambda_{J}=\lambda_{CJ}+\lambda_{BJ}$ as well as a constant variance
$\sigma$ for the random variables $\eta_{HJ}$.

In Figure 3, we plotted the results for $(\lambda,\sigma)=(0.075,0.150)$
(circle-solid, teal) and for $(\lambda,\sigma)=(0.001,0.002)$ (square-solid,
ochre): those choices correspond to a scaling regime of $\varepsilon=0.075$
and $\varepsilon=0.001$, respectively, with $\mu=2$. Finally, we assumed that
$m_{B}=0.75$ (left) and $m_{B}=1$ (right).

Figure 3: Test 1. Comparison of the competence distributions at the end of
epidemics for system (13) with the explicit Fokker-Planck solution (31) with
scaling parameters $\varepsilon=0.075,0.001$. We considered the case
$m_{B}=0.75$ (left) and $m_{B}=1$ (right).

Directly comparing the Boltzmann dynamics equilibrium with the explicit
analytic solution of the Fokker-Planck regime shows that if $\varepsilon$ is
small enough, Fokker-Planck asymptotics provide a consistent approximation of
the steady states of the kinetic distributions.

###  4.2 Test 2: Learning dynamics and fake news dissemination

For this test, we applied the structure-preserving scheme to system (22)–(25)
in a more realistic scenario featuring an interaction term dependent on the
competence level of the agents, as well as a competence-dependent delay during
which agents evaluate the information and decide how to act. In this setting,
we refer to the recent Survey of Adult Skills (SAS) made by the OECD [26]: in
particular, we focus on competence understood as a set of information-
processing skills, especially through the lens of literacy, defined [26] as
“the ability to understand, evaluate, use and engage with written texts in
order to participate in society”. One of the peculiarities that makes the SAS,
which is an international, multiple-year spanning effort in the framework of
the PIAAC (Programme for the International Assessment of Adult Competencies)
by the OECD, interesting in our case is that it was administered digitally to
more than 70% of the respondents. Digital devices are arguably the most
important vehicle for information diffusion in OECD countries, so that helps
to keep consistency.

Literacy proficiency was defined through $6$ increasing levels; we therefore
consider a population partitioned in $6$ classes based on the competence level
of their occupants, equated to the score of the literacy proficiency test of
the SAS, normalized. Thus, we chose a log-normal-like distribution

$f(x)=\frac{1}{(\tilde{\xi}-x)\tilde{\sigma}\sqrt{2\pi}}\cdot\mathrm{e}^{-\frac{(\log(\tilde{\xi}-x)-\tilde{\mu})^{2}}{2\tilde{\sigma}^{2}}},$

where $\tilde{\xi}=5$, $\tilde{\mu}\approx 0.85$ and $\tilde{\sigma}\approx
0.22$ to make $f(x)$ agree with the empirical findings in [26]. The
computational domain is restricted to $x\in[0,5]$ and stationary boundary
conditions have been applied as described in Appendix B.

Initial distributions for the epidemiological compartments were set as

$f_{S}(x,0)=\rho_{S}f(x),\quad f_{E}(x,0)=\rho_{E}f(x),\quad
f_{I}(x,0)=\rho_{I}f(x),\quad f_{R}(x,0)=\rho_{R}f(x),$

with $\rho_{I}=10^{-2}$, $\rho_{S}=1-\rho_{I}$ and $\rho_{E}=\rho_{R}=0$.

The contact rate $\beta(x,x_{*})$ was set as

$\beta(x,x_{*})=\frac{\beta_{0}}{(1+x^{2})(1+x_{*}^{2})}\chi(\lvert
x-x_{*}\rvert\leq\Delta)$ (32)

with $\Delta=2$ on the hypothesis that interactions occur more frequently
among people with a similar competence level and are higher for people with
lower competence levels.

The rate $\delta$ at which the information is evaluated by the agents, who
therefore exits the exposed class, was set to be

$\delta(x)=\delta_{R}+(\delta_{L}-\delta_{R})\frac{1}{1+\mathrm{e}^{a(b-x)}},$
(33)

with $\delta_{L}=1$, $\delta_{R}=5$, $a=2$ and $b=2.5$. Here, we are taking
into account that people with higher efficacy at identifying fake news spend
significantly more time on conducting their evaluations than people with lower
efficacy [23]. In this specific test case the time range for the evaluation of
the information spans between 1 day and about 5 hours. The values were
purposely chosen rather large compared to realistic values in order to
highlight also the behavior of the exposed compartment. Finally, we set
$\gamma=0.2$, which correspond to an average fake news duration of 5 days, and
$\alpha=0.2$, so that individuals have a moderate possibility to remember the
fake news and become immune to it, and assume $\eta=0.1$, namely in this test
we do not relate the decision to spread or not fake news to the level of
competence.

