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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

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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

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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.

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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.

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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

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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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