# Modelling the random spreading of fake news through a two-dimensional time-
inhomogeneous birth-death process111Paper accepted for publication on
Mathematical Methods in the Applied Sciences, $\copyright$ John Wiley & Sons
Ltd.

Antonio Di Crescenzo(1), Paola Paraggio(1)
(1) Department of Mathematics, Università degli Studi di Salerno
Via Giovanni Paolo II, 132, Fisciano (SA), Italy
Email: {adicrescenzo<EMAIL_ADDRESS>
## Abstract

We consider a two-dimensional time-inhomogeneous birth-death process to model
the time-evolution of fake news in a population. The two components of the
process represent, respectively, (i) the number of individuals (say spreaders)
who know the rumor and intend to spread it, and (ii) the number of individuals
(say inactives) who have forgotten the rumor previously received. We employ
the probability generating function-based approach to obtain the moments and
the covariance of the two-dimensional process. We also analyze a new
adimensional index to study the correlation between the two components. Some
special cases are considered in which both the expected numbers of spreaders
and inactives are equal to suitable sigmoidal curves, which are often adopted
in modelling growth phenomena. Finally, we provide an application based on
real data related to the diffusion of fake news, in which the optimal choice
of the sigmoidal curve that fit the datasets is based on the minimization of
the mean square error and of the relative absolute error.
Keywords: Two-dimensional birth-death process, probability generating
function, adimensional correlation index, fake-news spreading, mean square
error, relative absolute error.
MSC Classification: 60J80, 60J85, 91B62, 91B70.

##  1 Introduction

In recent years, the interest in issues related to the spread of fake news and
to information security in general has increased significantly. Mainly, the
researchers study the dynamics of online news diffusion aiming to warn users
against all those situations that can harm the security of their own data.
Along this line, governments of several states are allocating funds for
cybersecurity. Internet allows everyone to spread news, both true and false;
therefore, it is up to the user to recognize its truthfulness. Most of fake
news is often ambiguous and insidious enough to mislead even careful readers.
Moreover, it is important to remember that the dissemination of this type of
information is based on psychological assumptions and leverages the fear and
the emotion of public opinion. Often the authors of fake news cite real
events, selecting only evidence to support their thesis and exploit
advertising techniques to go viral. The consequences linked to the spread of
fake news can be multiple, such as: focusing attention only on certain issues,
diverting public opinion only on specific facts, denigrating entire ethnic
groups or triggering repercussions on choices in various contexts (think about
healthcare choices during the COVID-19 pandemic). Scientific research in this
sector branches out into various areas ranging from the recognition of visual
content modified by artificial intelligence and therefore false, to
mathematical modelling. Regarding this last topic, various mathematical models
have been proposed in the literature, represented both by simple curves
capable of mimicking the temporal diffusion of fake news and by systems of
differential equations useful to describe the temporal evolution of certain
categories of the population (see, for example, the papers of San Martin [1],
Di Crescenzo et al. [2] or Giorno and Spina [3]). Furthermore, to let the
models become more realistic, several investigations involve stochastic
processes which are solutions of special stochastic differential equations
defined by adding a random component in the deterministic models. The recent
works of Mahmoud [4] and Esmaeeli and Sajadi [5] can be viewed in this sense,
since the authors adopt discrete-time stochastic processes for the temporal
diffusion of fake news in a community. Many of these models are inspired by
epidemiological ones. Indeed, in several stochastic systems the population is
supposed to be divided into compartments according to the characteristics of
the individuals that compose them (see, for example, Martins and Pinto [6] for
a model for the diffusion of fake news, and also Giorno and Nobile [7] for an
epidemiological model).

In this work, along the lines of the previous studies we introduce a new
stochastic model for the diffusion of fake news in a community. More
precisely, we consider a two-dimensional birth-death process $(X(t),Y(t))$, in
which $X(t)$ represents the number of spreaders, i.e. individuals who, at time
$t$, possess the information and intend to spread it, while $Y(t)$ represents
the number of inactives, i.e. individuals who, at time $t$, have forgotten the
information previously received. The dynamics of the process $(X(t),Y(t))$ is
governed by two types of transitions. Specifically, if $(X(t),Y(t))=(n,k)$
then (i) the spreaders may increase with a time-dependent intensity,
$n\lambda(t)$, and (ii) the spreaders may decrease and become inactive with a
possibly different time-dependent intensity, $n\mu(t)$, too. The aim of this
paper is threefold:
(a) to provide exact results about the probability distribution and the
moments of the proposed model for the dynamics of the fake news spreading,
(b) to study conditions such that the mean trend of the population categories,
i.e. spreaders and inactives, correspond to typical growth curves adopted in
populations dynamics,
(c) to apply the proposed model to real datasets concerning the diffusion of
fake news.

In particular, regarding (a), we provide explicit expressions of the
probability generating function, of the conditional moments and of other
indexes of variation of both the components $X(t)$ and $Y(t)$. We also study
the mixed moment, the covariance and the correlation coefficient in some
special cases, namely when the spreader intensity $\lambda(t)$ and the
inactive intensity $\mu(t)$ are constant and when they are proportional, i.e.
$\lambda(t)=\rho\mu(t)$, $\rho>0$. Moreover, the correlation is studied using
a new adimensional correlation index defined as the ratio between the mixed
moment $\mathsf{E}\left[X(t)\cdot Y(t)\right]$ and the product of the means
$\mathsf{E}\left[X(t)\right]$ and $\mathsf{E}\left[Y(t)\right]$.

With reference to (b), we consider some special instances of the two-
dimensional process in which the intensities are proportional and are chosen
so that the spreaders (or the inactives) have conditional mean of sigmoidal
type. Specifically, we select 6 kinds of sigmoidal growth curves for the
spreaders, and two different types of sigmoidal growth curves starting from
the origin, since they turn out to be more suitable to model growth phenomena
subject to self-adaptive mechanisms, such as the diffusion of fake news.

Finally, regarding (c), we use the proposed two-dimensional birth-death
process with proportional intensities to fit real data concerning online fake
news propagation. The analysis of the mean square error and of the relative
absolute error allows us to assess the appropriateness of the considered
sigmoidal curves leading to the suitable choice for the datasets under
investigation.

We remark that two-dimensional birth-death processes have often been used in
applied contexts since they prove to be effective in modeling certain
phenomena evolving in a discrete state-space. For instance, in Di Crescenzo
and Martinucci [8], the authors consider a two-dimensional birth-death process
with constant rates, they analyze a property of quasi-symmetry and propose
applications in contexts linked to double-ended queues. Moreover, Gómez-Corral
and López-García [9], [10] introduce a two-dimensional continuous-time Markov
chain in which each component represents a species of a population and the
transition rates are proportional to the size of at least one of the two
components. They study, also, the number of births and deaths in an extinction
cycle and the number of direct descendants under the hypothesis of killing and
reproductive strategies. The contexts of application of two-dimensional
processes are multiple and range from epidemiology (see, for example,
Griffiths and Greenhalgh [11] and Billard [12]), to agriculture (see, for
instance, Abdullahi et al. [13], in which the authors review three existing
two-dimensional stochastic models showing their usefulness in applications
related to seed dispersals), from service systems (as in Krishna-Kumar et al.
[14] and Liang Luh [15]), to politics (see, for example, Mobilia [16] for a
generalized three-party constrained voter model accounting for the competition
between a persuasive majority and a committed minority) and also fake news
diffusion (see, for instance, Kapsikar et al. [17] in which the authors
propose a two-dimensional branching process to describe fake news diffusion
among populations composed by individuals capable to identify fake news).
Generally, the results regarding such multi-dimensional processes are of
numerical or approximation nature (as in Domenech-Benlloch [18], Servi [19],
Brandwajn [20], Fayolle et al. [21], Remiche [22] and Møller and Berthelsen
[23]).

This is the structure of the paper. An in-depth analysis of the stochastic
model is presented in Section 2 where the state-space of $(X(t),Y(t))$ is
determined, highlighting the existence of absorbing states, together with the
differential equations governing the dynamics of transition probabilities.
Sections 5 and Section 6 are devoted to the analysis of two specific cases.
More precisely, in Section 5 the case of constant rates is studied, whereas in
Section 6 the case of time-dependent and porportional rates is analyzed. In
these two particular cases, it is possible to determine the expression of the
probability generating function in closed form. In Section 3 the corresponding
probability generating function is obtained (for specific cases in closed
form). Moreover, Section 4 is devoted to the moments, so that in Section 4.1,
the moments (of the first and second order) of the first component $X(t)$ of
the process are determined. Section 4.2 contains similar results for the
moments of the second component $Y(t)$ of the process. In Section 4.3, we then
compute the mixed moments to analyze the correlation between $X(t)$ and
$Y(t)$. In particular, the correlation is studied using also the new
adimensional correlation index defined as the ratio between the mixed moment
and the product of the conditional means. In Section 7, we study special cases
in which the time-dependent rates $\lambda(t)$ and $\mu(t)$ are proportional,
when the mean of $X(t)$ and the mean of $Y(t)$ are equal to some sigmoidal
growth curves of interest in the literature, such as the Gompertz curve, both
in the classical and generalized versions, the logistic curve and certain
generalizations, and the modified Korf curve. The Korf and the Mitscherlich
curves are also employed for the mean of $Y(t)$ since they meet the customary
assumption $Y(0)=0$. Finally, in Section 8, we present an application based on
real data, related to the diffusion of 5 different fake news in an online
social network. We investigate the goodness of the proposed sigmoidal models
through their mean square errors and relative absolute errors.

##  2 The stochastic model

Let $\left\\{(X(t),Y(t));t\geq 0\right\\}$ be a two-dimensional birth-death
process, where $X(t)$ represents the number of spreaders, i.e. individuals
who, at time $t$, possess the information and have the intention to spread it.
Moreover, $Y(t)$ represents the number of inactives, i.e. individuals who, at
time $t$, have forgotten the information previously received and do not have
the intention of sharing the rumor anymore. We suppose that
$X(0)=j\in\mathbb{N}$, i.e. the number of informed individuals is initially
greater than $0$, otherwise there would be no diffusion of information.
Moreover, we assume $Y(0)=0$ so that at the beginning no one has already
forgotten the rumor. For $n,n^{\prime},k,k^{\prime}\in\mathbb{N}_{0}$, we
denote by

$q_{(n,k);(n^{\prime},k^{\prime})}(t):=\lim_{h\to
0}\frac{1}{h}\mathsf{P}\left[(X(t+h),Y(t+h))=(n^{\prime},k^{\prime})|(X(t),Y(t))=(n,k)\right]$

the transition intensities of the process $(X(t),Y(t))$. Thus, we assume that
the dynamics of the process are governed by the following transition rates:

$\displaystyle q_{(n,k);(n+1,k)}(t)=n\lambda(t),\qquad n,k\in\mathbb{N}_{0},$
(1) $\displaystyle q_{(n,k);(n-1,k+1)}(t)=n\mu(t),\qquad
n\in\mathbb{N},\;k\in\mathbb{N}_{0},$

where $\lambda(t),\mu(t)$ are two positive and integrable functions on any
interval $(0,\tau)$, $\tau>0$. Both intensities in (1) are linear in $n$,
since at each instant any spreader may transmit the information to a new
individual or, on the contrary, he/she may become inactive. The function
$\lambda(t)$ represents the intensity that, at time $t$, an informer shares
the rumor with a new individual. Whereas, the function $\mu(t)$ is the
intensity that, at time $t$, an informer forgets the rumor and becomes
inactive. These functions are both time-dependent since we assume that the
characteristics of the spread and of the forgetfulness of the news may change
over time owing to external factors. Hence, $\lambda(t)$ represents the
spreader intensity pro capite whereas $n\lambda(t)$ is the (cumulative)
spreader intensity. Similarly, $\mu(t)$ represents the inactivity density pro
capite and $n\mu(t)$ is the (cumulative) inactivity intensity. Note that the
choice of intensities with a linear dependence on $n$ rather than quadratic or
higher orders is owing to the necessity to define a non-explosive model, i.e.
one with finite time-limits. Note, additionally, that the intensities (1)
depend only on the current number of spreaders and not on the number of
inactives. These assumptions lead to a significant simplification of
calculations but are also dictated by modeling choices. The present model, in
fact, aims to consider an infinite population. Specifically, if one considers
a closed population of $N$ individuals divided into 3 categories (i.e.,
spreaders, inactives, and susceptibles, i.e. individuals who have never heard
the news), the spreaders’ rate of increase would naturally be represented by
$n(N-n-k)\lambda(t)$, where $n$ is the number of current spreaders, and $k$
that of current inactives. Consequently, we implicitly assume that $N$ is
large, so that the quantity $N-n-k$ is approximately equal to $N$. Moreover,
it is worth to notice that, for fixed $t$, the random variable $Y(t)$ counts
the number of departures from the category of spreaders up to time $t$. A
departure of a spreader can be interpreted as a recovering from the fake news.
For this reason, in our model the intensity with which inactives increase
depends exclusively on $n$. We remark that, owing to the Markovian nature of
the process,

$\pi(t):=\frac{\lambda(t)}{\lambda(t)+\mu(t)},\qquad t\geq 0,$

represents the probability that, if a new transition occurs at time $t$, it
increases the number of the spreaders by one.

According to Eqs. (1), the couples of $\mathbb{N}_{0}^{2}$ are not all
reachable by the process $(X(t),Y(t))$. Indeed, it is easy to note that the
set of the reachable states from the initial state $(j,0)$ is represented by

$S_{j}:=\left\\{(n,k)\in\mathbb{N}_{0}^{2}:n\geq
0,k\geq\max\\{0,j-n\\}\right\\},\qquad j\in\mathbb{N}.$

The state-space $S_{j}$ and the possible transitions are plotted in Figure 1.

$n$$k$$(0,0)$$(j,0)$$(0,j)$$\dots$$\dots$$\dots$ Figure 1: Representation of
the state-space $S_{j}$ for $j=2$. The solid dots indicate the absorbing
states, whereas the vectors represent all the admissible transitions.

By setting

$p_{n,k}(t):=\mathsf{P}\left[X(t)=n,Y(t)=k\right],\qquad t\geq 0,$

owing to Eqs. (1), the probabilities of the process $(X(t),Y(t))$ are governed
by the following Kolmogorov equations, for $t>0$ and $j\in\mathbb{N}$,

$\displaystyle\frac{d}{dt}p_{n,k}(t)=p_{n-1,k}(t)(n-1)\lambda(t)+p_{n+1,k-1}(t)(n+1)\mu(t)-n\left(\lambda(t)+\mu(t)\right)p_{n,k}(t),$
(2)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad\qquad\text{for
}n+k>j,\;k\geq 1,\;n\geq 1$
$\displaystyle\frac{d}{dt}p_{n,0}(t)=p_{n-1,0}(t)(n-1)\lambda(t)-n\left(\lambda(t)+\mu(t)\right)p_{n,0}(t),\qquad\quad\;\;\qquad\text{for
}n>j$
$\displaystyle\frac{d}{dt}p_{j,0}(t)=-j\left(\lambda(t)+\mu(t)\right)p_{j,0}(t),$
$\displaystyle\frac{d}{dt}p_{n,j-n}(t)=p_{n+1,j-n-1}(t)(n+1)\mu(t)-n\left(\lambda(t)+\mu(t)\right)p_{n,j-n}(t),$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad\qquad\quad\text{for
}1\leq n<j,\;1\leq k<j$
$\displaystyle\frac{d}{dt}p_{0,k}(t)=p_{1,k-1}(t)\mu(t),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\text{for
}k\geq j.$

Since the initial state of the two-dimensional process is given by
$(X(0),Y(0))=(j,0)$, the initial condition is

$p_{n,k}(0)=\delta_{n,j}\cdot\delta_{k,0},\qquad
n\in\mathbb{N},\;k\in\mathbb{N}_{0},$ (3)

being $\delta_{ij}$ the Kronecker symbol. Note that $X(t)$ is a linear time-
inhomogeneous birth-death process, owing to Eq. (1).

