Investigating the Effect of Uniform Random Distribution of
Nodes in Wireless Sensor Networks using an Epidemic
Worm Model
ChukwuNonso H. Nwokoye Virginia E. Ejiofor Rita Orji
Nnamdi Azikiwe University, Nnamdi Azikiwe University of Waterloo,
Awka, Nigeria. University, Awka, Nigeria. Canada.
explode2kg@yahoo.com virguche2004@yahoo.com purity.rita@gmail.com
Ikechukwu Umeh Njideka N. Mbeledogu
Nnamdi Azikiwe University, Nnamdi Azikiwe University,
Awka, Nigeria. Awka, Nigeria.
ikumeh1@gmail.com njidembeledogu@yahoo.com
ABSTRACT Random Deployment; Differential Equations; Stability
The emergence of malicious codes that attack Wireless Sensor
Networks (WSN) made it necessary to direct research attention 1. INTRODUCTION
to security. These attacks arising from worms pose devastating In recent times, Wireless Sensor Networks (WSNs) has enjoyed
threats to networks which can lead to substantial losses or considerable use in civilian applications for precision farming
damages. However, recent models developed for the purpose of [17] and the provision of smart and quality healthcare. In the
understanding worm transmission patterns and ensuring its military, WSNs are used to monitor rebel activities and to detect
containment did not account for the effect of uniform random enemy movements etc. It consists of large number of
deployment of sensor nodes on the Exposed and the Vaccinated communicating devices which are randomly deployed in
compartments. Therefore, in this paper we present a modified unreachable territories without an engineered or predetermined
Susceptible–Exposed–Infectious–Recovered–Susceptible with position for the nodes [9], [14]. These territories are basically
Vaccination (SEIRS-V) model for worm propagation dynamics unfriendly and unguarded.
in sensor networks. Our model applies the expression for
uniform distribution deployment of sensor nodes so as to study The sensor nodes are distributed in a sensor field where they are
the effect of distribution density and transmission range on the wirelessly connected to the sink. Although they have minimal
characterized compartments. Furthermore, we presented battery capacity they are able to monitor and collect data about a
solutions for the equilibrium points, the reproduction number given area. The data and information can be territorial
and proof of stability. Finally, we employed numerical methods parameters like pressure, position/condition of objects/humans,
to solve and simulate with real values the developed system of humidity, temperature etc. The collected data and information
differential equations. are sent back to the local sink through transmission between
neighboring nodes. This transmission is basically done in a
CCS Concepts “multihop infrastructureless” manner. Subsequently, analyses
Computer systems organizations Embedded and cyber- are performed on the collated data accessed by a remote user
physical systems Sensor networks. • Security and privacy through the internet for suitable decision making.
Intrusion/anomaly detection and malware mitigation
The constrained nature of sensor resources that gives rise to frail
Malware and its mitigation. • Mathematics of computing protective potential makes them suitable prey for self-replicating
Mathematical analysis Differential equations Ordinary malevolent codes (such as worms) that spread without human
differential equations. • Computing Methodologies involvement. In addition, these worms often tamper with the
modeling and simulation Model development and analysis, confidentiality, integrity and availability measures of
neighboring sensor nodes due to its distributed nature.
Simulation support systems.
Keywords With the proliferation of the use of network technologies,
Wireless Sensor Network; Worm; Epidemic Model; Uniform increasing efforts have been focused on developing appropriate
cyber protection structure in order to secure both stationary and
moving information. Wireless Sensor Network research believes
that achieving this objective is overly expedient. As a result,
several continuous (and discrete) equation-based models that
characterize, investigate and aid better comprehension of the
CoRI’16, Sept 7–9, 2016, Ibadan, Nigeria. behavioral tendencies of worm variants have been developed.
Predictions of worm behavior are largely dependent on the
presuppositions of the model characterization and analysis.
58
diagram); b. finding the equilibrium states (for the worm-free
2. RELATED WORKS and the endemic states); and c. deriving the Reproduction
It is unarguably true that propagation of malicious agents in number. Subsequent stages include; d. proof of stability and e.
cyberspace is similar to the spread of epidemic in the biological simulations experiments using software such as MatLab, Maple
world. Therefore, modeling and analysis enhance optimized etc.
