Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
Edge-based compartmental modelling for infectious disease
spread
Joel C. Miller, Anja C. Slim and Erik M. Volz
J. R. Soc. Interface published online 5 October 2011
Supplementary data
"Data Supplement"
http://rsif.royalsocietypublishing.org/content/suppl/2011/10/04/rsif.2011.0403.DC1.htm
l
References
This article cites 45 articles, 9 of which can be accessed free
http://rsif.royalsocietypublishing.org/content/early/2011/10/04/rsif.2011.0403.full.html#
ref-list-1
Article cited in:
http://rsif.royalsocietypublishing.org/content/early/2011/10/04/rsif.2011.0403.full.html#related-url
s
P<P
Published online 5 October 2011 in advance of the print journal.
Subject collections
Articles on similar topics can be found in the following collections
biomathematics (321 articles)
Email alerting service
Receive free email alerts when new articles cite this article - sign up in the box at the top
right-hand corner of the article or click here
Advance online articles have been peer reviewed and accepted for publication but have not yet appeared in
the paper journal (edited, typeset versions may be posted when available prior to final publication). Advance
online articles are citable and establish publication priority; they are indexed by PubMed from initial publication.
Citations to Advance online articles must include the digital object identifier (DOIs) and date of initial
publication.
To subscribe to J. R. Soc. Interface go to: http://rsif.royalsocietypublishing.org/subscriptions
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
J. R. Soc. Interface
doi:10.1098/rsif.2011.0403
Published online
Edge-based compartmental modelling
for infectious disease spread
Joel C. Miller1,2,*, Anja C. Slim and Erik M. Volz3
1
Center for Communicable Disease Dynamics, Department of Epidemiology, Harvard School
of Public Health, Boston, MA 02115, USA
2
Fogarty International Center, National Institutes of Health, Bethesda, MD 20892-2220, USA
3
Department of Epidemiology, University of Michigan, Ann Arbor, MI 48109-2029, USA
The primary tool for predicting infectious disease spread and intervention effectiveness is the
mass action susceptible – infected – recovered model of Kermack & McKendrick. Its usefulness
derives largely from its conceptual and mathematical simplicity; however, it incorrectly
assumes that all individuals have the same contact rate and partnerships are fleeting. In this
study, we introduce edge-based compartmental modelling, a technique eliminating these
assumptions. We derive simple ordinary differential equation models capturing social heterogeneity (heterogeneous contact rates) while explicitly considering the impact of partnership
duration. We introduce a graphical interpretation allowing for easy derivation and communication of the model and focus on applying the technique under different assumptions about
how contact rates are distributed and how long partnerships last.
Keywords: infectious disease; network; edge-based compartmental model
recipient is susceptible. Recovery to an immune state
happens at rate g. The flux of individuals from susceptible to infected to recovered is represented by a flow
diagram (figure 1), making the model conceptually
simple. This leads to a simple mathematical interpretation, the low-dimensional, ordinary differential
equation (ODE) system
1. INTRODUCTION
The conceptual and mathematical simplicity of Kermack
& McKendrick’s [1,2] mass action susceptible – infected –
recovered (SIR) model has made the model the most
popular quantitative tool to study infectious disease
spread for over 80 years. However, it ignores important
details of the fabric of social interactions, assuming
homogeneous contact rates and negligible partnership
duration. Improvements are largely ad hoc, spanning
the range between mild modifications of the model
and elaborate agent-based simulations [2 – 7]. Increased complexity allows us to incorporate more realistic
effects, but at a price. It becomes difficult to identify
which variables drive disease spread or to address sensitivity to changing the underlying assumptions. In this
study, we show that shifting our attention to the status
of an average partner rather than an average individual
yields a surprisingly simple mathematical description, expanding the universe of analytically tractable
models. This allows epidemiologists to consider more
realistic social interactions and test sensitivity to
assumptions, improving the robustness of public health
recommendations.
We motivate our approach using the standard mass
action (MA) SIR model. We are interested in the
susceptible S(t), infected I(t) and recovered R(t) proportions of the population as time t changes. Under
MA assumptions, an infected individual causes new
infections at rate b^ SðtÞ, where b^ is the per-infected
transmission rate and S is the probability that the
S_ ¼ b^IS;
and
_ ¼ gI :
R
The dot denotes differentiation in time. An ODE
system allows for easy prediction of details such as
early growth rates, final sizes and intermediate
dynamics. Using S þ I þ R ¼ 1, we re-express this as
S_ ¼ b^IS;
I ¼1S R
and
_ ¼ gI :
R
ð1:1Þ
The product IS measures the proportion of partnerships
that are from an infected individual to a susceptible
individual.
The MA model often provides a reasonable description of epidemics; however, it has well-recognized
flaws, which cause the frequency of infected to susceptible partnerships to vary from IS. We highlight two. It
neglects both social heterogeneity, variation in contact
rates which can be quite broad [8,9], and partnership
duration, implicitly assuming all partnerships are infinitesimally short. Because of these omissions, model
predictions can differ from reality. For example, owing
to social heterogeneity, early infections tend to have
more partnerships [10] and may cause more infections
than ‘average’ individuals, enhancing the early spread
over that predicted by the MA model [11– 15]. When
partnership duration is significant, an infected individual
may have already infected its partners, reducing its
*Author for correspondence ( joel.c.miller.research@gmail.com).
Electronic supplementary material is available at http://dx.doi.org/
10.1098/rsif.2011.0403 or via http://rsif.royalsocietypublishing.org.
Received 22 June 2011
Accepted 13 September 2011
I_ ¼ b^IS gI
1
This journal is q 2011 The Royal Society
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
2 Edge-based compartmental modelling
^
S
b IS
I
gI
J. C. Miller et al.
R
Figure 1. Mass action flow diagram. The flux of individuals
from susceptible to infected to recovered for the standard
mass action model. Each compartment accumulates and
loses probability at the rates given on the arrows. (Online version in colour.)
ability to cause new infections. Because of these assumptions, the MA model predicts the same results for a
sexually transmitted disease in a completely monogamous population, a population with serial monogamy,
and a population with wide variation in contact levels
with mean one. Intuitively, we expect these to produce
dramatically different epidemics, but no existing mathematical theory allows analytical comparisons.
Over the past 25 years, attempts have been made to
eliminate these assumptions without sacrificing analytical tractability. With few exceptions (notably Volz &
Meyers [16]), these make an ‘all-or-nothing’ assumption
about partnership duration: partnerships are fleeting
and never repeated or they never change. With fleeting
partnerships, social heterogeneity is introduced by
adding multiple risk groups to the MA model: in extreme
cases, there are arbitrarily many subgroups (the mean
field social heterogeneity model) [2,17– 20]. This model
is relatively well understood and can be rigorously
reduced to a handful of equations. With permanent
partnerships, social heterogeneity is introduced through
static networks [12– 14,21,22]. Static network results
typically give the final size but no dynamic information.
Some attempts to predict dynamics with static networks
use pair approximation techniques [23] relying on
approximation of network structures. More rigorous
approaches avoid these approximations, but are more
difficult [24– 27]. Of these, only Volz [24] yields a closed
ODE system, (see also [28,29]). This model lacks an
illustration like figure 1, hampering communication and
further development. Finite, non-zero partnership duration is typically handled through simulation, which is
often too slow to effectively study parameter space.
Although no coherent mathematical structure to
study social heterogeneity and partnership duration
exists, there have been studies collecting these data in
real-world contexts [9,10,30,31]. Typically, the resulting
measurements have been reduced to average contact
rates to make the mathematics tractable. Much of the
available and potentially relevant detail is therefore
discarded because existing models cannot capture the
detail collected.
We find that the appropriate perspective allows us
to develop conceptually and mathematically simple
models that incorporate social heterogeneity and (arbitrary) partnership duration. This provides a unifying
framework for existing models and allows an expanded
universe of models. Our goal in each case is to calculate
the susceptible, infected or recovered proportions of the
population, but we find that this can be answered more
easily using an equivalent problem. We ask the question, ‘what is the probability that a randomly chosen
test node u is susceptible, infected or recovered?’
Because u is chosen randomly, the probability that
J. R. Soc. Interface
u is susceptible equals the proportion susceptible S(t),
and similarly for I and R. If we know S(t), then the
_ ¼ gI , I ¼ 1 2 S 2 R determine
initial conditions and R
I(t) and R(t) as in equation (1.1).
The probability that u is susceptible is the probability
that no partner has ever transmitted infection to u. The
method to calculate this is the focus of this study.
This probability depends on how many partners u has,
the rate its partners change, and the probability that a
random partner is infected at any given time. Because
a random partner is likely to have more partners than a
random individual, knowing the infected fraction of the
population does not give the probability that a partner
of u is infected. We focus on the probability that a
random partner is infected rather than the probability
a random individual is infected. Once we calculate this,
it is straightforward to calculate that the probability u
is susceptible. The resulting edge-based compartmental
modelling approach significantly increases the effects
we can study compared with MA models, but with only
a small complexity penalty.