For notational convenience, in the remainder of the paper we denote by
$\mathsf{E}_{j}$ and $\mathsf{Var}_{j}$, respectively, the mean and the
variance conditional on the initial state $(X(0),Y(0))=(j,0)$.

##  3 The probability generating function

In order to obtain the probabilities $p_{n,k}(t)$ and the moments of the
process $(X(t),Y(t))$, we consider the corresponding probability generating
function (p.g.f.)

$G_{j}(z_{1},z_{2},t):=\mathsf{E}_{j}\left[z_{1}^{X(t)}z_{2}^{Y(t)}\right]=\sum_{n=0}^{+\infty}\sum_{k=\max\\{0,j-n\\}}^{+\infty}p_{n,k}(t)z_{1}^{n}z_{2}^{k},\qquad|z_{1}|\leq
1,\;|z_{2}|\leq 1,\;t\geq 0.$ (4)

In the following proposition we provide the partial differential equation
solved by the p.g.f. introduced in Eq. (4).

######  Proposition 3.1.

The p.g.f. $G_{j}(z_{1},z_{2},t)$ of the two-dimensional birth-death process
$(X(t),Y(t))$, with transition rates (1), solves the following partial
differential equation

$\frac{\partial}{\partial
t}G_{j}=\left[-z_{1}\lambda(t)\left(1-z_{1}\right)+\mu(t)(z_{2}-z_{1})\right]\frac{\partial}{\partial
z_{1}}G_{j},\qquad|z_{1}|\leq 1,\;|z_{2}|\leq 1,\;t\geq 0,$ (5)

with initial condition given by $G_{j}(z_{1},z_{2},0)=z_{1}^{j}$.

###### Proof.

The equation given in Eq. (5) comes from the definition of
$G_{j}(z_{1},z_{2},t)$ and Eqs. (2) after straightforward calculations. The
initial condition is an immediate consequence of Eq. (3). ∎

The PDE given in Eq. (5) generally does not have a closed-form solution.
However, as will be seen in Sections 5 and 6, it can be solved using classical
methods when the rates are constant, or temporally non-homogeneous and
proportional. Note that for the usual birth-death process, the probability
generating function is such that

$G_{j}(z_{1},z_{2},t)=\left(G_{1}(z_{1},z_{2},t)\right)^{j},\qquad|z_{1}|<1,\;|z_{2}|<1,\;t\geq
0,\;j\in\mathbb{N}.$ (6)

From Eq. (5), it follows from Eq. (6) then $G_{1}(z_{1},z_{2},t)$ satisfies
the same PDE and the same initial condition with $j=1$. Therefore, despite the
validity of the relation (6), the difficulty of the problem in question
remains unchanged. It is also observed that the validity of the identity (6)
will be further confirmed in Sections 5 and 6, where the probability
generating function is available in closed form.

######  Remark 3.1.

Clearly, when $z_{2}\to 1$ in Eq. (4) we return to p.g.f. of the the classical
one-dimensional birth-death process with linear rates. Indeed, for $z_{2}\to
1$, Eq. (5) becomes

$\frac{\partial}{\partial
t}G_{j}=(1-z_{1})(\mu(t)-z_{1}\lambda(t))\frac{\partial}{\partial
z_{1}}G_{j},\qquad|z_{1}|\leq 1,\;t\geq 0,$ (7)

which is the partial differential equation solved by the p.g.f. of a one-
dimensional birth-death process with linear rates (see Ricciardi [25] where
the explicit solution of Eq. (7) is available).

###  3.1 Extinction of the fake news

From the differential equations (2) and the related assumptions, it is easy to
notice the existence of absorbing states for the probabilities of the process
$(X(t),Y(t))$. More in detail, the set of the absorbing states is represented
by

$A_{j}:=\left\\{(0,k):k\geq j\right\\},\qquad X(0)=j\in\mathbb{N}.$ (8)

The absorption in $A_{j}$ corresponds to the eventuality that no individuals
are left willing to spread the news, and therefore the diffusion of
information ends. The absorption probability, namely the probability that the
process $(X(t),Y(t))$ reaches the absorbing set $A_{j}$, defined as

$p_{A_{j}}(t):=\mathsf{P}\left((X(t),Y(t))\in A_{j}\right).$

The probability $p_{A_{j}}(t)$ can be easily computed by means of the p.g.f.
(4), since

$p_{A_{j}}(t)=\left.G_{j}(z_{1},z_{2},t)\right|_{z_{1}\to 0,z_{2}\to 1},\qquad
t\geq 0.$ (9)

##  4 The conditional moments

Hereafter, we derive some explicit results about the moments of the process
$(X(t),Y(t))$, with special attention to the correlation.

###  4.1 The conditional moments of $X(t)$

We remark that $X(t)$ is a time non-homogeneous linear birth-death process
with individual rates given by $\lambda(t)$ and $\mu(t)$, whose moments are
already known in literature (see, for example, Eq. (7.48) and Eq. (7.49) of
[25]). Specifically, the linear time-inhomogeneous birth-death process $X(t)$
has the conditional expected value and the conditional variance given
respectively by

$m_{X}(t;j):=\mathsf{E}_{j}[X(t)]=j\eta(t),\qquad t\geq 0,$ (10)

and

$Var_{X}(t;j):=\mathsf{Var}_{j}[X(t)]=j\eta^{2}(t)\left[\frac{1}{\eta(t)}+2\phi_{X}(t)-1\right],\qquad
t\geq 0,$ (11)

with

$\eta(t):=\exp\left\\{\int_{0}^{t}(\lambda(\tau)-\mu(\tau))\textrm{d}\tau\right\\},\qquad\phi_{X}(t):=\int_{0}^{t}\frac{\lambda(\tau)}{\eta(\tau)}\textrm{d}\tau,\qquad
t\geq 0.$ (12)

###  4.2 The conditional moments of $Y(t)$

In this section, we provide the expected value and the variance of the
stochastic process $Y(t)$. We remark that $Y(t)$ represents the number of the
inactives, i.e. the individuals who have forgotten the information previously
received.

######  Proposition 4.1.

The stochastic process $Y(t)$ has the conditional expected value given by

$m_{Y}(t;j):=\mathsf{E}_{j}\left[Y(t)\right]=j\left[\phi_{Y}(t)-\eta(t)+1\right],\qquad
t\geq 0,$ (13)

where

$\phi_{Y}(t):=\int_{0}^{t}\lambda(\tau)\eta(\tau)\textrm{d}\tau,\qquad t\geq
0.$

###### Proof.

Differentiating both sides of Eq. (5) with respect to $z_{2}$, we obtain

$\frac{\partial^{2}}{\partial z_{2}\partial
t}G_{j}=\left[-z_{1}\lambda(t)(1-z_{1})+\mu(t)(z_{2}-z_{1})\right]\frac{\partial^{2}}{\partial
z_{2}\partial z_{1}}G_{j}+\mu(t)\frac{\partial}{\partial z_{1}}G_{j}.$ (14)

Hence, by evaluating Eq. (14) for $z_{1},z_{2}\to 1$, we get the following
ordinary differential equation

$\frac{d}{dt}m_{Y}(t;j)=m_{X}(t;j)\mu(t),\qquad m_{Y}(0;j)=0.$

Therefore, the result comes after straightforward calculations. ∎

In the following proposition we provide the second order moment of $Y(t)$
given the initial condition $(X(0),Y(0))=(j,0)$. To this aim, we will need the
conditional mixed moment given in Section 4.3.

######  Proposition 4.2.

The conditional second order moment of the stochastic process $Y(t)$ is given
by

$m_{2,Y}(t;j):=\mathsf{E}_{j}\left[Y(t)^{2}\right]=\int_{0}^{t}\mu(\tau)m_{X}(\tau;j)\left[1+2\gamma(\tau)\right]\textrm{d}\tau,\qquad
t\geq 0,$ (15)

where

$\gamma(t):=\int_{0}^{t}{\mu(\tau)\eta(\tau)}\left(2\phi_{X}(\tau)+j-1\right)\textrm{d}\tau,\qquad
t\geq 0.$ (16)

###### Proof.

By differentiating once again Eq. (14), we obtain

$\frac{\partial^{3}}{\partial z_{2}^{2}\partial
t}G_{j}\left[-z_{1}\lambda(t)(1-z_{1})+\mu(t)(z_{2}-z_{1})\right]\frac{\partial^{3}}{\partial
z_{2}^{2}\partial z_{1}}G_{j}+2\mu(t)\frac{\partial^{2}}{\partial
z_{2}\partial z_{1}}G_{j}.$ (17)

Combining Eqs. (14) and (17), and evaluating the resulting equality for
$z_{1},z_{2}\to 1$, we obtain

$\frac{d}{dt}m_{2,Y}(t;j)=\mu(t)m_{X}(t;j)+2\mu(t)m_{X\cdot Y}(t;j),\qquad
m_{2,Y}(0;j)=0,$ (18)

with $m_{X\cdot Y}(t;j)$ given in Eq. (19) below. Hence, Eq. (15) above easily
follows. ∎

###  4.3 The conditional mixed moments

Hereafter, we obtain the expression of the conditional expected value of
$X(t)\cdot Y(t)$ and some other indexes to study the correlation of the
processes $X(t)$ and $Y(t)$.

######  Proposition 4.3.

The conditional mixed moment of the first order of $(X(t),Y(t))$ is

$m_{X\cdot Y}(t;j)=\mathsf{E}_{j}[X(t)Y(t)]=m_{X}(t;j)\gamma(t),\qquad t\geq
0$ (19)

where $m_{X}(t;j)$ is given in Eq. (10) and $\gamma(t)$ is defined in Eq.
(16).

###### Proof.

By differentiating twice Eq. (5), once with respect to $z_{1}$ and once with
respect to $z_{2}$, we obtain the following equation

$\displaystyle\frac{\partial^{3}}{\partial z_{1}\partial z_{2}\partial
t}G_{j}$
$\displaystyle=\left[-z_{1}\lambda(t)(1-z_{1})+\mu(t)(z_{2}-z_{1})\right]\frac{\partial^{3}}{\partial
z_{2}\partial z_{1}^{2}}G_{j}$ (20)
$\displaystyle+\mu(t)\frac{\partial^{2}}{\partial
z_{1}^{2}}G_{j}+\left[-\lambda(t)-\mu(t)+2z_{1}\lambda(t)\right]\frac{\partial^{2}}{\partial
z_{2}\partial z_{1}}G_{j}.$

Therefore, by considering $z_{1},z_{2}\to 1$, from Eq. (20) we get the
following ordinary differential equation

$\frac{d}{dt}m_{X\cdot Y}(t;j)-m_{X\cdot
Y}(t;j)\left(\lambda(t)-\mu(t)\right)=\left(m_{2,X}(t;j)-m_{X}(t;j)\right)\mu(t),\qquad
m_{X\cdot Y}(0;j)=0.$

Hence, Eq. (19) comes after straightforward calculations and by considering
Eqs. (10) and (11). ∎

In order to get deeper information about the correlation of the random
processes $X(t)$ and $Y(t)$, we can introduce and study the following index
defined as the ratio between the mixed moment $m_{X\cdot Y}(t;j)$ and the
product of the expected values $m_{X}(t;j)$ and $m_{Y}(t;j)$.

######  Definition 4.1.

The correlation index of $(X(t),Y(t))$ is the ratio between $m_{X\cdot Y}(t)$
and the product $m_{X}(t)\cdot m_{Y}(t)$, i.e.

$r(t;j):=\frac{m_{X\cdot Y}(t;j)}{m_{X}(t;j)\cdot
m_{Y}(t;j)}=\frac{\gamma(t)}{m_{Y}(t;j)},\qquad t\geq 0,$ (21)

where $\gamma(t)$ is defined in Eq. (16) and $m_{Y}(t;j)$ in Eq. (13).

Clearly, the ratio $r(t;j)$ represents an adimensional correlation index, in
the sense that

$r(t;j)\begin{cases}<1,\qquad\text{if $X(t)$ and $Y(t)$ are negatively
correlated}\vspace{-0.2cm}\\\ =1,\qquad\text{if $X(t)$ and $Y(t)$ are
uncorrelated}\vspace{-0.2cm}\\\ >1,\qquad\text{if $X(t)$ and $Y(t)$ are
positively correlated.}\end{cases}$

For the computation of the ratio $r(t;j)$ we do not need the mixed moment
$m_{X\cdot Y}(t)$ but only the function $\gamma(t)$, given in (16), and the
conditional mean $m_{Y}(t;j)$. So the analysis of the correlation index
$r(t;j)$ is recommended especially when the determination of the mixed moment
$m_{X\cdot Y}(t)$ is complicated.

##  5 Constant intensities

In this section, we analyze the process $(X(t),Y(t))$ in the time-homogeneous
case, i.e. when the individual intensities are constant, i.e.

