Mathematical and Computer Modeling of a Dynamic System for
Effectively Combating Disinformation
NUGZAR KERESELIDZE
Faculty of Natural Sciences, Mathematics, Technologies and Pharmacy,
Sokhumi State University,
61, A. Politkovskaia St., Tbilisi 0186,
GEORGIA
Abstract: - The work investigated a mathematical and computer model of a dynamic system for
effectively combating disinformation. In the compartmental model of false information dissemination
in society, there are groups of citizens: - At risk, prone to the perception of misinformation; Adepts -
those who accepted false information and with Immunity - those who rejected false information from
the very beginning or future adepts. A barrier level for the number of adherents will be introduced as a
measure of the information security of society. As a result of a computer experiment, the possibility of
an uncontrolled growth in the number of adherents was identified, threatening the safety of society as
a whole.
Key-Words: - Dynamic system, mathematical and computer model, computer experiment, Internet
Technologies, disinformation, controllability, optimal control task.
Received: May 26, 2023. Revised: November 13, 2023. Accepted: December 17, 2023. Published: January 17, 2024.
1 Introduction
The article discusses mathematical and computer
modeling of social processes. In particular, one of
the types of Information Warfare - the fight against
disinformation. In the conditions of the universal
Internet, developed social networks, etc. information
dissemination has become publicly available. This
opportunity is often negatively used by some forces
to spread false information. The goals of
disseminating false information can be different,
including imposing on the part of society the
“needed” point of view and thereby determining the
desired model of implementation. We can recall the
scandal related to the US elections when several
false accounts created by external forces were
discovered on social media, through which
information was disseminated. In the consequences,
this fact was called “an attempt to influence the US
elections”. In the European Union, disinformation
has begun to be fought in an organized manner. In
2015, by decision of the Heads of State and
Government of the Council of Europe, a special
group “Strategic Communication with the East” was
established - EastStratCom Task Force, to counter
the disinformation campaign aimed at discrediting
the EU's Eastern Neighbourhood Policy. In less
than ten years, the group's budget has grown
from one to eleven million Euros. On the
group’s official website, you can find
information about how many articles containing
misinformation were reviewed and what responses
were prepared and sent out. Logically, a campaign
against disinformation can be planned and
conducted more effectively if there is a clear idea
and quantitative characteristics of the false
information spread. Naturally, the study of
disseminating false information using mathematical
and computer modeling, conducting computer
experiments along with other methods of studying
the task, make it possible to effectively describe the
process under study and plan measures against
disinformation campaigns. When constructing a
mathematical model for spreading false information
and combating it, we will use the compartmental
approach proposed in the works, [1], [2], [3], [4],
[5], [6], [7], [8], [9], [10].
2 Problem Formulation
Let us assume that in a society with a constant
number
N
of people, information flows are
distributed at each moment of time
0,tT
by two
operators. The second operator distributes false
information in the amount of
5
yt
, the first
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operator distributes anti-false information, in
contrast to the second operator, in the amount of
4
yt
. These two operators, spreading the
corresponding information in fact, in figurative
terms, lead the battle for the minds of members of
society. As a result of this, the following groups are
formed in society: -
1
Y
. Risk group with a number
1
yt
, that has not yet decided which operator to
follow; -
2
Y
group of Adherents of false information
with the number
2
yt
, who became followers of
the second operator; -
3
Y
Immunity group with the
number
3
yt
, rejected false information, and
thereby perceived the first operator.
We will assume that people who find
themselves in the Immunity group no longer leave
it. If we manage to determine the dynamics of the
transition of people from one group to another, we
can thereby establish the degree of influence of false
and anti-false information on society. Structurally,
the transition of people from one group to another
under the influence of false and anti-false
information can be depicted as follows, in Figure 1.
Fig. 1: Structure of transitions from one group to
another
As can be seen from Figure 1, operator 2 acts
only on the Risk group, and operator 1, in addition
to the Risk group, also acts on the group of Adepts.
Over time, the size of the Risk group decreases and
approaches zero, and this occurs as a result of the
mutual action of operators 1 and 2. Naturally, if the
size of the group of adherents is small, then false
information does not significantly affect society,
and it can adequately perceive and respond to
challenges that may arise for society. Operator 1
acts to reduce the number of Adepts by spreading
anti-false information. The more intensively anti-
false information is spread, the smaller the number
of Adepts is. However, the creation and
dissemination of one unit of anti-false information
requires certain financial and other resources. It is
quite possible that, given certain limited resources,
Operator 1 will not be able to reduce the size of the
group of Adepts. Therefore, the task arises of
determining the required amount of resources for
operator 1 to reduce the number of Adepts and at the
same time effectively use these resources. Those.
The problem of optimal control of combating false
information arises.