In this study, we consider the spread of epidemics in
two general classes of networks: actual degree networks
(based on configuration model networks [32 – 34]) and
expected degree networks (based on mixed Poisson network; commonly called Chung-Lu networks [35 – 37]).
In both cases, we can consider static and dynamic networks. In actual degree networks, a node is assigned k
stubs where k is a random non-negative integer assigned
independently for each node from some probability distribution. Edges are created by pairing stubs from
different nodes. In expected degree networks, a node is
assigned k, where k is a random non-negative real
number. Edges are assigned between two nodes u and v
with probability proportional to kukv. We develop
exact differential equations for the large population
limit, which we compare with simulations. Detailed
descriptions of the simulation techniques are given in
the electronic supplementary material.
For many assumptions about population structure,
the equations we derive are equivalent to those found
by others. These previous equations have not been
widely used. We speculate that this is because the
derivation underlying the earlier equations are not for
the faint of heart, and consequently the models have
been conceptually difficult as well. The main result
of this study is a conceptual shift, after which the
derivations become straightforward, and the equations
become simpler.
We summarize the populations we consider in table 1.
We begin by analysing the simplest edge-based compartmental model in detail, exploring epidemic spread in
a static network of known degree distribution, a configuration model (CM) network. To derive the equations,
we introduce a flow diagram that leads to a simple
mathematical formulation. We next consider disease
spreading through dynamic actual degree networks and
then static and dynamic expected degree networks. The
template shown here allows us to derive a handful of
ODEs for each of these populations. Unsurprisingly, the
stronger our assumptions, the simpler our formulation
becomes. We neglect heterogeneity within the population
other than the contact rates, assume that the disease has a
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
Edge-based compartmental modelling
J. C. Miller et al. 3
Table 1. Populations to which we apply edge-based
compartmental models.
model
population structure
section
configuration model
(CM)
static network with
specified degree
distribution, assigned
using the probability
mass function P(k).
dynamic network for which
each node’s degree
remains a constant value,
assigned using P(k).
dynamic fixed-degree
network incorporating
gaps between
partnerships; a node may
wait before replacing a
partner.
static network with
specified distribution of
expected degrees k
assigned using the
probability density
function r(k).
dynamic network with
degrees varying in time
with averages assigned
using r(k).
population with a
distribution of contact
rates assigned using P(k)
or r(k) and negligible
partnership duration.
2
dynamic fixeddegree (DFD)
dormant contact
(DC)
mixed Poisson
model (MP)
dynamic variabledegree (DVD)
mean field social
heterogeneity
(MFSH)
3.2
3.3
4.1
4.3
3.1 and
4.2
very simple structure and assume that the population is at
statistical equilibrium prior to disease introduction.
2. CONFIGURATION MODEL EPIDEMICS
We demonstrate our approach with CM networks. A CM
network is static with a known degree distribution (the
distribution of the number of partners). We create a CM
network with N nodes as follows: we assign each node u
its degree ku with probability P(ku) and give it ku stubs
(half-edges). Once all nodes are assigned stubs, we pair
stubs randomly into edges representing partnerships. The
probability a randomly selected node u has degree k is
P(k). In contrast, the probability a stub of u connects to
some stub of v is proportional to kv. So the probability a randomly selected partner (or neighbour) of u has degree k is
Pn ðkÞ ¼
kPðkÞ
:
kK l
See earlier studies [15,38] for more detail. A sample CM
network is shown in figure 2.
We assume that the disease transmits from an
infected node to a partner at rate b. If the partner is
susceptible, then it becomes infected. Infected nodes
recover at rate g. Throughout, we assume a large population, a small initial proportion infected, a small initial
proportion of stubs belonging to infected nodes and a
growing outbreak. Our equations become correct once
J. R. Soc. Interface
Figure 2. A sample configuration model network. Half the
nodes have degree 3 and half degree 1. Partners are chosen randomly. The probability a partner has degree 3 is three times the
probability it has degree 1. In general, the probability a partner
has degree k is Pn(k) ¼ kP(k)/kKl. (Online version in colour.)
the number of infections NI has grown large enough
to behave deterministically, whereas the proportion
infected I is still small. Until stochastic effects are unimportant, other methods such as branching process
approximations [39] (which apply in large populations
with small numbers infected) may be more useful.
To calculate S(t), I(t) and R(t), we note that these
are the probabilities that a random test node u is in
each state. We calculate S(t) by noting that it is also
the probability none of u’s partners has yet transmitted
to u. We would like to treat each partner as independent,
but the probability that one partner v has become
infected is affected by whether another partner w of u
has transmitted to u as u could infect v. Accounting for
this directly requires considerable bookkeeping. A simpler approach removes the correlation by assuming that
u causes no infections. This does not alter the state
of u: the probabilities we calculate for u will be the proportion of the population in each state under the
original assumption that u behaves as any other node,
and so this yields an equivalent problem. Further discussion of this modification is given in the electronic
supplementary material.
We define u(t) to be the probability that a randomly
chosen partner has not transmitted to u. Initially, u is
close to 1. For large CM networks, partners of u are
independent. So given its degree k, u is susceptible at
time tP with probability s(k,u(t)) ¼ u(t)k. Thus,
SðtÞ ¼ k PðkÞsðk; uðtÞÞ ¼ cðuðtÞÞ; where
X
cðxÞ ¼
PðkÞx k
k
is the probability generating function [40] of the degree
distribution (such functions have many useful
P properties,
k1
,
of which we use three: its derivative
is
k kPðkÞx
P
k2
kðk
1ÞPðkÞx
and
its second derivative is
k
c 0 (1) ¼ kKl). For many important probability distributions, c takes a simple form, which simplifies our
examples. Combining with the flow diagram for S, I
and R in figure 3, we have
_ ¼ gI ;
R
S ¼ cðuÞ and
I ¼ 1 S R:
To calculate the new variable u, we break it into
three parts: the probability that a partner v is susceptible at time t, fS; the probability that v is infected at
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
4 Edge-based compartmental modelling
(a)
fS =
y ¢(q)
y ¢(1)
fI
J. C. Miller et al.
fR
g fI
bf I
(b)
gI
S = y (q)
I
R
1−q
Figure 3. Edge-based compartmental modelling for configuration model (CM) networks. The flow diagram for a static CM network. (a) The fS, fI and fR compartments represent the probability that a partner is susceptible, infected or recovered and has
not transmitted infection. The 1 2 u compartment is the probability that it has transmitted. The fluxes between the f compartments result from infection or recovery of a partner of the test node and the fI to 1 2 u flux results from a partner transmitting
infection to the test node. (b) S, I and R represent the proportion of the population susceptible, infected and recovered. We can
find S explicitly, and I and R follow as in the mass action model. (Online version in colour.)
time t but has not transmitted infection to u, fI; the
probability that v has recovered by time t but did not
transmit infection to u, fR. Then u ¼ fS þ fI þ fR.
Initially, fS and u are approximately 1 while fI and fR
are small (they sum to u 2 fS). The flow diagram for
fS, fI, fR and 1 2 u (figure 3) shows the probability
fluxes between these compartments. The rate an infected
partner transmits to u is b; so the fI to 1 2 u flux is bfI.
We conclude u_ ¼ bfI . To find fI, we will use fI ¼ u 2
fS 2 fR and calculate fS and fR explicitly.
The rate at which an infected partner recovers is g. Thus
the fI to fR flux is gfI. This is proportional to the flux into
1 2 u with the constant of proportionality g/b. As fR
and 1 2 u both begin as approximately 0, we have fR ¼
g(1 2 u)/b. To find fS, recall a partner v has degree k
with probability Pn(k) ¼ kP(k)/kKl. Given k, v is susceptible with probability uk21 (we disallow transmission from
u; so k 2 1 nodes can infect v). A weighted average gives
fS ¼
X
k
Pn ðkÞuk1 ¼
X kPðkÞ
c 0 ðuÞ
k1
:
u
¼
k kK l
c 0 ð1Þ
0
c ðu Þ g
ð1 uÞ
c0 ð1Þ b
and u_ ¼ bfI :
Thus our equations become
0
c ðuÞ
þ gð1 uÞ;
u_ ¼ bu þ b 0
c ð1Þ
S ¼ cðuÞ
and
I ¼ 1 S R:
R0 ¼
b kK 2 K l
:
b þ g kK l
ð2:3Þ
We want the expected final size if an epidemic
occurs. We set u_ ¼ 0 in equation (2.1) and solve
uð1Þ ¼
g
b c 0 ðuð1ÞÞ
þ
b þ g b þ g c 0 ð1Þ
ð2:4Þ
2.2. Example
ð2:2Þ
This captures substantially more population
structure than the MA model with only marginally
more complexity. This is the system of Miller [28]
and is equivalent to that of Volz [24]. It improves on
approaches of earlier studies [25,26], which require
either O(M ) or O(M 2) ODEs, where M is the ( possibly
unbounded) maximum degree. This derivation is simpler than Volz [24] because we choose variables with a
conserved quantity, simplifying the bookkeeping.
