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Journal of Natural and Applied Sciences -Nasara Scientifique, Vol. 7 No. 1. pp 95-110, December 2018
95
MODELING POLITICAL RUMOUR WITH CONDITIONAL LATENT
PERIOD IN A VARYING POPULATION
*1Oduwole H. K., 2Joseph, I. K., & 3Ikani K. S.
1,2,3Department of Mathematical Sciences,
Nasarawa State University, Keffi
*Corresponding Author Email: kenresearch@yahoo.com
*Corresponding Author
ABSTRACT
In this paper a rumour propagation model with conditional latent period and varying
population is considered. In the literature, classical model assume that an ignorant
individual enters the latent period and decide whether to become a spreader or stifler.
In our model we introduce a new compartment called the blackmailers, another type
of spreaders who spread the rumour for selfish reason. The model equations are first
transformed into proportions, thus reducing the model equations from five to four
differential equations. The model exhibit two equilibra, namely the Rumour Free
Equilibrium (RFE) and the Rumour Endemic Equilibrium (REE). Using the method
of linearized stability, we establish that the RFE state exist and is locally
asymptomatically stable when and that when the endemic state exist.
The model allows us to discuss the relationship between spreaders and blackmailers,
and the effect of blackmailers on the stiflers. Finally, we present numerical
simulations that show the impact of political motivated rumours and how its control
can be achieved.
Keywords: Rumor propagation, Political motivated rumour, Epidemiological models;
stability analysis, Transition parameters
Introduction
Rumours are an integral part of human life
and its spread has significant impact in
society. Basically, Rumours are unverified
stories or report circulating in a community,
usually by word of mouth or via social
network and are accepted as facts, although
its original source may be unknown
(DiFonzo & Prashant, 2007). Rumours
reflect people desired to find meaning to
events around them. Rumours may contain
classified information, true or false
information about any issue and
information that may be detriment to the
society at a given time. Various types of
rumours exist for various reasons. Some
rumours are characterized by the channel of
communication, like social network
rumours, newspapers rumours, while others
are characterized by the classes of people
involved and the environment where the
thrive, like rumours that spread in an
organization (like hospital or schools about
management), rumours occurring among
Journal of Natural and Applied Sciences -Nasara Scientifique, Vol. 7 No. 1. pp 95-110, December 2018
96
specific group of people and political
rumours (rumours that spread for political
motivated reasons).
The classical model for the spread of
rumours was first introduced by Daley and
Kandal (1965) and Maki and Thompson
(1973). The approach in these models has
been used extensively for quantitative
studies. In their model, a closed and
homogeneous mixed population consisting
of three groups; the ignorant, spreaders and
the stifler was considered. The ignorant are
those individuals who are ignorant of the
rumour, the spreader are those who heard
the rumour and are actively involved in
spreading it and the stifler are those who
have heard the rumour but have ceased to
spread it. According to them, rumour spread
through the law of mass action within the
population by a pair-wise contact between
spreaders and between spreaders and
ignorant. Any spreader involved in a pair-
wise contact attempts to “infect” the other
individual with the rumour, thereafter
individual who was ignorant initially
becomes a spreader. In another case, either
one or both of those involved in spreading
the rumour decide not to tell the rumour to
anyone, thereby becoming a stifler (Daley
and Gani, 2000). Other non-political
rumours models include the works of
Oduwole et al.,(2010) who modeled the
spread of rumour in an academic institution
using markov chain model, Zanette, (2002),
Nekovee, et al.,(2007), Chunru and Zujun
(2015) and a host of others
Boris (2002) observed that rumours
function as a psychological process that
develops around opposing views or mutual
misconceptions. One of such psychological
function occurs mostly in political circles,
which eventually leads to political rumours.
According to Weeks and Garrett (2014),
political rumours often characterized
unsubstantiated claims about candidates and
issues that are often false, and are a
potentially important source of
misperception that may threaten democratic
outcomes and objectives. Political rumours
arise around topics of national importance
and are driven by feeling of uncertainty and
anxiety (Rosnow, 1998). This makes it
possible for political rumours to flourish
during political campaigns by disseminating
unverified claims concerning the opposition
party or candidates in a bid to mislead the
public and affect voting choice (Jamieson,
1992). The spread of political rumour for
selfish and negative reasons dates backs to
various form of elections in different
countries and continents, as seen in the
election of Thomas Jefferson as President of
the United States (Shibutani, 1996, Weeks
& Garrett, 2014) and has continued in recent
election in the United States (Garrett, 2011).