We investigate the relation between the dissemination-related component of the
model and the competence-related one, which entails that agents can learn,
i.e., increase their competence level, both from the background and from
direct binary interactions. Under the assumption that
$\lambda_{CJ}+\lambda_{BJ}=\lambda_{J}$, which is a conservative choice: the
expected value of the competence gained through interactions cancels out the
one lost due to forgetfulness, in this latter process two main parameters are
involved: $\lambda=\lambda_{J}$ (i.e., all compartments have the same learning
rate) and $m_{B}$, which is the mean of the background competence variable
$z$.

For what concerns the dissemination-related component, instead, the main
factor is the reproduction number $R_{0}$ (15). Hence, we measured the
differences on the spread of fake news varying these three parameters. In
Figure 4 (left) we show the highest portion of spreaders in relation to the
mean of background $m_{B}$ and to the reproduction number $R_{0}$; in the
right image $\lambda$ is opposed to $R_{0}$.

Figure 4: Test 2. Interplay between competence levels and dissemination
dynamics. Contour plots of the highest number of infected in relation to the
reproduction number (15) $R_{0}\in[1.1,10]$ and the background competence mean
$m_{B}\in[0.03125,0.25]$ (left) and learning rate $\lambda\in[0.0125,0.125]$
(right).

To perform the test, we leveraged the structure-preserving numerical scheme
[30] whose details are presented for convenience in Appendix B. In both images
of Figure 4 we see transition effects: the learning process triggered by the
competence dynamics is capable of slowing down the dissemination of fake news
in the population, even to the point of preventing it to take place. In the
first case, the mean of the background $m_{B}$, i.e., the mean of the
distribution of the background competence variable $z$, which we assumed
uniformly distributed, varies between $0.03125$ and $0.25$, while the
reproduction number $R_{0}$ varies between $1.1$ and $10$. In the second case,
we left untouched $R_{0}$, while $\lambda$ varies between $0.0125$ and $0.125$
with a background mean $m_{B}=0.125$. We can see that the mean of the
background has a more pronounced impact on the slowing of the diffusion of
fake news, with a steeper transition effect.

###  4.3 Test 3: Impact of the different competence levels

In this final test we considered how much of an impact the competence level
can have on the dissemination of fake news in the population. We simulated the
mean-field model (22)–(25) assuming the same competence-dependent contact rate
$\beta(x,x_{*})$ of Test 2, in this case with $\beta_{0}=4$, as well the same
delay rate $\delta(x)$ and the same $\gamma$, but we additionally assume that
the decision to spread or not a fake news is affected by the level of
competence. This is somewhat controversial in the literature since other
factors also affect this behavior like the age of individuals (tests carried
out on young people have shown independence from the competence in the
decision to share a fake news in contrast to what happens in in older people,
see [26, 23]). To emphasize this effect we assume

$\eta(x)=1-e^{-kx^{2}},$ (34)

with $k=0.1$. Thus individuals with high level of competence rarely decide to
spread fake news.

In Figure 5 we report the time evolution of the distributions of susceptible
(top left), exposed (top right), infected (bottom left) and removed (bottom
right) agents with competence parameters of $\lambda_{BJ}=\lambda_{CJ}=0.125$,
$\lambda_{J}=\lambda_{BJ}+\lambda_{CJ}$ and $m_{B}=0.125$, in the case
$\alpha=0.1$. In Figure 6, instead, we show the evolution with the same
parameters except for a larger probability $\alpha=0.9$ of remembering the
fake news.

Figure 5: Test 3. Time evolution of the competence distribution for the
kinetic model (13) with competence parameters
$\lambda_{BJ}=\lambda_{CJ}=0.125$, $\lambda_{J}=\lambda_{BJ}+\lambda_{CJ}$,
with a background competence mean of $m_{B}$ = 0.125. We considered
$\alpha=0.1$, $\beta_{0}=4$, $\gamma=0.2$. Top left: susceptible; top right:
exposed; bottom left: infected; bottom right: removed.