The edge-based compartmental modelling approach
we have introduced forms the basis of our study.
J. R. Soc. Interface
One of the most important parameters for an infectious
disease is its basic reproductive number R0, the average
number of infections caused by a node infected early in
an epidemic. When R0 , 1 epidemics are impossible,
while when R0 . 1 they are possible, though not guaranteed. For this model, we find that R0 is (see the
electronic supplementary material)
ð2:1Þ
and
_ ¼ gI ;
R
2.1. R0 and final size
for u(1). If R0 . 1 then this has two solutions, the larger
of which is u ¼ 1 (the pre-disease equilibrium). We want
the smaller solution. The total fraction of the population
infected in the course of an epidemic is R(1) ¼ 1 2
c(u(1)). These calculations of R0 and R(1) are in
agreement with previous observations [11–13,41,42].
When R0 , 1, our approach breaks down: full details
are given in the electronic supplementary material.
Thus
fI ¼ u fS fR ¼ u Depending on the network structure, some details
will change. However, we will remain as consistent as
possible.
We consider a disease with b ¼ 0.6 and g ¼ 1. Figure 4
compares simulations with solutions to our ODEs using
four different CM networks, each with 5 105 nodes
and average degree kK l ¼ 5, but different degree
distributions. In order from latest peak to earliest
peak, the networks are: every node has degree 5, the
degree distribution is Poisson with mean 5, half the
nodes have degree 2 and the other half degree 8 and
finally a truncated powerlaw distribution in which
P(k) / k 2ne2k/40, where n ¼ 1.418. We see that the
degree distribution significantly alters the spread, with
increased heterogeneity leading to an earlier peak, but
generally a smaller epidemic. Our predictions fit, while
the MA model using b^ ¼ bkK l fails.
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
Edge-based compartmental modelling
0.8
0.35
mass action
homogeneous
Poisson
bimodal
truncated powerlaw
mass action
homogeneous
Poisson
bimodal
truncated powerlaw
0.30
0.25
infections
cumulative infections
1.0
0.6
0.4
J. C. Miller et al. 5
0.20
0.15
0.10
0.2
0
–5
0.05
0
5
t
10
0
–5
15
0
5
t
10
15
Figure 4. Configuration model (CM) example (§2.2). Model predictions (dashed) match simulated epidemics (solid) of the same
disease on four CM networks with kKl ¼ 5 and 5 105 nodes, but different degree distributions. Each solid curve is a single simulation. Time is set so t ¼ 0 when there is 1% cumulative incidence. The corresponding mass action model (short dashes) based on
the average degree does not match. (Online version in colour.)
(a)
(b)
Figure 5. Sample mean field social heterogeneity network. (a) One moment in time and (b) an arbitrarily short time later. Half the
nodes have degree 3 and the others degree 1. The degrees remain the same, but the edges are all changed. (Online version in colour.)
3. ACTUAL DEGREE MODELS
For the CM networks, each node has a specific number of
stubs. Edges are created by pairing stubs, and no partner
changes occur. In generalizing to dynamic ‘actual degree’
models, we assign each node a number of stubs, but
allow edges to break and the freed stubs to create new
edges. We consider three limits. In the first, the mean
field social heterogeneity (MFSH) model, at every
moment a stub is connected to a new partner. In the
second, the dynamic fixed-degree (DFD) model, edges
last for some time before breaking. When an edge
breaks, the stubs immediately form new edges with
stubs from other edges that have just broken. In the
third, the dormant contact (DC) model, we assume
edges break as in the DFD model, but stubs wait before
finding new partners.
3.1. Mean field social heterogeneity
For the MFSH model, the number of stubs k an individual has is assigned using the probability mass function
P(k). At each moment in time, all stubs are rewired to
form new edges. A sample network at two times is
shown in figure 5.
We analyse the MFSH model similarly. We take u as
the probability a stub has never transmitted infection
to the test node u from any partner. To define fS, fI
and fR, we require that the stub has not transmitted
infection to u and additionally the current partner is
J. R. Soc. Interface
susceptible, infected or recovered. As at each moment
an individual chooses a new partner, the probability
of connecting to a node of a given status is the proportion of all stubs belonging to nodes of that status.
We must track the proportion of stubs that belong to
susceptible, infected or recovered nodes pS, pI and pR.
Because of the rapid turnover of partners, we find
that fS is the product of the probability that a stub
has not transmitted u with the probability it has just
joined with a susceptible partner pS so fS ¼ upS.
Similarly, fI ¼ upI and fR ¼ upR.
We create flow diagrams as before in figure 6. The S, I
and R diagram is unchanged, but the diagram for the f
variables and 1 2 u changes. There are no fS to fI or fI
to fR fluxes because of the explicit assumption that the
partners associated with a single stub any two times are
independent. The change in partner status is owing to
ending and forming partnerships rather than infection
or recovery of the partner. The flux into 1 2 u from fI is
bfI as before. We need a new flow diagram for pS, pI
and pR similar to that for S, I and R. Stubs belonging to
infected nodes become stubs belonging to recovered
nodes at rate g, thus p_ R ¼ gpI . We calculate pS explicitly:
the probability a stub belongs to a node of degree k is
the node is susceptible is uk.
Pn(k), and the probability
P
A weighted average kPn(k)uk gives pS ¼ uc 0 (u)/c 0 (1).
Finally, pI ¼ 1 2 pS 2 pR.
Combining these observations p_ R ¼ gpI ¼ gfI =u ¼
ðg=bÞu_ =u. So, pR ¼ 2( g/b) ln u (the constant of
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
6 Edge-based compartmental modelling
J. C. Miller et al.
(a)
(b)
f S = qp S
f I = qp I
gI
S = y (q)
f R = qp R
bf I
pS=
I
qy ¢(q)
y ¢(1)
pI
R
g pI
pR
1−q
Figure 6. Mean field social heterogeneity model. The flow diagram for the MFSH model (actual degree formulation). Because
partnerships are durationless, partners do not change status while joined to a node; so there is no flux between the f variables
(a). The new variables pS, pI and pR (bottom, b) represent the probability that a randomly selected stub belongs to a susceptible,
infected or recovered node. We can find pS in terms of u and then solve for pI and pR in much the same way we solve for I and R in
the configuration model. We then find that each f variable is u times the corresponding p variable. (Online version in colour.)
stubs belong to each class: P(1) = 25/31, P(5) ¼ 5/31
and P(25) = 1/31. Thus
integration is 0). We have
c ðuÞ
g
ln u
þ
pI ¼ 1 pS pR ¼ 1 u 0
c ð1Þ
b
0
cðxÞ ¼
and u_ ¼ bupI . Thus
u2 c 0 ðuÞ
ug ln u;
u_ ¼ bu þ b 0
c ð1Þ
ð3:1Þ
and
25x þ 5x 5 þ x 25
:
31
We set b ¼ g ¼ 1 and compare a simulation in a
population of 5 105 with theory in figure 7.
3.2. Dynamic fixed-degree
_ ¼ gI ;
R
S ¼ cðuÞ
and
I ¼ 1 S R:
ð3:2Þ
The MFSH model has been considered previously [2,17 –20], with the population stratified by
degree. Setting z to be the proportion of all stubs that
belong to infected nodes (equivalent to pI above), the
pre-existing system is
S_ k ¼ bkSk z;
and
I_ k ¼ bkSk z gIk
P
kPðkÞIk
;
z¼ k
kK l
where Sk and Ik are the probabilities a random individual
with k partners is susceptible or recovered. A known
change of variables reduces this to a few equations
equivalent to ours.
3.1.1. R0 and final size
We find
R0 ¼
b kK 2 l
;
g kK l
ð3:3Þ
consistent with previous observations [2]. The total
proportion infected is R(1) ¼ 1 2 c(u(1)), where
b
uð1Þc 0 ðuð1ÞÞ
:
ð3:4Þ
uð1Þ ¼ exp 1
c 0 ð1Þ
g
Full details are given in the electronic supplementary
material.
3.1.2. Example
We take a population with degrees 1, 5 and 25. The
proportions are chosen such that an equal number of
J. R. Soc. Interface
The DFD model interpolates between the CM
and MFSH models. We assign each node’s degree k
as before and pair stubs randomly. As time progresses, edges break. The freed stubs immediately join
with stubs from other edges that break, a process we
refer to as ‘edge swapping’. The rate an edge breaks
is h. A sample network at two times is shown in
figure 8.
We develop flow diagrams (figure 9) as before. The S,
I and R diagram is unchanged. We again track the
probabilities pS, pI and pR that a random stub belongs
to a susceptible, infected or recovered node, respectively. The diagram is unchanged. The diagram for u
and the f variables changes: we have fluxes from fS
to fI and fI to fR representing infection or recovery
of the partner as in the CM model, but we also have
fluxes from fS to fS, fI or fR resulting from edge swapping. We have similar edge swapping fluxes from fI and
fR. The flux into fS from edge swapping is hupS. The
flux out of fS from edge swapping is hfS. Similar
results hold for fI and fR.