$\lambda(t)=\lambda>0,\qquad\mu(t)=\mu>0,\qquad t\geq 0.$

Let us start from the problem of finding the solution of Eq. (5). Note that
Eq. (5) can be solved by using the method of characteristics (see Section 3.2
of Evans [26], for instance). Such method converts the partial differential
equation into an appropriate system of ordinary differential equations. More
in detail, for the problem under analysis, from Eq. (5) we get the following
system

$\begin{cases}\frac{\partial t}{\partial v}=1\\\ \frac{\partial
z_{1}}{\partial v}=z_{1}(\lambda+\mu)-z_{2}\mu-z_{1}^{2}\lambda=:P(z_{1})\\\
\frac{\partial z_{2}}{\partial v}=0\\\ \frac{\partial G}{\partial
v}=0.\end{cases}$ (22)

From the first equation of (22) we have $t=v$. Moreover, since
$(\lambda+\mu)^{2}-4z_{2}\lambda\mu>0$ for any $\lambda,\mu>0$, $|z_{2}|\leq
1$, then the polynomial $P(z_{1})$ has two different real roots, namely

$\xi_{1,2}:=\xi_{1,2}(z_{2})=\frac{\lambda+\mu\pm\sqrt{(\lambda+\mu)^{2}-4z_{2}\lambda\mu}}{2\lambda},\qquad\xi_{1}<\xi_{2}.$

Consequently, the solution of the second equation of (22) is given by

$z_{1}(t)=\begin{cases}\displaystyle\frac{\xi_{2}(z_{1}-\xi_{1})-\xi_{1}(z_{1}-\xi_{2})e^{\lambda
t(\xi_{2}-\xi_{1})}}{z_{1}-\xi_{1}-(z_{1}-\xi_{2})e^{\lambda
t(\xi_{2}-\xi_{1})}},\qquad&\lambda\neq\mu\\\
\displaystyle\frac{z_{1}\left(e^{2\mu
t\sqrt{1-z_{2}}}(1-\sqrt{1-z_{2}})-1-\sqrt{1-z_{2}}\right)+z_{2}\left(1-e^{2\mu
t\sqrt{1-z_{2}}}\right)}{1-z_{1}-e^{2\mu
t\sqrt{1-z_{2}}}(1-z_{1}+\sqrt{1-z_{2}})-\sqrt{1-z_{2}}},\qquad&\lambda=\mu.\end{cases}$
(23)

We remark that the above expression of $z_{1}(t)$, for $\lambda=\mu$, is
obtained by an application of L’Hôpital’s rule to case for $\lambda\neq\mu$.
This holds for various other equations given in the rest of the paper.
Finally, from the last equation of (22), one has

$G_{j}(z_{1},z_{2},t)=\left[\widetilde{G}(z_{1},z_{2},t)\right]^{j},\qquad
t\geq 0,$ (24)

where

$\displaystyle\widetilde{G}(z_{1},z_{2},t)=\begin{cases}\displaystyle\frac{\xi_{2}(z_{1}-\xi_{1})-\xi_{1}(z_{1}-\xi_{2})e^{\lambda
t(\xi_{2}-\xi_{1})}}{z_{1}-\xi_{1}-(z_{1}-\xi_{2})e^{\lambda
t(\xi_{2}-\xi_{1})}},\quad&\lambda\neq\mu\\\
\displaystyle\frac{z_{1}\left(e^{2\mu
t\sqrt{1-z_{2}}}(1-\sqrt{1-z_{2}})-1-\sqrt{1-z_{2}}\right)+z_{2}\left(1-e^{2\mu
t\sqrt{1-z_{2}}}\right)}{1-z_{1}-e^{2\mu
t\sqrt{1-z_{2}}}(1-z_{1}+\sqrt{1-z_{2}})-\sqrt{1-z_{2}}},\quad&\lambda=\mu,\end{cases}$
(25)

for any $t\geq 0$. It is worth to notice that, owing to Eq. (4),
$G_{j}(1,1,t)=1$ as expected. We remark that also in the time-homogeneous
case, expanding the p.g.f. as a power series with respect to $z_{1}$ and
$z_{2}$ is complicated, since the dependence on the variables $z_{i}$,
$i=1,2$, is quite intricate. Therefore, providing explicit formulae for the
probabilities $p_{n,k}(t)$ is not an easy task.

Although, as previously mentioned, finding the closed-form expression of the
probabilities $p_{n,k}(t)$ is complicated, it may be interesting to analyze
$p_{0,k}(t)$ for $t\to+\infty$. This quantity represents the portion of
individuals who have had contact with the fake news in the long term by the
end of the diffusion. It is analogous to the size of the outbreak which in the
epidemiological context represents the portion of individuals who have been
infected by the epidemic in the long term. From Eq. (4), we have

$G_{j}(0,z_{2},t)=\sum_{k=j}^{+\infty}p_{0,k}(t)z_{2}^{k},\qquad t\geq 0,$

hence

$\lim_{t\to+\infty}p_{0,k}(t)=\lim_{t\to+\infty}\frac{1}{k!}\left.\frac{\partial^{k}}{\partial
z_{2}^{k}}G_{j}(0,z_{2},t)\right|_{z_{2}\to 0},\qquad k\geq j.$

In this specific case, from Eq. (25), the following limit holds

$\lim_{t\to+\infty}G_{j}(0,z_{2},t)=\lim_{t\to+\infty}\left(\widetilde{G}(0,z_{2},t)\right)^{j}=\xi_{1}^{j}.$

By considering the expansion in power series of $\xi_{1}^{j}$, we finally
obtain

$\lim_{t\to+\infty}p_{0,k}(t)=\left(\frac{\lambda\mu}{(\lambda+\mu)^{2}}\right)^{k}\left(\frac{\lambda+\mu}{\lambda}\right)^{j}\left(\binom{2k-j-1}{k-1}-\binom{2k-j-1}{k}\right),\qquad
k\geq j.$ (26)

We remark that a different approach useful to come to similar results has been
developed in Artalejo et al. [27] for the analysis of the number of
proliferation events to reach the maximum clonal size in a stochastic model of
cell proliferation. Some plots of $\lim_{t\to+\infty}p_{0,k}(t)$ are given in
Figure 2.

Figure 2: The probability $\lim_{t\to+\infty}p_{0,k}(t)$ for $j=1$ and $k=2$.

Recalling Eq. (9), in this case the absorption probability is given by

$p_{A_{j}}(t)=\begin{cases}\displaystyle\left(\frac{\mu
e^{(\lambda-\mu)t}-\mu}{\lambda
e^{(\lambda-\mu)t}-\mu}\right)^{j},\qquad&\lambda\neq\mu\\\
\displaystyle\left(\frac{\mu t}{1+\mu
t}\right)^{j},\qquad&\lambda=\mu,\end{cases}$ (27)

In Figure 3 some plots of the absorption probability (27) are shown. Note that
$p_{A_{j}}(t)$ converges to 1 when $\lambda\leq\mu$, whereas when
$\lambda>\mu$, indeed the following limit holds

$\lim_{t\to+\infty}p_{A_{j}}(t)=\begin{cases}\displaystyle\left(\frac{\mu}{\lambda}\right)^{j},\qquad&\lambda>\mu\\\
1,\qquad&\lambda\leq\mu.\end{cases}$ (28)

Figure 3: The absorption probability $p_{A_{j}}(t)$ for $\lambda=1$ and $j=1$.

Note that Eq. (28) is in agreement with Eq. (26). Indeed, from Eq. (26), it
follows that
$\lim_{t\to+\infty}\sum_{k=j}^{+\infty}p_{0,k}(t)=\lim_{t\to+\infty}p_{A_{j}}(t),$
as expected. Moreover, we remark that the results given in Eqs. (27) and (28)
are not fully novel, indeed they correspond to $p_{0}(t)$ and
$\lim_{t\to+\infty}p_{0}(t)$ given Section 6.4.3 of Allen [24] for the time-
homogeneous linear birth-death process.

######  Example 5.1.

Disclosing an explicit expression for the probabilities $p_{0,k}(t)$,
$k\in\mathbb{N}$, is complex, since obtaing the derivative of order $k$ of the
p.g.f. $G_{j}(z_{1},z_{2},t)$ with respect to $z_{2}$ is a hard task. As
example, we consider the most simple case in which the process $(X(t),Y(t))$
starts from $(1,0)$ and it is then absorbed at $(0,1)$. In particular, owing
to Eq. (24), the probability $p_{0,1}(t)$ is given by:

$p_{0,1}(t)=\left.\frac{\partial}{\partial
z_{2}}G_{j}(z_{1},z_{2},t)\right|_{z_{1},z_{2}\to
0}=\displaystyle\frac{\mu(1-e^{-(\lambda+\mu)t})}{\lambda+\mu}$

for any $t\geq 0$. See Figure 4 for some plots of the probability
$p_{0,1}(t)$. Note that $p_{0,1}(t)$ is increasing with respect to $t$ and
$\mu$, whereas it is decreasing with respect to $\lambda$. Moreover, the
following asymptotic behaviour holds

$\lim_{t\to+\infty}p_{0,1}(t)=\frac{\mu}{\lambda+\mu}.$

Figure 4: The probability $p_{0,1}(t)$ for $\lambda=1$ and $j=1$.

###  5.1 The conditional moments of $X(t)$

Regarding the linear time-homogeneous birth-death process $X(t)$, recalling
Eqs. (10) and (11), the conditional expected value and the conditional
variance of $X(t)$ are given by

$m_{X}(t;j)=je^{(\lambda-\mu)t},\qquad
Var_{X}(t;j)=\begin{cases}\displaystyle\frac{j(\mu+\lambda)}{\lambda-\mu}e^{(\lambda-\mu)t}\left(e^{(\lambda-\mu)t}-1\right),\qquad&\lambda\neq\mu\\\
2j\mu t,\qquad&\lambda=\mu,\end{cases}$

for any $t\geq 0$. Note that the following limits hold

$\lim_{t\to+\infty}m_{X}(t;j)=\begin{cases}+\infty\qquad&\lambda>\mu\\\
j,\qquad&\lambda=\mu\\\
0,\qquad&\lambda<\mu,\end{cases}\qquad\lim_{t\to+\infty}Var_{X}(t;j)=\begin{cases}+\infty\qquad&\lambda>\mu\\\
0,\qquad&\lambda\leq\mu,\end{cases}$

and

$\lim_{j\to+\infty}m_{X}(t;j)=\lim_{j\to+\infty}Var_{X}(t;j)=+\infty.$

Moreover, let us now determine the Fano factor $X(t)$ defined as the ratio
between the variance and the mean. In this case, this index is given by

$D_{X}(t;j):=\frac{Var_{X}(t;j)}{m_{X}(t;j)}=\begin{cases}\displaystyle\frac{(\mu+\lambda)(e^{(\lambda-\mu)t}-1)}{\lambda-\mu},\qquad&\lambda\neq\mu\\\
2\mu t,\qquad&\lambda=\mu,\end{cases}$ (29)

for any $t\geq 0$. This is a dispersion measure that has the same units of the
related data and it is often used for comparison with the Poisson process, for
which the factor is equal to 1. We remark that $D_{X}(0,j)=0$, and that
$D_{X}(t;j)$ is increasing with respect to $t$. In particular, the following
limits hold

$\lim_{t\to+\infty}D_{X}(t;j)=\begin{cases}\displaystyle\frac{\lambda+\mu}{\mu-\lambda},\qquad&\lambda<\mu\\\
+\infty,\qquad&\lambda\geq\mu.\end{cases}$

From Eq. (29), we have that the process $X(t)$ is underdispersed, i.e.
$D_{X}(t;j)<1$, for all

$t<\begin{cases}\log\frac{2\lambda}{\lambda-\mu},\qquad&\lambda\neq\mu\\\
\frac{1}{2\mu},\qquad&\lambda=\mu.\end{cases}$

and it is overdispersed otherwise. Moreover, the coefficient of variation of
$X(t)$ is given by

$\sigma_{X}(t;j):=\frac{\sqrt{Var_{X}(t;j)}}{m_{X}(t;j)}=\begin{cases}\frac{e^{-\lambda
t/2}}{j}\displaystyle\sqrt{\frac{(\lambda+\mu)j}{\lambda-\mu}\left(e^{\lambda
t}-e^{\mu t}\right)},\qquad&\lambda\neq\mu\\\ \displaystyle\sqrt{\frac{2\mu
t}{j}},\qquad&\lambda=\mu,\end{cases}$

for any $t\geq 0$. We note that $\sigma_{X}(0;j)=0$ and that $\sigma_{X}(t;j)$
is increasing with respect to $t$. Moreover, the following limits hold

$\lim_{t\to+\infty}\sigma_{X}(t;j)=\begin{cases}0,\qquad&\lambda<\mu\\\
+\infty,\qquad&\lambda=\mu\\\
\displaystyle\sqrt{\frac{\lambda+\mu}{j(\lambda-\mu)}},\qquad&\lambda>\mu,\end{cases}\qquad\lim_{j\to+\infty}\sigma_{X}(t;j)=0.$

Therefore, for large times and for $\lambda<\mu$ or for large initial values,
the process $X(t)$ turns out to be less variable around its mean. On the
contrary, when $\lambda=\mu$, the process $X(t)$ is more variable around its
mean. Note that some results contained in this Section are well-known in the
literature and can be found, for instance, in Section 6.4.3 of Allen [24].

###  5.2 The conditional moments of $Y(t)$

Now, we consider the process $Y(t)$. The conditional expected value of $Y(t)$
is, in time-homogeneous case, given by

$m_{Y}(t;j)=\begin{cases}\displaystyle\frac{j\mu}{\lambda-\mu}\left(e^{(\lambda-\mu)t}-1\right),\qquad&\lambda\neq\mu\\\
j\mu t,\qquad&\lambda=\mu,\end{cases}$

for any $t\geq 0$. In Figure 5(a) some plots of the conditional expected value
$m_{Y}(t;j)$ are provided. Note that in any case, $m_{Y}(t;j)$ is an
increasing function but when $\lambda>\mu$ the concavity is upward, when
$\lambda<\mu$ the concavity is downward. Note that when $\lambda>\mu$, the
process $X(t)$ shows an exponential growth trend and thus the growth trend of
$m_{Y}(t;j)$ is increasing with upward concavity. Moreover, the following
limits hold

$\lim_{t\to+\infty}m_{Y}(t;j)=\begin{cases}+\infty\qquad&\lambda\geq\mu\\\
\displaystyle\frac{j\mu}{\mu-\lambda},\qquad&\lambda<\mu,\end{cases}\qquad\lim_{j\to+\infty}m_{Y}(t;j)=+\infty.$

Figure 5: The conditional expected value $m_{Y}(t;j)$ and the conditional
variance $Var_{Y}(t)$ for $j=1$ and $\lambda=1$.

Table 1 shows the sign of the difference between $m_{X}(t;j)$ and
$m_{Y}(t;j)$. Note that when the spreader intensity $\lambda$ is at least
equal to the double of the inactivity intensity $\mu$, then $m_{X}(t;j)$ is
greater than $m_{Y}(t;j)$ for any $t\geq 0$, meaning that the expected number
of spreaders is always greater than the expected number of forgetful
individuals. Whereas, after an initial time interval $(0,\widetilde{t})$ then
$m_{Y}(t;j)>m_{X}(t;j)$ for any $t>\widetilde{t}>0$ when $\lambda\geq\mu$,
otherwise $m_{Y}(t;j)<m_{X}(t;j)$ when $\lambda<\mu$ with

$\widetilde{t}=\begin{cases}\displaystyle\frac{1}{\mu},\qquad&\lambda=\mu\\\
\displaystyle\frac{1}{\lambda-\mu}\log\left(\frac{\mu}{2\mu-\lambda}\right),\qquad&\lambda\neq\mu.\end{cases}$
(30)

Table 1: The sign of the difference between the conditional means for different range of values of $\lambda$ and $\mu$ and $\widetilde{t}$ defined in Eq. (30). $0<\lambda<\mu$ | $\displaystyle m_{X}(t;j)>m_{Y}(t;j)\iff t>\widetilde{t}$
---|---
$\mu\leq\lambda<2\mu$ | $\displaystyle m_{X}(t;j)>m_{Y}(t;j)\iff t<\widetilde{t}$
$\lambda\geq 2\mu$ |  $\displaystyle m_{X}(t;j)>m_{Y}(t;j)\qquad\forall t\geq 0$.