3 Problem Solution
From the Risk group (
1
Y
) as a result of the influence
of false information (
5
yt
), its members can leave
it and go to the Adept group (
2
Y
), and under the
influence of anti-false information (
4
yt
) to the
Immunity group (
3
Y
). Interpersonal relationships of
members of Risk and Adept groups also affect the
transition from Risk to Adept group. The
interpersonal relationships of members of Risk and
Immunity groups affect the transition from Risk to
Immunity group. Note that the number of members
of the Risk group does not increase, the function
1
yt
does not grow. Thus, you can write in
mathematical relations the rate of change in the
number of Risk groups:
1
4 1 5 1
1 1 2 2 1 3
dy t t y t y t t y t y t
dt
t y t y t t y t y t
,
where
t
,
t
,
1t
,
2t
are non-
negative variable coefficients. Note that in the
resulting ratio, given a specific example, the number
of members of the Risk, Adept, and Immunity
groups can be determined by sociologists using the
appropriate methodology. In general, to derive the
relationship between the transition of people from
one group to another requires joint research by
scientists from different specialties. For example,
not all information disseminated by operator 2 may
be false; this may serve the purpose of increasing
the “reliability” of operator 2’s information.
Therefore, it is necessary to isolate false information
from the entire flow of information from operator 2,
in fact, in real time. Expert Systems can probably
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cope with this task. To determine the value of
t
,
t
,
1t
,
2t
and other coefficients in real
time, it is preferable to use, in addition to Expert
Systems, other capabilities of the artificial
intelligence system.
Similarly, we can approach other groups, taking
into account the principle of balance and the fact
that from the Adept group the transition to the
Immunity group can occur spontaneously. As a
result, we will get a dynamic system for combating
disinformation in society:
1
4 1 5 1
1 1 2 2 1 3
2
1 1 2 5 1
1 4 2 2
1 2 3
3
2 2 1 3
1 2 3 1 4 2
41
44
12 1
5
21
,
,
(1)
,
1,
1
dy t t y t y t t y t y t
dt
t y t y t t y t y t
dy t t y t y t t y t y t
dt
t y t y t t y t
t y t y t
dy t t y t t y t y t
dt
t y t y t t y t y t
t y t y t
dy t yt
t y t M
dt
dy t t y t
dt
5
2
.
yt
M
where, all the system coefficients (1) are positive
and variable. The parameters
1
M
and
2
M
correspond to the levels of those Internet
Technologies with the help of which information
flows by operators are distributed accordingly.
Suppose that at an initial point in time, the
numbers of all groups and the volume of operator
flows are known, i.e.
1 10 2 20 3 30
4 40 5 50
0 , 0 , 0 ,
0 0 .
y y y y y y
y y y y
(2)
Thus, a mathematical model was built for the
spread of disinformation and the fight against it in
the form of the Cauchy Problem (1), (2). Where (2)
is the initial conditions for the dynamic system (1).
The significance of functions
2
yt
determines
how much misinformation has penetrated society. If
the goal is set that during what is important voting,
society should either be completely freed from
misinformation, or the "carriers" of disinformation
should not exceed or equal five percent of the
number of voters. Usually, five percent is the barrier
in some European countries that political actors
must overcome in elections to enter the legislature.
If the elections are scheduled for time
T
, then by
this time the number of adherents of operator 2 -
2
yT
should be less than five percent of voters.
Let the number of voters coincide with the size of
society, then the following must be fulfilled:
2/ 20.y T N
(3)
Thus, the task arises - the dynamic system should
be transferred from state (2) to state (3), so that a
society, mainly free from disinformation, would
make a choice. To transfer the system from state (2)
to (3) in operation, [11], it is proposed to consider
the flow volume of the first operator as a control
parameter and set the task of optimal control of the
fight against disinformation:
4 1 1 4
0
,.
T
J y t J t M t y t dt inf
(4)
where
t
the price of spreading one unit of anti-
false information at a particular point in time
0,tT
. Since the flow of anti-false information is
determined by the levels of Internet Technologies
1
M
and the parameter
1t
, we actually have two
control parameters. Thus, the task of optimally
controlling the fight against disinformation - (4), (1)
- (3) makes the following sense. Operator 1 must
produce such an amount of anti-false information
that will satisfy the system (1), the boundary values
(2), (3) and the price for its creation will be
minimal.