Our earlier techniques to find fI break down. We
solve for fS and fI using ODEs. To complete the
system, we need the fS to fI flux. Consider a partner
v of our test node u such that: the stub belonging to u
never transmitted to u and the stub belonging to v
never transmitted to v prior to the u– v edge forming.
Given
P this, the probability v is susceptible is
q ¼ k kPðkÞuk1 =kK l ¼ c0 ðuÞ=c0 (1). Thus, given that
v is susceptible, v becomes infected at rate
_ 00 ðuÞ=c 0 ð1Þ
q_
uc
c 00 ðuÞ
¼
:
¼ 0
bf
I
q
c ðuÞ=c 0 ð1Þ
c 0 ðuÞ
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
1.0
0.25
0.8
0.20
infections
cumulative infections
Edge-based compartmental modelling
0.6
0.4
0.2
0
J. C. Miller et al. 7
0.15
0.10
0.05
0
5
10
0
15
5
0
t
10
15
t
Figure 7. Mean field social heterogeneity example (§3.1.2). Model predictions (dashed) match a simulated epidemic (solid) in a
population of 5 105 nodes. The solid curve is a single simulation. Time is set so t ¼ 0 when there is 1% cumulative incidence.
The degree distribution satisfies P(1) ¼ 25/31, P(5) ¼ 5/31 and P(25) ¼ 1/31. (Online version in colour.)
(a)
(b)
Figure 8. Sample dynamic fixed-degree network. (a) One moment in time and (b) a short time later. When edges break, the nodes
immediately form new edges with other nodes whose edges break simultaneously. In our simulations, edges break in pairs, but
that is not required for the theory. Half of the nodes have degree 3 and the other half have degree 1. (Online version in colour.)
hq
hqp I
hqp R
S = y (q)
hf S
pS =
qy ¢(q)
y ¢(1)
hf I
hqp S
hf R
fS
fI
bf I f S
y ¢¢(q)
y ¢(q)
fR
g fI
pI
I
g pI
gI
bf I
1−q
pR
R
Figure 9. Dynamic fixed-degree (DFD) model. The flow diagram for the DFD model. Unlike the configuration model case, we
cannot calculate fS explicitly, so we must calculate the fS to fI flux. (Online version in colour.)
Thus the fS to fI flux is the product of fS, the probability that a stub has not transmitted infection to the
test node and connects to a susceptible node, with
bfIc 00 (u)/c 0 (u), the rate the node becomes infected
given that the stub has not transmitted and connects
to a susceptible node. This completes figure 9.
The model requires more equations, but remains
relatively simple:
u_ ¼ bfI ;
J. R. Soc. Interface
ð3:5Þ
00
c ðuÞ
þ hupS hfS ;
f_ S ¼ bfI fS 0
c ðuÞ
ð3:6Þ
00
c ðuÞ
þ hupI ðb þ g þ hÞfI ;
f_ I ¼ bfI fS 0
c ðu Þ
p_ R ¼ gpI ;
pS ¼
uc 0 ðuÞ
c 0 ð1Þ
pI ¼ 1 pR pS
and
ð3:7Þ
ð3:8Þ
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
J. C. Miller et al.
1.0
0.12
0.8
0.10
infections
cumulative infections
8 Edge-based compartmental modelling
0.6
0.4
0.2
0.08
0.06
0.04
0.02
0
–5
0
5
t
10
15
0
–5
20
0
5
t
10
15
20
Figure 10. Dynamic fixed-degree example (§3.2.2). Model predictions (dashed) match the average of 102 simulated epidemics (solid)
in a population of 104 nodes. For each simulation, time is chosen so that t ¼
to 3% cumulative incidence. Then, they are
0 corresponds
kþr 1
ð1 pÞr pk for r ¼ 4, p ¼ 1/3. (Online version
averaged to give the solid curve. The degree distribution satisfies PðkÞ ¼
k
in colour.)
(a)
(b)
Figure 11. Sample dormant contact network. (a) One moment in time and (b) a short time later. Half the nodes have five stubs
and the other half have two stubs. When edges break, the stubs become dormant. Dormant stubs can join together to form edges.
(Online version in colour.)
and
_ ¼ gI ;
R
SðtÞ ¼ cðuÞ
and
I ðtÞ ¼ 1 S R:
ð3:9Þ
This is simpler than, but equivalent to, the model of
Volz & Meyers [16].
3.2.1. R0 and final size
We find
b
h þ g kK 2 K l h
:
þ
R0 ¼
ðb þ h þ g Þ
g
kK l
g
3.3. Dormant contacts
ð3:10Þ
We do not find a simple expression for final size.
Instead, we must solve the ODEs numerically. Full details
are given in the electronic supplementary material.
3.2.2. Example
We choose a population having negative binomial
degree distribution NB(4,1/3) with size parameter
r ¼ 4 and probability p ¼ 1/3. Thus PðkÞ ¼
kþr 1
ð1 pÞr pk . The mean is 2 and the vark
iance 3. For negative binomial distributions, c(x) ¼
[(1 2 p)/(1 2 px)]r; so
4
2
cðxÞ ¼
:
3x
J. R. Soc. Interface
We take b ¼ 5/4, g ¼ 1 and h ¼ 1/2. The equations
accurately predict the spread (figure 10).
Our simulations are slower because we must track
edges; so we have used smaller population sizes. To
reduce noise, we perform 250 simulations, averaging
the 102 that become epidemics.
We finally generalize the DFD model, allowing
stubs to enter a dormant phase after edges break.
This is the most general model we present. It reduces to
any model of this study in appropriate limits [43]. It is
appropriate for serial monogomy where individuals to
not immediately find a new partner.
A node is assigned km stubs using the
P probability
mass function P(km). We take cðxÞ ¼ km Pðkm Þx km .
A stub is dormant or active depending on whether it
is currently connected to a partner. The maximum
degree of a node is km and the ‘active’ and ‘dormant’
degrees are ka and kd, respectively, ka þ kd ¼ km. In
addition to fS, fI and fR, we add fD denoting the
probability that a stub is dormant and has never transmitted infection from a partner; so u ¼ fS þ fI þ fR þ
fD. Active stubs become dormant at rate h2 and dormant stubs become active at rate h1. A sample
network at two times is shown in figure 11.
We now develop flow diagrams (figure 12). The diagram for S, I and R is as before. The diagram for u and
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
Edge-based compartmental modelling
(a)
h1 pS f D
p
h1f D
fD
(c)
h2f S
h2f I
fS
fI
bf I f S
y ¢¢(q)
y ¢(q)
g fI
fR
pS =
f Dy ¢ (q)
y ¢ (1)
h2xS
xS=
(q – f D)y ¢ (q)
y ¢(1)
h1p S
S = y (q)
h1 p R f D
p
h1 p I f D
p
(b)
h2(q – f D)
J. C. Miller et al. 9
h2x I
h2f R
xI
I
gI
bf I
g xI
pI
h1p I
g pI
h2x R
1−q
pR
xR
R
h1p R
Figure 12. Dormant contact model. A flow diagram accounting for the dormant stage. (a) Movement of stubs between different
stages, including dormant. Stubs are classified by whether they have received infection and the status (or existence) of the current
partner. (b) The flow of individuals between different states. (c) Movement of stubs between states with stubs classified based on
the status of the node to which they belong. (Online version in colour.)
the f variables is similar to the DFD model, but with
the new compartment fD. The fluxes associated with
edge breaking are at rate h2 times fS, fI or fR and
go from the appropriate compartment into fD, for a
total of h2(u 2 fD). To describe fluxes owing to edge
creation, we generalize the definitions of pS, pI and
pR to give the probability that a stub is dormant
(and thus available to form a new partnership) and
belongs to a susceptible, infected or recovered node,
with p ¼ pS þ pI þ pR the probability a stub is dormant. The probability that a new partner is
susceptible, infected or recovered is pS/p, pI/p and
pR/p, respectively. The fluxes associated with edge
creation occur at total rate h1fD, with proportions
pS/p, pI/p and pR/p into fS, fI and fR, respectively.
The flow diagram for the p variables is related to
that for the DFD model, but we must account for
active and dormant stubs. We use jS, jI and jR to be
the probabilities that a stub is active and belongs to
each type of node, with 1 2 p ¼ j ¼ jS þ jI þ jR the
probability that a stub is active. The pS to jS and jS
to pS fluxes are h1pS and h2jS, respectively. Similar
results hold for the other compartments. We can use
this to show j_ ¼ h2 j þ h1 p ¼ h2 j þ h1 ð1 jÞ from
which we can conclude that (at equilibrium) p ¼ h2/
(h1 þ h2) and j ¼ h1/(h1 þ h2). The fluxes from jI
and pI to jR and pR, respectively, represent recovery
of the node the stub belongs to and so are gjI and
gpI, respectively.
