a dynamic e -epidemic model for the attack against … · 2019-02-24 · a dynamic e -epidemic...

Journal of Analysis and Computation (JAC) (An International Peer Reviewed Journal), www.ijaconline.com, ISSN 0973-2861

Volume XIII, Issue I, January 2019

Yerra Shankar Rao, Aswin Kumar Rauta, Tarini Charan Panda, Subash Chandra Mishra 1

A DYNAMIC e -EPIDEMIC MODEL FOR THE ATTACK AGAINST

THE SPREAD OF VIRUS IN COMPUTER NETWORK

Yerra Shankar Rao1, Aswin Kumar Rauta2, Tarini Charan Panda3, Subash Chandra Mishra4

1Assistant Professor Department of Mathematics Gandhi Institute of Excellent Technocrats, Ghangapatana

Bhubaneswar, Odisha, India.

2Lecturer, Department of Mathematics, S.K.C.G. College, Paralakhemundi, Odisha, India.

3Professor Department of Mathematics Ravenshaw University, Cuttack, India.

4Assistant Professor Department of EE & Electrical & Electronics Engineering Gandhi Institute of Excellent

Technocrats, Ghangapatana Bhubaneswar, Odisha, India.

ABSTRACT:

Internet worms/ viruses cause a serious threat to the Internet security. In order to successfully

defend against Internet worms/virus, vaccination is one of most effective measures for the minimize the

spared of computer virus. In this paper we develop a new e-epidemic (e-SVIR). Which we describe the

behaviour of the model and derive the reproduction number. We also analyze the stability of the model.

Though a mathematical analysis of this model, it is found that infection free equilibrium is asymptotically

stable when the basic reproduction number is less than one. Where as it is unstable if basic reproduction

number is more than one. Here also analysis of vaccination is power the network security. Based on these

results and parameters to eradicating the transmission of virus in the network.

Keywords: Reproduction number, stability analysis, epidemic model, vaccination, equilibrium

[1]INTRODUCTION

The growth in cyber world has brought sweeping changes in to human life with the rising

technology of internet. The usage has drastically increased offering functionality and facilities The

availability of reliable models of computer virus propagation would prove useful in a number of

ways, in order both to predict future threats, and to develop new containment measures.. Viruses were

once spread by sharing disk, now globally connectivity allows the malicious code to spread farther

and faster. The number of computer virus has-been increasing exponentially from their 1st

appearance in 1086 to over 74000 different strains identified today. The spread of malicious agent is

identical to that of spread of epidemic in biological world. A virus is a program that can `infect' other

programs by modifying them to include a, possibly evolved, version of it”. The exact definition of

computer virus that virus contains programme code that can explicitly copies itself and by doing so

that the ability to infect other programme by modifying them or their environment. In order for virus

A DYNAMIC e -EPIDEMIC MODEL FOR THE ATTACK AGAINST THE SPREAD OF VIRUS IN COMPUTER NETWORK


to propagate it typically needs to attach it to host programme. Virus attacks are considered by network

experts the highest security risk on computer network. Computer virus is built to propagate without

warning or user interaction, causing an increase the service request that will lead to cyber attack. To

stop or decrease the attack of virus, we need e-epidemic model that can contently capture the most

important objects as accepting the spread of virus in is critical for the most effective reactive

measures.

There are several computational techniques that look to biology for inspiration. The cost

caused by the damage of computer viruses can be potentially huge. Various approaches have been

proposed to address the computer virus problem theoretically. Adapting and applying mathematical

epidemiology to this problem is one such attempt. The hope is that a science of computer virus

epidemiology will benefit from the success of epidemiology in biology. Many researcher have taken

help of biological system to understand the behaviour of spread of malicious objects in computer

network and how to immune to computer system. Based on Kermack Mack end risk SIR classical

epidemic model [9-11]. Dynamic model for the malicious object propagation were proposed to

estimate for temporal evaluation of infected nodes depending upon the network. Parameter

considering topologically aspect of the network. Mishra & Saini present SEIRS model with latent and