This scenario is not different from what is
being experienced in recent election in
Africa.
According to the theory of partisan
motivated reasoning, individuals’ prior
attitudes will affect how they assess
political rumours; hence motivated
reasoning suggests that humans evaluate
new information in ways that are in line
with their prior beliefs (Kunda, 1990; Taber
& Lodge, 2006). This is in consonance with
politicians and the political landscape in
today modern democracies been practice in
different countries and continents. In most
political scenarios, individuals are often
driven to defend their prior political position
Journal of Natural and Applied Sciences -Nasara Scientifique, Vol. 7 No. 1. pp 95-110, December 2018
97
through biased evaluations of new
information. Thus, people are willing to
accept attitude consistent to their political
position with little evidence while rejecting
well supported attitude that are at
discrepancy to their political position (Taber
& Lodge, 2006). Recent works on
modelling political rumours include those
by Sunstein, (2009), Bullock, (2006) and
Weeks and Garreth (2014).
The fact that rumour acceptance in the
political arena is largely a function of
political attachment solicit an important
question that affect the decision whether to
spread a true information (true spreader) or
to spread a false information (blackmailers).
In this paper, we make a modification to
classical model of Huo et. al (2015) by
introducing another kind of spreaders called
the blackmailers. The new model allows us
to discuss the relationship between
spreaders and blackmailers (those spreading
the rumour for selfish reasons), and the
effect of blackmailers on the stiflers.
Finally, we present numerical simulations
that show the impact of political motivated
rumours and how its control can be
achieved.
Mathematical Formulation
The new rumour model is a prototype of the classical susceptible-infected-removed (SIR)
model that considers a homogeneous mixing of individuals with state variable as a function
of time. The population is sub-divided into five classes; the ignorant class , the latent
class, also called the exposed class true spreader those spreading the rumour for
selfish reasons, also known as the blackmailers and the stiflers . The individuals in
the ignorant class have not heard the rumour. Individuals in the latent class have heard
the rumour due to contact with the either the true spreader or the blackmailers, but need active
effort to discern between true and false rumour and decide to spread the rumour or not; a part
of them believe the rumour (either true or false) and decides either to spread it or become
stiflers. Those in the spreaders class either continue as true spreaders or as blackmailers (that
is, those who spread the rumour for selfish reasons). Both the true spreaders and the
blackmailers can become stiflers at a certain time after losing interest in spreading the
rumour. The total population at time is denoted by , where
. The model diagram is shown in figure 1 below:
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Figure 1. Flow diagram of population dynamics for the spread of rumour
There is no transition of rumour unless a spreader (either in the or class) contact
an ignorant individual or another spreader. A spreader contacts an ignorant individual and
transmits the rumour at a constant frequency and the ignorant individual now move to the
latent class or exposed class, where time is require to distinguished between true or false
rumour content and decides either to be in the or class or chose not to spread the
rumour and move stifler class . An ignorant individual become exposed to the rumour
at a rate of
within a small interval , where is a positive constant
number representing the product of the contact frequency and the probability of transmitting
the rumour. Those in the exposed class have heard the rumour, but are not yet spreaders. We
assume that is the rate at which an individual in the exposed class become spreaders
and other become stiflers at the rate of . Two or more spreaders (true spreader and
another true spreader, or true spreader and a blackmailer or a blackmailer and a blackmailer)
both transmit the rumour at a constant frequency after which they become get bored, losing
interest in the rumour and will eventually become stiflers at the rate
and
respectively for true spreaders and blackmailer respectively.
Assumptions:
a) There is no transition of rumour unless a spreader (either in the or class) contact
an ignorant individual or another spreader.
b) An Ignorant individual do not always become spreaders (either in the or class)
initially, but enters into the latent class with conditional transition parameter and
c) A constant immigration rate and a constant emigration rate in all classes are
considered.