We can calculate jS and pS explicitly. The probability a dormant stub belongs to a susceptible node
is pS ¼ fDc 0 (u)/c 0 (1), where fD is the probability
the dormant stub has never received infection, and
c 0 (u)/c 0 (1) is the probability that none of the other
stubs have received infection. Similarly, the probability
that an active stub belongs to a susceptible node is jS ¼
(u 2 fD)c 0 (u)/c 0 (1). We can use pI ¼ p 2 pS 2 pR
and jI ¼ j 2 jS 2 jR to simplify the system further.
Our new equations are
u_ ¼ bfI ;
J. R. Soc. Interface
ð3:11Þ
00
c ðu Þ
pS
þ h1 fD h2 fS ;
f_ S ¼ bfI fS 0
c ðu Þ
p
ð3:12Þ
00
c ðuÞ
pI
þ h1 fD ðh2 þ b þ gÞfI ; ð3:13Þ
f_ I ¼ bfI fS 0
c ðuÞ
p
f_ D ¼ h2 ðu fD Þ h1 fD ;
ð3:14Þ
0
c ðuÞ
j_ R ¼ h2 jR þ h1 pR þ gjI ; jS ¼ ðu fD Þ 0
c ð1Þ
and jI ¼ j jS jR ;
ð3:15Þ
0
p_ R ¼ h2 jR h1 pR þ gpI ; pS ¼ fD
c ðu Þ
c 0 ð1Þ
and pI ¼ p pS pR ;
h1
h2
j¼
and p ¼
h1 þ h2
h1 þ h2
ð3:16Þ
ð3:17Þ
and
_ ¼ gI ; S ¼ cðuÞ and I ¼ 1 S R:
R
ð3:18Þ
3.3.1. R0 and final size
We can show that
R0 ¼
b
b þ h2 þ g
2
kKm Km l h1 h2 þ g
h1 h2
:
þ
g ðg þ h 1 þ h 2 Þ
kKm l h1 þ h2 g
ð3:19Þ
However, we have not found a simple final size
relation. Details are given in the electronic supplementary
material.
3.3.2. Example
In figure 13, we consider the spread of a disease through
a network with dormant edges. The distribution of km is
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
10
Edge-based compartmental modelling J. C. Miller et al.
0.20
0.8
0.15
infections
cumulative infections
1.0
0.6
0.4
0.10
0.05
0.2
0
–5
0
5
t
10
15
0
–5
0
5
t
10
15
Figure 13. Dormant contact example (§3.3.2). The average of 155 simulated epidemics in a population of 5000 nodes (solid) with
a Poisson maximum degree distribution of mean 3 compared with theory (dashed). Simulations are shifted in time so that t ¼ 0
corresponds to 3% cumulative incidence and then averaged. Because stochastic effects are not negligible, the individual peaks are
not perfectly aligned, so the averaging has a small, but noticeable, effect to reduce and broaden the simulated peak. As population
size increases, this disappears. (Online version in colour.)
Poisson with mean 3
cðxÞ ¼ e3ð1xÞ :
The parameters are b ¼ 2, g ¼ 1, h1 ¼ g and h2 ¼ g/2.
Simulations are again slow; so we use a smaller population with 322 simulations, and average the 155 that
become epidemics.
4. EXPECTED DEGREE MODELS
We now consider SIR diseases spreading through
‘expected degree’ networks. In these networks, each individual has an expected degree k, which need not be an
integer. It is assigned using the probability density function r(k). Edges are placed between two nodes with
probability proportional to the expected degrees of the
two nodes. In the actual degree models, once a stub
belonging to u was joined into an edge, it became unavailable for other edges: the existence of a u– v edge reduced
the ability of u to form other edges. In contrast, for
expected degree models, edges are assigned independently: a u– v edge does not affect whether u forms
other edges. Similar to actual degree models, a partner
will tend to have larger expected degree than a randomly
chosen node. The probability density function for a
partner to have expected degree k is rn(k) ¼ kr(k)/kKl.
Our approach remains similar. We consider a randomly chosen test node u which cannot infect its
partners, and calculate the probability that u is susceptible. We first consider the spread of disease through
static ‘mixed Poisson’ (MP) networks (also called
Chung-Lu networks), for which an edge from u to v
exists with probability kukv/(N 2 1)kKl. We then
consider the expected degree formulation of MFSH.
Following this, we consider the more general dynamic
variable-degree (DVD) model, for which a node creates
and deletes edges as independent events, unlike the
DFD model where deleted edges were instantly replaced.
The MP model produces equations effectively identical
to the CM equations. The MFSH equations differ somewhat from the actual degree version, but may be shown
to be formally equivalent [43]. The DVD equations are
simpler than the DFD equations, and it may be more
J. R. Soc. Interface
Figure 14. Sample mixed Poisson model network. Half the
nodes have expected degree 3 and the other half 1.5. An
edge is assigned between two nodes with probability
proportional to the product of their expected degrees.
(Online version in colour.)
realistic because it does not enforce constant degree for
an individual.
4.1. Mixed Poisson
We now consider the MP model. In this model, each node
is assigned an expected degree k using the probability
density function r(k). A u– v edge exists with probability
kukv/(N 2 1)kKl independently of other edges. We use
the name ‘MP’ because at large N the actual degree of
nodes with expected degree k is chosen from a Poisson
distribution with mean k. The degree distribution is a
mixture of Poisson distributions. An example is shown
in figure 14.
Consider two nodes u and v whose expected degrees
are ku and kv ¼ ku þ Dk with Dk 1. Our question is,
how much does the additional Dk reduce the probability
that v is susceptible? At leading order, it contributes an
extra edge to v with probability Dk, and we may assume it
contributes at most one additional edge. We define Q
to be the probability that an edge has not transmitted
infection. With probability QDk there is an additional
edge that has not transmitted. The probability the
extra Dk either does not contribute an edge or contributes an edge that has not transmitted is 1 2 Dk þ
QDk ¼ 1 2 (1 2 Q)Dk. If s(k,t) is the probability
that a node of expected degree k is susceptible, then
we have s(k þ Dk, t) ¼ s(k, t)[1 2 (1 2 Q)Dk]. Taking
Dk ! 0, this becomes @ s/@ k ¼2(1 2 Q)s. Thus,
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
Edge-based compartmental modelling
J. C. Miller et al.
11
(a)
FS =
y¢Q
y ¢ (1)
FI
g FI
FR
b FI
(b)
gI
S = y (Q)
I
R
1−Q
Figure 15. Mixed Poisson (MP) model. (a) The flux of edges for a static MP network. (b) The flux of individuals through the
different stages. (Online version in colour.)
s(k,t) ¼ exp[2 k(1 2 Q)] and S(t) ¼ C(Q(t)), where
ð1
CðxÞ ¼
ekð1xÞ rðkÞ dk:
0
Note that this is the Laplace transform of r evaluated at
12x. As before, figure 15 gives
_ ¼ gI ;
R
S ¼ Cð Q Þ
and I ¼ 1 S R;
and we need Q(t).
We follow the CM approach almost exactly. The
value of Q is the probability that an edge has not transmitted to the test node u. We define FS, FI and FR to
be the probabilities that an edge has not transmitted to
u and connects to a susceptible, infected or recovered
node; so Q ¼ FS þ FI þ FR. To calculate FS, we
observe that a partner v of u with expected degree
k has the same probability of having an edge to any
w = u as any other node of expected degree k because
edges are created independently of one another. Thus
given k, v is susceptible with probability s(k,t).
Taking
the weighted
Ð1
Ð 1 average over all k gives FS ¼
r
(
k
)s(
k
,t)dt
¼
0 n
0 k exp[2 k(1 2 Q)]r(k)dk/kKl ¼
C 0 (Q)/C 0 (1). The same techniques as for the CM
networks give FR ¼ g(1 2 Q)/b. Our equations are
0
_ ¼ bQ þ b C ðQÞ þ gð1 QÞ;
Q
C 0 ð1Þ
_ ¼ gI ;
R
S ¼ CðQÞ and I ¼ 1 S R:
This is almost identical to CM epidemics except that C
and Q replace c and u . This similarity is explained
in [43].
4.1.1. R0 and final size
We find
2
R0 ¼
^ l
b kK
;
b þ g kK l
2
^ l is the average of k2 (which equals the averwhere kK
2
age of k 2 k) and kKl is the average of k (which equals
the average of k). The total proportion infected by an
epidemic is R(1) ¼ 1 2 C(Q(1)), where
Qð1Þ ¼
g
b C 0 ðQð1ÞÞ
:
þ
b þ g b þ g C 0 ð1Þ
Full details are given in the electronic supplementary
material.
J. R. Soc. Interface
4.1.2. Example
We consider a population whose distribution of expected
degrees satisfies
81
0k2
>
<4
1
rðkÞ ¼ 20 10 k 20
>
:
0
otherwise:
Half the individuals have an expected degree between 0
and 2 uniformly, and the other half have expected
degree between 10 and 20 uniformly. This gives
CðxÞ ¼
1
e20ðx1Þ e10ðx1Þ
:
e2ðx1Þ 1 þ
5
4ðx 1Þ
We take b ¼ 0.15 and g ¼ 1, and perform simulations with a population of size 5 105 generated
using the O(N ) algorithm of Miller & Hagberg [44].