Furthermore, the conditional second order moment of $Y(t)$ is given by

$\displaystyle
m_{2,Y}(t;j)=\begin{cases}\displaystyle\frac{j\mu}{(\lambda-\mu)^{3}}\left(-\lambda^{2}-\lambda\mu+j\lambda\mu-j\mu^{2}+e^{2(\lambda-\mu)t}\mu(\lambda+j\lambda+\mu-j\mu)\right.\\\
\displaystyle\left.+e^{(\lambda-\mu)t}(\mu-\lambda)\left((2j-1)\mu+\lambda(4\mu
t-1)\right)\right),\qquad&\lambda\neq\mu\\\ \displaystyle\frac{1}{3}j\mu
t\left(3+3(j-1)\mu t+2\mu^{2}t^{2}\right),\qquad&\lambda=\mu,\end{cases}$

for any $t\geq 0$. In this case, the expression of the conditional variance of
$Y(t)$ is also available in closed form and is given by

$\displaystyle Var_{Y}(t;j):=\mathsf{Var}_{j}\left(Y(t)\right)$
$\displaystyle=\begin{cases}\displaystyle\frac{j\mu}{(\lambda-\mu)^{3}}\left(-\lambda(\lambda+\mu)+e^{2(\lambda-\mu)t}\mu(\lambda+\mu)+e^{(\lambda-\mu)t}(\mu-\lambda)(-\mu+\lambda(4\mu
t-1))\right),\quad&\lambda\neq\mu\\\ \displaystyle\frac{j\mu t}{3}\left(3-3\mu
t+2\mu^{2}t^{2}\right),\quad&\lambda=\mu,\end{cases}$

for any $t\geq 0$. Moreover, note that the following limits hold

$\lim_{t\to+\infty}Var_{Y}(t;j)=\begin{cases}\displaystyle\frac{j\lambda\mu(\lambda+\mu)}{(\mu-\lambda)^{3}},\qquad&\lambda<\mu\\\
\displaystyle+\infty,\qquad&\lambda\geq\mu,\end{cases}\qquad\lim_{j\to+\infty}Var_{Y}(t;j)=+\infty.$

Hence, the conditional mean $m_{Y}(t;j)$ is a significant index for the
description of the evolution of $Y(t)$ when $\lambda<\mu$, since in this case
the corresponding conditional variance is bounded. Some plots of the
conditional variance $Var_{Y}(t;j)$ are shown in Figure 5(b). In this case,
the Fano factor $D_{Y}(t;j)$ is given by

$D_{Y}(t;j)=\begin{cases}\displaystyle\frac{e^{2(\lambda-\mu)t}\mu(\lambda+\mu)-e^{(\lambda-\mu)t}(\lambda-\mu)(\lambda(4\mu
t-1)-\mu)-\lambda(\lambda+\mu)}{(\lambda-\mu)^{2}(e^{(\lambda-\mu)t}-1)},\qquad&\lambda\neq\mu\\\
\displaystyle\frac{3-3\mu t+2\mu^{2}t^{2}}{3},\qquad&\lambda=\mu,\end{cases}$

for any $t\geq 0$. Moreover the following limit holds

$\lim_{t\to+\infty}D_{Y}(t;j)=\begin{cases}+\infty\qquad&\lambda\geq\mu\\\
\displaystyle\frac{\lambda(\lambda+\mu)}{(\mu-\lambda)^{2}},\qquad&\lambda<\mu.\end{cases}$

Furthermore, the coefficient of variation is given by

$\sigma_{Y}(t;j)=\begin{cases}\sqrt{\frac{\displaystyle
e^{2(\lambda-\mu)t}\mu(\lambda+\mu)+e^{(\lambda-\mu)t}(\mu-\lambda)(\lambda(4\mu
t-1)\lambda-\mu)-\lambda(\lambda+\mu)}{(\lambda-\mu)j\mu(e^{(\lambda-\mu)t}-1)^{2}}},\quad&\lambda\neq\mu\\\
\displaystyle\sqrt{\frac{3-3\mu t+2\mu^{2}t^{2}}{3j\mu
t}},\quad&\lambda=\mu,\end{cases}$

for any $t\geq 0$. We point out that $\sigma_{Y}(t;j)$ is convergent for
$\lambda\neq\mu$ as $t\to+\infty$, indeed we have

$\lim_{t\to+\infty}\sigma_{Y}(t;j)=\begin{cases}\sqrt{\frac{\lambda(\lambda+\mu)}{j\mu(\mu-\lambda)}},\qquad&\lambda<\mu\\\
+\infty,\qquad&\lambda=\mu\\\
\sqrt{\frac{\lambda+\mu}{j(\lambda-\mu)}},\qquad&\lambda>\mu.\end{cases}$

###  5.3 Correlation between $X(t)$ and $Y(t)$

In this section, we analyze the correlation between the two components of the
process $(X(t),Y(t))$. Considering that $X(t)$ represents the number of
spreaders and $Y(t)$ the number of inactives, we expect that at the beginning
of the diffusion of the fake news the two processes are negatively correlated,
since it is reasonable that in the initial diffusion phase all individuals
behave as gullible and therefore they have the intention of spreading the
rumor. This intention is then gradually lost by the individuals and therefore
we expect that the two processes ultimately are positively correlated. To this
aim, let us start from the conditional expected value of $X(t)\cdot Y(t)$
which is given by $m_{X\cdot Y}(t;j)=j\mu\mathcal{M}(t;j)$, where

$\mathcal{M}(t;j)=\begin{cases}\displaystyle\frac{e^{(\lambda-\mu)t}}{\mu-\lambda}\left(2\lambda
t-\frac{(e^{(\lambda-\mu)t}-1)(\lambda+j\lambda+\mu-j\mu)}{\lambda-\mu}\right),\qquad&\lambda\neq\mu\\\
\displaystyle t(j+\mu t-1),\qquad&\lambda=\mu,\end{cases}$

for $t\geq 0$. In this case, it is possible to derive an explicit expression
for the conditional covariance, which is given by

$Cov(t;j):=\mathsf{Cov}_{j}(X(t),Y(t))=\mathsf{Cov}\left[(X(t),Y(t))|(X(0),Y(0))=(j,0)\right]=j\mu\mathcal{C}(t),\qquad
t\geq 0$

with

$\displaystyle\mathcal{C}(t):=\begin{cases}\displaystyle
e^{(\lambda-\mu)t}\left(\frac{e^{(\lambda-\mu)t}(\lambda+\mu)-(\lambda+\mu+2\lambda(\lambda-\mu))t}{(\lambda-\mu)^{2}}\right),\qquad&\lambda\neq\mu\\\
\mu t-1,\qquad&\lambda=\mu,\end{cases}$

and the conditional correlation is given by

$\displaystyle Corr(t;j)$
$\displaystyle:=\mathsf{Corr}_{j}((X(t),Y(t)))=\mathsf{Corr}\left[(X(t),Y(t))|(X(0),Y(0))=(j,0)\right]$
$\displaystyle=\begin{cases}\displaystyle\left[(\lambda+\mu)\left(e^{(\lambda-\mu)t}-1\right)-2(\lambda-\mu)t\right]\sqrt{\frac{\mu
e^{(\lambda-\mu)t}}{(\lambda+\mu)(\alpha_{1}(t)+\alpha_{2}(t))}},\quad&\lambda\neq\mu\\\
\displaystyle\frac{\sqrt{3}(\mu t-1)}{\sqrt{6-6\mu
t+4\mu^{2}t^{2}}},\qquad&\lambda=\mu,\end{cases}$

with

$\alpha_{1}(t):=\lambda(\lambda+\mu)+e^{3(\lambda-\mu)t}\mu(\lambda+\mu)+e^{2(\lambda-\mu)t}(\lambda^{2}-\lambda\mu-2\mu^{2}+4\lambda\mu(\mu-\lambda)t),\qquad
t\geq 0$

and

$\alpha_{2}(t):=e^{(\lambda-\mu)t}\left(\mu^{2}+\lambda^{2}(4\mu
t-2)-\lambda\mu(1+4\mu t)\right),\qquad t\geq 0.$

Note that in this case $Corr(t;j)$ does not depend on $j$. Moreover, the
following limits hold

$\displaystyle\lim_{t\to+\infty}Cov(t;j)=\begin{cases}0,\qquad&\lambda<\mu\\\
+\infty,\qquad&\lambda\geq\mu,\end{cases}\qquad$
$\displaystyle\lim_{t\to+\infty}Corr(t;j)=\begin{cases}0,\qquad&\lambda<\mu\\\
\frac{\sqrt{3}}{3}\qquad&\lambda=\mu\\\ 1,\qquad&\lambda>\mu,\end{cases}$
$\displaystyle\lim_{t\to 0}Cov(t;j)=0,\qquad$ $\displaystyle\lim_{t\to
0}Corr(t;j)=-\sqrt{\frac{\mu}{\lambda+\mu}}.$

Note that, for $t\to 0$, $Cov(t;j)\to 0$ whereas
$Corr(t;j)\to-\sqrt{\frac{\mu}{\lambda+\mu}}$ since $Var_{X}(t;j)\to 0$ and
$Var_{Y}(t;j)\to 0$. Hence, at the beginning the processes $X(t)$ and $Y(t)$
are negatively correlated. In Figure 6, some plots of the conditional
covariance $Cov(t;j)$ and of the conditional correlation coefficient
$Corr(t;j)$ are provided. Note that, in any case, the processes $X(t)$ and
$Y(t)$ are negatively correlated for small $t$, and are positively correlated
or uncorrelated for large $t$. More in detail, when $t\to+\infty$ the
processes are positively correlated if $\lambda\geq\mu$, and uncorrelated
otherwise. These results can also be justified from a modeling point of view.
At the beginning of the spread of fake news, it is reasonable to believe that
all individuals in the population are gullible and are therefore inclined to
spread the news. On the contrary, when the rumor has spread sufficiently,
individuals tend to be skeptical and lose interest in spreading the fake news.
Regarding the covariance, we note that at the beginning it tends to 0 and when
$t\to+\infty$ and $\lambda<\mu$, the covariance tends to 0, otherwise it tends
to $+\infty$. Furthermore, we notice that when $\mu\leq 1/2$ or when
$\mu>1/2,\lambda<\mu(2\mu+1)/(2\mu-1)$ it results that
$\left.\frac{\mathrm{d}}{\mathrm{d}t}Cov(t;j)\right|_{t\to 0}<0$, otherwise
$\left.\frac{\mathrm{d}}{\mathrm{d}t}Cov(t;j)\right|_{t\to 0}>0$. Hence, when
$\mu\leq 1/2$ or when $\mu>1/2,\lambda<\mu(2\mu+1)/(2\mu-1)$, the covariance
between the processes is negative in a time interval close to 0.

Figure 6: The conditional covariance $Cov(t;j)$ and the conditional
correlation coefficient $Corr(t;j)$ for $j=1$ and $\lambda=1$.

The coefficient $r(t;j)$, introduced in Definition 4.1 has the following
expression

$r(t;j)=\begin{cases}\displaystyle
1+\frac{1}{j}\left[\frac{\lambda+\mu}{\lambda-\mu}-\frac{2\lambda
t}{e^{(\lambda-\mu)t}-1}\right],\qquad&\lambda\neq\mu\\\ \displaystyle
1+\frac{\mu t-1}{j},\qquad&\lambda=\mu,\end{cases}$ (31)

for any $t\geq 0$. Some plots of the ratio $r(t;j)$ are given in Figure 7.

Figure 7: The ratio $r(t;j)$ for $j=1$ and $\lambda=1$.

Moreover, the following limits hold

$\displaystyle\lim_{t\to+\infty}r(t;j)=\begin{cases}\displaystyle
1+\frac{\lambda+\mu}{j(\lambda-\mu)},\qquad&\lambda>\mu\\\
+\infty,\qquad&\lambda\leq\mu,\end{cases}\qquad$ $\displaystyle\lim_{t\to
0}r(t;j)=1-\frac{1}{j},\qquad\lim_{j\to+\infty}r(t;j)=1.$

Note that

$r(t;1)=2\lambda\left(\frac{1}{\lambda-\mu}-\frac{t}{e^{(\lambda-\mu)t}-1}\right),\qquad
t\geq 0.$

We note that initially the processes $X(t)$ and $Y(t)$ are negatively
correlated, for $j>1$ since $r(t;j)$ for $t\to 0$ is smaller than 1. Moreover,
since $\lim_{t\to 0}r(t;j)\leq 1<\lim_{t\to+\infty}r(t;j)$ and the function
$f(t):=-\frac{\lambda+\mu}{\lambda-\mu}+\frac{2\lambda
t}{e^{(\lambda-\mu)t}-1}$ is decreasing for any $t\geq 0$, owing to (31),
there exists a single time instant $t^{*}$ in which $r(t^{*};j)=1$ or,
equivalently, $Corr(t^{*};j)=0$. We remark that the correlation of $X(t)$ and
$Y(t)$ does not depend on $j$, as we have already pointed out, whereas the
index $r(t;j)$ depens on $j$ and for great values of $j$ it is near 1.

##  6 Proportional intensities

In this section, we consider the case of proportional and time-dependent
spreader and inactivity rates, i.e.

$\lambda(t)=\rho\mu(t),\qquad\rho>0,\quad t\geq 0.$

In the context related to the growth of biological populations, the assumption
that the birth and death rates are linked by a proportionality coefficient
$\rho>0$ reflects an interaction between population growth and the regulation
of its dynamics. A self-regulation system of this kind can also find an
analogue in the context of spreading of fake news. In particular, there are
mechanisms and processes through which the dissemination of fake news can be
self-regulated over time through, for instance, information filtering by
recipients (i.e., the recipient’s ability to discern between accurate and fake
news), technological interventions (such as the implementation of algorithms
and technological tools to identify and limit the spread of fake news),
education, and awareness (the greater the public’s awareness of the risks and
effects of fake news, the greater their propensity to exclusively disseminate
accurate news).

Also in this case, it is possible to use the method of characteristics. The
corresponding system of ordinary differential equations is given by (22), with
$P(z_{1})=-\mu(t)\left[z_{1}^{2}\rho-z_{1}(\rho+1)+z_{2}\right]$. With a
reasoning similar to that of Section 5, we get the following expression for
the p.g.f.

$G_{j}(z_{1},z_{2},t)=\begin{cases}\displaystyle\left(\frac{\xi_{2}(z_{1}-\xi_{1})-\xi_{1}(z_{1}-\xi_{2})\exp\left((\xi_{2}-\xi_{1})M(t)\right)}{z_{1}-\xi_{1}-(z_{1}-\xi_{2})\exp\left((\xi_{2}-\xi_{1})M(t)\right)}\right)^{j},\;&\rho\neq
1,\\\
\displaystyle\left(\frac{z_{1}\left(e^{2M(t)\sqrt{1-z_{2}}}(1-\sqrt{1-z_{2}})-1-\sqrt{1-z_{2}}\right)+z_{2}\left(1-e^{2M(t)\sqrt{1-z_{2}}}\right)}{1-z_{1}-e^{2M(t)\sqrt{1-z_{2}}}(1-z_{1}+\sqrt{1-z_{2}})-\sqrt{1-z_{2}}}\right)^{j},\;&\rho=1,\end{cases}$

where

$M(t):=\int_{0}^{t}\mu(\tau)\mathrm{d}\tau,\qquad t\geq 0,$ (32)

represents the cumulative inactivity intensity, $|z_{1}|\leq 1$, $|z_{2}|\leq
1$, $t\geq 0$, and

$\xi_{1,2}:=\xi_{1,2}(z_{2})=\frac{\rho+1\mp\sqrt{(\rho+1)^{2}-4z_{2}\rho}}{2\rho},\qquad\xi_{1}<\xi_{2}.$

Note that $G_{j}(1,1,t)=1$, as expected. Also in this case, as in Section 5,
it is hard to obtain the probabilities $p_{n,k}(t)$ from
$G_{j}(z_{1},z_{2},t)$. Recalling Eq. (9), in this case we get

$p_{A_{j}}(t)=\begin{cases}\displaystyle\left(\frac{e^{\frac{\rho-1}{\rho}M(t)}-1}{\rho
e^{\frac{\rho-1}{\rho}M(t)}-1}\right)^{j},\qquad&\rho\neq 1\\\
\displaystyle\left(\frac{M(t)}{1+M(t)}\right)^{j},\qquad&\rho=1.\end{cases}$
(33)

Note that the probabilities $p_{A_{j}}(t)$ given in Eq. (27) and in Eq. (33)
coincide with the probabilities of extinction of a birth-death process given,
for example, in Eq. (4.41) and Eq. (7.42a) of [25].