temporary immune period which reveal common virus propagation [13-15] .Hyman& Li proposed a

biological SIR model that describes the transmission dynamic of an infectious diseases assuming

susceptible population divided into different group is distinct. In SI, SIS, SEI, SIR model are do not

explicitly describe the transmission between the nodes individuals i.e. the force of infection is not

expressed in terms of transmission parameter and number of imperfection person[16-20]. So an

attempt has been made to minimize the attack of worm in the computer network. Vaccination is a one

of the effective measures for minimize the spread of computer virus. It plays the vital role in isolation

of computer virus by which susceptible computer would temporary immunity. Dynamic modelling of

the spread process of computer virus is an effective approach to understanding of behaviour of

computer virus due to on this basis some effective measure can be posed to private infection [21-26].

The subsequent of this paper is organised as follows section -2Nomenclatuer,section-3 Mathematical

Assumptions and formulation of the model calculation of vaccinate basic reproduction number ,

section -4 equilibrium condition, stability analysis and calculation of basic reproduction number,

section -5 Discussion of effect of parameter in the models and section-6. Summarizes the work as

well as discussion of the simulated results.

[2]NOMENCLATURE

N: Total number of nodes interacted with the network under consideration.

S: The number of susceptible nodes at time t

V: the number of vaccinated nodes at time t

I: The number of infectious nodes at time t

R: The number of recovered nodes at time t after using antivirus software

: The constant number of new nodes attached in the network

: Proportion o the nodes attached in the network

: The new nodes attached in the network.




: The contact rate

: Natural death

: Crash of nodes other than attack

: Rate of vaccinated from susceptible to infected nodes

: Rare if infection from vaccinated to infected nodes

: Rate of recovery from infected nodes to recovered after using antivirus

[3] MATHEMATICAL MODEL AND ASSUMPTIONS

To avoid the total crash the network, we divide the total number N in to four sub group or

class which are susceptible , vaccinated, infective and recovery S, V, I, R respectively Which are

varies from time to time . We the new dynamic model using mass action law. Therefore flow of the

worm/virus shown in the figure 1. The transmission of the virus either susceptible or infective.

Scanning of the computer before use of internet it can some nodes are vaccinated. Using the internet

the some vaccinated nodes transferred in to infect again us of antivirus technology the infected node

becomes recovered. These mechanism are shown in the below figure-1.



(1 ) ( )

( )

( )

dSSI S

dt

dVS VI V

dt

dISI VI I

dt

dRI R

dt

(1)

By adding all the classes

We have N S V I R

And so dN dS dV dI dR

dt dt dt dt dt (2)

Hence dN

N Idt

This above equation can be written as by replacing S by N-V-I-R as

( ) ( )

( (1 ) ) ( )

dVN I R VI V

dt

dIN V I R I I

dt

dRI R

dt

dNN I

dt

(3)

The above equation can linearize the in the matrix form as

( ) ( )

( 1) (2 ) ( (1 ) )( , , , )

0 0

0 0

I V

I I N V R I IJ V I R N

[4]EQUILIBRIUM CONDITIONS

For the steady state condition the above equation (3) to be zero

( ) ( ) 0

( (1 ) ) ( ) 0

0

0

N I R VI V

N V I R I I

I R

N I

(4)

From last two equations of (4)




IN

IR

Solving the equation for V and substuiting the values of N and R we have

( ( ) )

( )

IV

I

Eliminating N,V, and R by substuiting the values in (4) of 2nd equation as in the form of cubic

equation as in I form

3 2

1 2 3 4

2

1

2 2

2

3 2

3

2 2

4

0

( )

( )

( ) ( ( )) ( )

( ) ( )

a I a I a I a

where

a

a

a

a

(5)

Vaccine Reproduction number

For the disease free equilibrium I=0

So R=0 and N

Hence for solving the equation (4) for V by putting the values of N and R as

( )

( )V

For the disease free equilibrium ( )

( , , , ) ,0,0,( )

V I R N

Theorem The infection free equilibrium is asymptotical stable when 0 1VR

.

if it is unstable when 0 1VR

Proof:

The Jocobian matrix for infection free equilibrium as



( )( ) ( )

( )

( )0 ( ) ( (1 ) ) 0 0( ,0,0, )

( )

0 0

0 0

INFJ V N

The Eigen values are

1

2

3

4

( )

( )( ) ( (1 ) )

( )

Since first three Eigen values are negative, other can be calculated as

( ). . ( ) ( (1 ) ) 0

( )

( )( (1 ) ) ( )

( )

( )( )( )

( )

( )( )1

( )( )

i e

Hence the vaccination reproduction number as

0

( )( )

( )( )

VR

This id infection free equilibrium is locally asymptotical stable iff 0 1VR

In the absence of vaccine we define the reproduction number as

0

( )

( )R

When infective immigrants are zero

[5] DISCUSSION OF EFFECT OF PARAMETER IN THE MODELS

Let’s consider the three cases to examine the endemic equilibrium for the above model

Case-1. 1& 0 i.e. when the vaccination is not reactive and there i no infective immigrants




The model becomes

( )

( )

dSSI S

dt

dVS VI V

dt

dISI VI I

dt

dRI R

dt

and N S V I R .

Here the vaccination reproduction number (R0V) reduce to be basic reproduction number (R0).

These can solve by simultaiously for endemic equilibrium as * ( ) ( )

( )I

Which exist only when 0

( )1

( )R

Case- 2 When 0 & 0

The model becomes

( )

( )

dSSI S

dt

dVS V

dt

dISI I

dt

dRI R

dt

And N S V I R

By calculating the endemic equilibrium for infective immigrants as

* ( ) ( )( )

( )I

also 0

( )1

( )R

Case-3 When 0 1& 0

This model becomes



( )

( )

dSSI S

dt

dVS VI V

dt

dISI VI I

dt

dRI R

dt

And N S V I R .From equilibrium point condition (5) the infection free equilibrium

regardless of different parameter values. Factoring the I we have

2 0AI BI C

Where

2

2 2

3 2

( )

( )

( ) (( ) ) ( )

A

B

C

So by AB>C

As per Routh Hurwitz satiability condition endemic equilibrium is stable.

Figure-2 Dynamic Behaviour of the e-SVIR model

[6]CONCLUSION




In this paper we formulated an e-epidemic model with vaccination. We discuss

the vaccination reproduction number and reproduction number. Vaccination

reproduction number plays the vital role for isolation of infective nodes. The

behaviour, simulation the system of equations developed. The mathematical analysis

and the stability of the proposed model is discussed. Which reflect the effects of the

anti-virus software.. The initial parameter values were chosen in such a way that it

better suit a real worm/virus attack scenario .Infection free equilibrium stable when

the reproduction number below the unity.when we increases with constant vaccination

, while decrease the infection rate in the network. The main vaccination

recommendation is to increase the constant vaccination effort as much as possible.

The use of vaccine for the computer network should be benefitted for long time

immunity against infection. This process will apply in computer security in the

software organisation leds to security. The simulated results agree with real

parameter. The simulated results show that, for the chosen numbers of vaccinated

nodes and for the given value of parameters, recovery of nodes is very high. So it is

recommended to the software organization to maintain the value of the parameters for

anti-virus software.

REFERENCE

[1]. Amichai-Hamburger Y, Hayat Z., “The impact of the Internet on the social

lives of users: A representative sample from 13 countries. Computers in Human

Behavior”.; vol.27,issue.1,pp90.585–589(2011).

[2]. Aswin Kumar Rauta , Yerra.Sankar Rao, T. C. Panda, Hemraj Saini , “A

Probabilistic Approach Using Poisson Process for Detecting the Existence of Unknown

Computer Virus in Real Time,” The International Journal Of Engineering And

Science,Vol. 4 ( 6) PP.47-51.,(2015)

[3]. Chenquean gan,Xiaofan yang,W Liu,X Zhang .,”Propagation of computer

virus under human Intervention,” Discreat dynamic in nature and society .pp.1-8.,(2012)

[4]. Gan, X. Yang, Q. Zhu, J. Jin, and L. He, “The spread of computer virus under

the effect of external computers,” Nonlinear Dynamics., vol.733,pp.-1615–1620.(2013)

[5]. Gil, S., Kott, A. & Barabasi, A.-L. “A genetic epidemiology approach to cyber-

security.” Sci. Rep. 4, 5659, (2014)

[6]. Saini H, Rao Y. S, Panda T. C. “Cyber-Crimes and their Impacts: A review.”