Figure 6: Test 3. Time evolution of the competence distribution for the
kinetic model (13) with competence parameters
$\lambda_{BJ}=\lambda_{CJ}=0.125$, $\lambda_{J}=\lambda_{BJ}+\lambda_{CJ}$,
with a background competence mean of $m_{B}$ = 0.125. We considered
$\alpha=0.9$, $\beta_{0}=4$, $\gamma=0.2$. Top left: susceptible; top right:
exposed; bottom left: infected; bottom right: removed.

In Figure 7 are shown the relative numbers of susceptible, exposed, infected
and removed agents, on the left for $\alpha=0.1$ and on the right for
$\alpha=0.9$.

Figure 7: Test 3. Evolution in time of the densities for susceptible, exposed,
infectious and removed agents. Left: $\alpha=0.1$; right: $\alpha=0.9$.

To measure the effects of the competence, we considered the curve of the
infected agents depending on their levels accordingly to [26] for $x\in[0,5]$:

  * •

below level 1: scoring less then 175/500, ($x<1.75$);

  * •

level 1: scoring between 176/500 and 225/500, ($x>1.75$ and $x<2.25$);

  * •

level 2: scoring between 226/500 and 275/500, ($x>2.25$ and $x<2.75$);

  * •

level 3: scoring between 276/500 and 325/500, ($x>2.75$ and $x<3.25$);

  * •

level 4: scoring between 326/500 and 375/500, ($x>3.25$ and $x<3.75$);

  * •

level 5: scoring more than 375/500, ($x>3.75$).

Figure 8: Test 3. Evolution of the fraction of infectious agents with
different levels of competence as defined in [26]. Left: $\alpha=0.1$; right:
$\alpha=0.9$.

Figure 8 shows clearly that the more competent the individual, the lesser they
contribute to the spread of fake news, in perfect agreement with the
transition effects observed in Test 4.2 due to the interplay between
competence and the dissemination dynamics. Moreover, we can see how the
probability $\alpha$ of detecting fake news influences its dissemination in
the population: a lower probability implies a higher peak of infected agents
for each competence level, as well as a slower spread overall.

##  5 Concluding remarks

In this paper, we introduced a compartmental model for fake news dissemination
that also considers the competence of individuals. In the model, the concept
of competence is not introduced as a static feature of the dynamic, but as an
evolutionary component that takes into account both learning through
interactions between agents and interventions aimed at educating individuals
in the ability to detect fake news.

From a mathematical viewpoint the competence dynamics has been introduced as a
Boltzmann interaction term in the corresponding system of differential
equations. A suitable scaling limit, permits to recover the corresponding
Fokker-Planck models and then the resulting stationary states in terms of
competence. These, in agreement with [29], are given explicitly by Gamma
distributions.

The numerical results demonstrate the model’s ability to correctly describe
the interplay between fake news dissemination and individuals’ level of
competence, highlighting transition phenomena at the level of expertise that
allow fake news to spread more rapidly.

Future developments of the model will be considered in particular in the case
of networks, in order to describe the spread of fake news on social networks
and present plausible scenarios useful to limit the spread of false
information. This can be done by following an approach similar to that of
kinetic models for opinion formation on networks [1]. Another challenging
aspect concerns the matching of the model with realistic data that requires
the introduction of quantitative aspects not always easy to identify [25, 36].
One of the main applications will be related to combating misinformation in
the vaccination campaign against COVID-19.

## Acknowledgments

This work was partially supported by MIUR (Ministero dell’Istruzione,
dell’Università e della Ricerca) PRIN 2017, project “Innovative numerical
methods for evolutionary partial differential equations and applications”,
code 2017KKJP4X.

##  Appendix A Proof of Theorem 1

We provide here a proof for Theorem 1. The proof is identical to [15]; we
develop it here, too, for completeness, only in the case $\alpha=\eta=0$. If
$H\in\\{S,E,I\\}$ we have

$\displaystyle\sum_{\mathclap{J\in\\{S,E,I\\}}}\hat{Q}(\hat{f}_{H},\hat{f}_{J})(\xi,t)$
$\displaystyle=\sum_{\mathclap{J\in\\{S,E,I\\}}}\langle\hat{f}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)\rangle\hat{f}_{J}(\lambda_{CJ}\xi,t)-\sum_{\mathclap{J\in\\{S,E,I\\}}}\hat{f}_{H}(\xi,t)J(t)$
(35) $\displaystyle=\hat{Q}_{+}(\hat{f}_{H})(\xi,t)-\hat{H}(\xi,t)$

where the second equality follows from mass conservation. In (35), we have
defined $\hat{Q}_{+}(\hat{f}_{H})(\xi,t)$ as

