The emergence of malicious codes that attack Wireless Sensor Networks (WSN) made it necessary to direct research attention to security. These attacks arising from worms pose devastating threats to networks which can lead to substantial losses or damages. However, recent models developed for the purpose of understanding worm transmission patterns and ensuring its containment did not account for the effect of uniform random deployment of sensor nodes on the Exposed and the Vaccinated compartments. Therefore, in this paper we present a modified Susceptible-Exposed-Infectious-Recovered-Susceptible with Vaccination (SEIRS-V) model for worm propagation dynamics in sensor networks. Our model applies the expression for uniform distribution deployment of sensor nodes so as to study the effect of distribution density and transmission range on the characterized compartments. Furthermore, we presented solutions for the equilibrium points, the reproduction number and proof of stability. Finally, we employed numerical methods to solve and simulate with real values the developed system of differential equations.
physical systems Sensor networks. • Security and privacy
differential equations. •
modeling and simulation
In recent times, Wireless Sensor Networks (WSNs) has enjoyed
considerable use in civilian applications for precision farming
[17] and the provision of smart and quality healthcare. In the
military, WSNs are used to monitor rebel activities and to detect
enemy movements etc. It consists of large number of
communicating devices which are randomly deployed in
unreachable territories without an engineered or predetermined
position for the nodes [
The sensor nodes are distributed in a sensor field where they are wirelessly connected to the sink. Although they have minimal battery capacity they are able to monitor and collect data about a given area. The data and information can be territorial parameters like pressure, position/condition of objects/humans, humidity, temperature etc. The collected data and information are sent back to the local sink through transmission between neighboring nodes. This transmission is basically done in a “multihop infrastructureless” manner. Subsequently, analyses are performed on the collated data accessed by a remote user through the internet for suitable decision making.
The constrained nature of sensor resources that gives rise to frail protective potential makes them suitable prey for self-replicating malevolent codes (such as worms) that spread without human involvement. In addition, these worms often tamper with the confidentiality, integrity and availability measures of neighboring sensor nodes due to its distributed nature. With the proliferation of the use of network technologies, 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 behavioral tendencies of worm variants have been developed. Predictions of worm behavior are largely dependent on the presuppositions of the model characterization and analysis.
It is unarguably true that propagation of malicious agents in
cyberspace is similar to the spread of epidemic in the biological
world. Therefore, modeling and analysis enhance optimized
containment by providing better understanding of the factors
that aid faster propagation of malicious codes in networks. The
Susceptible-Infectious-Recovered (SIR) model by [
In this paper we modify the Susceptible–Exposed–Infectious–
Recovered–Susceptible with a Vaccination compartment
(SEIRS-V) epidemic model of [
In this study, we basically perform modeling and simulation.
Specifically, we employ a widely applied method for
investigating network epidemic [
During model formulation, the analyst presents the equations that represent a real world phenomenon (in this case worm propagation in WSNs). Some authors also present a conceptual (or schematic) diagram at this point. Next, the analyst will derive the equilibrium points by equating the model (or system of differential equation) to zero. The Reproduction number is then derived to establish a threshold for disease/infection extinction in the network. Furthermore, stability analyses (using several renowned methods in literature) and simulation experiments are performed. The simulations experiments are more like sensitivity analyses. They are done by first solving the proposed model using a numerical method and applying real values for the simulation. Depending on the modeler’s intention the method can take different turns for analyses. But stages such as model formulation and simulation experiments are significantly part of this methodology.
We characterize worm attack in wireless sensor network using the Susceptible–Exposed–Infectious–Recovered–Susceptible with a Vaccination compartment (SEIRS-V). Our assumptions include addition of nodes in the network and removal (i.e. death) of nodes as a result of worm attack or due to hardware/software failure. All sensor nodes are susceptible to potential of worm attack and with time (probably) get infected (i.e. forming nodes in the Infectious compartment). Some sensor nodes before becoming infectious exists in the Exposed stage where the worm is latent and the nodes cannot transmit the infection. A symptom of this stage is slower data transmission for affected nodes. However, due to the existence of several worm variants in cyberspace the sensor nodes never acquire a permanent immunity i.e. they become susceptible to worm infection with time.
The total population N (t) represents the nodes in the Wireless Sensor Network which is subdivided into Susceptible, Exposed (latent), Infectious (contagious), Recovered (temporarily immune), Vaccinated (immunized) denoted by S( t), E(t), I(t), R(t) and V(t). This implies that S (t) + E (t) + I (t) + R (t) + V (t) = N (t).
Specifically, the sensor nodes are uniformly and randomly
deployed with a distribution density of σ and a transmission
range of , this implies that the effective contact with an
infected node for transfer of infection is in the order of .
Other parameters include which is the inclusion rate of nodes
into the sensor network population, is the Infectivity contact
rate, is the mortality or the death rate of nodes due to hardware
or software failure, is the crashing rate due to attack of
malicious objects (in this case worm), is the rate at which
exposed nodes become infectious, is the recovery rate, is the
rate at which recovered nodes become susceptible to infection,
is the rate of vaccination for susceptible sensor nodes and is
the rate of transmission from the Vaccinated compartment to the
Susceptible compartment.
(
(
(
(
(
(
(
)
)
)
E
V
S
[
We equate the modified system of differential equations (1) to zero i.e. solutions which are the Worm-free equilibrium and the Endemic equilibrium points. The Worm-free equilibrium describes the absence of worms while the Endemic equilibrium describes the presence of worms in the
Wireless Sensor Network using formulated mathematical model.
The solutions of equilibrium points are Worm-free equilibrium
;
to
obtain
two
) i.e.
and Endemic equilibrium
= ( ,
(3)
A cursory look at the symbolic solutions of the endemic
equilibrium in [
Reproduction number commonly denoted as
threshold quantity defined as “the expected number of secondary
cases produced in a completely susceptible population, by a
typical infective individual” [
) at the endemic equilibrium [
We solved the system of differential equation using a numerical
method i.e. Runge-Kutta Fehlberg method of order 4 and 5.
Subsequently we performed simulation experiments using this
following initial values for the Wireless Sensor network: S=100;
E=3; I=1; R=0; V=0. Other values used for the simulation
include =0.33; =0.003; =0.07; =0.25 =0.4;
=0.3; =0.3 =0.06; adapted from the time history of [
Infectious Sensor Nodes against Time
From our simulation experiments we noticed that at transmission
range of 1 and density of 0.3 depicted in Figure 8 the dynamical
responses were close to Figure 3 of [
In this study, we discovered that the increase in density and
transmission range increased the Exposed and Infectious
compartments and decreased and Vaccinated compartments.
This study is consistent with [
Due to the effect of uniform random distribution on Vaccinated sensor nodes, it is only wise that the rate at which nodes are immunized is increased as density and range increase in order to reduce high susceptibility to infection. In future we would focus our analyses on how to achieve increased vaccination rate for vulnerable sensor nodes (to ensure reduced susceptibility to infection) in the light of increased density and increased communication/transmission range.
Furthermore, we would also perform analysis and simulation
experiments to observe the effect of uniform random distribution
on Quarantine models. In addition, pursuit of other mathematical
objectives such as extending analyses to the global stability at
the endemic equilibrium using the geometrical approach of [