Note that, since $M(t)$ is an increasing and positive function,
$\lim_{t\to+\infty}M(t)\in(0,+\infty]$. Hence, when
$\lim_{t\to+\infty}M(t)=M\in\mathbb{R}_{+}$, we have

$\lim_{t\to+\infty}p_{A_{j}}(t)=\begin{cases}\displaystyle\left(\frac{e^{\frac{\rho-1}{\rho}M}-1}{\rho
e^{\frac{\rho-1}{\rho}M}-1}\right)^{j},\qquad&\rho\neq 1\\\
\displaystyle\left(\frac{M}{1+M}\right)^{j},\qquad&\rho=1.\end{cases}$

Otherwise, when $\lim_{t\to+\infty}M(t)=+\infty$, we get

$\lim_{t\to+\infty}p_{A_{j}}(t)=\begin{cases}\displaystyle\left(\frac{1}{\rho}\right)^{j},\qquad&\rho>1\\\
1,\qquad&0<\rho\leq 1.\end{cases}.$

Note that when $\lim_{t\to+\infty}M(t)=+\infty$, with a reasoning similar to
that used in Section 5, it is possible to show that

$\lim_{t\to+\infty}p_{0,k}(t)=\left(1+\frac{1}{\rho}\right)^{j}\left(\frac{\rho}{(1+\rho)^{2}}\right)^{k}\left[\binom{2k-j-1}{k-1}-\binom{2k-j-1}{k}\right].$

Also in this case,
$\lim_{t\to+\infty}\sum_{k=j}^{+\infty}p_{0,k}(t)=\lim_{t\to+\infty}p_{A_{j}}(t)$
as expected.

######  Example 6.1.

The probability $p_{0,1}(t)$, i.e. the probability that the process
$(X(t),Y(t))$ starting from $(0,1)$ is then absorbed in $(1,0)$ by time $t$,
is given by

$p_{0,1}(t)=\left.\frac{\partial}{\partial
z_{2}}G_{j}(z_{1},z_{2},t)\right|_{z_{1},z_{2}\to
0}=\begin{cases}\displaystyle\frac{1-e^{-\frac{1+\rho}{\rho}M(t)}}{1+\rho},\qquad&\rho\neq
1\\\ \displaystyle\frac{1-e^{-2M(t)}}{2},\qquad&\rho=1,\end{cases}$

where $M(t)$ is defined in (32).

###  6.1 The conditional moments of $X(t)$

Under the assumptions $\lambda(t)=\rho\mu(t)$ with $\rho>0$, the conditional
mean and the conditional variance of $X(t)$ are given by

$m_{X}(t;j)=\displaystyle je^{(\rho-1)M(t)},\qquad t\geq 0$

and

$Var_{X}(t;j)=\begin{cases}\displaystyle\frac{j(\rho+1)}{\rho-1}e^{(\rho-1)M(t)}\cdot\left[e^{(\rho-1)M(t)}-1\right],\qquad&\rho\neq
1\\\ \displaystyle 2jM(t),\qquad&\rho=1,\end{cases}$ (34)

for any $t\geq 0$. Moreover, the Fano factor is given by

$D_{X}(t;j)=\begin{cases}\displaystyle\frac{\rho+1}{\rho-1}\left[e^{(\rho-1)M(t)}-1\right],\qquad&\rho\neq
1\\\ \displaystyle 2M(t),\qquad&\rho=1,\end{cases}$

and the coefficient of variation by

$\sigma_{X}(t;j)=\begin{cases}\displaystyle\frac{e^{-\rho/2M(t)}}{j}\sqrt{\frac{(\rho+1)j}{\rho-1}\left[e^{\rho
M(t)}-e^{M(t)}\right]},\qquad&\rho\neq 1\\\
\displaystyle\sqrt{\frac{2M(t)}{j}},\qquad&\rho=1.\end{cases}$

In Figure 8 we provide some plots of $m_{X}(t;j)$ and $Var_{X}(t;j)$ by
considering $\mu(t)=\mu+\alpha\cos\left(\frac{2\pi}{Q}t\right)$ with $Q>0$,
$t\geq 0$, $\mu>|\alpha|>0$. Such choice may represent the periodical
fluctuation of the inactivity rate owing to, for example, periodic denies of
the rumor.

Figure 8: The conditional mean $m_{X}(t;j)$ and the conditional variance
$Var_{X}(t;j)$ for $\lambda=\rho\mu(t)$ and
$\mu(t)=\mu+\alpha\cos\left(\frac{2\pi}{Q}t\right)$ with $\mu=j=1$ and
$Q=2.5$.

Note that, considering that $\lim_{t\to+\infty}M(t)\in[0,+\infty]$, when
$\lim_{t\to+\infty}M(t)=M\in\mathbb{R}$ then $m_{X}(t;j)$, $Var_{X}(t;j)$,
$D_{X}(t;j)$ and $\sigma_{X}(t;j)$ converge, otherwise when
$\lim_{t\to+\infty}M(t)=+\infty$ the following limits hold

$\displaystyle\lim_{t\to+\infty}m_{X}(t;j)=\begin{cases}0,\qquad&\rho<1\\\
j,\qquad&\rho=1\\\
+\infty,\qquad&\rho>1\end{cases},\qquad\lim_{t\to+\infty}Var_{X}(t;j)=\begin{cases}0,\qquad&\rho<1\\\
+\infty,\qquad&\rho\geq 1\end{cases},$
$\displaystyle\lim_{t\to+\infty}D_{X}(t;j)=\begin{cases}\frac{\rho+1}{1-\rho},\qquad&\rho<1\\\
+\infty\qquad&\rho\geq
1\end{cases},\qquad\lim_{t\to+\infty}\sigma_{X}(t;j)=\begin{cases}+\infty,\qquad&\rho\leq
1\\\ \sqrt{\frac{\rho+1}{j(\rho-1)}},\qquad&\rho>1.\end{cases}$

###  6.2 The conditional moments of $Y(t)$

In the case of proportional intensities, the conditional expected value of
$Y(t)$ is given by

$m_{Y}(t;j)=\begin{cases}\displaystyle\frac{j}{\rho-1}\left(e^{(\rho-1)M(t)}-1\right),\qquad&\rho\neq
1\\\ \displaystyle jM(t),\qquad&\rho=1,\end{cases}$ (35)

where $M(t)$ is defined in Eq. (32) for any $t\geq 0$. Considering that the
sign of $\mu(t)$ is always positive, the conditional expected value
$m_{Y}(t;j)$ is increasing. The concavity of $m_{Y}(t;j)$ depends on the sign
of $\rho-1$ and on the monotonicity of $\mu(t)$. Indeed, since

$\frac{d^{2}}{dt^{2}}m_{Y}(t;j)=je^{(\rho-1)M(t)}\left(\frac{d}{dt}\mu(t)+\mu^{2}(t)(\rho-1)\right),\qquad
t\geq 0,$

then the concavity of $m_{Y}(t;j)$ is upward when $\mu(t)$ is an increasing
function and $\rho>1$, whereas it is downward when $\mu(t)$ is a decreasing
function and $0<\rho<1$. In Table 2 the sign of the difference between
$m_{X}(t;j)$ and $m_{Y}(t;j)$ is analyzed. Note that when the spreader
intensity $\lambda(t)$ is at least equal to the double of the inactivity
intensity $\mu(t)$, then $m_{X}(t;j)$ is greater than $m_{Y}(t;j)$ for any
$t\geq 0$, similarly to the result of Section 5.

Table 2: The sign of the difference between the conditional means for different range of values of $\rho$. $0<\rho<1$ | $\displaystyle m_{X}(t;j)>m_{Y}(t;j)\iff M(t)>\frac{1}{\rho-1}\log\left(\frac{1}{2-\rho}\right)$
---|---
$\rho=1$ | $\displaystyle m_{X}(t;j)>m_{Y}(t;j)\iff M(t)<1$
$1<\rho<2$ | $\displaystyle m_{X}(t;j)>m_{Y}(t;j)\iff M(t)<\frac{1}{\rho-1}\log\left(\frac{1}{2-\rho}\right)$
$\rho\geq 2$ |  $\displaystyle m_{X}(t;j)>m_{Y}(t;j)\qquad\forall t\geq 0$.

The conditional second order moment of $Y(t)$ is given by

$\displaystyle
m_{2,Y}(t;j)=\begin{cases}\displaystyle\frac{j(\rho+1-2j)}{(\rho-1)^{2}}\left(e^{(\rho-1)M(t)}-1\right)-\frac{4\rho
j}{(\rho-1)^{2}}M(t)\\\ \displaystyle\times
e^{(\rho-1)M(t)}+\frac{j(1+j(\rho-1)+\rho)}{(\rho-1)^{3}}\left(e^{2(\rho-1)M(t)}-1\right),&\rho\neq
1\\\
\displaystyle\frac{j}{3}M(t)\left(3+3(j-1)M(t)+2M^{2}(t)\right),\qquad&\rho=1,\end{cases}$
(36)

where $M(t)$ is defined in Eq. (32) for any $t\geq 0$. The expressions of the
variance and of the other indexes are also available in closed form but they
are omitted for brevity. We remark that, if
$\lim_{t\to+\infty}M(t)=M\in\mathbb{R}_{+}$, then $m_{Y}(t;j)$,
$Var_{Y}(t;j)$, $D_{Y}(t;j)$ and $\sigma_{Y}(t;j)$ are finite. Otherwise, if
$\lim_{t\to+\infty}M(t)=+\infty$, then the following limits hold

$\lim_{t\to+\infty}m_{Y}(t;j)=\begin{cases}\frac{j}{1-\rho},\quad&\rho<1\\\
+\infty,\quad&\rho\geq
1\end{cases},\quad\lim_{t\to+\infty}m_{2,Y}(t;j)=\begin{cases}\displaystyle\frac{j(j(\rho-1)+(3-5\rho)\rho)}{(\rho-1)^{3}},\quad&\rho<1\\\
+\infty,\quad&\rho\geq 1.\end{cases}$

###  6.3 Correlation between $X(t)$ and $Y(t)$

The conditional expected value of $X(t)\cdot Y(t)$ is given by

$\displaystyle m_{X\cdot Y}(t;j)=\begin{cases}\displaystyle
je^{(\rho-1)M(t)}\left[\frac{2\rho}{1-\rho}M(t)+\left(j-1-\frac{2\rho}{1-\rho}\right)\frac{e^{(\rho-1)M(t)}-1}{\rho-1}\right],\quad&\rho\neq
1\\\ \displaystyle jM(t)\left(j-1+M(t)\right),\quad&\rho=1,\end{cases}$

where $M(t)$ is defined in Eq. (32) for any $t>0$. In this case, the study
regarding the covariance $Cov(t;j)$ and the correlation $Corr(t;j)$ is omitted
since such indexes possess cumbersome expressions. Hence, we focus on the
determination and analysis of the adimensional correlation index $r(t;j)$.
Recalling Definition 4.1, the coefficient $r(t;j)$ is given by

$r(t;j)=\begin{cases}\displaystyle
1+\frac{1}{j}+\frac{2}{\rho-1}-\frac{2M(t)\rho}{j(e^{(\rho-1)M(t)}-1)},\qquad&\rho\neq
1\vspace{0.3cm}\\\ \displaystyle 1+\frac{M(t)-1}{j},\qquad&\rho=1,\end{cases}$

where $M(t)$ is defined in Eq. (32) for any $t\geq 0$. In this case
$\lim_{j\to+\infty}r(t;j)=1$, meaning that when the number of initial
spreaders is high, then the processes $X(t)$ and $Y(t)$ are uncorrelated. We
note that, when $\lim_{t\to+\infty}M(t)=M\in\mathbb{R}$, both the mean
$m_{X\cdot Y}(t)$ and the adimensional correlation index $r(t;j)$ are finite.
Otherwise, when $\lim_{t\to+\infty}M(t)=+\infty$, the following limits hold

$\lim_{t\to+\infty}m_{X\cdot Y}(t;j)=\begin{cases}0,\qquad&\rho<1\\\
+\infty,\qquad&\rho\geq
1,\end{cases}\lim_{t\to+\infty}r(t;j)=\begin{cases}+\infty,\qquad&\rho\leq
1\\\ 1+\frac{1}{j}+\frac{2}{\rho-1},\qquad&\rho>1.\end{cases}$

Moreover, if $0<\rho\leq 1$ then $r(0,j)<1$, otherwise $r(0,j)>1$, meaning
that if $0<\rho\leq 1$, the processes are negatively correlated at the
beginning of the time-evolution. This result can be interpreted from a
modeling point of view as follows. When spreaders grow more slowly than
inactives (i.e. $0<\rho\leq 1$), then all individuals behave as skeptical from
the beginning and they therefore do not intend to spread the news. Similarly,
when spreaders grow faster than inactive ones (i.e. $\rho>1$), then the
population initially driven to spread the news being influenced by the rapid
growth of the number of spreaders. Some specific cases are discussed in the
following section.

##  7 Special cases of the conditional means

In this section, we analyze some special situations in which the spreader and
the inactivity intensities are proportional, i.e. $\lambda(t)=\rho\mu(t)$,
$\rho>0$ (as in Section 6) and the conditional means $m_{X}(t;j)$ and
$m_{Y}(t;j)$ are set equal to some very well-known growth curves. Considering
the application to the context of fake news, it is expected a non-explosive
behavior from the category of spreaders and consequently from the inactives.
Indeed, it usually happens that after an initial period of explosion in the
spread of a fake news, it gradually tends to disappear, thus reaching only a
limited portion of the population. Therefore, it is plausible to expect an
average trend for spreaders and inactives of sigmoidal type. This assumption
is also appropriate for taking into account that the members of the population
are generally not particularly inclined to follow fake news irrationally,
although (as detailed in Section 2) the total size $N$ of the population is
seen as very large. In addition, it is worth mentioning that even if the
present model can be applied, in general, not only for the diffusion of fake
news but for any kind of news, the specific assumption that the conditional
means of the components follow a sigmoidal pattern is appropriate specifically
in the context of the dissemination of fake news which, by their nature, are
not intended to achieve widespread diffusion in the population. Hence, it
seems reasonable to give attention to the cases when the means $m_{X}(t;j)$
and $m_{Y}(t;j)$ are equal to specific sigmoidal growth functions, which are
often used to model real growth phenomena (for a deep study of sigmoidal
models see, for instance, Albano et al. [28]). Regarding the process $Y(t)$,
it is known that $Y(0)=0$, hence we investigate the conditions under which
$m_{Y}(t;j)$ is equal to sigmoidal growth curves starting from $0$. It seems
reasonable to assume that the initial size of the inactive individuals is
equal to 0, since at the beginning of the diffusion all the individuals
believe in the rumor and have the intention of spreading it. Likewise, we
assume that the initial size of spreaders is positive (i.e. $X(0)=j>0$);
otherwise, the rumor diffusion would not ensue.