$\hat{Q}_{+}(\hat{f}_{H})(\xi,t)=\sum_{\mathclap{J\in\\{S,E,I\\}}}\langle\hat{f}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)\rangle\hat{f}_{J}(\lambda_{CJ}\xi,t),$

where $A_{HJ}$ has been defined originally in (17). Thus, now system (16)
reads

$\left\\{\begin{aligned} \frac{\partial\hat{f}_{S}(\xi,t)}{\partial
t}+\hat{f}_{S}(\xi,t)&=-\beta
I(t)\hat{f}_{S}(\xi,t)+\gamma\hat{f}_{I}(\xi,t)+\hat{Q}_{+}(\hat{f}_{I})(\xi,t)\\\
\frac{\partial\hat{f}_{E}(\xi,t)}{\partial t}+\hat{f}_{E}(\xi,t)&=\beta
I(t)\hat{f}_{S}(\xi,t)-\delta\hat{f}_{E}(\xi,t)+\hat{Q}_{+}(\hat{f}_{E})(\xi,t)\\\
\frac{\partial\hat{f}_{I}(\xi,t)}{\partial
t}+\hat{f}_{I}(\xi,t)&=\delta\hat{f}_{E}(\xi,t)-\gamma\hat{f}_{I}(\xi,t)+\hat{Q}(\hat{f}_{I})(\xi,t),\end{aligned}\right.$
(36)

To ensure positivity of all coefficients on the right-hand side (since
$I(t)<1$ and $\beta,\gamma,\delta<1$ we can add $\hat{f}_{J}(\xi,t)$ to each
side, where $J$ equals $S$, $E$ and $I$ in the first, second and third
equation, respectively, to resort to the equivalent system

$\left\\{\begin{aligned} \frac{\partial\hat{f}_{S}(\xi,t)}{\partial
t}+2\hat{f}_{S}(\xi,t)&=(1-\beta
I(t))\hat{f}_{S}(\xi,t)+\gamma\hat{f}_{I}(\xi,t)+\hat{Q}_{+}(\hat{f}_{I})(\xi,t)\\\
\frac{\partial\hat{f}_{E}(\xi,t)}{\partial t}+2\hat{f}_{E}(\xi,t)&=\beta
I(t)\hat{f}_{S}(\xi,t)(1-\delta)\hat{f}_{E}(\xi,t)+\hat{Q}_{+}(\hat{f}_{E})(\xi,t)\\\
\frac{\partial\hat{f}_{I}(\xi,t)}{\partial
t}+2\hat{f}_{I}(\xi,t)&=\delta\hat{f}_{E}(\xi,t)(1-\gamma)\hat{f}_{I}(\xi,t)+\hat{Q}(\hat{f}_{I})(\xi,t),\end{aligned}\right.$
(37)

in which all coefficients on the right-hand side are positive.

Now, let $\hat{f}_{J}(\xi,t)$ and $\hat{g}(\xi,t)$ be two solutions of the
system (37). We look at the time behavior of the $d_{2}$ metric of their
difference, where the Fourier-based metric was defined in (19); therefore we
define

$h_{J}(\xi,t)=\frac{\hat{f}_{J}(\xi,t)-\hat{g}_{J}(\xi,t)}{\lvert\xi\rvert^{2}}.$

We see that the metric and $h_{J}$ are related by

$d_{2}(f_{J},g_{J})=\lVert h_{J}\rVert_{\infty}$ (38)

By its definition, the $h_{J}$ are solutions of

$\left\\{\begin{aligned} \frac{\partial\hat{h}_{S}(\xi,t)}{\partial
t}+2\hat{h}_{S}(\xi,t)&=(1-\beta
I(t))\hat{h}_{S}(\xi,t)+\gamma\hat{h}_{I}(\xi,t)+L_{+}(\hat{f}_{S})(\xi,t)\\\
\frac{\partial\hat{h}_{E}(\xi,t)}{\partial t}+2\hat{h}_{E}(\xi,t)&=\beta
I(t)\hat{h}_{S}(\xi,t)(1-\delta)\hat{h}_{E}(\xi,t)+L_{+}(\hat{f}_{E})(\xi,t)\\\
\frac{\partial\hat{h}_{I}(\xi,t)}{\partial
t}+2\hat{h}_{I}(\xi,t)&=\delta\hat{h}_{E}(\xi,t)+(1-\gamma)\hat{h}_{I}(\xi,t)+L_{+}(\hat{f}_{I})(\xi,t),\end{aligned}\right.$
(39)

where, with $H\in\\{S,E,I,R\\}$ are defined as

$L_{+}(\hat{f}_{H})(\xi,t)=\frac{\hat{Q}_{+}(\hat{f}_{H})(\xi,t)-\hat{Q}_{+}(\hat{g}_{H})(\xi,t)}{\lvert\xi\rvert^{2}}.$