3.1 Stationary Solutions - Singular Points
Let
12345
( ) ( ), ( ), ( ), ( ), ( ) T
y t y t y t y t y t y t
,
12345
( ( )) ( ( )), ( ( )), ( ( )), ( ( )), ( ( )) T
F y t f y t f y t f y t f y t f y t
then the autonomous system (1) can be
represented in the following form-
() ( ( ))
d y t F y t
dt
(5)
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Let the coefficients in (1) be constant and equal
-
0,015;
10,011;
0,009;
10,013;
20,014;
10,013;
0,0092
10,018;
20,017;
145;M
260M
. For these coefficient
values, we find stationary solutions of system (1) or
(5), i.e. let's solve a system of nonlinear equations:
( ) 0.Fy
(6)
Note that system (6) has a trivial solution
0,0,0,0,0 T
y
. To find other stationary solutions,
we will use the MatLab application package,
namely the fsolve function. To do this, we will
create two m-files:
adsys2.m
function F=adsys2(y)
l=0.015; l1=0.011; k=0.009; a1=0.031; a2=0.014;
b=0.013;g=0.0092;
o1=0.018;o2=0.017; m1=45; m2=60;
F = [-l*y(1).*y(4)-k*y(1).*y(5)-a1*y(1).*y(2)-
a2*y(1).*y(3)
a1*y(1).*y(2)+k*y(1).*y(5)-l1*y(2).*y(4)-
g*y(2)-b*y(2).*y(3)
g*y(2)+a2*y(1).*y(3)+b*y(2).*y(3)+l1*y(2).*y(4)+
l*y(1).*y(4)
o1*y(2).*(1-y(4)/m1)
o2*y(1).*(1-y(5)/m2)]
end znles.m
y0 = [9; 13; 11; 10; 8];
[y,fval] =
fsolve(@adsys2,y0,optimset('Display','off','TolFun',
1e-4))
As a result of a computer experiment, when the
system initially approaches zero, y0 = [9; 13; 11; 10;
8], we obtain a stationary solution:
0;0;9.6806;4.8259;20.6643 T
y
(7)
For each stationary solution, its nature should be
determined. To do this, you need to find the
eigenvalues of the Jacobian matrix of system (5):
12345
1 2 3 4 5
, , , , ,
, , ,
i
j
D f f f f f f
D y y y y y y
, 1,2,3,4,5;ij
(8)
The Jacobian matrix (8) for a stationary point
(solution) (7) has the form
0.3939 0 0 0 0
0.1860 0.1881 0 0 0
0.2079 0.1881 0 0 0
0 0.0161 0 0 0
0.0111 0 0 0 0
j
(9)
Now let's calculate the eigenvalues of the
Jacobian (9) in MatLab using eig(j) . We get L= [0,
0, 0, -0.1881, -0.3939] real eigenvalues. In this case,
the Jacobian determinant is equal to zero det(j)=0.
Since there are zeros among the eigenvalues, the
first Lyapunov methods for stability are not
applicable. In our case, there is a critical singular
point and for its further study we should look for the
Lyapunov function, and apply Lyapunov’s theorems
on stability, asymptotic stability and Chetaev’s
theorem on instability. However, we will refrain
from doing so at this stage. What is interesting for
us is not so much asymptotic stability, i.e. does the
system transition to a stationary position as time
tends to infinity, and the behavior of the system over
a finite period, is it possible to transfer it to the
required position during this time? Because, over a
finite period of time, events may unfold in such a
way and the system will find itself in such a state
that it can cause upheavals in society.
3.2 Controllability Problem
We examine the task of optimal control of the fight
against disinformation (4), (1) - (3) for
controllability. I.e. find out whether there is such a
function that satisfies (2) to (3). We use the MatLab
Application Package for this, in the system (1) the
coefficients will be considered constant. Let us have
the following boundary conditions:
15T
,
10 400,y
20 50,y
30 10,y
40 1,y
50 17y
,
215 460 / 20y
. Let the coefficients
of system (1) be previously determined values. With
such coefficients of the autonomous system (1), the
conditions for the existence and uniqueness of a
solution to the Cauchy Problem(1), (2) in the class
of smooth functions are satisfied.
To solve the Cauchy problems (1), (2) with the
values indicated above, we use the ode15s solver,
the program code is compiled - Listing
[X,Y]=ode15s(@adsys,[0 15],[400 50 10 1 17]);
plot(X,Y,'linewidth',2)
legend('y1','y2','y3','y4','y5')
function dydx=adsys(x,y)
l=0.01; l1=0.015; k=0.01; a1=0.03; a2=0.017;
b=0.015;g=0.001; o1=0.019; o2=0.79; m1=40;
m2=55;
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dydx=[-l*y(1)*y(4)-k*y(1)*y(5)-a1*y(1)*y(2)-
a2*y(1)*y(3)
a1*y(1)*y(2)+k*y(1)*y(5)-l1*y(2)*y(4)-
g*y(2)-b*y(2)*y(3)
g*y(2)+a2*y(1)*y(3)+b*y(2)*y(3)+l1*y(2)*y(4)+l*
y(1)*y(4)
o1*y(2)*(1-y(4)/m1)
o2*y(1)*(1-y(5)/m2)]
end
A computer experiment shows, for the above
data, that operator 1, even with little effort, manages
to neutralize the influence of false information by
the end of the time interval - the number of
adherents is zero, although it is enough that the
number of adherents is less than five percent of the
population, in Figure 2.