containment by providing better understanding of the factors
that aid faster propagation of malicious codes in networks. The During model formulation, the analyst presents the equations
Susceptible-Infectious-Recovered (SIR) model by [5], [4] and that represent a real world phenomenon (in this case worm
[6] initiated the journey into developing mathematical models propagation in WSNs). Some authors also present a conceptual
for worm/virus propagation. Building on the SIR model, other (or schematic) diagram at this point. Next, the analyst will
extensions such as Susceptible-Exposed-Infectious-Recovered derive the equilibrium points by equating the model (or system
(SIER) [13], [12], Susceptible-Exposed-Infectious-Vaccinated of differential equation) to zero. The Reproduction number is
(SIEV) [11], Susceptible-Exposed-Infectious–Recovered- then derived to establish a threshold for disease/infection
Vaccinated (SEIR-V) [9] etc., were developed to address several extinction in the network. Furthermore, stability analyses (using
concerns arising in a real world network environment. several renowned methods in literature) and simulation
experiments are performed. The simulations experiments are
In this paper we modify the Susceptible–Exposed–Infectious– more like sensitivity analyses. They are done by first solving the
Recovered–Susceptible with a Vaccination compartment proposed model using a numerical method and applying real
(SEIRS-V) epidemic model of [9] by applying [15] expression values for the simulation. Depending on the modeler’s intention
for uniform random distribution of sensor nodes with the aim of the method can take different turns for analyses. But stages such
investigating the effect of both distribution density (σ) and as model formulation and simulation experiments are
transmission range ( ) on the characterized compartments. We significantly part of this methodology.
discovered that though [15] represented distribution density and
transmission range their SIR model did not include analyses for 3.1 The Modified SEIRS-V Model
the Exposed and the Vaccinated compartments. On the other We characterize worm attack in wireless sensor network using
hand, though [9] included these two compartments their the Susceptible–Exposed–Infectious–Recovered–Susceptible
analyses did not involve distribution density and transmission with a Vaccination compartment (SEIRS-V). Our assumptions
range. Wang and Yang [16] also applied Tang and Mark’s include addition of nodes in the network and removal (i.e. death)
formulation but their model used the Susceptible-Infectious (SI) of nodes as a result of worm attack or due to hardware/software
compartments for their analyses; and didn’t discuss the Exposed failure. All sensor nodes are susceptible to potential of worm
and Vaccinated compartments. Considering the argument by [3] attack and with time (probably) get infected (i.e. forming nodes
that standard incidence “presents a more reasonable and in the Infectious compartment). Some sensor nodes before
practical scenario of contact than the simple mass action becoming infectious exists in the Exposed stage where the worm
incidence”, we would modify the above model using the former. is latent and the nodes cannot transmit the infection. A symptom
of this stage is slower data transmission for affected nodes.
In epidemiology, the Exposed class contains nodes that are However, due to the existence of several worm variants in
infected but not infectious. These nodes which are in a latent cyberspace the sensor nodes never acquire a permanent
phase possess different infectivity rate when compared to the immunity i.e. they become susceptible to worm infection with
Infectious nodes. Common symptom for nodes in this latent time.
stage is slow data transmission speed [9]. On the hand, The total population N (t) represents the nodes in the Wireless
vaccination (or immunization) is a known countermeasure in Sensor Network which is subdivided into Susceptible, Exposed
epidemiology. It is aimed at fortifying a fraction of the total (latent), Infectious (contagious), Recovered (temporarily
sensor node population prior to the outset of an epidemic. This immune), Vaccinated (immunized) denoted by S( t), E(t), I(t),
study is necessary since there is a strong likelihood that sensor R(t) and V(t). This implies that S (t) + E (t) + I (t) + R (t) + V
nodes can exist in the latent stage and that network managers (t) = N (t).
can employ vaccination strategies to ensure security. In addition, Specifically, the sensor nodes are uniformly and randomly
the analyses of distribution density and transmission range using deployed with a distribution density of σ and a transmission
worm models can positively impact sensor deployment activities range of , this implies that the effective contact with an
for institutions. infected node for transfer of infection is in the order of .
Other parameters include which is the inclusion rate of nodes
3. METHODOLOGY into the sensor network population, is the Infectivity contact
In this study, we basically perform modeling and simulation. rate, is the mortality or the death rate of nodes due to hardware
Specifically, we employ a widely applied method for or software failure, is the crashing rate due to attack of
investigating network epidemic [10], [9] and [11] etc. This malicious objects (in this case worm), is the rate at which
method is called modeling and analysis of dynamical systems. exposed nodes become infectious, is the recovery rate, is the
Here, the WSN is treated like a dynamical system and rate at which recovered nodes become susceptible to infection,
equilibrium positions are studied. The methodology starts with; is the rate of vaccination for susceptible sensor nodes and is
a. model formulation (and optionally drawing the schematic the rate of transmission from the Vaccinated compartment to the
Susceptible compartment.
