Journal of Theoretical Biology 283 (2011) 136–144

Contents lists available at ScienceDirect

Journal of Theoretical Biology

0022-51

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/yjtbi

An individual-based approach to SIR epidemics in contact networks

Mina Youssef �, Caterina Scoglio

K-State Epicenter: Center for Complex Network Approach to EpiModeling, Department of Electrical and Computer Engineering, Kansas State University,

2061 Rathbone Hall, Manhattan, KS 66506, USA

a r t i c l e i n f o

Article history:

Received 27 July 2010

Received in revised form

29 April 2011

Accepted 23 May 2011Available online 6 June 2011

Keywords:

Spread of epidemics

Contact networks

Individual-based approach

93/$ - see front matter Published by Elsevier

016/j.jtbi.2011.05.029

esponding author. Tel.: þ1 785 532 4690.

ail addresses: mkamel@ksu.edu (M. Youssef), c

a b s t r a c t

Many approaches have recently been proposed to model the spread of epidemics on networks. For

instance, the Susceptible/Infected/Recovered (SIR) compartmental model has successfully been applied

to different types of diseases that spread out among humans and animals. When this model is applied

on a contact network, the centrality characteristics of the network plays an important role in the

spreading process. However, current approaches only consider an aggregate representation of

the network structure, which can result in inaccurate analysis. In this paper, we propose a new

individual-based SIR approach, which considers the whole description of the network structure. The

individual-based approach is built on a continuous time Markov chain, and it is capable of evaluating

the state probability for every individual in the network. Through mathematical analysis, we rigorously

confirm the existence of an epidemic threshold below which an epidemic does not propagate in the

network. We also show that the epidemic threshold is inversely proportional to the maximum

eigenvalue of the network. Additionally, we study the role of the whole spectrum of the network,

and determine the relationship between the maximum number of infected individuals and the set of

eigenvalues and eigenvectors. To validate our approach, we analytically study the deviation with

respect to the continuous time Markov chain model, and we show that the new approach is accurate for

a large range of infection strength. Furthermore, we compare the new approach with the well-known

heterogeneous mean field approach in the literature. Ultimately, we support our theoretical results

through extensive numerical evaluations and Monte Carlo simulations.

Published by Elsevier Ltd.

1. Introduction

Epidemics are usually studied via compartmental models, whereindividuals can be in different states, such as susceptible, infected, orrecovered. For infections like common colds among humans andmalware infections in computer networks, the susceptible/infected/susceptible (SIS) epidemic model (Gomez et al., 2010; Wang et al.,2003; Van Mieghem et al., 2009) is considered appropriate, sinceindividuals are again susceptible after contracting the infection andrecovering. However, the susceptible/infected/recovered (SIR) epi-demic model fits better influenza-like and other human contagiousdiseases, where immunity can be obtained after contracting thedisease. These compartmental models can be used to estimate theinfection parameters given the number of confirmed infected casesduring pandemics and epidemics. They can also be used to predictthe peak of the infected cases and to define and test efficientmitigation strategies (Towers and Feng, 2009).

To take into account the heterogeneous contact process ina social context, contact networks (Watts and Strogatz, 1998;Amaral et al., 2000; Liljeros et al., 2001; Girvan and Newman,

Ltd.

aterina@ksu.edu (C. Scoglio).

2002; Strogatz, 2001; Jeong et al., 2000, 2001; Fell and Wagner,2000; Williams and Martinez, 2000; Montoya and Sole, 2002;Gehring et al., 2010) have been considered in modeling epidemics.A contact network is the substrate where the spread of epidemicstakes place. It is composed of nodes representing individuals andlinks representing the contact between any pair of individuals. Thesimplest representation of a contact network is the binary networkin which the contact level takes two values, either 1 when a pair ofindividuals are in contact, or 0 otherwise. Most of the researchworks consider static contact networks where nodes and links donot change with time. The rational behind this assumption is thatthe dynamic of the epidemic is faster than the change in theexistence of the nodes due to the longer average lifetime ofindividuals as compared to the lifetime of the epidemic. Therefore,we consider that the contact network is static, and the population(number of individuals in the network) is fixed. In such modeling,the role of the network structure is critical in the dynamics of theepidemic and is therefore the focus of our work.

In the SIR model, each individual can be in one of the threestates, namely, S susceptible, I infected, and R recovered (cured). Toclarify, an infected individual infects its susceptible neighbors withrate b. Also, an infected individual can cure itself with a curerate m. The curing process represents either the death (removal)or the complete recovery of the individual after the infection.