We can rewrite $L_{+}(\hat{f}_{H})(\xi,t)$ in full:

$L_{+}(\hat{f}_{H})(\xi,t)=\sum_{\mathclap{J\in\\{S,E,I,R\\}}}\frac{\langle\hat{f}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)\hat{f}_{J}(\lambda_{CJ}\xi,t)-\hat{g}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)\hat{g}_{J}(\lambda_{CJ}\xi,t)\rangle}{\lvert\xi\rvert^{2}},$
(40)

with the expectation $\langle\cdot\rangle$ put outside for convenience. As
shown, e.g., in [27], and since $f_{J}$ and $g_{J}$ are solution of the SEIS
system for the masses and the mean values, we can profitably bound the addends
in the sum on the right-hand side of (40) in terms of the functions $h_{H}$
and $h_{J}$

$\begin{multlined}\left\lvert\frac{\langle\hat{f}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)\hat{f}_{J}(\lambda_{CJ}\xi,t)-\hat{g}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)\hat{g}_{J}(\lambda_{CJ}\xi,t)\rangle}{\lvert\xi\rvert^{2}}\right\rvert\\\
\leq\left\langle\lvert\hat{f}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)\rvert\left\lvert\frac{\hat{f}_{J}(\lambda_{CJ}\xi,t)-\hat{g}_{J}(\lambda_{CJ}\xi,t)}{\lvert\lambda_{CJ}\xi\rvert^{2}}\right\rvert\lambda_{CJ}^{2}\right\rangle\\\
+\left\langle\lvert\hat{g}_{J}(\lambda_{J}\xi,t)\rvert\left\lvert\frac{\hat{f}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)-\hat{g}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)}{\lvert
A_{HJ}\xi\rvert^{2}}\right\rvert A_{HJ}^{2}\right\rangle\\\ \leq
H(t)\sup_{\xi}\left\lvert\frac{\hat{f}_{J}(\xi,t)-\hat{g}_{J}(\xi,t)}{\lvert\xi\rvert^{2}}\right\rvert\lambda_{CJ}^{2}+J(t)\sup_{\xi}\left\lvert\frac{\hat{f}_{H}(\xi,t)-\hat{g}_{H}(\xi,t)}{\lvert\xi\rvert^{2}}\right\rvert\langle
A_{HJ}^{2}\rangle\\\ =\lambda_{CJ}^{2}H(t)\lVert h_{J}\rVert_{\infty}+\langle
A_{HJ}^{2}\rangle\lVert h_{H}\rVert_{\infty}.\\\
\end{multlined}\left\lvert\frac{\langle\hat{f}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)\hat{f}_{J}(\lambda_{CJ}\xi,t)-\hat{g}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)\hat{g}_{J}(\lambda_{CJ}\xi,t)\rangle}{\lvert\xi\rvert^{2}}\right\rvert\\\
\leq\left\langle\lvert\hat{f}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)\rvert\left\lvert\frac{\hat{f}_{J}(\lambda_{CJ}\xi,t)-\hat{g}_{J}(\lambda_{CJ}\xi,t)}{\lvert\lambda_{CJ}\xi\rvert^{2}}\right\rvert\lambda_{CJ}^{2}\right\rangle\\\
+\left\langle\lvert\hat{g}_{J}(\lambda_{J}\xi,t)\rvert\left\lvert\frac{\hat{f}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)-\hat{g}_{H}(A_{HJ}\xi-\lambda_{BH}z,t)}{\lvert
A_{HJ}\xi\rvert^{2}}\right\rvert A_{HJ}^{2}\right\rangle\\\ \leq
H(t)\sup_{\xi}\left\lvert\frac{\hat{f}_{J}(\xi,t)-\hat{g}_{J}(\xi,t)}{\lvert\xi\rvert^{2}}\right\rvert\lambda_{CJ}^{2}+J(t)\sup_{\xi}\left\lvert\frac{\hat{f}_{H}(\xi,t)-\hat{g}_{H}(\xi,t)}{\lvert\xi\rvert^{2}}\right\rvert\langle
A_{HJ}^{2}\rangle\\\ =\lambda_{CJ}^{2}H(t)\lVert h_{J}\rVert_{\infty}+\langle
A_{HJ}^{2}\rangle\lVert h_{H}\rVert_{\infty}.\\\ $