In particular, for the process $X(t)$ we consider:

  * (i)

The classical Gompertz growth curve (cf. Román-Román et al. [29]), which has
been used in several contexts during recent years, especially related to tumor
growth (see for example Román-Román et al. [30]).

  * (ii)

The generalized Gompertz growth curve introduced in Asadi et al. [31], which
is related to the generalized Pareto distribution. In particular, such growth
function can have different kinds of behaviour (it can be sigmoidal,
increasing with infinite limit value, bell-shaped) depending on the values of
the parameters. For our interest, we consider only the case in which the
generalized Gompertz curve has a sigmoidal shape.

  * (iii)

The classical logistic growth curve, which is one of the most popular growth
models characterized by a finite carrying capacity. It was originally derived
by Verhulst to describe the self-limiting growth of a biological population
and during the years it has been deeply analyzed (see, for example, Di
Crescenzo and Paraggio [32]).

  * (iv)

The extended logistic growth curve (cf. Di Crescenzo et al. [2] and San
MartÃ¬n [1]), which has been proposed to model the rumor spread among
restricted populations.

  * (v)

The multisigmoidal logistic growth curve (studied in Di Crescenzo et al. [33]
and [34]), which exhibits multiple inflections.

  * (vi)

The modified Korf growth curve (analyzed in Di Crescenzo and Spina [35]), that
captures certain features of both the Gompertz and Korf growth laws.

Whereas, for the process $Y(t)$ we consider:

  * (I)

The classical Korf model (cf. Eq. (9) of Di Crescenzo and Spina [35]), which
starts from the state $0$ at time $0$, then evolving as a sigmoidal function.

  * (II)

The Mitscherlich model (cf. Eq. (2) of Ware et al. [36]) which starts from the
state $0$ at time $0$ and has a finite limit. Such growth curve has been
proposed to model the plant growth and to determine critical nutrient
deficiency levels.

Note that all the aforementioned curves are S-shaped, namely they start with
an explosion of exponential type, then an inflection (or more) occurs and
finally they flattens up to a constant value, known as carrying capacity.

Table 3 summarizes the most relevant information considering cases
(i)$\div$(vi), such as the expression of the conditional mean $m_{X}(t;j)$,
which corresponds to the expression of the corresponding growth curve, the
expression of the cumulative inactivity intensity $M(t)$, the time intervals
in which the sign of the difference $m_{X}(t;j)-m_{Y}(t;j)$ is positive, the
sign of the correlation coefficient $Corr(t;j)$, the Fano factor, the time
intervals of overdispersion and the coefficient of variation for both the
conditional processes $X(t)$ and $Y(t)$ given the initial state
$(X(0),Y(0))=(j,0)$. To shorten the formulas contained in Table 3, for case
(iv), we set

$m_{X}^{(el)}(t;j):=\frac{(\varepsilon-1)N(N-j)+e^{(1+\varepsilon)t}N(2\varepsilon
j+N-\varepsilon N)}{2\varepsilon(N-j)+e^{(1+\varepsilon)t}(2\varepsilon
j+N-\varepsilon N)},\qquad t\geq 0,$

and

$M^{(el)}(t):=\frac{1}{\rho-1}\log\left(\frac{(\varepsilon-1)N(N-j)+e^{(1+\varepsilon)t}N(N(\varepsilon-1)-2\varepsilon
j)}{2\varepsilon j(N-j)+e^{(1+\varepsilon)t}j(2\varepsilon j+N-\varepsilon
N)}\right),\qquad t\geq 0.$

Table 3: Some relevant characteristic of the conditional processes $X(t)$ and
$Y(t)$ given the initial state $(X(0),Y(0))=(j,0)$ by considering different
growth models for $m_{X}(t;j)$. Note that $\alpha,\beta,A,b,C,j>0$, $\rho>1$.
For the logistic case one has $0<j<C$, for the multisigmoidal logistic
$Q_{\beta}(t):=\sum_{k=1}^{n}\beta_{k}t^{k}$, with $\beta_{n}<0$.

growth curve | $m_{X}(t;j)$ | $M(t)$ | $\mathop{\rm sgn}(m_{X}(t;j)-m_{Y}(t;j))=+1,\;t\geq\widetilde{t}$ | $\mathop{\rm sgn}(Corr(t;j))$
---|---|---|---|---
(i) Gompertz | $j\exp\left(\alpha(1-e^{-\beta t})\right)$ | $\frac{\alpha\left(1-e^{-\beta t}\right)}{\rho-1}$ | $\widetilde{t}=\begin{cases}-\frac{1}{\beta}\left(1-\frac{1}{\alpha}\log(\rho-2)\right),&\text{ if }1<\rho<2\\\ 0,&\text{ otherwise}\end{cases}$ | $+1,t\geq 0$
(ii) generalized Gompertz | $j\exp\left(Ab\left(1-\frac{b}{t+b}\right)\right)$ | $\frac{Abt}{(\rho-1)(t+b)}$ | $\widetilde{t}=\begin{cases}0,&\text{ if }\rho>2\\\ +\infty&\text{ otherwise}\end{cases}$ | changing
(iii) logistic | $\frac{Cj}{j+(C-j)e^{-rt}}$ | $\frac{1}{\rho-1}\left({rt+\log\frac{C}{C+j(e^{rt}-1)}}\right)$ | $\widetilde{t}=\begin{cases}\frac{1}{r}\log\left(\frac{C-j}{C(2-\rho)-j}\right),&\text{ if }1<\rho<2,C>\frac{j}{2-\rho},\\\ 0,&\text{ otherwise}\end{cases}$ | changing
(iv) extended logistic | $m_{X}^{(el)}(t;j)$ | $M^{(el)}(t)$ | NA in closed form | changing
(v) multisigmoidal logistic | $\frac{Cj}{j+(C-j)e^{Q_{\beta}(t)}}$ | $\frac{1}{\rho-1}\left({Q_{\beta}(t)+\log\frac{C}{C+j(e^{Q_{\beta}(t)}-1)}}\right)$ | NA in closed form | changing
(vi) modified Korf | $j\exp\left[\frac{\alpha}{\beta}\left(1-(1+t)^{-\beta}\right)\right]$ | $\frac{A\left(1-(1+t)^{-b}\right)}{b(\rho-1)}$ | $\widetilde{t}=\begin{cases}e^{-1/\beta}\left(1+\frac{\beta}{\alpha}\log(2-\rho)\right),&\text{ if }\alpha+\beta\log(2-\rho)>0\\\ 0,&\text{ otherwise}\end{cases}$ | changing

growth curve | process |  Fano factor $D_{Z}(t;j)$ | underdispersion conditions |  coefficient of variation $\sigma_{Z}(t;j)$
---|---|---|---|---
(i) Gompertz | $X$ | $\frac{\rho+1}{\rho-1}\left(\exp\left(\alpha-\alpha e^{-\beta t}\right)-1\right)$ | $t\geq\widetilde{t}=\begin{cases}\frac{1}{\beta}\log\frac{\alpha}{\alpha-\log 2+\log{1+\rho}}&\text{ if }\alpha>\log\frac{2}{1+\rho}\\\ 0&\text{ otherwise }\end{cases}$ | $\frac{e^{-\alpha/2}}{j}\sqrt{\frac{j(\rho+1)}{\rho-1}\left(e^{\alpha}-\exp\left(\alpha e^{-\beta t}\right)\right)}$
$Y$ | omitted | NA in closed form | omitted
(ii) generalized Gompertz | $X$ | $\frac{\rho+1}{\rho-1}\left(\exp\frac{Abt}{t+b}-1\right)$ | $t\leq\widetilde{t}=\begin{cases}\frac{b\log\frac{2\rho}{\rho+1}}{\log\frac{2\rho}{\rho+1}-Ab},\text{ if }Ab-\log\frac{2\rho}{\rho+1}<0\\\ +\infty\text{ otherwise }\end{cases}$ | $\sqrt{\frac{j(\rho+1)}{\rho-1}\left(1-\exp\frac{Abt}{b+t}\right)}$
$Y$ | omitted | NA in closed form | omitted
(iii) logistic | $X$ | $\frac{\rho+1}{\rho-1}\left(\frac{C}{j+e^{-rt}(C-j)}-1\right)$ | $t<\frac{1}{r}\log\left(\frac{(C-j)(\rho-1)}{\rho(C-j)+C+j}\right)$ | $\frac{e^{-rt}\left(C+(e^{rt}-1)j\right)}{Cj}\sqrt{\frac{Ce^{rt}(e^{rt}-1)(\rho+1)(C-j)j}{(\rho-1)\left(C+j(e^{rt}-1)\right)^{2}}}$
$Y$ | omitted | NA in closed form | omitted
(iv) extended logistic | $X$ | $\frac{(e^{(1+\varepsilon)t}-1)(N-j)(\rho+1)(2\varepsilon j+N(1-\varepsilon))}{j(2\varepsilon(N-j)+e^{(1+\varepsilon)t}(2\varepsilon j+N(1-\varepsilon)))(\rho-1)}$ | NA in closed form | omitted
$Y$ | omitted | NA in closed form | omitted
(v) multisigmoidal logistic | $X$ | $\frac{\rho+1}{\rho-1}\left(\frac{C}{j+e^{Q_{\beta}(t)}(C-j)}-1\right)$ | NA in closed form | $\frac{e^{Q_{\beta}(t)}\left(C+(e^{-Q_{\beta}(t)}-1)j\right)}{Cj}\sqrt{\frac{Ce^{-Q_{\beta}(t)}(e^{-Q_{\beta}(t)}-1)(\rho+1)(C-j)j}{(\rho-1)\left(C+j(e^{-Q_{\beta}(t)}-1)\right)^{2}}}$
$Y$ | omitted | NA in closed form | omitted
(vi) modified Korf | $X$ | $\frac{\rho+1}{\rho-1}\left(\exp\left\\{\frac{\alpha}{\beta}\left[1-(1+t)^{-\beta}\right]\right\\}-1\right),$ | $t\geq\widetilde{t}=\begin{cases}0,\text{ if }\alpha-\beta\log\left(\frac{2\rho}{\rho+1}\right)<0\\\ \left[1-\frac{\beta}{\alpha}\log\left(\frac{2\rho}{1+\rho}\right)\right]^{-1/\beta}-1,\text{ otherwise}\end{cases}$ | $\sqrt{\frac{1+\rho}{j(\rho-1)}\left[1-\exp\left\\{\frac{\alpha}{\beta}\left[(1+t)^{-\beta}-1\right]\right\\}\right]}$
$Y$ | omitted | NA in closed form | omitted

Note that knowing $M(t)$ gives the possibility to get $Var_{X}(t;j)$,
$m_{Y}(t;j)$ and $Var_{Y}(t;j)$ by means of Eqs. (34), (35) and (36),
respectively. Moreover, we remark that when $m_{X}(t;j)-m_{Y}(t;j)>0$ then the
expected number of spreaders is greater than the expected number of inactive
individuals. Other interesting limit behaviours are given in Table 5 where the
function $\mathcal{V}_{i,j}$, $\mathcal{C}_{i,j}$ and $\mathcal{R}_{i,j}$, for
$i=1,2,\dots,6$, $j=1,2,3$ are given in Table 6. We point out that in the
Gompertz, generalized Gompertz, extended logistic, modified Korf cases, all
the available limits with respect to the time are finite, whereas in the
logistic case the variance $Var_{Y}(t;j)$ diverges as $t\to+\infty$. In
Figures 9(a)$\div$(d), we provide the conditional means $m_{X}(t;j)$ and
$m_{Y}(t;j)$, the conditional variance $Var_{X}(t;j)$ and $Var_{Y}(t;j)$.
Moreover, Figures 9(e)-(f) show the conditional correlation coefficient
$Corr(t;j)$ and the conditional coefficient $r(t;j)$. Note that in the
Gompertz case, the processes $X(t)$ and $Y(t)$ given the initial state
$(X(0),Y(0))=(j,0)$ are always positively correlated, whereas in the logistic
case the correlation coefficient is close to 0. Comparing the results related
to cases (i)-(vi) shown in Figure 9, we remark that expected number of
spreaders and the expected number of inactive individuals tend to the
corresponding carrying capacities (i.e. the maximum reachable number of
spreaders/inactives in the population) but with different speed. The slowest
case is the generalized Gompertz, whereas the fastest is the multisigmoidal
logistic. Hence, in the multisigmoidal logistic case (generalized Gompertz
case), the fake news exhibits the fastest (the slowest) diffusion among the
population and the fastest (slowest) growth of the inactives. Moreover, the
correlation between spreaders and inactives is the highest (from the beginning
to the end of the diffusion) in the Gompertz case, whereas it is the lowest in
the multisigmoidal logistic case at the beginning of the propagation of the
rumor and in the logistic case for large times. Similarly, the adimensional
correlation index (21) is always greater than 1 (corresponding to highly
positive correlation) in the Gompertz case whereas in all the other cases such
index is smaller than 1 (corresponding to negative correlation) at the
beginning, and it is greater than 1 for large times.

Moreover, Table 4 summarizes the most relevant information considering the
case in which $m_{Y}(t;j)$ is set equal to the Korf and to the Mitscherlich
growth function, such as the expression of the conditional mean $m_{Y}(t;j)$,
the expression of the cumulative inactivity intensity $M(t)$, the time
intervals in which the sign of the difference $m_{X}(t;j)-m_{Y}(t;j)$ is
positive, the sign of the correlation coefficient $Corr(t;j)$, the Fano
factor, the time intervals of overdispersion and the coefficient of variation
for both the conditional processes $X(t)$ and $Y(t)$ given the initial state
$(X(0),Y(0))=(j,0)$, the limit behaviors of the means, of the variances, of
the covariance and of the adimensional correlation index $r(t;j)$. Note that
if $t$ is large, the expected number of spreaders is greater than the expected
number of inactives. Whereas, if $t$ is close to $0$ and the mean of $Y(t)$ is
the Mitscherlich growth function, then the process $X(t)$ is underdispesed. On
the other hand, if $m_{Y}(t;j)$ is the Korf function, then the process $X(t)$
is overdispersed for $t$ close to $0$. The coefficient of variation of $X(t)$
tends to a finite value for $t\to+\infty$ both in Case (I) and Case (II).
Moreover, for both Case (I) and Case (II), the means $m_{X}(t;j)$ and
$m_{Y}(t;j)$ are significant indexes since the corresponding variances are
finite. Considering Figure 10, we note that for the specific values of
parameters, the expected number of spreaders/inactives is higher in Case (I).
Also for the variances of $X(t)$ and $Y(t)$, we note that in Case (I) we
obtain the greatest values. Whereas, at the beginning of the propagation of
the rumor, the correlation between spreaders and inactive individuals is
higher in Case (I), but for large times it is higher in Case (II). Regarding
the adimensional correlation index $r(t;j)$, we note that it is always higher
in Case (II). Moreover, for that specified choice of parameters, one has
$r(t;j)<1$ for any $t\geq 0$ in Case (I), meaning that when the expected
number of inactive individuals is the Mitscherlich growth function, the
components $X(t)$ and $Y(t)$ are negatively correlated.