We compare simulation and prediction in figure 16.
4.2. Mean field social heterogeneity
For the expected degree formulation of the MFSH model,
the probability that an edge exists between u and v
at time t is kukv/(N 2 1)kKl. Whether this edge exists
at one moment is independent of whether it exists at
any other moment and what other edges exist.
A sample network at two times is shown in figure 17.
As before, we consider two nodes whose expected
degrees differ by Dk and ask how much the additional
Dk reduces the probability of being susceptible. The
definition of Q is slightly more problematic here because
having an edge at one moment is independent of having
one later. So it does not make sense to discuss the probability that an edge did not transmit previously because
the edge did not exist previously. Instead, we note that
in the MP case, (1 2 Q)Dk could be interpreted as the
probability that the additional amount of Dk ever contributed an edge that had transmitted infection.
Guided by this, we define Q so that as Dk ! 0, the
extra amount of Dk has at some time contributed an
edge that transmitted to the node with probability
(1 2 Q)Dk. This again leads to
ð1
CðxÞ ¼
ekð1xÞ rðkÞdk:
e
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
Edge-based compartmental modelling J. C. Miller et al.
1.0
0.10
0.8
0.08
infections
cumulative infections
12
0.6
0.4
0.06
0.04
0.2
0
–5
0.02
0
5
10
15
0
–5
20
0
5
10
t
15
20
t
Figure 16. Mixed Poisson (MP) example (§4.1.2). Model predictions (dashed) match a simulated epidemic (solid) in a MP network with 5 105 nodes. The solid curve is a single simulation. Time is chosen so that t ¼ 0 corresponds to 1% cumulative
incidence. With probability 0.5, expected degrees are chosen uniformly from [0,2], and with probability 0.5 they are chosen uniformly from [10,20]. (Online version in colour.)
(a)
(b)
Figure 17. Sample mean field social heterogeneity network. (a) One moment in time and (b) an arbitrarily short time later. Half
the nodes have expected degree 4 and the others expected degree 1. All edges change, and the actual degree varies with time.
(Online version in colour.)
The flow diagram for S, I and R (figure 18) is
unchanged:
_ ¼ gI ;
R
S ¼ CðQÞ and
I ¼ 1 S R:
To find the evolution of Q, we define FS, FI and FR
to be the probabilities that a current edge connects u
to a susceptible, infected or recovered node, respectively. The probability that a small extra amount Dk
currently contributes an edge and previously had a
different edge that transmitted scales like Dk2(1 2 Q).
As Dk2 Dk, this is negligible compared with the probability that there is a current edge. We conclude that
at leading order, FSDk , FIDk and FRDk give the probability that the Dk contributes a current edge
connected to a susceptible, infected or recovered node
and there has never been a transmission owing to this
extra Dk.
We can construct a flow diagram between FS Dk ,
FIDk, FRDk and (1 2 Q)Dk. Because all of these
have Dk in them, we factor it out to just use FS, FI,
FR and 1 2 Q. Because edges have no duration, there
is no FS to FI or FI to FR flux. Instead, there is flux
in and out of these compartments representing the continuing change of edges. The FI to 1 2 Q flux is bFI.
Because edges have no duration, the probability
that an edge connects to an individual of a given type
is the probability a new edge connects to an individual
of that type: FS ¼ PS , FI ¼ PI and FR ¼ PR where
PS, PI and PR are the probabilities that a newly
J. R. Soc. Interface
formed edge connects to a susceptible, infected or recovered node, respectively.1 We have PS ¼ C 0 (Q)/C 0 (1)
_ =b.
_ R ¼ gQ
_ R ¼ gPI . As PI ¼ FI, this means P
and P
Integrating gives PR ¼ g (1 2 Q)/b. So
PI ¼ 1 C 0 ðQÞ gð1 QÞ
:
b
C0 ð1Þ
_ ¼ bFI ¼ bPI , finally, we have
Since Q
0
_ ¼ b þ b C ðQÞ þ gð1 QÞ;
Q
C 0 ð1Þ
and
_ ¼ gI ;
R
S ¼ CðQÞ
and
I ¼ 1 S R:
Under appropriate limits, the MFSH equations in
k reduce to those in k and vice versa; so the models are
equivalent [43]. Surprisingly, this system differs from the
_ equation.
MP equations only in the first term of the Q
4.2.1. R0 and final size
We find
R0 ¼
^ 2l
b kK
;
g kK l
and the final size is R(1) ¼ 1 2 C(Q(1)), where Q(1)
1
Unlike the actual degree formulation, we do not need a factor of Q in
these because the smallness of Dk allows us to assume that there has
never been a previous transmission.
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
Edge-based compartmental modelling
(a)
J. C. Miller et al.
13
(b)
gI
S = y (Q)
FI = PI
FS = P S
I
R
FR = PR
b FI
PS =
1−Q
y ¢ (Q)
y ¢ (1)
PI
g PI
PR
Figure 18. Mean field social heterogeneity (MFSH) model. The flow diagram for a MFSH population (expected degree formulation). This is similar to the actual degree formulation in figure 6. The new variables PS, PI and PR (bottom, b) represent
the probability that a newly formed edge connects to a susceptible, infected or recovered node; they can be thought of as the
relative rates at which each group forms edges. As the test node does not cause infections, and the probability that a partnership
is with a node of a given k is equal to the probability that a new partner will be with a node of the given k, the probability that a
current partner has a given state is equal to the probability that a new partner has that state. Thus each F variable equals the
corresponding P variable. (Online version in colour.)
0.0007
0.0006
0.08
0.0005
infections
cumulative infections
0.10
0.06
0.04
0.0004
0.0003
0.0002
0.02
0.0001
0
–100 –50
0
50
100 150 200 250 300
t
0
–100 –50
0
50
100 150 200 250 300
t
Figure 19. Expected degree mean field social heterogeneity (MFSH) example (§4.2.2). Model predictions (dashed) match a simulated
epidemic (solid) in a MFSH network with 15 106 nodes. The solid curve is a single simulation. Time is chosen so that t ¼ 0 corresponds
to 0.5% cumulative incidence. Expected degrees are chosen with r(k) ¼ ek/(e3 21) for 0, k ,3. (Online version in colour.)
solves
Q¼
0
b
C ðQ Þ
þ 1:
1þ 0
C ð1Þ
g
Full details are given in the electronic supplementary
material.
4.2.2. Example
For our example, we take a population with r(k) ¼ ek/
(e3 2 1) for 0 , k , 3 and 0 otherwise, giving
CðxÞ ¼
e3x 1
:
xðe3 1Þ
We take g ¼ 1 and b ¼ 0.435 and compare simulation with theory in figure 19. We choose these
parameters to demonstrate that the approach remains
accurate for small R0 ¼ 1.04. We use a population of
15 106. There is noise as the epidemic does not
infect a large number of people.
4.3. Dynamic variable-degree
The DVD model interpolates between the MP model and
the expected degree formulation of the MFSH model.
Each node is assigned k using r(k) and creates edges at
rate kh ( joining to another node also creating an edge).
J. R. Soc. Interface
Existing edges break at rate h. Thus, a node has expected
degree k, though its value varies around k. In fact, it is
Poisson-distributed over time. A sample network at
two times is shown in figure 20.
We define Q such that the probability that a small Dk
has ever contributed an edge that has transmitted infection is (1 2 Q)Dk . We again have
_ ¼ gI ;
R
S ¼ CðQÞ and
I ¼ 1 S R:
We generalize the earlier definitions and define FS,
FI and FR to be the probabilities a current edge
has never transmitted infection and connects to a
susceptible, infected or recovered node.
We define PS, PI and PR to be the probabilities a
new edge connects to a susceptible, infected or recovered node. We have PS ¼ C 0 (Q)/C 0 (1), PI ¼ 1 2
_ R ¼ gPI . We build the flow diagram
PS 2 PR and P
for FSDk, FIDk, FRDk and (1 2 Q)Dk. There is flux
into FSDk at rate hPSDk because this is the rate at
which the Dk leads to edge creation. There is flux out
of FSDk at rate hFSDk because existing edges break
at rate h, and the probability that such an edge exists
is FSDk. Similar fluxes exist for FI and FR. The flux
out of FIDk into FRDk is gFIDk as before, and the
flux into (1 2 Q)Dk is bFIDk. We factor out Dk and
the flow diagrams (figure 21) are defined.
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
14
Edge-based compartmental modelling J. C. Miller et al.
