Mathematical Model on Distributed Denial of Service Attack in the

Computer Network

YERRA SHANKAR RAO1, ASWIN KUMAR RAUTA2*, SATYA NARAYAN KUND3,

BHAGIRATHI SETHI4, JANGYADATTA BEHERA5

1Department of Mathematics, NIST (Autonomous) College,

Berhampur - 761008, Odisha,

INDIA

2Department of Mathematics, SKCG (Autonomous) College,

Paralakhemundi-761200, Odisha,

INDIA

3Controller of Examinations, Berhampur University,

Berhampur -760007, Odisha,

INDIA

4Department of Mathematics, Khemundi Degree College,

Digapahandi- 761012, Odisha,

INDIA

5Department of Mathematics, Roland Engineering College,

Berhampur- 761008, Odisha,

INDIA

*Corresponding Author

Abstract: - In this paper, an electronic- epidemic two-folded mathematical model is formulated with help of

non-linear ordinary differential equations. Distributed Denial of Service (DDoS) attacks in the computer

network are studied. The modeling of both attacking nodes and targeting nodes is performed. Botnet based

malicious devices and their threats on computer networks are addressed using appropriate parameters. The

basic reproduction numbers for both the attacking and the targeting population are calculated and interpreted.

Local and global stability analysis is carried out for the infection-free and endemic equilibrium points.

Differential equations are solved with the help of the Runge-Kutta 4th order numerical method and graphs are

analyzed using MATLAB software. Simulation shows that the success or failure depends on the number of

initially infected computers in the attacking group. The proposed model exhibits the phenomenon of backward

bifurcation for different values of transmission parameters. This model gives the theoretical base for controlling

and predicting the DDoS attack. This shows the way to minimize the attack in the network. This study will be

helpful to identify the botnet devices and run the latest version of antivirus in the network to protect against

DDoS attacks from attacking sources. The application of this study is to ascertain online crime and locate the

attacking nodes in the field of online transactions of real-life problems that involve the internet and computer

networking systems. Moreover, our model can play an important role in policy-making against the distributed

attack.

Key-Words: - Basic Reproduction Number, Bifurcation, Cyber-Crime, DDoS attack, Eigen Value, Malware,

Mathematical Modeling, Simulation, Stability Analysis, Virus.

Received: July 23, 2022. Revised: September 15, 2023. Accepted: November 8, 2023. Published: December 31, 2023.

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1 Introduction

The computer connects every person in the world,

controlling their daily business through internet

networking or browsing. The development of

internet technology has thrown a number of

challenges in the form of necessity in day-to-day

life. Digital technology has brought a big change in

the society. But, it is being the heaven of crimes

using computer networks by the trained

intelligentsia. The new type of such crime called

cybercrime. It is a current research topic for the

investigators. Many researchers have studied, [1],

[2], [3], [4], [5], [6], [7], [8], this new crime using

mathematical modeling. Among different types of

internet-based crimes, the Denial of Service (DoS)