Table 4: Some relevant characteristic and some limit behaviors of the
conditional processes $X(t)$ and $Y(t)$ given the initial state
$(X(0),Y(0))=(j,0)$ by considering the Korf and the Mitscherlich growth models
for $m_{Y}(t;j)$. Note that $\alpha,\beta,j>0$, $\rho>1$.

growth curve for $Y(t)$ | $m_{Y}(t;j)$ | $M(t)$ | $\mathop{\rm sgn}(m_{X}(t;j)-m_{Y}(t;j))=+1,\;t\geq\widetilde{t}$ | $\mathop{\rm sgn}(Corr(t;j))$
---|---|---|---|---
(I) Korf | $\frac{j}{\rho-1}\exp\left(-\frac{\alpha}{\beta}t^{-\beta}\right)$ | $\frac{1}{\rho-1}\log\left[1+\exp\left(-\frac{\alpha}{\beta}t^{-\beta}\right)\right]$ | $\widetilde{t}=\begin{cases}\left(\frac{\alpha}{\beta\log\frac{2-\rho}{\rho-1}}\right)^{1/\beta},&\text{ if }\frac{3}{2}<\rho\leq 2\\\ 0,&\text{ otherwise}\end{cases}$ | changing
(II) Mitscherlich | $\beta\left(1-e^{-\alpha t}\right)$ | $\frac{1}{\rho-1}\log\left(2-e^{-\alpha t}\right)$ | $\widetilde{t}=\begin{cases}0,\;\text{if }\rho>\frac{2\beta-j}{\beta}\\\ \frac{1}{\alpha}\log\left(\frac{\beta(2-\rho)}{\beta(2-\rho)-j}\right),\;\text{otherwise}\end{cases}$ | changing

growth curve for $Y(t)$ | process |  Fano factor $D_{Z}(t;j)$ | underdispersion conditions |  coefficient of variation $\sigma_{Z}(t;j)$
---|---|---|---|---
(I) Korf | $X$ | $\frac{\rho+1}{\rho-1}\exp\left(-\frac{\alpha}{\beta}t^{-\beta}\right)$ | $t\geq\widetilde{t}=\left(-\frac{\alpha}{\beta\log\frac{\rho-1}{\rho+1}}\right)^{1/\beta}$ | $\frac{1}{j(\rho-1)\sqrt{1+\exp\left(\frac{\alpha}{\beta}t^{-\beta}\right)}}$
$Y$ | omitted | NA in closed form | omitted
(II) Mitscherlich | $X$ | $\frac{\rho+1}{j}\beta\left(1-e^{-\alpha t}\right)$ | $t\leq\widetilde{t}=\frac{1}{\alpha}\log\frac{\beta(\rho+1)}{\beta(\rho+1)-j}$ | $\sqrt{\frac{\rho+1}{j\left(\rho-1+\frac{e^{\alpha t}j}{\beta(e^{\alpha t}-1)}\right)}}$
$Y$ | omitted | NA in closed form | omitted

growth curve | conditions | $m_{X}(t;j)$ | $m_{Y}(t;j)$ | $Var_{X}(t;j)$ | $Var_{Y}(t;j)$ | $Cov(t;j)$ | $r(t;j)$
---|---|---|---|---|---|---|---
(I) Korf | $t\to+\infty$ | $2j$ | $\displaystyle\frac{j}{\rho-1}$ | $\displaystyle\frac{2j(\rho+1)}{\rho-1}$ | $<+\infty$ | $\frac{2j(1+\rho-\rho\log 4)}{(\rho-1)^{2}}$ | $\frac{j-1+\rho+j\rho-\rho\log 4}{j(\rho-1)}$
$\alpha\to 0$ | $2j$ | $\frac{j}{\rho-1}$ | $\frac{2j(\rho+1)}{\rho-1}$ | $<+\infty$ | $\frac{2j(1+\rho-\rho\log 4)}{(\rho-1)^{2}}$ | $\frac{j-1+\rho+j\rho-\rho\log 4}{j(\rho-1)}$
$\alpha\to+\infty$ | $j$ | $0$ | $0$ | $0$ | $0$ | $\frac{(\rho+1)(j-1)}{j(\rho-1)}$
$\beta\to 0$ | $j$ | $0$ | $0$ | $0$ | $0$ | $\frac{j-1+\rho j+\rho}{j(\rho-1)}$
$\beta\to+\infty$ | $2j$ | $\displaystyle\frac{j}{\rho-1}$ | $\displaystyle\frac{2j(\rho+1)}{\rho-1}$ | $<+\infty$ | $\frac{2j(1+\rho-\rho\log 4)}{(\rho-1)^{2}}$ | $\frac{j-1+\rho+j\rho-\rho\log 4}{j(\rho-1)}$
(II) Mitscherlich | $t\to+\infty$ | $j+\beta(\rho-1)$ | $\beta$ | $\displaystyle\frac{\beta(\rho+1)(j+\beta(\rho-1))}{j}$ | $<+\infty$ | $<+\infty$ | $<+\infty$
$\alpha\to 0$ | $j$ | $0$ | $0$ | $0$ | $0$ | $\frac{j-1}{j}$
$\alpha\to+\infty$ | $j+\beta(\rho-1)$ | $\beta$ | $\displaystyle\frac{\beta(\rho+1)(j+\beta(\rho-1))}{j}$ | $<+\infty$ | $<+\infty$ | $<+\infty$
$\beta\to 0$ | $j$ | $0$ | $0$ | $0$ | $0$ | $\frac{j-1}{j}$

Figure 9: The conditional means $m_{X}(t;j)$ and $m_{Y}(t;j)$, the conditional
variances $Var_{X}(t;j)$ and $Var_{Y}(t;j)$, the conditional correlation
coefficient $Corr(t;j)$ and the conditional coefficient $r(t;j)$ with
$\rho=1.5$, $j=1$, $\alpha=3,\beta=2$ for Gompertz case, $A=\log 20,b=1$ for
generalized Gompertz case, $r=2,C=20$ for logistic case,
$N=20,\varepsilon=0.5$ for the extended logistic case,
$Q_{\beta}(t)=t-t^{2}+10t^{3}-t^{4},C=20$ for the multisigmoidal logistic case
and $\alpha=2\log 20,\beta=2$ for the Korf case.

Table 5: The limit behaviour of the conditional means $m_{X}(t;j)$ and $m_{Y}(t;j)$, the conditional variance $Var_{X}(t;j)$ and $Var_{Y}(t;j)$, the conditional covariance $Cov(t;j)$ and the conditional correlation index $r(t;j)$. growth curve | conditions | $m_{X}(t;j)$ | $m_{Y}(t;j)$ | $Var_{X}(t;j)$ | $Var_{Y}(t;j)$ | $Cov(t;j)$ | $r(t;j)$
---|---|---|---|---|---|---|---
(i) Gompertz | $t\to+\infty$ | $je^{\alpha}$ | $\displaystyle\frac{j(e^{\alpha}-1)}{\rho-1}$ | $\displaystyle\frac{je^{\alpha}(\rho+1)(e^{\alpha}-1)}{\rho-1}$ | $\mathcal{V}_{1,1}(j)$ | $\mathcal{C}_{1,1}(j)$ | $\mathcal{R}_{1,1}(j)$
$\alpha\to 0$ | $j$ | $0$ | $0$ | $\mathcal{V}_{1,2}(t)$ | $0$ | $1+\frac{1}{j}$
$\alpha\to+\infty$ | $+\infty$ | $+\infty$ | $+\infty$ | $0$ | $+\infty$ | $1+\frac{3}{j}+\frac{2}{j(\rho-1)}$
$\beta\to 0$ | $j$ | $0$ | $0$ | $0$ | $0$ | $1+\frac{1}{j}$
$\beta\to+\infty$ | $je^{\alpha}$ | $\displaystyle\frac{j(e^{\alpha}-1)}{\rho-1}$ | $\displaystyle\frac{je^{\alpha}(\rho+1)(e^{\alpha}-1)}{\rho-1}$ | $+\infty$ | $\mathcal{C}_{1,1}(j)$ | $\mathcal{R}_{1,1}(j)$
(ii) generalized Gompertz | $t\to+\infty$ | $je^{Ab}$ | $\frac{j}{\rho-1}\left(e^{Ab}-1\right)$ | $\frac{j(\rho+1)e^{Ab}(e^{Ab}-1)}{\rho-1}$ | $\mathcal{V}_{2,1}(b;j)$ | $\mathcal{C}_{2,1}(j)$ | $\mathcal{R}_{2,1}(b;j)$
$A\to+\infty$ | $+\infty$ | $+\infty$ | $+\infty$ | $+\infty$ | $+\infty$ | $\frac{1+j(\rho-1)+\rho}{j(\rho-1)}$
$A\to 0$ | $j$ | $0$ | $0$ | $0$ | $0$ | $\frac{j-1}{j}$
$b\to+\infty$ | $je^{At}$ | $\frac{j(e^{At}-1)}{\rho-1}$ | $\frac{j(\rho+1)e^{At}(e^{At}-1)}{\rho-1}$ | $\mathcal{V}_{2,1}(t;j)$ | $\mathcal{C}_{2,2}(t;j)$ | $\mathcal{R}_{2,1}(t;j)$
$b\to 0$ | $j$ | $0$ | $0$ | $0$ | $0$ | $\frac{j-1}{j}$
(iii) logistic | $t\to+\infty$ | $C$ | $\frac{C-j}{\rho-1}$ | $\displaystyle\frac{C(\rho+1)(C/j-1)}{\rho-1}$ | $+\infty$ | $\mathcal{C}_{3,1}(j)$ | $\mathcal{R}_{3,1}(j)$
$C\to+\infty$ | $je^{rt}$ | $\frac{j(e^{rt}-1)}{\rho-1}$ | $\displaystyle\frac{je^{rt}(e^{rt}-1)(1+\rho)}{\rho-1}$ | $\mathcal{V}_{3,1}(t;j)$ | $\mathcal{C}_{3,2}(j)$ | $\mathcal{R}_{3,2}(t;j)$
$r\to 0$ | $j$ | $0$ | $0$ | $\frac{j(1+\rho+j(\rho-1))}{(\rho-1)^{3}}$ | $0$ | $1-\frac{1}{j}$
$r\to+\infty$ | $C$ | $\displaystyle\frac{C-j}{\rho-1}$ | $\displaystyle\frac{C(\rho+1)(C/j-1)}{\rho-1}$ | $+\infty$ | $\mathcal{C}_{3,1}(j)$ | $\mathcal{R}_{3,1}(j)$
(iv) extended logistic | $t\to+\infty$ | $N$ | $\displaystyle\frac{N-j}{\rho-1}$ | $\displaystyle\frac{N(N-j)(1+\rho)}{j(\rho-1)}$ | $<+\infty$ | NA | NA
$\varepsilon\to+1$ | $\displaystyle\frac{jNe^{2t}}{(e^{2t}-1)j+N}$ | $\displaystyle\frac{(e^{2t}-1)j(N-j)}{(\rho-1)(N+j(e^{2t}-1))}$ | $\displaystyle\frac{e^{2t}(e^{2t}-1)jN(N-j)(1+\rho)}{(\rho-1)((e^{2t}-1)j+N)^{2}}$ | $\mathcal{V}_{4,1}(t;j)$ | NA | NA
$\varepsilon\to-1$ | $\displaystyle\frac{N(j-2jt+2Nt)}{N-2jt+2Nt}$ | $\displaystyle\frac{2(N-j)^{2}t}{(\rho-1)(N-2jt+2Nt)}$ | $\displaystyle\frac{2(N-j)^{2}N(1+\rho)t(2Nt-j(2t-1))}{j(\rho-1)(N-2jt+2Nt)^{2}}$ | $<+\infty$ | NA | NA
$N\to+\infty$ | $+\infty$ | $+\infty$ | $+\infty$ | $+\infty$ | NA | NA
(v) multisigmoidal logistic | $t\to+\infty$ | $C$ | $0$ | $0$ | $<+\infty$ | $<+\infty$ | $<+\infty$
$\beta_{i}\to-\infty$ | $C$ | $0$ | $0$ | $<+\infty$ | $\frac{Cj}{\rho-1}$ | $<+\infty$
$\beta_{i}\to+\infty$ | $0$ | $\frac{C-j}{\rho-1}$ | $\frac{C(C-j)(\rho+1)}{j(\rho-1)}$ | $<+\infty$ | $<+\infty$ | $<+\infty$
$C\to+\infty$ | $je^{Q_{\beta}(t)}$ | $\frac{j(e^{Q_{\beta}(t)-1})}{\rho-1}$ | $\displaystyle\frac{je^{-Q_{\beta}(t)}(e^{-Q_{\beta}(t)}-1)(1+\rho)}{\rho-1}$ | $<+\infty$ | $<+\infty$ | $\frac{\frac{1+j(\rho-1)+\rho}{j}-\frac{2\rho Q_{\beta}(t)}{e^{Q_{\beta}(t)}-1}}{\rho-1}$
(vi) modified Korf | $t\to+\infty$ | $je^{\alpha/\beta}$ | $\frac{j(e^{\alpha/\beta}-1)}{\rho-1}$ | $\frac{e^{\alpha/\beta}(e^{\alpha/\beta}-1)j(\rho+1)}{\rho-1}$ | $\mathcal{V}_{6,2}(j)$ | $\mathcal{C}_{6}(j)$ | $\mathcal{R}_{6,1}(j)$
$\alpha\to 0$ | $j$ | $0$ | $0$ | $0$ | $0$ | $\frac{j-1}{j}$
$\alpha\to+\infty$ | $+\infty$ | $+\infty$ | $+\infty$ | $+\infty$ | $+\infty$ | $\frac{1+j(\rho-1)+\rho}{j(\rho-1)}$
$\beta\to 0$ | $j(1+t)^{\alpha}$ | $\frac{j(-1+(1+t)^{\alpha})}{\rho-1}$ | $\mathcal{V}_{1,1}(t;j)$ | $\mathcal{V}_{6,3}(t;j)$ | $\mathcal{C}_{7}(t;j)$ | $\mathcal{R}_{6,2}(t;j)$
$\beta\to+\infty$ | $j$ | $0$ | $0$ | $+\infty$ | $0$ | $\frac{j-1}{j}$

Table 6: Limit functions of the conditional variances, of the correlation
coefficient and of the coefficient $r(t;j)$. Note that such auxiliary
functions are needed only for certain cases.

