The spread of infectious disease on complex networks with household-structure

Abstract

In this paper, we study the household-structure SIS epidemic spreading on general complex networks. The household structure gives us the way to distinguish the inner and the outer infection rates. Unlike household-structure models on homogenous networks, such as regular and random networks, we consider heterogeneous networks with arbitrary degree distribution p(k). Through mean field analysis and computer simulations, we find some unique phenomena only existing in divergent network with household structure. In our model, disease can spread on the scale-free network even when local recovery rate is greater than inner infection rate in every household. This implies that no disease is spreading in every single household, but for the whole network disease is spreading. Since our society network seems like this structure, maybe this conclusion remind us that during disease spreading we should pay more attention to network structure than local cure condition.

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arXiv:q-bio/0402019v2 [q-bio.PE] 25 Feb 2004
The Spread of Infectious Disease with
Household-Structure on the Complex
Networks
Jingzhou Liu1∗,Jinshan Wu2and Z.R. Yang1
1Department of physics, Beijing Normal University, Beijing 100875, China
2Department of Physics, Simon Fraser University, Burnaby, B.C., Canada, V5A
1S6
Abstract
In this paper we study the household-structure SIS epidemic spreading on general
complex networks. The household structure gives us the way to distinguish inner
and the outer infection rate. Unlike household-structure models on homogenous
networks, such as regular and random networks, here we consider heterogeneous
networks with arbitrary degree distribution p(k). First we introduce the epidemic
model. Then rate equations under mean field appropriation and computer simula-
tions are used here to analyze our model. Some unique phenomena only existing in
divergent network with household structure is found, while we also get some simi-
lar conclusions that some simple geometrical quantities of networks have important
impression on infection property of infectous disease. It seems that in our model
even when local cure rate is greater than inner infection rate in every household,
disease still can spread on scale-free network. It implies that no disease is spreading
in every single household, but for the whole network, disease is spreading. Since
our society network seems like this structure, maybe this conclusion remind us that
during disease spreading we should pay more attention on network structure than
local cure condition.
Key words: Infectious Disease; SIS Model; Networks
PACS: 89.75.-Hc; 05.70.Ln; 02.10.Yn; 87.23.Ge; 64.70.-p
∗Corresponding author. E-mail address: liujz606@hotmail.com
Preprint submitted to Elsevier Science9 February 2008