attack is one of them. This attack is a very critical

and continuous threat to cyber security. DoS is a

cyber-attack in which cybercriminals search

network resources or an IP address

or machine from thousands of hosts infected with

malware to make it unavailable to the intended

users by interrupting the services indefinitely. It is

done by notification of superficial requests when

the computer of the user is turned on in an attempt

to prevent some or all legal services from being

fulfilled or slow down the system to hamper the

services. When the DoS attack originates from

many different sources, it is called a Distributed

Denial of Service (DDoS) attack. So, it is difficult

to locate the error and may not be possible to block

the source of the attack. The DoS attacks are

targeted by consuming resources, and forcing a

computer to reset. e.g., network bandwidth, CPU

cycles memory, etc. so that the network does not

work properly that leads to the site. If someone

uses the same connection for internal software,

employees notice slowness issues. The TTL (time

to live) on a ping request timed out and the victim’s

server responds with service outages. DDoS attacks

can last as long as 24 hours and the cost of business

is minimized while the user remains

under attack. In this attack different kits like

Stacheldraht, Trinoo, Mstream, Tribe Flood

Network (TFN), etc. are launched to other

computers by DDoS attackers. DDoS attacks are

performed in two ways; (i) the crafted packets are

sent to crash a system that causes a reboot or

freezing of some operating system. (ii) Exhausted

the resources like operating system, data structures,

computing power, network bandwidth etc. of the

targeted computer. Due to DDoS attacks, the

quality of service is disabled or interrupted to the

intended users. It is tedious work to deal with the

second form of attack rather than the first form of

attack. A botnet is the usual medium of DDoS

attacks. Intelligent criminals make a network of

computers called BOTNET to launch an effective

DDoS attack. The people who control a botnet are

called botnet owners or botnet masters. The

software applications that are programmed to run

automatically according to their instruction without

users needing to start them are known as zombies

or bots. The source of the botnet is called the

control server. The most effective methods to

control, respond, and prevent the spread of DDoS

attacks are updating the operating system, data

mining, firewall, auto patching, etc. To reduce

transmission of botnet infective nodes, buy more

bandwidth, build redundancy into your

infrastructure, configure your network hardware

against DDoS attacks, deploy anti-DDoS hardware

and software modules, and deploy a DDoS

protection appliance and DNS servers. The visitor’s

information could be stolen using the attacks. They

are often used to make 'political' statements against

the targeted organization or just as a form of

malicious vandalism. For example, the criminals

demand a ransom amount from the website

owners to stop the attack. So, it is an emerging

attention for the researchers to investigate and

locate the attacking sources. Many authors have

presented their investigation reports in this regard

for locating the attacking node and providing the

security system to the network, [9], [10], [11], [12].

The connection to the internet increases the

complexity of interconnected networks.

Mathematical modeling is used as a tool to identify

and understand the problem of DDoS attacks. In

order to provide better defense mechanisms, many

researchers have used epidemic models. Dynamic

models for infectious diseases are mostly based on

compartment structures that were initially proposed

for several areas of Mathematical Biology, [13],

[14], [15]. It was developed later by many other

mathematicians in the modeling of cybercrimes or

computer related malicious objects. These epidemic

models are dynamic in nature. Therefore,

transmission of malicious objects is epidemic in

nature. So, many mathematical models have been

developed that specify the comprehensible view of

attacking behavior as well as the spread of the

malware objects in the network, [16], [17], [18],

[19], [20], [21], [22], [23]. The use of vaccination

and quarantine effects were studied for the DDoS

attack and spread of malware in the computer

network, [24], [25], [26], [27], [28]. Presently, this

type of cybercrime is a new, global issue and draws

serious attention. But currently, less study has been

conducted in this field. Therefore, we have

developed this model to formulate the attacking

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nodes and targeted nodes for finding the basic

reproduction number, and investigated the stability

analysis to control DDoS attacks through analysis,

simulation, and interpretation of data obtained from

different sources.

2 Formulation of Mathematical

Model and Assumptions

The whole population is divided into two sections

namely attacking and targeting populations. The

entire targeted system is divided into four

compartments: Susceptible (St), Exposed (Et),

Infected (It), and Recovered (Rt) classes. Similarly,

the attacking nodes are divided into two classes

Susceptible (S), and Infected (I). Once the

malicious objects enter into the network, the

susceptible nodes of the targeted group after some

time become exposed (Et) at the rate β>0, and then

it gets infectious (It) at the rate α>0. Again, after

running the anti-malicious software at the rate γ>0,

infected nodes get recovered (Rt). The rate at which

the recovered population becomes susceptible is

taken as εt >0. Each susceptible node for attacking

and targeted nodes becomes infected at the rate

β>0. The model takes essential dynamics in each of

the attacking nodes born or dies at the rate μ >0.

The rate at which infected classes become

susceptible in the attacking network is taken as ‘ε’.

Based on these assumptions we have developed an

e-epidemic mode as shown in the schematic

diagram.

εtRt

βStI αEt γIt

Targeted Population

μ βSI εI

µ µ

Attacking Population

Fig. 1: Compartmental Model for Targeted

Population and Attacking Population.

Using the schematic diagram in Figure 1, the

rate of change of each class size is given by the set

of ordinary differential equations (ODEs):

The targeted classes have the following ODEs

tt t t

ttt

tt t t

dS S I R

dE S I E

dI EI

dR IR









  



(1)

The attacking classes are governed by the following

ODEs.

dS SI S I

dI SI I I

   

  

   

  

(2)

Here, the entire population of targeted compartment

is assumed as one unit i.e.,󰇛󰇜 󰇛󰇜 󰇛󰇜

󰇛󰇜  .The entire population of attacking class

is also assumed as one unit i.e.,    

The reduced form of above equations is;



    󰇛    󰇜



    



   



  󰇛      󰇜

(3)

Where,      

        

3 Calculation and Interpretation of

Basic Reproduction Number and

Equilibrium Points

Basic reproduction number is an important

threshold quantity that play a significant role in

epidemiology. It is defined as the average number

of secondary infections in a susceptible class

produced by a single infectious device during the

whole infection period. Two basic reproduction

numbers are derived for two types of populations.