M. Youssef, C. Scoglio / Journal of Theoretical Biology 283 (2011) 136–144 137

Additionally, the ratio between b and m is called the effectiveinfection rate. An epidemic threshold t is a specific value of theeffective infection rate above which an epidemic outbreak takesplace. It is a function of the network characteristics. In epidemiol-ogy, the epidemic threshold is a function of a quantity called basicreproductive number R0 (Anderson and May, 1992; Kermack andMcKendrick, 1927; Macdonald, 1952; May and Anderson, 1979;Anderson, 1981). The basic reproductive number is defined as theaverage number of secondary infected individuals when a singleindividual is infected initially. Mathematically, the reproductivenumber is /kSb=m where /kS is the average connectivity in thenetwork. Therefore, if R0 is greater than 1, the epidemic spreads onthe network and vice versa. Many networks with individualscharacterized by a wide range of contacts (i.e. some individualshave few contacts and other individuals have many contacts) canhave the same average level of contact (Mahadevan et al., 2005);(Mahadevan et al., ; Doyle et al., 2005). The information on theaverage contact does not provide any indication of the contact leveldistribution. Being R0 only dependent on the average level ofcontact, it will be the same for all networks with the same averagelevel of contact, independent of their contact level distribution.Recent works showed that the epidemic threshold is a function ofnon-trivial network characteristics. In this paper, we confirm thatthe epidemic threshold is a function of the maximum eigenvalue ofthe matrix representing accurately the contact level among theindividuals (Wang et al., 2003; Van Mieghem et al., 2009).

Different SIR models are applied to some classes of contactnetworks (Scoglio et al., 2010; Boguna, 2003; Newman, 2002;Moreno et al., 2002; Manzano et al., 2010; Barthelemy et al.,2005; Volz and Meyers, 2009; Frank et al., 2009; Kenah andRobins, 2007; May and Lloyd, 2001; Yang et al., 2007; Black et al.,2009; Tuckwell et al., 1998) depending on the network character-istics. For example, an early SIR approach uses the homogeneousmean field approximation in which all individuals have the sameprobability of being infected and infectious. On the other hand,the SIR heterogeneous mean field approach has also been appliedto structured networks considering the local connectivity of thenetwork’s individuals. For example, scale-free networks, whichare networks owning power-law node degree distributionPðkÞ � k�n, show a high level of vulnerability to the spreading ofepidemics due to highly heterogeneous node degree distributionwhen the minimum node degree is greater than two (Boguna,2003). In addition, the spread of epidemics has been studied oncorrelated networks and uncorrelated networks separately. How-ever, the SIR epidemic approaches studied in the literature do notconsider the whole structure of the network but only localconnectivity. For instance, the heterogeneous mean fieldapproach only considers the node degree distribution, which isan aggregate representation of the network. However, it has beenshown that networks with distinct topological properties can becharacterized by the same node degree distribution (Li et al.,2004). Consequently, these approaches cannot distinguish amongindividuals having the same node degrees because they neglectthe centrality properties of the individuals. Therefore, the SIRepidemic approaches presented in the literature are not accuratefor studying the effect of the network in the spreading process,and hence there is a need for a generalized SIR epidemic approachthat considers not only the network properties (i.e. average nodedegree and node degree distribution), but also the whole networkstructure, and that represents every individual independently.

In this paper, we propose an individual-based SIR approach,which is inspired by the continuous-time Markov chain modeland which represents the network in the most accurate way.The individual-based SIR approach is developed extending theN-intertwined SIS model proposed by Van Mieghem et al. (2009).We separately study the state of each individual during the

infection process, revealing the role of the individual’s centralityproperties in spreading the infection across the network.Although the continuous time SIR Markov model, based on theMarkov chain stochastic process, describes the global change inthe state probabilities of the network, it is limited to smallnetworks due to the exponential divergence in the number ofpossible network states 3N with the growth of network size N.Instead, our approach aims to reduce the complexity of theproblem to O(N) and to offer insights into the epidemic spreadingmechanism. Through the new SIR approach, we study the spreadof epidemics on any type of network regardless of its topologicalstructure. We analytically derive the epidemic threshold for thenew approach, which is inversely proportional to the spectralradius lmax (the supremum eigenvalue within the eigenvaluespectrum) of the network. We perform Monte Carlo simulationsto validate the new SIR approach, and we compare it with the SIRheterogeneous mean field approach in the literature. We show thatthe individual-based approach outperforms the heterogeneousmean field approach when the effective infection rate is close to theepidemic threshold. Analytically, our study shows the role of thecentrality properties of the network in the spreading of epidemics.Moreover, we analyze the deviation between the individual-basedapproach and the continuous time Markov chain model, and we alsoshow that the new approach is accurate for a large range of infectionstrength. We summarize the contribution of the paper as follows:

�

Proposing an individual-based SIR approach. � Outlining the role of the eigenvalues and the eigenvector

centrality of the contact network.

� Validating the new approach and providing guidelines on its

accuracy.

The paper is organized as follows: In Section 2, we reviewbasic approaches that are applied to SIR model in the literature. InSection 3 we review the continuous time Markov chain model,and in Section 4, we present the individual-based approach indetails and we show simulative and numerical results. Theproperties of the individual-based approach are discussed inSection 5. Moreover, the theoretical deviation between theindividual-based approach and the continuous time Markov chainmodel is analyzed in Section 6. We summarize the theoretical andnumerical findings in Section 7. Finally, we conclude and discussfuture work in Section 8.