International Journal of Engineering Research and Applications. ; 2(2):202–209. (2012)

[7]. J.M.Wong, N.Pino, B.Hallahan , “On modeling the spread of a computer

virus”. Mathematical modeling, springer. (2014)



[8]. J. Ren, X. Yang, Q. Zhu, L.-X. Yang, and C. Zhang, “A novel computer virus

model and its dynamics,” Nonlinear Analysis: Real World

Applications.,vol.13,issue.1,pp. 376–384, (2012)

[9]. Kermack WO, McKendrick AG, “Contributions of mathematical theory to

epidemics,” Proceedings of the Royal Society of London – Series A, vol.115,pp.700–721.

(1927)


epidemics,” Proceedings of the Royal Society of London – Series A, vol.138.pp.55–

83.(1932)


epidemics,” Proceedings of the Royal Society of London – Series A, vol.141pp.94–122.

(1933)

[12]. Lu Xing Yang,X-Yang, “A new epidemic model in computer virus,

,Nonlinear Science and numerical simulation,” Elsevier., vol.19, issue. 6,pp. 1935–

1944.(2014)

[13]. Mishra Bimal Kumar, Samir Kumar Pandey, “Effect of Anti-virus Software

on infectious nodes in Computer Network A Mathematical Model,” Physics Letters A,

vol.376, pp.2389-2393, (2012)

[14]. Mishra Bimal Kumar, Navneet Jha, “SEIQRS model for the transmission of

malicious objects in computer network”, Applied Mathematical Modeling,vol.34,pp.710-

715. (2010),

[15]. Munna Kumar , Mishra Bimal Kumar , Panda T.C.,”Stability analysis of

quarantine of epidemic model with latent and breaking out over the internet”,

International Journal of Hybrid Information technology., vol.8,issue7,pp.133-148, (2015)

[16]. O. A. Toutonji, S. M. Yoo, and M. Park ”Stability analysis of VEISV

propagation modeling for net-work worm attack," Applied Mathematical Modelling,vol.

36, no. 6, pp. 2751-2761. (2012)

[17]. Rao, Shankar. Yerra., Panda, T. C., & Saini, H. “Mathematical Analysis on

computer Virus in the computer network” In Souvenir 43rd Annual conference Orissa

mathematical society (p. 41)., (2016).

[18]. Rao, Shankar. Yerra., Rauta, A. K., Saini, H., & Panda, T. C. “Mathematical

model for cyber attack in computer network.” International Journal of Business Data

Communications and Networking, 13(1), 58–65. doi:10.4018/IJBDCN.2017010105,

(2017).

[19]. Subramani, M. & Walden, E. “The impact of e-commerce announcements on

the market value of firms”. Information Systems Research, 12(2), 135-154. (2001).

[20]. Saini Dinesh Kumar “Mathematical Model for the Effect of Malicious Object

on Computer Network Immune System”, Applied Mathematical Modelling.,

vol.35,pp.3777-3787(2011),

http://www.sciencedirect.com/science/journal/10075704/19/6




[21]. X. Yang and L.-X. Yang, “Towards the epidemiological modeling of

computer viruses,” Discrete. Dynamics in Nature and Society., vol.259671,pp.1- 11,

(2012).

[22]. Yang,L. X.Yang,X.Zhuq, Wen L, “A computer virus model with graded cure

rate,” Non linear Analysis, real world Application .,vol.14, issue.1,pp.414-422. (2013)

[23]. Yuan Hua, Chen Guoqing, “Network virus-epidemic model with the point-to-

group information propagation.”, Applied Mathematics and Computation, vol.206 ,pp.

357–367, (2008).