The basic reproduction number of the targeted

population is derived as;

0()







(4)

Similarly, the basic reproduction number of the

attacking population is calculated as;

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0()







(5)

The single basic reproduction number for the entire

system is defined as the geometric mean of

equations (4) and (5) which is,

0( )( )



   



(6)

In this article, it is observed that the basic

reproduction number for the attacking population

determines the overall risk of the attack on the

targeted group. The basic reproduction number of

the targeting population determines the

effectiveness of attack and infection of the system.

Theorem-1:

System (2) is infection-free equilibrium in the

region and admits the endemic equilibrium in the

given region.

Proof

Consider the right-hand side of equations with zero

to obtain the attack-free equilibrium points.

(1 ) 0

t t t t t

S I S E I

S I E

I I I I







  

     



   

(7)

For, attack free, we get I=0, E=0, R=0.

Therefore, S=S0

After solving the above equations simultaneously,

we get the endemic equilibrium point for the attack

to be persisting.

()



       

   

       

   

       

  



     



     



     





(8)

Theorem-2

The system (6) is locally asymptotically stable at

attacking free equilibrium in the given region if

R

and it is unstable when

R

Proof

Linearization of the system (3) around the infection

free equilibrium point (1, 0,0,0), the Jacobian

Matrix is

0 0 0

000

ttt

IFE







  





















Therefore, the characteristic roots are given by,

when

()

  





()









, i.e.

R

As all the eigen values have negative real parts at

infection free equilibrium point, so by Routh-

Hurwitz criteria, the system is locally

asymptotically stable for

R

and is unstable

when

R

i.e.

()

  



Theorem-3

The system is local asymptotically stable at the

endemic equilibrium point when

R

Proof:

Linearization of (3) at the endemic equilibrium, we

get the following Jacobian Matrix

0 0 0 2

t t t t

    

  



   



    













  



One of the Eigen values is given by

    󰇛  󰇜

 󰇛 󰇛    󰇜󰇜

  



R

,i.e.,

( ) 0

  

   

,then  is negative.

Other eigen values are also determined from the

following cubic equation;

     

Where,        

        

        







   



  

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Therefore, AB>C

Again, as per Routh-Hurwitz stability criteria, the

system is local asymptotically stable.

4 Global Stabilities for Infection-Free

Equilibrium

Theorem-4

The system is globally asymptotically stable for Roa

< 1 and is unstable when Roa>1 at the infection-free

equilibrium point.

Proof: Consider a Lyapunov function, [29], [30],

[31], as

(1 ) ( )

()

1( 1)

()

1( 1)

()

t t t t

t t a

V E I I

dV S I E E I I I I

dt S I I I I I

S I I I I

S I I I R I

      

    



    

 

  

       

     

     

    



    



R

, then

dt 

Using LaSalle’s maximum invariant principle, it is

globally asymptotically stable at the infection-free

equilibrium point for

R

5 Numerical Simulation and

Discussion

In this research, the differential equations are

solved using Runge-Kutta 4th order method and

numerical simulations are carried out by MATLAB

software in support of theoretical analysis

discussed in the previous section. Some parameters

can be used for infection-free equilibrium. The

basic reproduction number depends on the values

of contact rate β. Further, it is seen that if we

increase the contact rate β then increase the

infection that leads to more infection of the

network, and the system remains unstable. Thus, if

the recovery rate is higher from the attack, then the

system remains stable. The graphs are plotted for

initial values of St=0.8585, Et=0.4718, It=0.1415,

S=0.1847, I=0.1888 when R0a>1 and initial values

St=1000, Et=0.2773, It=0, S=0.1754, I=0 when

R0a<1. The interpretations of the numerical results

are discussed below.

5.1 Dynamic Behaviour of Nodes with

Respect to Times when R0a>1

Fig. 2: All nodes versus time graph when Ra = 4.9,

Rt =1.2.

Fig. 3: All nodes versus time graph when

Ra=5.2,Rt=1.3.

When the basic reproduction number  ,

then I increase to a peak and S decreases for time

being then I decrease for endemicity. Figure 2 and

Figure 3 explain that all the nodes St, Et, It, Rt ,S and

I approach to its steady state values as time goes to

infinity for R0a>1, due to being endemic in nature.

5.2 Dynamic Behaviour of Nodes with

Respect to Times when R0a<1

Fig. 4: All nodes versus time graph when Ra = 0.57,

Rt =1.1.