2. Epidemic models background

The science of the spread of epidemics is based on compart-mental models that assume individuals are classified into non-intersecting sets (Anderson and May, 1992; Murray, 1993). Thus,the classical susceptible/infected/removed SIR model charac-terizes diseases that lead to either immunization or death ofindividuals. The infected individuals are in the infected set, thehealthy ones are in the susceptible set, and the cured or removedones are in the removed set. Initially, a small number of infectedindividuals exist that try to infect their susceptible (healthy)neighbors. After receiving the infection, susceptible individualsbecome infected, and later they try to infect their susceptibleneighbors. In such cases, infected individuals are infectious.Subsequently, every infected individual is either cured due toimmunization or removed due to death.

2.1. SIR homogeneous mean field approach

In the SIR homogeneous mean field approach (Anderson andMay, 1992; Kermack and McKendrick, 1927), s(t), i(t), and r(t)

SS

RR

SIIS

SRRS II

IRRI

Fig. 1. The Markov chain state diagram for a simple network composed of two

individuals (N¼2) that are in contact. Number of states equals 32¼ 9. The state SS

does not have any transition with other states because all the individuals are

susceptible and there is no infection incidence presented in the network to cause

the spreading of epidemics.

M. Youssef, C. Scoglio / Journal of Theoretical Biology 283 (2011) 136–144138

represent, respectively, the fraction of susceptible, infected, andrecovered populations. As mentioned in the Introduction, weassume that the population is fixed, i.e. sðtÞþ iðtÞþrðtÞ ¼ 1. Inaddition, the approach approximates the representation of thenetwork and assumes that, on the average, every individual isconnected with /kS neighbors neglecting the heterogeneity ofthe node degrees. Depending on the fixed population assumptionand the network representation, the homogeneous mean fieldapproach describes the change in the susceptible, infected andrecovered population fractions over time. The infected fraction i(t)infects the fraction of susceptible neighbors /kSsðtÞ with infec-tion rate b, and simultaneously, a fraction of the infected popula-tion recovers miðtÞ. The rates of changes in s, i and r fractions aregoverned by the following continuous time differential equations:

dsðtÞ

dt¼�/kSbiðtÞsðtÞ, ð1Þ

diðtÞ

dt¼�miðtÞþ/kSbiðtÞsðtÞ, ð2Þ

drðtÞ

dt¼ miðtÞ: ð3Þ

These differential equations interpret the infection and cureprocesses. Initially, the spreading process starts with a smallinfected fraction ið0ÞC0, a susceptible fraction of almost onesð0ÞC1, and the removed fraction of zero r(0)¼0. Every infectedindividual infects on average /kS susceptible neighbors, eachwith an infection rate b, where /kS¼

PN�1k ¼ 1 kpðkÞ is the average

node degree (average number of contacts), and p(k) is theprobability of having an individual with degree k. Followingdifferential equation (2), an infected individual is removed at arate m. The removed fraction increases with time until it reaches acertain fraction level depending on the strength of the epidemic.A non-zero epidemic threshold exists and it is equal to mðR0�1Þwhere R0 is the reproductive number and equals /kSb=m. If R0 isgreater than 1, the epidemic prevails in the network. On the otherhand, if R0 is less than 1, the initially infected individuals totallyrecover without infecting other susceptible individuals (Barratet al., 2008). Since on average, every infected individual infects aconstant number of neighbors, the homogeneous approach doesnot account for heterogeneity in the node degrees of individualsin the network.

2.2. SIR heterogeneous mean field approach

Another approach in the literature is the heterogeneous meanfield (also called heterogeneous mixing) SIR approach (Morenoet al., 2002; Boguna, 2003; Yang et al., 2007), which was proposedto overcome the shortcomings of the homogeneous approach. Inthis approach, individuals are classified according to their nodedegrees. Thus, for a given node degree k, the states’ fractions sk(t),ik(t) and rk(t) evolve with time t, and their sum is constant, suchthat skðtÞþ ikðtÞþrkðtÞ ¼ 1. The rates of changes in the three statesfor a given node degree k are governed by the following set ofdifferential equations:

dskðtÞ

dt¼�kbskðtÞykðtÞ, ð4Þ

dikðtÞ

dt¼�mikðtÞþkbskðtÞykðtÞ, ð5Þ

drkðtÞ

dt¼ mikðtÞ: ð6Þ

This approach was applied to both uncorrelated and corre-lated networks, leading to further analysis of the epidemicthreshold. For uncorrelated networks, the epidemic threshold is

tucr ¼/kS=ð/k2S�/kSÞ, where /k2S is the second moment ofthe node degree distribution, and yðtÞ, representing the probabil-ity that a link is pointing to an infected individual, is found to beP

kðk�1ÞpðkÞikðtÞ=/kS. On the other hand, the epidemic thresholdfor correlated networks is tcr ¼ 1=Lm, where Lm is the maximumeigenvalue of the connectivity matrix C kk0 ¼ bðkðk0�1Þ=k0Þpðk0jkÞ,and ycr

k ðtÞ equalsP

k0 ik0 ðtÞððk0�1Þ=k0Þpðk0jkÞ.