Journal of Natural and Applied Sciences -Nasara Scientifique, Vol. 7 No. 1. pp 95-110, December 2018
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The table below shows the variables and parameters used in the new model.
Table 1 Model variables and parameters
Variable
Description
Number of Ignorant individuals at time
Number of Exposed individuals at time
Number of true spreaders individuals at time
Number of Blackmailers individuals at time
Number of Stiflers individuals at time
Parameter
Description
Rumour propagation coefficient
Rate at which individual at the latent class becomes stiflers
Rate at which individual at the latent class becomes true spreaders
Rate at which individual at the latent class becomes false spreaders (blackmailers)
Change rate for the exposed/latent class who become true spreaders
Change rate for the exposed/latent class who becomes false spreaders (blackmailers)
True-spreaders stifler coefficient
Blackmailers-stifler coefficient
Change rate for true-spreaders.
Change rate for blackmailers.
Emigration rate
Immigration rate
Believe and spread rate, = Believe and spread rate for the exposed class,
Total population
From the assumptions and the flow diagram above, the following model equations are
derived.
. . . (1)
. . . (2)
. . . (3)
. . . (4)
. . . (5)
We transform the model equations (1) – (5) above to proportion as follows:
Let
,
,
,
so that and let
,
,
,
,
,
,
Hence adding equation (1) – (5), we have that
Journal of Natural and Applied Sciences -Nasara Scientifique, Vol. 7 No. 1. pp 95-110, December 2018
100
. . . (6)
Therefore,
. . . (7)
But
and
, Then equation (7) becomes
. . . (8)
Similarly,
. . . (9)
and
, Then equation (9) becomes
. . . (10)
But based on the assumption that the latent class will spread the rumour and equation
(10) becomes
. . . (11)
Similarly,
. . . (12)
and
, Then equation (12) becomes
. . . (13)
Similarly,
. . . (14)
and
, Then equation (14) becomes
. . . (15)
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Finally,
. . . (16)
and
, Then equation (16) becomes
. . . (17)
Rewriting as and letting , equation (8), (11), (13) and (15)
becomes
. . . (18)
. . . (19)
. . . (20)
. . . (21)
Mathematical Treatment and Analysis
Equilibrium States of the Model
We discuss the Rumour Free Equilibrium (RFE) and the Rumour Endemic Equilibrium
(REE) of the model equations (18) – (21). Equating the left hand side of equations (18) –
(21) to zero and letting we have that
. . . (22)
Hence the Rumour free equilibrium (RFE) is
For the Rumour Endemic Equilibrium (REE), we equate equation (18) – (21) to zero, letting
, and solving simultaneously we have
. . . (23)
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Computation of Basic Reproduction Number
In this section, we compute the Basic Reproduction Number using the next generation
method as applied by Diekmann et al., and Van den Driessche and Watmough, (2002). From
equation (18) – (21) using their approached we have that
and
where and are the rate of appearances of new infections (rumour) in compartment and
the transfer of individuals into and out of compartment by all other means respectively.
Using the linearization method, the associated matrices at rumour-free equilibrium ()
and after taking partial derivatives as defined by
and
where is nonnegative and is a non-singular matrix, in which both are the
matrices defined by
and
, with and is the number of infected
classes.
In particular we have
and
If the inverse of is given as
Then the next generation matrix denoted by is given as
. . . (24)
We find the eigenvalues of by setting the determinant
with characteristics polynomial
. . . (25)
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103
and characteristics equation given as
. . . (26)
Solving the characteristic equation (26) for the eigenvalues , , and , we obtained
and
is the maximum of the three eigenvalues , and . Hence the Basic Reproductive
number is the dominant eigenvalues of . Thus we have that
. . . (27)
Stability Analysis of the Rumour Free Equilibrium of the Model
To study the behaviour of the system (18) – (21) around the Rumour-free equilibrium state
, we resort to the linearized stability approach.