[24]. Y. Michel, H. Smith, and L. Wang, “Global dynamics of an seir epidemic

model with vertical transmission,” SIAM Journal on Applied Mathematics, vol. 62, no. 1,

pp. 58–69, (2001)..

[25]. Yerra Shankar Rao, Prasant Kumar Nayak ,Hemraj Saini, Tarini Charan

Panda, “Behavioral Modeling of Malicious Objects in a Highly Infected Network Under

Quarantine Defence”, International Journal of Information Security and Privacy Volume

13 • Issue 1 • January-March 2019, pp- 17-19 (2019)

[26]. Z. Zhang, and H. Yang, “Dynamics of a delayed model for the transmission of

malicious objects in computer network," The Scientific World Journal, , Article ID

194104, 14. (2014 ).

BIBLIOGRAPHY OF AUTHORS

Yerra Shankar Rao is presently working as an Assistant Professor in Department of

Mathematics, Gandhi Institute of Excellent Technocrats (GIET) Bhubaneswar, Odisha. He

has about 12 years of academic including 7 years of research experience. He was completed

his Ph.D. degree at Siksha O Anusandhan University, Bhubaneswar Odisha and received his

master’s degree in Mathematics from Berhampur University, Odisha. He has published more

than 10 research papers in Journals of repute and conference proceedings in the area of cyber

security, mathematical modelling and nonlinear analysis.

PROFESSOR (DR) YERRA SHANKAR RAO

ASSISTANT PROFESSOR, DEPARTMENT OF MATHEMATICS GANDHI INSTITUTE

OF EXCELLENT TECHNOCRATS, GHANGAPATANA BHUBANESWAR, ODISHA,

INDIA.PIN 752054, PHONE NO 09337231875, E-MAIL [email protected]

Aswin Kumar Rauta was born in village Khallingi of district Ganjam; Odisha, India in

1981.He obtained his M.Sc. and M.Phil. degree in Mathematics from Berhampur University,

Berhampur, Odisha, India. He has qualified NET in 2009 conducted by CSIR-UGC,

government of India. He joined as a lecturer in Mathematics in the Department of

Mathematics, S.K.C.G.College, Paralakhemundi, and Odisha, India in 2011 and is continuing

his research work since 2009 and work till now.

DR ASWIN KUMAR RAUTA LECTURER, DEPARTMENT OF MATHEMATICS, S.K.C.G. COLLEGE,

PARALAKHEMUNDI, GAJAPATI, ODISHA, INDIA, PIN 761200, PHONE NUMBER

7008669552E MAIL [email protected]..

mailto:[email protected]



Tarini Charan Panda is a Fellow of Royal Astronomical Society and is presently a Visiting

Professor at Ravenshaw University, Cuttack, India. He has completed 14 Sponsored National

& International Funding Projects and acted as reviewer of 07 International Journals of repute

and supervised 25 candidates leading to Ph.D & D.Sc degrees. He was the Professor & Head

of Dept. of Mathematics & Computer Science at Mizoram Central University & Berhampur

University and also held the position of President, Orissa Mathematical Society, India.

TARINI CHARAN PANDA PROFESSOR DEPARTMENT OF MATHEMATICS RAVENSHAW UNIVERSITY,

CUTTACK, INDIA. PHONE NUMBER 9437261364E MAIL [email protected],

Subash Chandra Mishra is presently working as an Assistant Professor in Department of

EE & EEE, Gandhi Institute of Excellent Technocrats GIET Bhubaneswar, Odisha. He has

about 10 years of academic and 3 years of research experience.

SUBASH CHANDRA MISHRA ASSISTANT PROFESSOR, EE & ELECTRICAL & ELECTRONICS ENGINEERING

GANDHI INSTITUTE OF EXCELLENT TECHNOCRATS, GHANGAPATANA

BHUBANESWAR, ODISHA, INDIA.PIN 752054, PHONE NO 7008044456, E-MAIL

[email protected]

mailto:[email protected]

a dynamic e -epidemic model for the attack against … · 2019-02-24 · a dynamic e -epidemic...

Documents