Although this approach considers the heterogeneous connec-tivity in the networks, it does not reveal the state of eachindividual in the network. It only reflects the evolution of thefractions over time for a given node degree, while neglecting thestates of individuals within that node degree.

3. Continuous time Markov chain SIR epidemic model

In this section, we discuss the exact continuous time Markovchain model and its complexity. The continuous time Markovchain model describes the epidemic spread process accurately,based on the fact that each individual in the network is eithersusceptible, infected, or recovered. An infected individual infectsits susceptible neighbors, each with a rate b. Additionally, theinfected individual cures itself with cure rate m. Therefore, thereare 3N different states for any network with N individuals. Inaddition, there are ðNj Þ2

N�j different states with j infected indivi-duals. Fig. 1 shows an example of the state diagram when thespread of epidemics takes place on a network with two indivi-duals that are in contact.

To analytically describes the exact Markov chain model, let usdefine the network state X with 3N configurations, i.e.X1,X2, . . . ,X3N . Each network state describes the state of eachindividual X ¼ fx1,x2, . . . ,xNg where x¼ fs,i,rg. Additionally, theprobability that the network is in state X is given by WX ¼

pðX ¼ fx1,x2, . . . ,xNgÞ. To be more precise, the rate of change amongthe different states is described by the infinitesimal Q matrix,which has dimensions of 3N

�3N. Let qX,Y be an element in the Q

matrix representing the rate of change from network state X tonetwork state Y. The Q matrix is described as follows:

�

qX,Y ¼ m whenever network state X has an infected individualthat is cured in network state Y, while other individuals do notchange their states.P � qX,Y ¼ b N

l ¼ 1 am,ldil ,1 whenever a susceptible individual m innetwork state X is connected with infected neighbors lAN andit becomes infected in state Y, while other individuals do notchange their states. Note that am,l equals 1 when individuals m

M. Youssef, C. Scoglio / Journal of Theoretical Biology 283 (2011) 136–144 139

and l are connected, and dil ,1 is the Kronecker delta functionand equals 1 if individual l is infected ðil ¼ 1Þ.P

� qX,X ¼� Y aXqX,Y . � qX,Y ¼ 0 otherwise.

At any time t, the network is in any state X, with a givenprobability WX(t) such that

PX A fX1 ,...,X

3N gWXðtÞ ¼ 1. The rate of

change of every network state governs by the following differ-ential equation:

dWTðtÞ

dt¼WT ðtÞQ : ð7Þ

The solution of the differential equation is as follows:

WT ðtÞ ¼WT ð0ÞeQt , ð8Þ

where WT ðtÞ is the transpose of the state probability vector W(t)at time t, and WT ð0Þ is the transpose of the initial state probabilityvector.

4. Individual-based SIR approach

In this paper, we present an individual-based SIR approach tomodel the spread of epidemics in networks. A network iscomposed of nodes and links. A node represents an individualand the link represents the contact between a pair of individuals.The new approach overcomes the shortcomings of the homo-geneous and heterogeneous mean field approaches. In the indi-vidual-based approach, each individual can be either susceptibleS, infected I or recovered R with a given probability for each state.The new approach is inspired by the continuous-time Markovchain SIR model, which is discussed in Section 3, and it aims todecrease the complexity of the solution from exponential O(3N) topolynomial O(N). Therefore, instead of considering the combina-torial states of the individuals in the network, we study eachindividual specifically (Van Mieghem et al., 2009), by decompos-ing Q3N

�3N matrix to N infinitesimal matrices, each with threestates as follows:

qvðtÞ ¼

�bP

zav,zdizðtÞ,1 b

Pz

av,zdizðtÞ,1 0

0 �m m0 0 0

2664

3775,

where av,z represents the contact level between individual v andindividual z in a weighted or unweighted network, and theKronecker delta function dizðtÞ,1 ¼ 1 represents the event thatindividual z is infected and zero otherwise. In exact Markov chainmodel, the infection event represents a condition given theneighbor individual is infected, and conditioning on every indivi-dual in the network leads to 3N states in Markov chain (reviewSection 3). In individual-based approach, instead of conditioningon the state of every individual, we replace the actual randominfection rate with its effective average infection rate,

E bX

z

av,zdizðtÞ,1

" #¼ b

Xz

av,zE½dizðtÞ,1�, ð9Þ

where the infection rate b and the network topology are constant.Therefore, E½dizðtÞ,1 ¼ izðtÞ ¼ 1� ¼ pðizðtÞ ¼ 1Þ is the probability thatthe neighbor individual z is infected. Replacing the actual randominfection rate with its effective rate is basically a mean fieldapproximation, and therefore the effective qeff

v ðtÞ infinitesimalmatrix is obtained and has the following expression:

qeffv ðtÞ ¼

�bP

zav,zpðizðtÞ ¼ 1Þ b

Pz

av,zpðizðtÞ ¼ 1Þ 0

0 �m m0 0 0

2664

3775:

For every individual v, we derive a system of differentialequations by applying the general differential equation in (7)using the effective qeff

v ðtÞ infinitesimal matrix as follows:

dStateTv ðtÞ

dt¼ StateT

v ðtÞqeffv ðtÞ, ð10Þ

where StateTv ðtÞ ¼ ½SvðtÞ IvðtÞ RvðtÞ� is the vector of the state prob-

abilities of individual v. The obtained differential equations are asfollows:

dSvðtÞ

dt¼�SvðtÞb

XzAN

av,zIzðtÞ, ð11Þ

dIvðtÞ

dt¼ SvðtÞb

XzAN

av,zIzðtÞ�mIvðtÞ, ð12Þ

dRvðtÞ

dt¼ mIvðtÞ: ð13Þ

At any time t, each individual v will be in any of the states withtotal probability of 1, SvðtÞþ IvðtÞþRvðtÞ ¼ 1. In addition, the sum ofrates of changes in the state probabilities is zerodSvðtÞ=dtþdIvðtÞ=dtþdRvðtÞ=dt¼ 0. Therefore, we only solve 2N

simultaneous differential equations instead of 3N.We performed Monte Carlo (MC) simulation to evaluate the

accuracy of the individual-based SIR approach (numerical solu-tion (NS)). In Monte Carlo approach, every infected individualtries to infect each of its susceptible neighbors with infection rateb, and it cures itself with rate m. Fig. 2 represents the totalincidences with respect to the number of individuals in thenetworks when an epidemic spreads on random scale-free net-works (Barabasi and Albert, 1999) with N¼104 and exponents nequal 2.6, 2.9, 3.3 and 3.6 given different cure rates to infectionrate m=b for both MC and NS. The results are averaged over 100runs. For every cure rate m and infection rate b, we compute thenormalized total incidences r so that we obtain the relationshipbetween m=b and r. In MC, each simulation starts with a singleinfected incidence that is randomly chosen among the indivi-duals, while in NS, each individual is initially infected withprobability 10�4. The error bar represents the standard deviationof MC trials. We notice that NS upper-bounds MC simulations fora large range of m=b. In Section 6, we analyze the deviationbetween the individual-based approach and Markov chain modeland we show that the individual-based approach upper-boundsMarkov model results. The insets in Fig. 2 show that for0:7lmaxrm=brlmax, NS results approach MC simulations.

Additionally, we compare the individual-based SIR approachwith the heterogeneous mean field approach discussed in Section2 given the same range of cure rate to infection rate m=b values.We use the same random scale-free networks with the sameexponent values. We also assume that the initial probability ofinfection for every individual is 10�4. In Fig. 2, the results showthat the heterogeneous mean field approach does not show anyinfection incidence for higher values of m=b that are near thereciprocal of the epidemic threshold. On the other hand, theresults of the individual-based approach are closer to MC simula-tion results, and both show the existence of incidences with non-zero values. Basically, the difference between the two approachescomes from the epidemic thresholds and the network representa-tion in each approach. The range of m=br1=t in the individual-based approach is larger than the range in the heterogeneousmean field approach, which is observed in Fig. 2. Therefore, theindividual-based approach captures more chances for the spreadof epidemics with a broader range of m=b. Additionally, in theheterogeneous mean field approach, individuals are representedthrough their node degrees, which means that all individuals withthe same node degree have the same probability of infection,

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

� 7 8 9 10 110

0.02

0.04

�

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

�

6 8 100

0.05

0.1

�

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

�

5 6 7 80

0.05

0.1

�

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

� 4 5 60

0.10.20.3

�

� /� � /�

� /� � /�

� /�� /�

� /�� /�

Fig. 2. Normalized total incidences r given different cure rate and infection rate m=b on random scale-free networks with different exponents n. Each network has 104

individuals. The results are averaged over 100 runs. The dashed line represents the numerical solution (NS) of the individual-based SIR approach, the ‘o’ symbol represents

the heterogeneous mean field approach results, and the ‘n’ symbol represents the average of Monte Carlo (MC) results. The error bar is the standard deviation of MC results.

The insets show how the individual-based approach outperforms the heterogeneous mean field approximation for the values m=b that are close to the reciprocal of the

epidemic threshold ð1=tÞ. (a) n¼ 2:6; (b) n¼ 2:9; (c) n¼ 3:3; and (d) n¼ 3:6.

M. Youssef, C. Scoglio / Journal of Theoretical Biology 283 (2011) 136–144140

while in the individual-based approach, every individual isstudied separately. Thus, the heterogeneous mean field approachneglects the centrality properties of the individuals in the net-work, while the individual-based approach not only considers theindividual probability of infection, but also distinguishes amongthe individuals’ topological characteristics.

5. Properties of the individual-based SIR approach

In this section, we derive some useful properties from theindividual-based SIR approach. Those properties describe theinfection process more fully, and they relate the infection processwith the contact network topology over which an outbreak takesplace. First, we derive the epidemic threshold, and we show howit is related to the network topological properties. We also derivethe condition under which the infection reaches its peak valuewith time, and how the peak infection value is related to theepidemic threshold. Moreover, we study the role of the networktopology on the spread of epidemics.