Let
. . . (28)
. . . (29)
. . . (30)
. . . (31)
100
010
10
01
bs
ss
bs
RFE
J
. . . (32)
The Determinant and the Trace of the Jacobian matrix of equation (32) is given as
. . . (33)
. . . (34)
Theorem 1
The Rumour free equilibrium state of the model (18) – (21) is locally
asymptotically stable if , otherwise is unstable.
Proof
Then Jacobian matrix of the model equations (18) – (21)at the RFE is given by
100
010
10
01
bs
ss
bs
RFE
J
. . . (32)
If the Jacobian matrix is evaluated at the Rumour-Free equilibrium state (RFE), then the
required criteria for stable equilibrium are that the Determinant of the Jacobian is positive
and the Trace of the Jacobian is negative.
Journal of Natural and Applied Sciences -Nasara Scientifique, Vol. 7 No. 1. pp 95-110, December 2018
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From the Determinant and the Trace of the Jacobian matrix of equation (32), we have that
where
provided that
Also
where
, provided that
Results
Numerical Experiments of the Model
The Rumour propagation model (18) - (21) was solved numerically using Runge-Kutta-
Fehllberg4th order method and implemented using Maple 17 (Maplesoft, Waterloo)
The following experiments were carried out
Experiment 1: Effect of prevalence of rumour spread by and with constant
transition rate when (Case 1) and (Case 2)
Experiment 2: Effect of prevalence of rumour spread by and with variable
transition rate when (Case 1) and (Case 2)
Experiment 3: Effect of prevalence of rumour spread by and with variable
transition rate when (Case 1) and (Case 2)
Table 2 Estimated values of the parameters used for simulation of the rumour model
Parameter
Values
1.0
0.5
0.2
0.2
0.08
0.1
0.1
Sources
Assumed
Assumed
Assumed
Assumed
Assumed
Huo et al
(2015)
Huo et al
(2015)
Parameter
Values
0.7
0.11**
0.11**
0.2**
0.2
0.08
0.05
Sources
Huo et al
(2015)
Assumed
Assumed
Huo et al
(2015)
Assumed
Assumed
Huo et al
(2015)
Assumed: Hypothetical data use for research purpose.** Based on parameter value
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Experiment 1 Case 1,
Experiment 1 Case 2,
Figure 2 Effect of prevalence of rumour spread by and with constant transition rate (
)
when (Case 1) and (Case 2)
Experiment 2 Case 1,
Experiment 2 Case 2,
Figure 3 Effect of prevalence of rumour spread by and with variable transition rate
when
when (Case 1) and (Case 2)
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Discussion of Results
As mentioned earlier in this paper, we make a modification to classical model of Huo et. al
(2015) by introducing another kind of spreaders called the blackmailers. We discuss he
relationship between spreaders and blackmailers (those spreading the rumour for selfish
reasons), and the effect of blackmailers on the stiflers.
In figure 2, we examined the effect of prevalence of rumour spread by true-spreaders
and blackmailers when there is constant transition rate . We observed that
when the change rate of true-spreaders is greater than the change rate of blackmailers, the
rate at which rumours spreads by blackmailers will be higher than those spread by true-
spreaders, and the contents in the rumour spread by blackmailers will stay on for some time
before diminishing. The change rate for true-spreaders is the rate at which the a member in
the true-spreaders class becomes a blackmailers or a stifler . This is similar
to the second observation in case 2, when the change rate of true-spreaders is less than the
change rate of blackmailers leading to an increase in the spread of rumour content of the true-
spreaders class. Generally, both the members in the exposed class and the stifler
becomes either a true-spreader when being convinced of the truth of the rumor and then
decides to inform others or a blackmailer by twisting the contents of the unverified
information for selfish reason and then decides to spread the rumour. It should be noted that
a member of the exposed class after coming in contact with the rumour from either a true
spreader or a blackmailer can possibly refuse to spread the rumor, or alternately a spreader
(true-spreaders or blackmailers) can lose interest in the rumor and then decide not to spread
the rumor any further. Hence the controlling parameter must be able to change the believe
concept of the person carrying the rumour.