59
𝝆𝑺 and Endemic equilibrium =( , , , , ) i.e.
( )( )
𝝃V
𝝇𝝅𝒓𝟐𝟎 ( )( )(
( )( )( )
)
𝜷𝑺𝑰 𝜽𝑬 𝝂𝑰 ( )
𝝀 𝑵 R
=
S E I V
( ) ( ) ( )( )
( )( )( )
( )( )
( )
(3)
( ) ( ) ( )( )
𝝉 𝝉+𝝎 𝝉 ( )( )( )
𝝉 𝝉 (
( )
)
𝝋𝑹
( ) ( ) ( )( )
Figure 1. Schematic diagram for the flow of worms in sensor
network ( )( )
( )
The schematic diagram for the dynamical transmission of worms A cursory look at the symbolic solutions of the endemic
in a Wireless Sensor Network given our assumption is depicted equilibrium in [9] shows the differences. It is observed here that
in Fig. 1. The system of differential equation (1) is adapted from the expression for uniform distribution deployment formed part
[9] but modified to capture distribution density and transmission of the solutions; this is absent in the solutions of [9].
range.
3.3 The Basic Reproduction Number
The modified SEIRS-V model is represented using the following The Reproduction number commonly denoted as is a
system of differential equations; threshold quantity defined as “the expected number of secondary
cases produced in a completely susceptible population, by a
typical infective individual” [1] or the spectral radius which can
also be referred to as the “dominant eigenvalue of the matrix G
( ) = FV-1” [2], [9]. Some authors also refer to it as the inverse of
the susceptible ( ) at the endemic equilibrium [10]. The
Reproduction number ( ) is .
( ) (1) ( )( )
(4)
( ) In this study, the Reproduction number is different from what
was obtained in [15] and [9] . Our Reproduction number can be
used to determine the possible contained/endemic dynamics of
( ) worm propagation in WSNs considering distribution density and
transmission range.
3.2 Solutions of Equilibrium Points
We equate the modified system of differential equations (1) to 3.4 Stability of the Worm-free
zero i.e. Equilibrium point
; to obtain two We show the proof of local asymptotic stability at the Worm-
solutions which are the Worm-free equilibrium and the Endemic free Equilibrium using the jacobian method. This is done by
equilibrium points. The Worm-free equilibrium describes the showing that “the eigen-values of the jacobian matrix all have
absence of worms while the Endemic equilibrium describes the negative real parts” [9] or that the “characteristic equation of the
presence of worms in the Wireless Sensor Network using jacobian matrix” derived from the system of equations has
formulated mathematical model. negative roots [8].
The solutions of equilibrium points are Worm-free equilibrium
Theorem: The worm-free equilibrium is locally asymptotically
( ) i.e.
stable if < 1 and unstable if > 1.
( )
( ) Proof: Using - the characteristic equation of system (1) at
worm-free equilibrium is
(2)
( )
60
( )
–( )
( )
| ( ) |
–( )
( )
det = 0 (5)
–( )
| –( ) |
–( )
which equates to; ( )( )(
)(( )( ) ) . (6)
The roots of the characteristic equation all have negative real
parts i.e. – , , (
√( ) ) (
√( ) );
Therefore the worm free equilibrium is locally asymptotically
stable.
4. NUMERICAL RESULTS AND
DISCUSSION
We solved the system of differential equation using a numerical Figure 3. Dynamical behaviour of the system for different
method i.e. Runge-Kutta Fehlberg method of order 4 and 5. compartments of the equivalent model
Subsequently we performed simulation experiments using this Source: [9]
following initial values for the Wireless Sensor network: S=100;
E=3; I=1; R=0; V=0. Other values used for the simulation Figure 4 shows the dynamical behavior of the Exposed
include =0.33; =0.003; =0.07; =0.25 =0.4; compartment with respect to changes in the distribution density
=0.3; =0.3 =0.06; adapted from the time history of [9]. We and transmission range. At 0.5 and 2.0 for density and range
compared our results with the results of similar models in respectively depicted with red, there was still an increase in the
literature for the purposes of verification and validation. number of Exposed sensor nodes even though the range was
kept constant (like in first response depicted with green). The
Figure 2 shows the time history of the compartments used for effect of both density and range was again evident in the third
the analyses. Note that the transient response of Figure 2 response (0.7 and 2.5 for density and range respectively)
simulated with values for distribution density and transmission depicted with blue when compared with the first response where
range differs from the time history of [9]. For the sake of clarity 0.3 and 2.0 for density and range respectively.
and ambiguity reduction we prepared the simulation experiment
of Figure 2 using the same colors used in Figure 3. It is evident Exposed Nodes against Time
90
that our model showed increase in both the Exposed and density = 0.3, range =2
80
Infected compartments and a reduction in both the Susceptible density = 0.5, range =2
density = 0.7, range =2.5
70
and Vaccinated compartments.