growth curve | variance limits
---|---
(i) Gompertz | $\displaystyle\mathcal{V}_{1,1}(j):=\frac{\beta\left(-\alpha(1+\alpha)+(1+\alpha)e^{2j}+e^{j}(-1+\alpha^{2}-4\alpha j)\right)}{(\alpha-1)^{3}}$
$\displaystyle\mathcal{V}_{1,2}(t;j):=\beta\exp\left\\{j-2je^{-\rho
t}\right\\}\left(\exp\left\\{je^{-\rho t}\right\\}-e^{j}\right)$
(ii) generalilzed Gompertz | $\displaystyle\mathcal{V}_{2,1}(x;j)=\frac{j\left(e^{2Ax}(\rho+1)-\rho(\rho+1)+e^{Ax}\left(-1-4Ax\rho+\rho^{2}\right)\right)}{(\rho-1)^{3}}$
(iii) logistic | $\displaystyle\mathcal{V}_{3,1}(t;j):=j\frac{-\rho(1+\rho)+e^{rt}(\rho^{2}-4r\rho t-1)+e^{2rt}(2-j+\rho(2+j))}{(\rho-1)^{3}}$
(iv) extended logistic | $\displaystyle\mathcal{V}_{4,1}(t;j):=\frac{j}{(\rho-1)^{3}(N+j(e^{2t}-1))^{2}}\left[(e^{2t}-1)(N-j)(1+\rho)((N-j)\rho\right.$
$\left.+e^{2t}(N+j\rho))\right.\left.+4e^{2t}N\left((e^{2t}-1)j+N\right)\rho\left(-2t+\log\frac{(e^{2t}-1)j+N}{N}\right)\right]$
(vi) modified Korf | $\displaystyle\mathcal{V}_{6,1}(t;j):=\frac{j(1+\rho)(1+t)^{\alpha}((1+t)^{\alpha}-1)}{\rho-1}$
$\displaystyle\mathcal{V}_{6,2}(j):=\frac{j\left[-4\alpha
e^{\alpha/\beta}\rho+\beta(e^{\alpha/\beta}-1)(1+\rho)(e^{\alpha/\beta}+\rho)\right]}{\beta(\rho-1)^{3}}$
$\displaystyle\mathcal{V}_{6,3}(t;j):=\frac{j\left[(1+\rho)((1+t)^{\alpha}-1)(\rho+(1+t)^{\alpha})-4\alpha\rho(1+t)^{\alpha}\log(1+t)\right]}{(\rho-1)^{3}}$

growth curve | correlation limits
---|---
(i) Gompertz | $\displaystyle\mathcal{C}_{1,1}(j):=je^{\alpha}\frac{1-\rho(3+2\alpha)+e^{\alpha}(3\rho-1)}{(\rho-1)^{2}}$
(ii) generalilzed Gompertz | $\displaystyle\mathcal{C}_{2,1}(j):=\frac{je^{Ab}\left(-1+2j-(1+2Ab+2j)\rho+2^{Ab}(1+2j(\rho-1)+\rho)\right)}{(\rho-1)^{2}}$
$\displaystyle
C_{2,2}(t;j):=\frac{je^{At}\left(-1+e^{At}(\rho+1)-\rho(1+2At)\right)}{(\rho-1)^{2}}$
(iii) logistic | ${C}_{3,1}(j):=\frac{C\left((\rho+1)(C-j)-2\rho j\log C/j\right)}{j(\rho-1)^{2}}$
$\displaystyle\mathcal{C}_{3,2}(t;j):=je^{rt}\frac{e^{rt}(\rho+1)-1-\rho(1+2rt)}{(\rho-1)^{2}}$
(vi) modified Korf | $C_{6,1}(j):=\frac{e^{\alpha/\beta}j(-2\alpha\rho+\beta(e^{\alpha/\beta}-1)(\rho+1))}{\beta(\rho-1)^{2}}$
$\displaystyle\mathcal{C}_{6,2}(t;j):=\frac{j(1+t)^{\alpha}\left((1+\rho)(-1+(1+t)^{\alpha})-2\alpha\rho\log(1+t)\right)}{(\rho-1)^{2}}$

growth curve |  $r(t;j)$ limits
---|---
(i) Gompertz | $\displaystyle\mathcal{R}_{1,1}(j):=\frac{1+j-(3+2\alpha+j)\rho+e^{\alpha}(-1+j(\rho-1)+3\rho)}{(e^{\alpha}-1)(\rho-1)}$
(ii) generalilzed Gompertz | $\displaystyle\mathcal{R}_{2,1}(x;j):=\frac{-2Ax\rho+(e^{Ax}-1)(1+j(\rho-1)+\rho)}{j(\rho-1)(e^{Ax}-1)}$
(iii) logistic | $\displaystyle\mathcal{R}_{3,1}(j):=1+\frac{1+\rho}{j(\rho-1)-\frac{2\rho\log C/j}{(\rho-1)(C-j)}}$
$\mathcal{R}_{3,2}(t;j):=\frac{1-j+\rho\left(1-\frac{2rt}{e^{rt}-1}+j\right)}{j(\rho-1)}$
(vi) modified Korf | $\displaystyle\mathcal{R}_{6,1}(j):=\frac{1+j(\rho-1)+\rho+\frac{2\alpha\rho}{\beta(1-e^{\alpha/\beta})}}{j(\rho-1)}$
$\mathcal{R}_{6,2}(t;j):=\frac{1+j(\rho-1)+\rho-\frac{2\alpha\rho\log(1+t)}{(1+t)^{\alpha}-1}}{j(\rho-1)}$

Figure 10: The conditional means $m_{X}(t;j)$ and $m_{Y}(t;j)$, the
conditional variances $Var_{X}(t;j)$ and $Var_{Y}(t;j)$, the conditional
correlation coefficient $Corr(t;j)$ and the conditional coefficient $r(t;j)$
with $\rho=2$, $j=1$, $\alpha=1,\beta=0.5$.

##  8 Application to real data

In this section we consider an application to real data. The datasets come
from [40] and concern the temporal diffusion of 5 different fake news on
$\mathbb{X}$/Twitter. For each fake news, the number of tweets and retweets
relating to it were counted with the relative time reference. Regarding the
time scale, we use the time instant in which the first tweet was posted as the
initial time, and number of days since the initial instant as the time unit.
Once the temporal evolution of the fake news has been constructed, we use the
growth curves analyzed in detail in Section 7 as candidate models. The
parameters involved in the various definitions were estimated by imposing the
minimization of the mean square error (MSE) or of the relative absolute error
(RAE), as done in [34]. Regarding the multisigmoidal logistic model, we chose
$Q_{\beta}(t)$ as a fourth degree polynomial, since with repeated empirical
tests this degree shows to be the right trade off between goodness of estimate
and reduced number of parameters to estimate. We remember, in fact, that the
presence of many parameters entails a high computational cost and the
possibility of having overfit problems. Once all the parameters of interest
are estimated, we select the best model as the one leading to the smallest MSE
(or to the smallest RAE). The temporal evolution of fake news and the related
estimating models are shown in Figure 11.

Figure 11: The fake news time evolution and the corresponding fitting models
$m_{X}(t;j)$.

It is clear that, for all the considered datasets, the model that guarantees
the best results is the multisigmoidal logistic one, as also confirmed by the
quantitative results reported in Table 7. However, using the multisigmoidal
logistic model imply the estimation of a quite large number number of
parameters (given by the degree of the polynomial + 2). Hence, one can be
interested in fitting the data with a more simple model, i.e. one of the
others studied in Section 7. Considering this type of data approximation, the
model which returns the best fit (i.e. the one with the smallest MSE or the
smallest RAE) varies depending on the analyzed fake news. More specifically,
for the dataset no. 1 the best model is the Gompertz one, for the dataset no.
2 it is the modified Korf, for the datasets no. 3-4 it is the generalized
Gompertz, for the dataset no. 5 it is the logistic.

Table 7: The mean square error (MSE) and the relative absolute error (RAE) between the data and the fitted models for dataset (DS) no. 1$\div$5\. The minimum MSE and the minimum RAE for each dataset is underlined. |  | case (i) | case (ii) | case(iii) | case (iv) | case (v) | case (vi)
---|---|---|---|---|---|---|---
|  | Gompertz | gen. Gompertz | logistic | ext. logistic | multis. logistic | mod. Korf
DS no. 1 | MSE | $2.71523$ | $3.53550$ | $2.86095$ | $2.99282$ | $0.48732$ | $2.96641$
| RAE | $0.99484$ | $0.01460$ | $0.01192$ | $0.01190$ | $0.00467$ | $0.01327$
DS no. 2 | MSE | $6.94419\cdot 10^{-5}$ | $6.91156\cdot 10^{-5}$ | $6.96783\cdot 10^{-5}$ | $1.22240\cdot 10^{-4}$ | $4.49001\cdot 10^{-5}$ | $6.83370\cdot 10^{-5}$
| RAE | $0.00695$ | $0.00698$ | $0.00693$ | $0.00935$ | $0.00526$ | $0.00703$
DS no. 3 | MSE | $0.30543$ | $0.21796$ | $0.31618$ | $0.31420$ | $0.06927$ | $0.24032$
| RAE | $0.96458$ | $0.00663$ | $0.00798$ | $0.00797$ | $0.00422$ | $0.00697$
DS no. 4 | MSE | $0.00958$ | $0.00733$ | $0.01274$ | $0.15590$ | $0.00501$ | $0.00594$
| RAE | $0.05806$ | $0.04751$ | $0.06657$ | $0.76616$ | $0.03131$ | $0.04303$
DS no. 5 | MSE | $0.02391$ | $0.04795$ | $0.00838$ | $0.03883$ | $0.00280$ | $0.03449$
| RAE | $0.55334$ | $0.57466$ | $0.34989$ | $0.57569$ | $0.09370$ | $0.58865$

Finally, for any fake news we also calculate the corresponding mean
$m_{Y}(t;j)$ considering different values for $\rho$. The plots are shown in
Figure 12. Note that in certain cases, it was not possible to proceed with the
determination of the mean $m_{Y}(t;j)$ using the multi-sigmoidal logistic
approximation which, we remember, is the best for any fake news. The reason is
owing to numerical causes, in fact the multi-sigmoidal logistic curve involves
the calculations of very small quantities which are mistakenly treated by the
computer as $-\infty$. Therefore, in these cases, it is preferred to use as
the approximating curve the one that determines the smallest MSE or the
smallest RAE after the multi-sigmoidal logistics. It can be interesting, as a
future development, studying a procedure to approximate also the coefficient
of proportionality $\rho$.

As mentioned before, in the analysis conducted in this section both the MSE
and the RAE decrease with increasing model complexity, expressed in terms of
the number of involved parameters. Precisely, the high number of parameters
produces models that tend to fit the data better. However, this characteristic
can lead to models prone to overfitting, i.e. models that do not generalize
well to new data. Therefore, measures of goodness that take into account the
complexity of the model in addition to its ability to fit the data are often
used in applied contexts. This is the case of the Akaike Information Criterion
and the Bayesian Information Criterion goodness-of-fit measures that introduce
a penalty based on the complexity of the model. This line of inquiry based on
such information criteria has been successfully pursued in [34]. To use the
aforementioned criteria, however, the expression of the likelihood function is
needed. Unfortunately, it is not available in the present case. Calculating
the likelihood function for the considered model is quite hard, and thus it is
beyond the specific scope of this work. However, one can consider proposing a
diffusive approximation of the current model, as performed in other studies on
similar topics. This can lead to a more comprehensive statistical analysis in
a future research work.

Figure 12: The approximating $m_{Y}(t;j)$ for different values of $\rho$.

##  9 Conclusions

The paper is devoted to modeling the temporal propagation of fake news in
online social networks. In particular, we introduced a probabilistic model
represented by a two-dimensional birth-death process whose first component
shows the number of individuals who knows the fake news and intends to spread
it, while the second component shows the number of individuals who has
forgotten the fake news or has lost the intention to spread it. The
transitions which regulate the dynamics of the two-dimensional process are
determined by temporal non-homogeneous individual birth and death rates and
they are proportional to the size of the first component. The process,
introduced in this paper for the first time, was then analyzed through the
study of the p.g.f. The partial differential equation whose solution is
represented by the p.g.f. is not easily solvable in the general case but has
been solved in the case of temporal homogeneous rates and in the case of
temporal non-homogeneous proportional rates. Starting from the p.g.f., the
first and second order moments of both the first and second components, the
mixed moments and the covariance of the two-dimensional process were
determined. Furthermore, to study the correlation between the two components,
a new index was studied, represented by the ratio between the mean of the
product of the two components and the product of the means. The definition of
such adimensional index is one of the novelties introduced by the paper.
Considering that the first component of the process represents the number of
individuals who knows the news and intends to spread it in a large population,
we considered the conditions that make the mean of this first component equal
to sigmoidal growth curves well known in the literature. Finally, we proposed
an application to real data considering the data linked to the temporal
diffusion of 5 fake news. In particular, by comparing the mean square error
and the relative absolute error we determined the model that best fits the
data of the case study. Such application makes the introduced two-dimensional
process interesting from a modeling point of view. In certain real situations
relating to the spread of fake news, the information of interest can be
obtained efficiently only via numerical methods (see, for instance, D’Ambrosio
et al. [37]). With the idea of providing closed-form results, which is one of
the main strength of the paper, the proposed stochastic process is a
significantly simplified model.

In order to make the model more realistic for the analyzed datasets, it may be
useful to consider cases where the mean of $X(t)$ coincides with other growth
curves, for example the T-growth model (cf. Barrera et al. [38]). Moreover,
analyzing the time that the process takes to reach a critical size is
interesting in certain application contexts; therefore, an analysis aimed at
the first passage time (as done in Albano et al. [39]) represents one of the
possible future developments. The model can be reformulated by giving a
different interpretation to the components $X(t)$ and $Y(t)$. Thus, the
stochastic process may be used to analyze the trend of other data sets, coming
from different real contexts, such as the epidemiological one. Furthermore, it
will be of interest to divide the population into more disjoint compartments
and analyze the resulting model both from a deterministic and a stochastic
point of view.

We conclude by remarking that in order to stop the spread of the fake news it
is necessary to make the number of inactives large and the number of spreaders
tending to 0. In the present model, such behavior emerges when the intensity
of the spreaders $\lambda$ is smaller than the intensity of the inactives
$\mu$, i.e. when $\rho<1$ in the case of proportional intensities.

Moreover, we point out that the model characterized by intensities (1) can
also be useful for describing the spread of news in general (not necessarily
just fake news) within an infinite population. In this case, one might expect
a non-sigmoidal growth in average, since the news could potentially spread
throughout the entire population which is assumed to be large. Such
generalization is out of the scope of the present paper but can be the object
of a future research.

## Acknowledgements

The authors are members of the research group GNCS of INdAM (Istituto
Nazionale di Alta Matematica). This work is partially funded by the ‘European
Union – Next Generation EU’ through MUR-PRIN 2022, project 2022XZSAFN
“Anomalous Phenomena on Regular and Irregular Domains: Approximating
Complexity for the Applied Sciences”, and MUR-PRIN 2022 PNRR, project
P2022XSF5H “Stochastic Models in Biomathematics and Applications”.

## Data availability statement

The data that support the findings of Section 8 of this study are available in
Zenodo at https://zenodo.org/records/5225338, reference number [40].

## Conflict of interest

The authors declare no potential conflict of interests.

## ORCID

A. Di Crescenzo: http//orcid.org/0000-0003-4751-7341
P. Paraggio: https://orcid.org/0000-0002-3308-7937

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