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Fig. 5: All nodes versus time graph when Ra = 0.62,

Rt =1.2.

When the basic reproduction number

R

i.e. when an infective computer replaces itself with

less than one new infective node, then the attack

dies out. It indicates that the susceptible class goes

to its full capacity because each system becomes

susceptible when the attack will disappear. So, in

Figure 4 and Figure 5; St Et, It, Rt ,S and I approach

to steady state as time goes to infinity for R0a<1.

5.3 Effect of Susceptible Nodes with

Infective Nodes of Targeting

Population

Fig. 6: Targeted susceptible node versus targeted

infective node phase plane analysis graph when Ra

=5.2, Rt=1.3.

If, the basic reproduction number   i.e.

the attack is persisting, infective class ‘I’ will

increase first then decrease just as for an epidemic.

Therefore, the susceptible class slowly starts to

increase due to the installation of anti DDoS

devices or new updated anti-malware software. As

time evolutions, the susceptible class reaches large

enough, and again due to new attacks every time,

there is a possibility of a second smaller epidemic.

Continuing in this process, we will get the path

spirals approaching to the equilibrium point and

trajectories that appear to be asymptotically stable

to the endemic equilibrium point that is shown in

Figure 6.

5.4 Bifurcation Diagram of the Model

Fig. 7: Targeted susceptible node versus targeted

infective node phase plane analysis graph when Ra

=4.7, Rt=1. 1.

The model with DDoS protection appliances

under certain conditions admit the backward

bifurcation. A backward bifurcation is predicted

when the targeted group is prevented from DDoS

attack.

Figure 7 exhibits the coexistence of two stable

equilibriums of model form

R

. If there is no

recovery case then bifurcation is reversed. As

infected computers are recovered every time due to

protection against DDoS attack, that shows a

backward bifurcation and sign of epidemic control.

6 Conclusion

In this study, we have presented a dynamic model

to control DDoS attacks in the computer network

by considering two sections. We started by

showing a nonnegative solution to the model. We

proved both infection free equilibrium points to be

locally and globally asymptotically stable. It is

observed that, if the basic reproduction

number then attack would continue.

Similarly, if

R

then the attacking

population would die out. The success and failure

of the attack is demonstrated graphically. Due to

the latent time between attacking susceptible and

infectious nodes, the model is more appropriate for

DDoS attacks. The attacking population of DDoS

attacks is very high approximately, when antivirus

software is not run at regular intervals of time.

These simulated results supported by the theoretical

approach show the malicious objects died out or

persisted.

In the stability analysis of the model, it is

shown that the attack dies out whenever R0<1.

Figure 2 to Figure 5 exhibit that the susceptible

  

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class remains at a steady state as time continues.

This indicates that the susceptible class is stable. It

is also interpreted from the figures that the

susceptible class, and recovered class remain the

same if no new infected cases arise in the later

stage. That is, as long as a new infected case does

not occur, then the size of the susceptible

compartment remains the same as the total

population.

The future scope of this study may be the

extension of the model by considering more

parameters. This study may also be extended by

including more compartments like quarantine

compartments to ascertain the global cyber threat

and provide security in the network. In addition to

this, the model can be used for modeling of

contagious diseases in the biological systems in

real-life problems.

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Nomenclatures:

St - Number of Susceptible Targeted nodes.

Et - Number of Exposed Targeted nodes.

It - Number of Infected Targeted nodes.

Rt - Number of Recovered targeted nodes.

S - Number of Susceptible attacking nodes.

I - Number of Infected attacking nodes.

β - Rate of contact both attacking and targeted

nodes.

α - Rate of contact from exposed to infected

targeted nodes.

γ - Rate of recovered from infected to recovered

targeted nodes.

εt - Rate at which from recovered to Susceptible

targeted nodes.

µ- Rate of death and newborn in attacking

compartment.

ε - Infection rate in the attacking class.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

- Yerra Shankar Rao has carried out the

formulation of the problem, derived the

mathematical equations and proved the theorems.

- Aswin Kumar Rauta has executed the

experiment, organized the manuscript, analyzed

and interpreted the results. He has also acted as

corresponding author.

- Satya Narayan Kund envisaged the theme of

research and motivated for the investigation.

- Bhagirathi Sethi verified the calculated results

and was responsible for the literature review.

- Jangyadatta Behera has implemented the

computer software MATLAB for numerical

simulation and plotted the graphs.

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 authors have 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.e

n_US

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Volume 22, 2023