Fig. 2: Manageability of the model for combating
disinformation
A similar result is achieved for other values of
system parameters - information security is ensured
by the end of the interval. However, information
security is nevertheless vulnerable, but not at the
end of the interval, but at the beginning. If we pay
attention to the value of the number of adherents at
the beginning of the interval, it turns out that, for
example, in our case, its maximum value is 322 at
time 0.3, which is 80,5% of the society population,
in Figure 3.
Fig. 3: Information security vulnerability
At the same time, and over a sufficiently long
period, the number of adherents is more than 50% of
the public. For the information security of society, it
is not sufficient to reduce the number of
disinformation adherents only at a certain point in
time, for example, before any voting. Since at some
points, the influence of disinformation on societies
can be so strong that society as a whole can
legitimize revolutionary, early parliamentary,
presidential elections, etc. Therefore, in the optimal
control problem of combating disinformation,
adjustments should be made and inequality in (3)
should be satisfied over the entire period, [11].
20,05y t N
, for
0;tT
. (10)
The issue of controllability in the optimal
control problem (4), (1), (2), and (5) is studied using
a computer experiment. By changing the control
parameters -
1t
and
1
M
, increasing from by an
order of magnitude, it is not possible to achieve the
fulfillment of condition (10) throughout the entire
section
0;T
, i.e. there are a sufficient number of
adherents of false information in society. The
question arises of attracting new control parameters,
these could be:
1t
- the intensity of interpersonal
contacts between the Immunity and Adept groups;
30
y
- initial conditions for the Immunity group;
40
y
- initial conditions for the dissemination of anti-false
information by the first operator. By varying the
values of these new control parameters, it is possible
to significantly reduce the maximum value of the
number of adherents, for example for
10.23t
,
30 40y
,
40 20y
- [X,Y]=ode15s(@adsys,[0 15],
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[400 50 40 20 17]), which is clearly seen in Figure
4.
Fig. 4: Result of control new parameters
The maximum number of adherents is 63,48 -
reached at the beginning of the time interval and
then not for long - soon the number of adherents
sharply decreases and becomes less than 24, which
is no more than five percent of the population of
society. At the same time, the number of adherents
at the beginning of the time interval is growing, and
for this not to happen, the inequality must be
fulfilled
2
0
0
t
dy t
dt
,
1 10 20 50 10
00y y y y
1 40 20 20 1 20 30
0 0 0 0.y y y y y
Let us rewrite the last inequality assuming that,
10 10 0 50 10 20 10 40 0 10 30
/.y y y y y y
(11)
In inequality (11) there are at least two control
parameters - , and with their help - by selecting the
appropriate values, it is possible to achieve the
fulfillment of condition (11). And this means that
the number of adherents from the very beginning
will decrease.
4 Conclusion
For the information security of society, it is not
sufficient to reduce the number of disinformation
adherents only at a certain point in time, for
example, before any vote. As it was shown, at some
points, the influence of disinformation on societies
can be so strong that society as a whole can
legitimize revolutionary, early parliamentary,
presidential elections, etc. Therefore, in the
proposed, [11], adjustments should be made to the
task of optimal control of the fight against
disinformation, specifically, the number of
adherents should be controlled throughout the entire
period. To effectively combat disinformation, it is
necessary to increase the number of control
parameters and the target control functionality and
its minimization has the form:
1 1 1 30 40 4
0
, , , , .
T
J t M y y t y t dt inf
(12)
1 10 30 40
; , , ,t C M y y R
Thus, the problem of optimal control of
effective combat against disinformation will take the
form (12), (1), (2), (10). If in the initial condition (2)
there is inequality
20 0.5yN
, then condition (10)
is not satisfied at the beginning of the time interval,
which means the vulnerability of society in terms of
Information Security. Mathematical and computer
modeling of the effective fight against
disinformation allows us to conclude that permanent
control of the number of adherents and information
flows of operator 2 makes it possible to ensure the
stability of society in Information Warfare.
The creation of an Automated System for
Effectively Combating Disinformation should occur
based on generalized principles and its application
will be possible in different countries and different
areas of activity. However, it can be assumed that it
is especially relevant for my country. Georgia
recently received the status of a candidate country
for joining the European Union with the expectation
that it will implement nine recommendations of the
European Commission in the future, among which
the first is to "fight disinformation and foreign
information manipulation and interference against
the EU and its values".
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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