60
Exposed Nodes
50
40
Popluation of Compartments against Time
100 30
Susceptible
Population of Compartments S, E, I, R,V
20
Exposed
80 Infected 10
Recovered
E Vaccinated 0
0 10 20 30 40 50 60 70 80 90 100
60 Time in Minutes
I R
40 V
S Figure 4. Dynamical behaviour of Exposed Compartment
20 versus Time w.r.t. to σ and
Figure 5 shows the dynamical behavior of the Infectious
0
0 10 20 30 40 50 60 70 80 90 100 compartment with respect to changes in the distribution density
Time in Minutes
and transmission range. At 0.5 and 2.0 for density and range
respectively depicted with red, there was still an increase in the
Figure 2. Dynamical behaviour of the system for different number of Infectious sensor nodes even though the range was
compartments of the modified model kept constant (like in the first response depicted with green).
61
The effect of both density and range was again evident in the Graph of Susceptible Nodes Plotted against Vaccinated Nodes
100
third response (i.e. 0.7 and 2.5 for density and range density=0.3, range=2.0
0
90
respectively) depicted with blue when compared with the first density=0.5, range=2.0
0
response where 0.3 and 2.0 for density and range respectively. 80
density=0.5, range=2.5
0
Susceptible Sensor Nodes
70
Infectious Sensor Nodes against Time 60
30
50
density = 0.3, range = 2.0
density = 0.5, range = 2.0 40
25
density = 0.7, range = 2.5
30
Infectious Sensor Nodes
20
20
10
15
0
0 5 10 15 20 25
10 Vaccinated Sensor Nodes
5
Figure 7. Dynamical behaviour of Susceptible Compartment
0 versus Vaccinated Compartment w.r.t. to σ and
0 50 100 150
Time in Minutes
Figure 5. Dynamical behaviour of Infectious Compartment
versus Time w.r.t. to σ and
Figure 6 shows the dynamical behavior of the Infectious
compartment plotted against the Exposed compartment with
respect to changes in the distribution density and transmission
range. This figure showed how both the Exposed and Infectious
compartments increased with increase in density and
transmission range. The increase was observed when density
was kept constant (at 0.5) and when range was kept constant (at Figure 7a. Dynamical behaviour of Susceptible
0.2). The difference between the Exposed and Infectious sensor Compartment versus Vaccinated Compartment of the
nodes is in line with the real world because even though both
equivalent model as adapted from [9]
have contacted the infection, only the Infectious sensor nodes
can transmit the infection to susceptible nodes.
From our simulation experiments we noticed that at transmission
Graph of Infectious Nodes Plotted against Exposed Nodes
range of 1 and density of 0.3 depicted in Figure 8 the dynamical
30
density=0.3,range=2.0
0
responses were close to Figure 3 of [9] for the Exposed and
density=0.5,range=2.0
0 Vaccinated compartments. Increasing the range to 2 (as evident
25 density=0.5,range=2.5
0 in Figure 2) visibly changed the behavior of the compartments
(most especially Exposed and the Vaccinated). The Exposed
Infectious Sensor Nodes
20
nodes moved from above 30 nodes to above sixty nodes while
15 Vaccinated nodes reduced to slightly above 20 nodes from
above 40 nodes.
10
Poplation CompartmentsS,E, I, R, V against Time
100
5 Susceptible
90 Exposed
Infectious
Population Compartments S, E, I, R, V
80
0 Recovered
0 10 20 30 40 50 60 70 80 Vaccinated
70
Exposed Sensor Nodes
60
Figure 6. Dynamical behaviour of Infectious Compartment 50 V
E
versus Exposed Compartment w.r.t. to σ and
40
R
30 I S
Figure 7 shows the dynamical behavior of the Susceptible 20
compartment plotted against the Vaccinated compartment with 10
respect to changes in the distribution density and transmission 0
0 10 20 30 40 50 60 70 80 90 100
range. This figure showed a decrease in both the Susceptible and Time in Minutes
the Vaccinated compartments. The decrease was observed when
density was kept constant (at 0.5) and when range was kept Figure 8. Dynamical behaviour of Infectious Compartment
constant (at 0.2); this is clearly visible when compared to Figure versus Exposed Compartment w.r.t. to σ=0.3 and =1
7a.
62
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