5.1. Epidemic threshold

The epidemic threshold is the value of the effective infectionrate above which the epidemic prevails in the network. Tocompute the threshold, we follow the analysis presented inBoguna (2003). We assume that the initial fraction of infectedindividuals is very small and therefore Svð0ÞC1. The differentialequation (12) is written as follows:

dIvðtÞ

dtCX

z

~Lv,zIzðtÞ, ð14Þ

where the element ~Lv,z ¼ bav,z�mdv,z is the entry of the Jacobianmatrix ~L ¼ f ~Lv,zg ¼ bA�mIN�N , and dv,z is the Kronecker deltafunction and equals 1 for all v¼z. Since any element av,z of thesymmetric adjacency matrix A of the binary contact network iseither 0 or 1, and according to the Frobenius theorem, themaximum eigenvalue lmax,A of A is positive and real, the eigen-values of the matrix ~L have the form of bli,A�m, and theeigenvectors are the same as those for the adjacency matrix A.Thus, the stability condition of the solution I¼0 of the differentialequation (14) is �mþblmax,Ao0, and the SIR individual-basedepidemic threshold t for any undirected network becomes:

1

lmax,A¼ t: ð15Þ

As a consequence, an outbreak will occur if and only if b=m4t.

5.2. The existence of a maximum number of infected individuals

The number of infected individuals increases in time following acertain profile (Barrat et al., 2008) depending on the infection strain.Below, we derive the condition for which a maximum number ofinfected individuals occurs, and how the condition is related to theepidemic threshold. Let uT IðtÞ ¼

PvIvðtÞ be the total number of

infected individuals in the network, where uT is the transpose of avector of ones uT ¼ ½1 1 . . . 1�. The existence of a maximum valuefor I(t) is determined through duT IðtÞ=dt¼

PvdIvðtÞ=dt¼ 0, and we

obtain:

Xv

SvðtÞbX

z

av,zIzðtÞ�mIvðtÞ

" #¼ 0: ð16Þ

S I R

Fig. 3. The state transition diagram of an individual v from susceptible state sv ¼ 1

to infection state sv ¼ 0.

M. Youssef, C. Scoglio / Journal of Theoretical Biology 283 (2011) 136–144 141

By rewriting Eq. (16) in the matrix form, we obtain thefollowing equation:

½bST ðtÞA�muT �IðtÞ ¼ 0: ð17Þ

There are two possible solutions for Eq. (17): (1) I(t) equalszero, which happens when the network becomes cured, or (2)bST ðtÞA�muT equals zero. The second solution derives a conditionfor the existence of a positive maximum value of I(t). Conse-quently, the second solution ASðtÞ ¼ ðm=bÞu has the form ofMx¼ rx, where x and r are an eigenvector and an eigenvalue ofthe arbitrary matrix M, respectively. The vector S(t) is equal to thevector u only if m=b is equal to the maximum eigenvalue lmax,A ofA, which follows the Frobenius theorem and takes place fort-0 and Sð0Þ-1. Moreover, this solution proves the existenceof the epidemic threshold shown in inequality (15) wheneverm=bolmax,A, and therefore the epidemic spreads in the network,SvðtÞr1 for all v, and the maximum number of incidences takesplace before the network becomes cured.

5.3. The effect of the network spectrum

This subsection addresses the effect of the network topologyon the spread of epidemics. Rigorously, we show that theeigenvalues and the corresponding eigenvectors reveal the roleof the centrality properties of the individuals in spreading theepidemics in networks. Below, we mathematically derive theeffect of the centrality properties of the networks on the spreadof epidemics followed by an interpretation for the final mathe-matical formula (21), and we show how the eigenvector corre-sponding to the maximum eigenvalue can predict the probabilityof infection of the individuals in the network.

Recall that uT is the transpose of a vector of ones(uT ¼ ½1 1 . . . 1�), and A to be the adjacency matrix of a binarycontact network. Thus, we can write the total rate of change ofinfected individuals uT IðtÞ as follows:

duT IðtÞ

dt¼ bðuT�IT ðtÞ�RT ðtÞÞAIðtÞ�muT ðtÞIðtÞ: ð18Þ

If we denote the vector of node degrees D¼ uT A, and theeigenvalue decomposition of the adjacency matrix A¼ULUT , wecan rewrite the differential equation (18) as follows:

duT IðtÞ

dt¼ ðbD�muÞT IðtÞ�bðUT IðtÞÞTLðUT IðtÞÞ

�bðUT RðtÞÞTLðUT IðtÞÞ: ð19Þ

Let xz be the zth element in the vector UT IðtÞ, and let yz be thezth element in the vector UT RðtÞ. Now we can rewrite thedifferential equation as follows:

duT IðtÞ

dt¼ ðbD�muÞT IðtÞ�b

XN

z ¼ 1

lzx2z�b

XN

z ¼ 1

lzxzyz: ð20Þ

To relate Imax with the spectrum lz and the eigenvectors U, letduT IðtÞ=dt equal zero, and therefore we can obtain the followingequation:

XN

v ¼ 1

dv�mb

� �Ivmax ¼

XN

z ¼ 1

lzx2z�XN

z ¼ 1

lzxzyz: ð21Þ

The fact that the matrix A is symmetric and therefore lmax is apositive eigenvalue and elements of the corresponding eigenvectorare positive as well is used to understand Eq. (21) as follows: As bincreases, both the LHS and the RHS increase too. We can also seethat the vector I(t) is proportional to the eigenvectors of theadjacency matrix A, while the coefficients of the proportion arethe eigenvalues; however, the dominant term in the RHS is

lmaxx2lmax¼ lmaxðUT

lmaxIðtÞÞ2 since both the maximum eigenvalue

lmax and the elements of the corresponding eigenvector UTlmax

arepositive. Therefore, the vector I(t) is increasingly more aligned withthe eigenvector corresponding to the maximum eigenvalue on theRHS. Thus, the elements of the eigenvectors corresponding to themaximum eigenvalue provide an estimate of the probability ofinfection for each individual in the network. To conclude, thisproperty reveals the role of the centrality properties of the indivi-duals in spreading the infection across the contact network.

6. Markov chain model and mean field approximation

In this section, we analytically show the differences betweenthe continuous time Markov chain epidemic model that isdiscussed in Section 3 and the individual-based SIR approach,and the effect of the mean field approximation on the outcome ofthe model.

Let us first consider the state transition of an individual fromsusceptible to infection. Let sv ¼ 1 if individual v is susceptible,and sv ¼ 0 if it is infected. In the SIR model, the transition happensfrom susceptible state to infection state, while the opposite doesnot happen as shown in Fig. 3. Thus, a susceptible individual v

becomes infected with rate bP

lav,ldsl ,0 (see Fig. 3), where dsl ,0 isthe Kronecker delta function and it equals 1 if the neighbor l isinfected, i.e. sl ¼ 0. Therefore, the susceptible individuals do notincrease with time. In other words, the change in the susceptiblestate sv ¼ 1 over a small time interval Dt is defined as follows:

svðtþDtÞ�svðtÞ

Dt¼�dsvðtÞ,1b

XN

l ¼ 1

av,ldslðtÞ,0: ð22Þ

By taking the average on both sides, and consideringE½svðtÞ� ¼ pðsv ¼ 1Þ ¼ SvðtÞ, we obtain the following equation:

SvðtþDtÞ�SvðtÞ

Dt¼�E dsvðtÞ,1b

XN

l ¼ 1

av,ldsvðtÞ,0

" #: ð23Þ

The above equation is only applied for any va l. Since weassume that the infection rate does not change with time, and thetopology is static, we process the expectation only on theKronecker delta functions as follows:

E½dsvðtÞ,1:dslðtÞ,0� ¼ E½ðsvðtÞ ¼ 1Þ \ ðslðtÞ ¼ 0Þ�

¼ pðsvðtÞ ¼ 1,slðtÞ ¼ 0Þ

¼ pðsvðtÞ ¼ 1jslðtÞ ¼ 0ÞpðslðtÞ ¼ 0Þ:

In Eq. (23), let Dt-0, and we obtain the following differentialequation:

dSvðtÞ

dt¼�b

XN

l ¼ 1

av,lpðsvðtÞ ¼ 1jslðtÞ ¼ 0ÞpðslðtÞ ¼ 0Þ: ð24Þ

The conditional probability represents the probability that indivi-dual v is susceptible given that the neighbor individual l is infected.Recalling Eq. (11), which is dSvðtÞ=dt¼�SvðtÞb

PNl ¼ 1 av,lIlðtÞ in the

individual-based approach, and comparing it with Eq. (24), wenotice that the mean field theory assumes that events are

M. Youssef, C. Scoglio / Journal of Theoretical Biology 283 (2011) 136–144142

independent as follows:

pðsvðtÞ ¼ 1jslðtÞ ¼ 0ÞpðslðtÞ ¼ 0Þ ¼ pðsvðtÞ ¼ 1,slðtÞ ¼ 0Þ

¼ pðsvðtÞ ¼ 1ÞpðslðtÞ ¼ 0Þ ¼ SvðtÞIlðtÞ:

Additionally, the following inequality

pðsvðtÞ ¼ 1jslðtÞ ¼ 0ÞrpðsvðtÞ ¼ 1Þ ð25Þ

is valid since an infected neighbor l does not increase the probabilityof an individual v to remain susceptible. Therefore, in the Markovchain model, the absolute rate at which a susceptible individualchanges its state to infection state is lower than the correspondingrate in the individual-based approach.

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Time

Nor

mal

ized

pop

ulat

ion

Susceptible (Markov Chain)Infected (Markov Chain)Recovered (Markov Chain)Susceptible (Individual−based)Infected (Individual−based)Recovered (Individual−based)

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Time

Nor

mal

ized

pop

ulat

ion

0 100 200 3000

0.2

0.4

0.6

0.8

1

Time

Nor

mal

ized

pop

ulat

ion

Fig. 4. Comparison between the continuous time Markov chain model and the individ

and 10�2. (a) m=b¼ 102; (b) m=b¼ 1:6667; (c) m=b¼ 1:25; (d) m=b¼ 0:5; and (e) m=b¼

We conclude that for any individual v, the susceptible prob-ability in the individual-based SIR approach lower-bounds thesusceptible probability in the Markov chain model. Therefore, wealso conclude that the infection probability in the individual-based SIR approach upper-bounds the infection probability in theMarkov chain model. Consequently, the probability of recovery inthe individual-based SIR approach also upper-bounds the prob-ability of recovery in the Markov chain model.