Experiment 3 Case 1,
Experiment 3 Case 2,
Figure 4 Effect of prevalence of rumour spread by and with variable transition rate
when
(Case 1) and (Case 2)
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107
In figure 3, we examined the effect of prevalence of rumour spread by true-spreaders
and blackmailers when there is a varying transition rate . We observed that
when the change rate of true-spreaders is greater than the change rate of blackmailers
, the frequency of the rumour content spread by blackmailers is high, but when the
change rate of true-spreaders is less than the change rate of blackmailers , then the
rumour frequency between both populations is reasonably control. This infers to the allusion
that political opponents must learn to respond or counteract their opponent view before the
public.
According to Huo, et. al (2015), individuals in the exposed/latent and stiflers class must be
provided with clear background knowledge that will assist them raise some reasonable
questions in order to present logical arguments in order to assess the credibility or the validity
of the rumour as they come in contact with either the true-spreaders or with the blackmailers.
This form of knowledge is a determining factor that contributes to the termination of the
spread of rumour in political environment.
In figure 4, we examined the effect of prevalence of rumour spread by true-spreaders
and blackmailers when there is a varying transition rate . We observed that
when the change rate of true-spreaders is less than the change rate of blackmailers ,
the frequency of the rumour content spread by blackmailers is low, but when the change rate
of true-spreaders is greater than the change rate of blackmailers , then the rumour
frequency between both populations is reasonably control.
According to Kostka et al. (2008), the starting point for rumor dissemination in the society
must be identified. According Kostka et al. (2008)), different starting point in the dynamic
of rumour network leads to different spreading behavior for a rumor. For our discussion on
political rumour have interesting starting point during election seasons and to score political
antagonism among the political class. Also it should be noted that the structure of society
and its homogeneity are two important factors on how rumor statements mutate and how
many people can hear and change the propositions of the rumor, and this might lead to an
equilibrium point between the true-spreaders and the blackmailers. Therefore, rumor of
spreading in a society can reveal belief indicators of the society, especially in the political
arena.
From the foregoing we have clearly seen that the controlling parameter in political rumours
must go around factors that affect the rate at which individual in the latent class become
either a true spreader or a blackmailer that distort information for selfish reasons. These
parameters and , together with and are of great importance in controlling political
rumours. The rate of all these parameter are govern by the believe rate . There are certain
factors that determine the believe rate of an individual in the population. These include the
psychology of the hearers in line with his/her political leaning, educational background,
benefits of accepting the rumour as true (and for blackmailer) and ability to cipher the
Journal of Natural and Applied Sciences -Nasara Scientifique, Vol. 7 No. 1. pp 95-110, December 2018
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intentions of the source of the unverified information. Naturally, people are willing to accept
attitude consistent to their political position with little evidence while rejecting well
supported attitude that are at discrepancy to their political position, nevertheless logical
reasons with verifiable fact and evidence must substantiate and rebuff all false rumour.
Politicians must learn to use all available means including democratic institutions to set all
record right when unverified information about the actions of the government of the day or
its actors in terms of their personal involvements or activities in government to neutralize the
negative impact of political motivated rumours or fake news.
While we do not obtain the optimal control parameter for this model in terms of equations or
formula, we have nevertheless shows the impact of political motivated rumours and how it
control can be achieve.
Conclusion
Rumours are an integral part of human life and its spread has significant impact in society.
Political rumours in recent times are negatively affecting the society and if not check can
lead to greater problems in the future. We have consider a rumour propagation model with
conditional latent period and varying population is considered. In our model we introduce a
new compartment called the blackmailers, another type of spreaders who spread the rumour
for selfish reason. There is a direct relationship between true-spreaders and blackmailers.
The model exhibit two equilibra, namely the rumour free equilibrium (RFE) and the rumour
endemic equilibrium (REE). Using the method of linearized stability, we establish that the
RFE state exist and is locally asymptomatically stable when , thus helping us control
the spread of the rumour and that when the endemic state exist leading to the rumour
persisting indefinitely through-out the political space.
Political rumours must not be left without providing measures to control it in the political
space. Political opponents must learn to use democratic institution to control the negative
impact or political motivated rumour. Though acceptance of rumour in the political arena is
largely a function of political attachment, there is still the need to use available means to
verify the unverified information. Every member of the population including the agencies
that spread information must do so with high level of professionalism and ethic.
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