So we obtain that

$\lVert L_{+}(\hat{f}_{H})(t)\rVert_{\infty}\leq
H(t)\sum_{\mathclap{J\in\\{S,E,I\\}}}\lambda_{CJ}^{2}\lVert
h_{J}(t)\rVert_{\infty}+\lVert
h_{H}(t)\rVert_{\infty}\sum_{\mathclap{J\in\\{S,E,I\\}}}\langle
A_{HJ}^{2}J(t)\rangle.$

Multiplying both sides of (39) by $\mathrm{e}^{2t}$ we have

$\left\\{\begin{aligned} \frac{\partial[h_{S}(\xi,t)\mathrm{e}^{2t}]}{\partial
t}&\leq(1-\beta I(t))\lVert
h_{I}(t)\mathrm{e}^{2t}\rVert_{\infty}+\gamma\lVert
h_{I}(t)\mathrm{e}^{2t}\rVert_{\infty}+\lVert
L_{+}(\hat{f}_{S})(t)\mathrm{e}^{2t}\rVert_{\infty}\\\
\frac{\partial[h_{E}(\xi,t)\mathrm{e}^{2t}]}{\partial t}&\leq\beta I(t)\lVert
h_{I}(t)\mathrm{e}^{2t}\rVert_{\infty}+(1-\delta)\lVert
h_{E}(t)\mathrm{e}^{2t}\rVert_{\infty}+\lVert
L_{+}(\hat{f}_{E})(t)\mathrm{e}^{2t}\rVert_{\infty}\\\
\frac{\partial[h_{I}(\xi,t)\mathrm{e}^{2t}]}{\partial t}&\leq\delta\lVert
h_{E}(t)\mathrm{e}^{2t}\rVert_{\infty}+(1-\gamma)\lVert
h_{I}(t)\mathrm{e}^{2t}\rVert_{\infty}+\lVert
L_{+}(\hat{f}_{I})(t)\mathrm{e}^{2t}\rVert_{\infty},\end{aligned}\right.$

If we integrate from $0$ to $t$ and take the suprema we get

$\left\\{\begin{aligned} \lVert
h_{S}(\xi,t)\mathrm{e}^{2t}\rVert_{\infty}&\leq\lVert
h_{S}(0)\rVert_{\infty}+\int_{0}^{t}\biggl{[}(1-\beta I(t))\lVert
h_{I}(t)\mathrm{e}^{2t}\rVert_{\infty}+\gamma\lVert
h_{I}(t)\mathrm{e}^{2t}\rVert_{\infty}+\lVert
L_{+}(\hat{f}_{S})(t)\mathrm{e}^{2t}\rVert_{\infty}\biggr{]}\,\mathrm{d}s\\\
\lVert h_{E}(\xi,t)\mathrm{e}^{2t}\rVert_{\infty}&\leq\lVert
h_{E}(0)\rVert_{\infty}+\int_{0}^{t}\biggl{[}\beta I(t)\lVert
h_{I}(t)\mathrm{e}^{2t}\rVert_{\infty}+(1-\delta)\lVert
h_{E}(t)\mathrm{e}^{2t}\rVert_{\infty}+\lVert
L_{+}(\hat{f}_{E})(t)\mathrm{e}^{2t}\rVert_{\infty}\biggr{]}\,\mathrm{d}s\\\
\lVert h_{I}(\xi,t)\mathrm{e}^{2t}\rVert_{\infty}&\leq\lVert
h_{S}(0)\rVert_{\infty}+\int_{0}^{t}\biggl{[}\delta\lVert
h_{E}(t)\mathrm{e}^{2t}\rVert_{\infty}+(1-\gamma)\lVert
h_{I}(t)\mathrm{e}^{2t}\rVert_{\infty}+\lVert
L_{+}(\hat{f}_{I})(t)\mathrm{e}^{2t}\rVert_{\infty}\biggr{]}\,\mathrm{d}s.\end{aligned}\right.$