We compare the individual-based approach with the continuoustime Markov chain model to study the deviation between the twoapproaches. In Figs. 4(a)–(e), we numerically evaluate the spread ofepidemics on a six-node ring network where initially a single

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Time

Nor

mal

ized

pop

ulat

ion

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Time

Nor

mal

ized

pop

ulat

ion

400 500 600

ual-based approach given a six-node ring network and m=b¼ 102 ,1:6667,1:25,0:5,

10�2.

M. Youssef, C. Scoglio / Journal of Theoretical Biology 283 (2011) 136–144 143

individual is infected and m=b equals 102, 1.6667, 1.25, 0.5, and10�2, respectively. Note: the Markov chain model can only bemodeled on smaller networks since the number of states exponen-tially increases with the number of individuals in the network. InFig. 4(a), we notice that the initially infected individuals recoveredand the epidemic die out, and there is no new infection since m=b isabove the reciprocal of the epidemic threshold (lring

max ¼ 2). However,a deviation is noticeable in Figs. 4(c) and (d) between the individual-based approach and the Markov model, and it is due to the meanfield approximation used in the individual-based approach. Inparticular, we observe in the same figures that the susceptiblepopulation in the individual-based approach lower-bounds thecorresponding population in the Markov chain model, while theinfected population in the individual-based approach upper-boundsthe corresponding population in the Markov chain model. InFig. 4(e), the individual-based approach almost coincides with thecontinuous time Markov chain model. The simulations in Fig. 2show that total incidences in the individual-based approach upper-bounds the total incidences in Monte Carlo simulations. Such aresult agrees with the theoretical prediction that the probability ofinfection in the individual-based approach upper-bounds the prob-ability of infection in Markov chain model.

7. Summary

We summarize our findings as follows:

�

The epidemic threshold is the reciprocal of the spectral radius of

the network: The theoretical analysis shows the role of thespectral radius in the epidemic spreading in networks, and thesimulations verify the accuracy of the theoretical epidemicthreshold.

� A condition exists for the occurrence of a peak value of the

infection: The theoretical analysis provides a condition for theexistence of a peak value for the infection incidence, which isrelated to the epidemic threshold.

� The individual-based SIR approach well approximates the con-

tinuous time Markov chain model: The individual-basedapproach is accurate for all m=b41=t and m=b51=t. Forvalues of m=bo1=t, the individual-based approach providesan upper-bound for the infection incidence.

� The accuracy of the individual-based approach increases as the

number of individuals increases: For large networks, the Markovchain closely approximates any stochastic process given suffi-ciently large numbers of states (i.e. 3N different states inour case).

� The range of effective infection rates leading to an epidemic is larger

for the individual-based approach than for the heterogeneous mean

field approach: For 0:7lmaxrm=brlmax, the individual-basedapproach agrees with Monte Carlo simulations showing theexistence of infection incidence, while the heterogeneous meanfield approach does not show any infection incidence.

8. Conclusion and future work

In this paper, we have reviewed the well-known homogeneousand heterogeneous mean field approaches to the SIR model, and wehave shown how both approaches do not evaluate the state of everyindividual in networks. To account for this, we have presented a newindividual-based SIR approach that is derived from the continuoustime Markov chain model. The new approach evaluates the prob-ability of infection of every individual separately considering theprobability of infection of the individual’s neighbors. Unlike previous

approaches in the literature that only consider the node degreedistribution of the network, the individual-based approach considersthe whole network structure. Additionally, we have reviewed thecontinuous time Markov chain model, and through numericalanalysis and evaluation we have studied the deviation betweenthe Markov chain model and the individual-based approach. Wehave also performed Monte Carlo simulations to show the accuracyof the new approach. Moreover, we have derived the epidemicthreshold above which an epidemic prevails in the network. Wefound that the reciprocal of the spectral radius of the contactnetwork is the epidemic threshold showing the role of networkcharacteristics in the spread of an epidemic. Furthermore, we haveshown the condition for the existence of a maximum number ofnew infected individuals, and how it is related to the epidemicthreshold. We have also addressed the role of the spectrum of thecontact network and the effective infection rate in determining themaximum number of the new infected individuals. Finally, throughthe numerical evaluations, we have compared the individual-basedapproach with the heterogeneous mean field approach.

Since the individual-based approach addresses the state prob-ability of every individual specifically, our future work will focuson finding the optimal mitigation strategies for both weightedand unweighted networks. Also the new approach will allow us torank the individuals according to their responsibility in bothspreading the infection and curing the network making sure thatthe approach does not neglect the centrality properties of theindividuals in the network.

Acknowledgment

This work is supported by National Agricultural BiosecurityCenter NABC at Kansas State University.

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