(a)
(b)
Figure 20. Sample dynamic variable-degree network. (a) One moment in time and (b) a short time later. The formation or deletion of
an edge has no impact on other edges. Half the nodes have expected degree 4 and the other half 1. (Online version in colour.)
hP S
hF S
hP I
hF I
hP R
hF R
S = y (Q)
y ¢ (Q)
FS =
y ¢ (1)
FI
g FI
I
gI
R
FR
b FI
PS =
y ¢ (Q)
y ¢ (1)
PI
g PI
PR
1−Q
Figure 21. Dynamic variable-degree (DVD) model. The flow diagrams for the DVD model. (Online version in colour.)
Because the existence of an edge from the test node u
to the partner v has no impact on any other edges v
might have, the partners v has—aside from u—are indistinguishable from the partners of another node with the
same k, and so they are susceptible with the same probability: FS ¼ PS. The fluxes into and out of FS from
edge creation–deletion balance, and the FS to FI flux
is simply Ḟ S . Using this and the other fluxes for FI,
we conclude Ḟ I ¼ Ḟ S þhPI ðh þ g þ bÞFI . As
Ṗ R ¼ gPI and Q̇ ¼ bFI , we can integrate this and
find
FI ¼ C 0 ðQÞ h
ðb þ h þ gÞ
hþg
þ PR þ
Q
:
C 0 ð1Þ
b
g
b
previous transmission occurred for the Dk, whereas for
a discrete stub, we cannot neglect previous transmission.
4.3.1. R0 and final size
We find
R0 ¼
^ 2l
b
h þ g kK
;
b þ h þ g g kK l
The total proportion infected is R(1) ¼ 1 2 C(Q(1)),
where
b
h þ g C 0 ðQð1ÞÞ h þ g h
þ
Qð1Þ ¼
:
bþhþg
g
b
g
C 0 ð1Þ
_ ¼ bFI can be written in terms of Q and PR. We
So Q
arrive at
Full details are given in the electronic supplementary
material.
C 0 ðQ Þ
þ gð1 QÞ
Q̇ ¼ bQ þ b 0
C ð1Þ
b
þ h 1 Q PR ;
g
C 0 ðQ Þ
and
Ṗ R ¼ gPI ; PS ¼ 0
C ð1Þ
4.3.2. Example
We choose the same distribution of k as of k in the DFD
example, NB(4,1/3). We find
4
2
CðxÞ ¼
:
3 ex1
ð4:1Þ
ð4:2Þ
PI ¼ 1 PS PR
and
_ ¼ gI ;
R
S ¼ CðQÞ and
I ¼ 1 S R:
ð4:3Þ
This is simpler than the DFD case because the arbitrary smallness of Dk allowed us to assume that no
J. R. Soc. Interface
We take the same parameters b ¼ 5/4, g ¼ 1 and h ¼
1/2. Figure 22 shows that the equations accurately predict
the spread. As in the DFD and DC case, we use an average
as the comparison point, taking 240 simulations and
averaging the 92 resulting in epidemics.
The final size is larger than for the DFD model.
Although the average numbers of partners are all the
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
Edge-based compartmental modelling
0.8
infections
cumulative infections
1.0
0.6
0.4
0.2
0
–4
–2
0
2
4
6
8
10
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0
–4
–2
0
t
2
4
J. C. Miller et al.
6
8
15
10
t
Figure 22. Dynamic variable-degree example (§4.3.2). Model predictions (dashed) match the average of 92 simulated epidemics
(solid) in a population of 104 nodes. For each simulation, time is chosen so that t ¼ 0 corresponds to 3% cumulative incidence.
Then, they are averaged to give the solid curve. k is chosen from the same distribution as k in figure 10. (Online version in colour.)
same, the increase in transmission routes when an individual has more partners than expected outweighs the
decrease when the number is less than expected. The
net effect is to increase the final size.
5. DISCUSSION
We have introduced a new approach to study the spread
of infectious diseases. This edge-based compartmental
modelling approach allows us to simultaneously consider the impacts of partnership duration and social
heterogeneity. It is conceptually simple and leads to
equations of comparable simplicity to the MA model.
It produces a broad family of models that contains several known models as special cases. It further allows us
to investigate the effect of many behaviours that have
previously been inaccessible to analytical study.
A significant contribution of this work is that it
allows us to study the spread of a disease in a population
for which some individuals have different propensity
to form partnerships while allowing us to explicitly
incorporate the impact of partnership duration. The
interaction of partnership duration and numbers of overlapping partnerships plays a significant role in the spread
of many diseases, and in particular may play an important role in the spread of HIV [45]. These techniques
open the door to studying these questions analytically
rather than relying on simulation.
The edge-based compartmental modelling approach
has a simple, graphical interpretation through flow
diagrams. This simplifies model description, and guides
generalizations. We propose that this is the correct perspective to study the deterministic dynamics of SIR
epidemics on random networks because the derivations
are straightforward, we need to track only a small
number of compartments and we do not require any closure approximations. Other existing techniques rely on
approximations [23] or produce complicated or large systems of equations [24– 26]. This approach does not offer
immediate insight into stochastic effects where methods
such as branching processes are more appropriate. In
later studies, we use this approach to derive other generalizations, including different correlations in population
structure, different disease dynamics and populations
that change structure in response to the spreading
disease.
J. R. Soc. Interface
Our approach is limited by the assumption that
infection of one partner of u can be treated as independent of that of another partner. This is a strong
assumption, and prevents us from applying this model
to susceptible – infected– susceptible (SIS) diseases for
which individuals return to a susceptible state. In this
case, the assumption that u does not infect its partners
can alter the future state of u. In the real population, if
u becomes infected, it can infect partners who then
infect u when u returns to a susceptible state. Thus if
partnership duration is nonzero, our predictions may be
significantly altered. This limitation is often not recognized but may lead to important failures of mean-field
or MA models when applied to a population for which
partnership duration is important [46].
Treating partners as independent also means that if v
and w are partners of u, we assume no alternate short
path between v and w exists. In particular, we assume
that clustering [47,48] is negligible: partners are unlikely
to see one another. This assumption applies equally to
most existing analytical epidemic models, but it can
be eliminated in very special cases using techniques
similar to those of earlier studies [49– 52].
J.C.M. was supported by the RAPIDD programme of the
Science and Technology Directorate, Department of Homeland
Security and the Fogarty International Center, National
Institutes of Health and the Center for Communicable Disease
Dynamics, Department of Epidemiology, Harvard School of
Public Health under award number U54GM088558 from the
National Institute Of General Medical Sciences. Much of this
work was a result of ideas catalysed by the China–Canada
Colloquium on Modelling Infectious Diseases held in Xi’an,
China in September 2009. E.M.V. was supported by NIH K01
AI091440. The content is solely the responsibility of the
authors and does not necessarily represent the official views of
the National Institute of General Medical Sciences or the
National Institutes of Health. We thank S. Bansal,
M. Lipsitch, R. Meza, B. Pourbohloul, P. Trapman and
J. Wallinga for useful conversations.
REFERENCES
1 Kermack, W. O. & McKendrick, A. G. 1927 A contribution
to the mathematical theory of epidemics. Proc. R. Soc.
Lond. A 115, 700–721. (doi:10.1098/rspa.1927.0118)
2 Anderson, R. M. & May, R. M. 1991 Infectious diseases of
humans. Oxford, UK: Oxford University Press.
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
16
Edge-based compartmental modelling J. C. Miller et al.
3 Lloyd, A. L. & May, R. M. 2001 How viruses spread
among computers and people. Science 292, 1316–1317.
(doi:10.1126/science.1061076)
4 Kiss, I. Z., Green, D. M. & Kao, R. R. 2006 The network
of sheep movements within Great Britain: network
properties and their implications for infectious disease
spread. J. R. Soc. Interface 3, 669–677. (doi:10.1098/rsif.
2006.0129)
5 Rohani, P., Zhong, X. & King, A. A. 2010 Contact network
structure explains the changing epidemiology of pertussis.
Science 330, 982– 985. (doi:10.1126/science.1194134)
6 Eubank, S., Guclu, H., Anil Kumar, V. S., Marathe, M. V.,
Srinivasan, A., Toroczkai, Z. & Wang, N. 2004 Modelling
disease outbreaks in realistic urban social networks.
Nature 429, 180 –184. (doi:10.1038/nature02541)
7 Germann, T. C., Kadau, K., Longini Jr, I. M. & Macken,
C. A. 2006 Mitigation strategies for pandemic influenza
in the United States. Proc. Natl Acad. Sci. USA 103,
5935–5940. (doi:10.1073/pnas.0601266103)
8 Liljeros, F., Edling, C. R., Amaral, L. A. N., Stanley, H. E.
& Åberg, Y. 2001 The web of human sexual contacts.