Thanks to mass conservation we can also write

$\displaystyle\sum_{\mathclap{H\in\\{S,E,I\\}}}\lVert
L_{+}(\hat{f}_{H})(t)\rVert_{\infty}$
$\displaystyle=\sum_{\mathclap{H,J\in\\{S,E,I\\}}}\lambda_{CJ}^{2}H(t)\lVert
h_{J}(t)\rVert_{\infty}+\sum_{\mathclap{H,J\in\\{S,E,I\\}}}\langle
A^{2}_{HJ}\rangle J(t)\lVert h_{H}(t)\rVert_{\infty}$
$\displaystyle=\sum_{\mathclap{H,J\in\\{S,E,I\\}}}\bigl{[}\lambda_{CJ}^{2}+\langle
A^{2}_{HJ}\rangle\bigr{]}H(t)\lVert h_{J}(t)\rVert_{\infty}$
$\displaystyle=\max_{H,J\in\\{S,E,I,R\\}}\bigl{[}\lambda_{CJ}^{2}+\langle
A^{2}_{HJ}\rangle\bigr{]}\sum_{\mathclap{J\in\\{S,E,I\\}}}\lVert
h_{J}(t)\rVert_{\infty}.$

Now we can take advantage of condition (18) to get

$\sum_{\mathclap{H\in\\{S,E,I\\}}}\lVert
L_{+}(\hat{f}_{H})(t)\mathrm{e}^{2t}\rVert_{\infty}\leq\nu\sum_{\mathclap{J\in\\{S,E,I\\}}}\lVert
h_{J}(t)\mathrm{e}^{2t}\rVert_{\infty},$

with $\nu<1$. If we sum the equations of the system we therefore have

$\sum_{\mathclap{J\in\\{S,E,I\\}}}\lVert
h_{J}(t)\mathrm{e}^{2t}\rVert_{\infty}\leq\sum_{\mathclap{J\in\\{S,E,I\\}}}\lVert
h_{J}(0)\rVert_{\infty}+\int_{0}^{t}\sum_{\mathclap{\quad
J\in\\{S,E,I\\}}}(1+\nu)\lVert h_{J}(s)\mathrm{e}^{2s}\rVert_{\infty}.$

Now Gronwall’s Lemma implies

$\sum_{\mathclap{J\in\\{S,E,I\\}}}\lVert
h_{J}(t)\mathrm{e}^{2t}\rVert_{\infty}\leq\lVert
h_{J}(0)\rVert_{\infty}\mathrm{e}^{(1+\nu)t}$

which is equivalent to

$\sum_{\mathclap{J\in\\{S,E,I\\}}}\lVert h_{J}(t)\rVert_{\infty}\leq\lVert
h_{J}(0)\rVert_{\infty}\mathrm{e}^{-(1-\nu)t}.$

Recalling the relation (38) we obtain the thesis of Theorem 1.

##  Appendix B Structure-preserving methods

Here we provide some details on the structure-preserving numerical scheme [30]
for the general class of nonlinear Fokker-Planck equations of the form

$\left\\{\begin{aligned} \frac{\partial g(x,t)}{\partial
t}&=\nabla_{x}\cdot\bigl{[}\mathscr{B}[g](x,t)g(x,t)+\nabla_{x}(D(x)g(x,t))\bigr{]},\\\
g(x,0)&=g_{0}(x),\end{aligned}\right.$ (41)

where $t\geq 0$, $x\in X\subseteq\mathbb{R}^{d}$, $d\geq 1$ and $g(x,t)\geq 0$
is the unknown distribution function. As mentioned above, $\mathscr{B}[g]$ is
a bounded aggregation operator and $D(\cdot)$ models diffusion.

The scheme [30] follows the work of Chang and Cooper [9] to construct a
numerical method which can preserve features of the solution such as its large
time behavior.

If we examine system (22)–(25), we notice it has a structure like the
following

$\frac{\partial\bm{f}(x,t)}{\partial
t}=\frac{\partial\mathscr{F}[\bm{f}](x,t)}{\partial
x}+\mathscr{E}(\bm{f}(x,t)),$ (42)

where $\bm{f}(x,t)=(f_{S}(x,t),f_{E}(x,t),f_{I}(x,t),f_{R}(x,t))^{T}$,
$\mathscr{E}(\bm{f}(x,t))$ is a vector accounting for dissemination dynamics