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1Introduction
The spread of disease has been one of the focuses in the field of statisti-
cal physics for many years. The dynamical behavior of so-called susceptible-
infected-susceptible (SIS) model and susceptible- infected- removed (SIR) model
have been widely investigated on regular network and complex networks[1-12].
Within the studying, individuals are modeled as sites and possible contacts
between individuals are linked by edges between the sites. It is easy to see
that both the properties of disease and topological character of network de-
termine the dynamics of the spread of disease. Studies have showed that there
is an epidemic threshold λcon regular networks. If the effective spreading rate
λ>λc, the infection spreads and becomes endemic; otherwise the infection will
die out. While the threshold disappears on scale-free networks[4].
Usually, infectious diseases, such as HIV and computer virus, have the simi-
lar spreading property. They not only can spread in one household, but also
can spread from one household to another. To study this spreading charac-
ter, there have been of considerable interests to epidemic models spreading
among a community of households[12-17]. These studies were concerned with
SIR model, which cannot appear endemic behavior. In 1999, Ball introduced
the SIS household-structure model[18], in which the population is partitioned
into m households with N members in each household. A threshold param-
eter R∗ was defined. It is shown that for the household with 2 members,
if R∗ < 1 then the epidemic die out; if R∗ > 1 the epidemic will exist at
an endemic equilibrium. This model has also been studied on homogeneous
network by the mean of self-consistent field[19,20]. The similar results have
been obtained. These previous studies about household-structure epidemic
model were mainly on regular networks. However, studies have showed that
a large number of systems, such as Internet, world-wide-web, physical, bio-
logical, and social networks, exhibit complex topological properties[21-23]. In
particular, small-world properties[24] and scale-free degree distributions[25]
appear in many real network systems. In this paper, we will analyze the SIS
household-structure epidemic model on complex networks. The outline is as
follows: 1) introduction; 2) description of the model; 3) mean-field equations;
4) steady-state solutions; 5) simulation; 6) summary.
2Model
In complex networks with degree distribution p(k), which is the probability
that a given site has k connections (links) that connect it with other k sites
(We say that the given site’ degree is k.), there are N individuals that are
grouped as a household on every site. We assume that these N individuals
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contact each other fully. A healthy individual may get infected from within
the household and from outside its household. The parameters λ and β are the
infection rates from outside and from within the household respectively. We
give each site x a number i(i ∈ [0,N]), which means that there are i infected
individuals in the household at site x. The number of infected individuals at
a given site x changes according to the following transition rates:
0 −→ 1 at rate λ?
i −→ i + 1 at rate iβ for 1 ≤ i ≤ N − 1
?x,y?iy
i + 1 −→ i at rate γ for 0 ≤ i ≤ N − 1
In the above expressions ?x,y?means that site x and site y are nearest neigh-
bors, and we suppose there is a (connection) link between them. Infected
individuals may infect healthy individuals in their household with rate β,and
also can infect healthy individuals in their nearest neighbors with rate λ.We
assume that once a site is infected, infections within the site are much more
likely than infections from outside, so we can neglect the latter. And also, an
infected individual in a site can recover with rate γ.We suppose that all the
individuals in a household have the same external connectivity and do not
take the birth and death into account.
3Mean-field equations
We now solve the above model with mean-field method. Let uk,ibe the density
of individuals whose household has i infected individuals and the correspond-
ing site’s degree is k, which means this site has k nearest neighbors. It is worth
noticing that?N
above section, the evolution equations of uk,iare written as below[4,6]:
i=0uk,i= 1. According to the transitions rate described in the
∂uk,0(t)
∂t
∂uk,1(t)
∂t
∂uk,i(t)
∂t
∂uk,N(t)
∂t
=γuk,1− λkΘk(t)uk,0
(1)
=λkΘk(t)uk,0− βuk,1+ γuk,2− γuk,1
(2)
=(i − 1)βuk,i−1− iβuk,i+γuk,i+1− γuk,i (i ∈ [2,N − 1])(3)
=(N − 1)βuk,N−1− γuk,N
(4)
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In equations(1)-(4), Θk(t) is the probability that a link from a site points to
another site with at least one infected individual. And the expression of Θk(t)
is:
Θk(t) =
?
k′
p(k
′/k)
N
?
j=1
juk′,j(t)(5)
where p(k
site with k
′/k) is the probability that a site with k degrees points to another
′degrees. For uncorrelated networks the expression of p(k
′/k) is[7]:
p(k
′/k) =
k
′p(k
′)
?
k′ k
′p(k
′)=
1
?k?k
′p(k
′)(6)
Substituting (6) to (5), we get Θk(t) = Θ(t) independent of k:
Θk(t) = Θ(t) =
1
?k?
?
k′
k
′p(k
′)
N
?
j=1
juk′,j(t)(7)
Now we are going to get steady solutions of Eqs.(1)-(4)
4Steady-state solutions
4.1For N=2
The evolution equations are simplified as:
∂uk,1(t)
∂t
∂uk,2(t)
∂t
=λkΘ(t)(1 − uk,1− uk,2) + γuk,2− βuk,1− γuk,1
(8)
=βuk,1− γuk,2
(9)
In Eq.(8), we have used the equality: uk,0+ uk,1+ uk,2= 1. When t → ∞,
∂uk,1
∂t
= 0 and
∂t
= 0, then the steady-state solutions are:
∂uk,2
uk,1=
λγkΘ
γ2+ (γ + β)λkΘ
λβkΘ
γ2+ (γ + β)λkΘ
(10)
uk,2= (11)
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Substituting (10)and (11) to (7), we get the self-consistent equation of Θ:
Θ =
1
?k?
?
k
kp(k)(uk,1+ 2uk,2) =
1
?k?
?
k
kp(k)
(γ + 2β)λkΘ
γ2+ (γ + β)λkΘ
(12)
Clearly, Θ = 0 is a solution of Eq.(12), which implies that uk,0= 1, uk,1= 0
and uk,2= 0 is a steady-state solution of Eqs.(8) and (9). A nonzero steady-
state solution Θ (That is: uk,i?= 0, for i > 0) is obtained when γ, β and λ
satisfy the following inequality:
1
?k?
?
k
kp(k)(γ + 2β)λk
γ2
≥ 1
Then we can get the spreading threshold:
λc=
γ2
γ + 2β
?k?
?k2?
(13)
where ?k? =?
out when λ < λc; otherwise the disease will pervade the system. Clearly, the
threshold λcis the function of β, γ and
networks plays an important role on λc.
kkp(k), ?k2? =?
kk2p(k). In other words, the disease will die
<k>
<k2>. So the degree distribution of
For p(k) = δk,kc, the network is homogeneous and λc=
λcto prevent infection in terms of increasing the recover rate γ or decreasing
the site degree kc.
γ2
γ+2β
1
kc. We can lift
For p(k) = Ck−υ(υ ∈ (2,3]), the networks are scale-free[19]. When k → ∞,
?k?
?k2?→ 0, the threshold is absent. This fact implies that for any positive value
of λ the infection can pervade the system, which is the same as the standard
SIS model[4].
4.2N>2
Let
Considering
∂uk,i
∂t
= 0(i = 1,2,···N). Suppose Uk=(uk,1,uk,2,··· ,uk,N)Tand V =(1,0,··· ,0)T.
?N
SUk= − λkΘV
j=0uk,j= 1,then Eqs.(1)-(4) can be written as:
(14)
The matrix S is:
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