Nature 411, 907 –908. (doi:10.1038/35082140)
9 Mossong, J. et al. 2008 Social contacts and mixing patterns
relevant to the spread of infectious diseases. PLoS Med. 5,
381– 391. (doi:10.1371/journal.pmed.0050074)
10 Christakis, N. A. & Fowler, J. H. 2010 Social network
sensors for early detection of contagious outbreaks. PLoS
ONE 5, e12948. (doi:10.1371/journal.pone.0012948)
11 Andersson, H. 1997 Epidemics in a population with social
structures. Math. Biosci. 140, 79 –84. (doi:10.1016/S00255564(96)00129-0)
12 Newman, M. E. J. 2002 Spread of epidemic disease on networks. Phys. Rev. E 66, 016128. (doi:10.1103/PhysRevE.
66.016128)
13 Miller, J. C. 2007 Epidemic size and probability in populations with heterogeneous infectivity and susceptibility.
Phys. Rev. E 76, 010101. (doi:10.1103/PhysRevE.76.
010101)
14 Kenah, E. & Robins, J. M. 2007 Second look at the spread
of epidemics on networks. Phys. Rev. E 76, 036113.
(doi:10.1103/PhysRevE.76.036113)
15 Meyers, L. A., Pourbohloul, B., Newman, M. E. J., Skowronski, D. M. & Brunham, R. C. 2005 Network theory and
SARS: predicting outbreak diversity. J. Theor. Biol. 232,
71–81. (doi:10.1016/j.jtbi.2004.07.026)
16 Volz, E. M. & Meyers, L. A. 2007 Susceptible–infected –
recovered epidemics in dynamic contact networks.
Proc. R. Soc. B 274, 2925–2933. (doi:10.1098/rspb.
2007.1159)
17 May, R. M. & Anderson, R. M. 1988 The transmission
dynamics of human immunodeficiency virus (HIV). Phil.
Trans. R. Soc. Lond. B 321, 565 –607. (doi:10.1098/rstb.
1988.0108)
18 May, R. M. & Lloyd, A. L. 2001 Infection dynamics on
scale-free networks. Phys. Rev. E 64, 066112. (doi:10.
1103/PhysRevE.64.066112)
19 Moreno, Y., Pastor-Satorras, R. & Vespignani, A. 2002 Epidemic outbreaks in complex heterogeneous networks. Eur.
Phys. J. B 26, 521–529. (doi:10.1140/epjb/e20020122)
20 Pastor-Satorras, R. & Vespignani, A. 2001 Epidemic
spreading in scale-free networks. Phys. Rev. Lett. 86,
3200–3203. (doi:10.1103/PhysRevLett.86.3200)
21 Diekmann, O., De Jong, M. C. M. & Metz, J. A. J. 1998 A
deterministic epidemic model taking account of repeated
contacts between the same individuals. J. Appl. Probab.
35, 448 –462. (doi:10.1239/jap/1032192860)
22 Andersson, H. 1999 Epidemic models and social networks.
Math. Sci. 24, 128 –147.
J. R. Soc. Interface
23 Eames, K. T. D. & Keeling, M. J. 2002 Modeling dynamic
and network heterogeneities in the spread of sexually
transmitted diseases. Proc. Natl Acad. Sci. USA 99,
13 330–13 335. (doi:10.1073/pnas.202244299)
24 Volz, E. M. 2008 SIR dynamics in random networks with
heterogeneous connectivity. J. Math. Biol. 56, 293–310.
(doi:10.1007/s00285-007-0116-4)
25 Lindquist, J., Ma, J., van den Driessche, P. &
Willeboordse, F. H. 2010 Effective degree network disease
models. J. Math. Biol. 62, 143– 164. (doi:10.1007/s00285010-0331-2)
26 Ball, F. & Neal, P. 2008 Network epidemic models with
two levels of mixing. Math. Biosci. 212, 69– 87. (doi:10.
1016/j.mbs.2008.01.001)
27 Noël, P.-A., Davoudi, B., Brunham, R. C., Dubé, L. J. &
Pourbohloul, B. 2009 Time evolution of disease spread
on finite and infinite networks. Phys. Rev. E 79, 026101.
(doi:10.1103/PhysRevE.79.026101)
28 Miller, J. C. 2011 A note on a paper by Erik Volz: SIR
dynamics in random networks. J. Math. Biol. 62,
349– 358.
29 House, T. & Keeling, M. J. 2010 Insights from unifying
modern approximations to infections on networks.
J. R. Soc. Interface 8, 67 –73. (doi:10.1098/rsif.2010.0179)
30 Wallinga, J., Teunis, P. & Kretzschmar, M. 2006
Using data on social contacts to estimate age-specific
transmission parameters for respiratory-spread infectious
agents. Am. J. Epidemiol. 164, 936. (doi:10.1093/aje/
kwj317)
31 Salathé, M., Kazandjiev, M., Lee, J. W., Levis, P., Feldman,
M. W. & Jones, J. H. 2010 A high-resolution human contact
network for infectious disease transmission. Proc. Natl
Acad. Sci. USA 107, 22 020–22 025. (doi:10.1073/pnas.
1009094108)
32 Molloy, M. & Reed, B. 1995 A critical point for random
graphs with a given degree sequence. Random Struct.
Algorithms 6, 161–179. (doi:10.1002/rsa.3240060204)
33 Newman, M. E. J., Strogatz, S. H. & Watts, D. J. 2001
Random graphs with arbitrary degree distributions and
their applications. Phys. Rev. E 64, 026118. (doi:10.
1103/PhysRevE.64.026118)
34 van der Hofstad, R.. 2010 Random graphs and complex
networks. Eindhoven, The Netherlands: Eindhoven
University of Technology.
35 Chung, F. & Lu, L. 2002 Connected components in
random graphs with given expected degree sequences.
Ann. Comb. 6, 125–145. (doi:10.1007/PL00012580)
36 Norros, I. & Reittu, H. On a conditionally Poissonian
graph process. Adv. Appl. Probab. 38, 59–75. (doi:10.
1239/aap/1143936140)
37 Britton, T., Deijfen, M. & Martin-Löf, A. 2006 Generating
simple random graphs with prescribed degree distribution.
J. Stat. Phys. 124, 1377–1397. (doi:10.1007/s10955-0069168-x)
38 Feld, S. L. 1991 Why your friends have more friends than
you do. Am. J. Sociol. 96, 1464–1477. (doi:10.1086/
229693)
39 Diekmann, O. & Heesterbeek, J. A. P. 2000 Mathematical
epidemiology of infectious diseases. London, UK: Wiley
Chichester.
40 Wilf, H. S. 2005 Generatingfunctionology, 3rd edn. Natick,
MA: AK Peters Ltd.
41 Andersson, H. 1998 Limit theorems for a random graph
epidemic model. Ann. Appl. Probab. 8, 1331 –1349.
(doi:10.1214/aoap/1028903384)
42 Trapman, P. 2007 On analytical approaches to epidemics
on networks. Theor. Popul. Biol. 71, 160–173. (doi:10.
1016/j.tpb.2006.11.002)
Downloaded from rsif.royalsocietypublishing.org on October 20, 2014
Edge-based compartmental modelling
43 Miller, J. C. & Volz, E. M. Submitted. Model hierarchies
in edge-based compartmental modeling for infectious disease spread.
44 Miller, J. C. & Hagberg, A. 2011 Efficient generation of networks with given expected degrees. In Proc. Algorithms and
Models for the Web Graph—8th International Workshop,
‘WAW 2011’, Atlanta, GA, USA, 27– 29 May 2011. Lecture
Notes in Computer Science (eds M. Frieze, P. Horn,
P. Pralat), vol. 6732, pp. 115 –126. Berlin, Germany:
Springer.
45 Morris, M. & Kretzschmar, M. 1995 Concurrent partnerships and transmission dynamics in networks. Soc.
Networks
17, 299 –318. (doi:10.1016/0378-8733(95)
00268-S)
46 Chatterjee, S. & Durrett, R. 2009 Contact processes on
random graphs with power law degree distributions have
critical value 0. Ann. Probab. 37, 2332 –2356. (doi:10.
1214/09-AOP471)
J. R. Soc. Interface
J. C. Miller et al.
17
47 Watts, D. J. & Strogatz, S. H. 1998 Collective dynamics of
‘small-world’ networks. Nature 393, 409–410. (doi:10.
1038/30918)
48 Newman, M. E. J. Properties of highly clustered networks.
Phys. Rev. E 68, 026121. (doi:10.1103/PhysRevE.68.026121)
49 Volz, E. M., Miller, J. C., Galvani, A. & Meyers, L. A.
2011 The effects of heterogeneous and clustered contact
patterns on infectious disease dynamics. PLoS Comput.
Biol. 7, e1002042. (doi:10.1371/journal.pcbi.1002042)
50 Miller, J. C. 2009 Percolation and epidemics in random
clustered networks. Phys. Rev. E 103, 020901(R).
(doi:10.1103/PhysRevE.80.020901)
51 Newman, M. E. J. 2009 Random graphs with clustering.
Phys. Rev. Lett. 103, 058701. (doi:10.1103/PhysRevLett.
103.058701)
52 Karrer, B. & Newman, M. E. J. 2010 Random graphs
containing arbitrary distributions of subgraphs. Phys.
Rev. E 82, 066118. (doi:10.1103/PhysRevE.82.066118)