$\mathscr{E}(\bm{f}(x,t))=\begin{pmatrix}-K(x,t)f_{S}(x,t)+(1-\alpha(x))\gamma(x)f_{I}(x,t)\\\
K(x,t)f_{S}(x,t)-\delta(x)f_{E}(x,t)\\\
\delta(x)(1-\eta(x))f_{E}(x,t)-\gamma(x)f_{I}(x,t)\\\
\delta(x)\eta(x)f_{E}(x,t)+\alpha(x)\gamma(x)f_{I}(x,t),\end{pmatrix}$

and $\mathscr{F}[\bm{f}](x,t)$ is the Fokker-Planck component

$\mathscr{F}[\bm{f}](x,t)=\biggl{(}\frac{\partial}{\partial
x}[(x\lambda_{J}-\overline{m}(t)-4\lambda_{BJ}m_{B})f_{J}(x,t)]+\frac{\sigma}{2}\frac{\partial^{2}}{\partial
x^{2}}(x^{2}f_{J}(x,t))\biggr{)}^{T}_{J\in\\{S,E,I,R\\}}.$

Here we recognize that the $J$-th entry of $\mathscr{F}[\bm{f}](x,t)$ is
precisely the right side of equation (41) in dimension $d=1$, with the choices

$\mathscr{B}[f](x,t)=\lambda_{J}x-\overline{m_{x}}(t)-4\lambda_{BJ}m_{B}$

when $\alpha>0$ and $D(x)=\sigma/2x^{2}$. Hence, if we consider system
(22)–(25) in the form (42) we can apply the structure-preserving numerical
scheme [30] to it: if we consider a spatially-uniform grid $x_{i}\in X$, such
that $x_{i+1}-x_{i}=\Delta x$, and denoting $x_{i\pm 2}=x_{i}\pm\Delta x/2$,
we have that the discretization of the $J$-th component of (42) can be
obtained by [30]

$\displaystyle\frac{\mathrm{d}f_{J,i}(t)}{\mathrm{d}t}$
$\displaystyle=\frac{\mathscr{F}_{i+1/2}(t)-\mathscr{F}_{i-1/2}(t)}{\Delta
x}+\mathscr{E}_{J,i}(t),$ (43)

where

$\mathscr{F}_{i+1/2}=\mathscr{C}_{i+1/2}\tilde{f}_{i+1/2}+D_{i+1/2}\frac{f_{i+1}-f_{i}}{\delta
x},$

having defined

$\mathscr{C}_{i+1/2}=\frac{D_{i+1/2}}{\Delta
x}\int_{x_{i}}^{x_{i+1}}\frac{\mathscr{B}[f](x,t)+\partial_{x}D(x)}{D(x)},$

and

$\tilde{f}_{i+1/2}=(1-\delta_{i+1/2})f_{i+1}+\delta_{i+1/2}f_{i},$

where

$\delta_{i+1/2}=\frac{1}{\lambda_{i+1/2}}+\frac{1}{1-\exp(\lambda_{i+1/2})},$

and finally

$\lambda_{i+1/2}=\int_{x_{i}}^{x_{i+1}}\frac{\mathscr{B}[f](x,t)+\partial_{x}D(x)}{D(x)}\,\mathrm{d}x.$

For what concerns integration with respect to the competence level was
performed using a Gauss-Legendre quadrature with 6 points. Notice also that we
need to truncate the domain of computation for $x>0$: following [30] we
imposed on the last grid point $x_{N+1}$ the _quasi-stationary condition_ [30]

$\frac{f_{N+1}(t)}{f_{N}(t)}=\exp\left\\{\int_{x_{N}}^{x_{N+1}}\frac{\mathscr{B}[f](x,t)+\partial_{x}D(x)}{D(x)}\,\mathrm{d}x\right\\}$

Time integration of (43) was performed using a semi-implicit scheme

$f_{i}^{n+1}=f_{i}^{n}+\Delta
t\frac{\hat{\mathscr{F}}_{i+1/2}^{n+1}-\hat{\mathscr{F}}_{i-1/2}^{n+1}}{\Delta
x}+\Delta t\,\mathscr{E}_{J,i}^{n},$

where

$\hat{\mathscr{F}}_{i+1/2}^{n+1}=\tilde{\mathscr{C}}_{i+1/2}^{n}\Bigl{[}(1-\delta_{i+1/2}^{n})f_{i+1}^{n+1}+\delta_{i+1/2}^{n}f{i+1}^{n+1}\Bigr{]}+D_{i+1/2}\frac{f_{i+1}^{n+1}-f_{i}^{n+1}}{\Delta
x},$

which, upon choosing $\Delta t=\mathcal{O}(\Delta x)$, preserves the
nonnegativity of the solution (see [30]).

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