The Future of Modelling
Networks
Matt Keeling
Warwick Infectious Disease Epidemiological Research
(WIDER) Centre
The Future of
Modelling Networks
1) Need for multiple model types
– beyond simulations.
2) Approximation models
– successes & failures.
3) Looking to the future.
Simulations are Superb!
The are many situations when we ‘know’ the network. In these cases it is
theoretically trivial although computationally demanding to run multiple
simulations on the network.
Simulations are Superb!
The are many situations when we ‘know’ the network. In these cases it is
theoretically trivial although computationally demanding to run multiple
simulations on the network.
Ferguson 2005
Simulations are Superb!
The are many situations when we ‘know’ the network. In these cases it is
theoretically trivial although computationally demanding to run multiple
simulations on the network.
Longini 2005-7
Simulations are Superb!
The are many situations when we ‘know’ the network. In these cases it is
theoretically trivial although computationally demanding to run multiple
simulations on the network.
Keeling 2009
So what’s the problem?
The problem comes when we don’t really know the network.
All measured networks are approximations to reality and in general tell us
about the past. What we usually need is:
•
predictions about the future, so we need future networks
•
some way to capture the uncertainties in network structure
•
an intuitive way to access the impact of networks.
Essentially we’d like all the tools and understanding we’ve gained over the
last 100 years for ODE models of epidemics applied to networks.
Approximation Models
Essentially we’d like all the tools and understanding we’ve gained over the
last 100 years for ODE models of epidemics applied to networks.
Approximation models fill this role, by constructing ODE models that capture
elements of network structure. Four main approaches:
1. Branching theory / Susceptiblity sets (Diekmann, Ball)
2. PGF models (Volz)
3. Probabilistic edges (Sharkey)
4. Pairwise models (Rand, Keeling, Eames, House, Kiss)
All of these produce equivalent results for the standard SIR model on an
unclustered / configuration network.
Formulating Models
In the standard equations for epidemics, the only
term that involves interaction is the transmission
term.
This transmission can only occur when there is
an infected individual connected to a
susceptible individual. It we label the number
of such connected pairs [SI], then the
equations become:
dS = B – b S I – dS
dt
dI = b S I – gI – dI
dt
dR = gI – dR
dt
Formulating Models
In the standard equations for epidemics, the only
term that involves interaction is the transmission
term.
This transmission can only occur when there is
an infected individual connected to a
susceptible individual. We label the number of
such connected pairs [SI], then the equations
become:
dS = B – t [S I] – dS
dt
dI = t [S I] – gI – dI
dt
dR = gI – dR
dt
These new equations are exact for a disease spreading through a network,
the only difficulty is that we do not know [SI].
We are therefore left with two choices:

we approximate [SI] in terms of what we do know – leads to standard
eqns.

we formulate a new equation for [SI].
Formulating Pair-wise Models
Destruction
[SI] PAIRS
Creation
S
R
Recovery of the infected
member of the SI pair.
I
S
S
S
I
Infection of an SS pair
by an external source.
Requires an SSI triple.
I
I
I
Infection of susceptible
from within the pair
I
S
I
I
Infection of susceptible
from outside the pair
d[SI] = t [SSI]  g [SI]  t [SI]  t [ISI]
dt
Pair-wise: Formulation of Triples
THE TRIPLE APPROXIMATION
1) In the simplest scenario, we can conceptualise a triple as two overlapping pairs, which share the middle individual.
S
I
R
[SI] [IR]
thus [SIR] = n-1 [SI][IR] when all individuals have n connections.
n
[I]
Pair-wise: Formulation of Triples
THE TRIPLE APPROXIMATION
1) In the simplest scenario, we can conceptualise a triple as two overlapping pairs, which share the middle individual.
S
I
R
[SI] [IR]
thus [SIR] = n-1 [SI][IR] when all individuals have n connections.
n
[I]
2) This enables us to close the equations and simulate an epidemic.
3) This has been shown to be exact for the SIR model on a regular
unclustered network.
Pair-wise: Formulation of Triples
Results for a
network of ten
thousand, and
exactly 4 contacts
per individual.
3) This has been shown to be exact for the SIR model on a regular
unclustered network.
So what’s the problem?
Obviously these approximation methods aren’t perfect, so what are the
challenges for the future:
1) SIS dynamics for sexually transmitted infections
2) Clustering
3) Extreme levels of heterogeneity
4) Dynamic networks
I’ll look at each of these in turn and suggest possible avenues of attack.
SIS models
Studying sexually transmitted infections (STIs) is an ideal use of networks – given
that the network is often well-known and easier to define. However the SISmodel formulation used for STIs does not approximate well:
SIS models
Studying sexually transmitted infections (STIs) is an ideal use of networks – given
that the network is often well-known and easier to define. However the SISmodel formulation used for STIs does not approximate well:
Results for a
network of ten
thousand, and
exactly 3 contacts
per individual.
SIS models
Studying sexually transmitted infections (STIs) is an ideal use of networks – given
that the network is often well-known and easier to define. However the SISmodel formulation used for STIs does not approximate well:
Degree k=3
Degree k=3
15
0.8
0.6
0.4
0
2
4
6
Transmission Rate, t
8
0.04
Simulation
ODE
Pairwise
New Pairwise
Star Model
5
2
4
6
8
10
4
6
8
10
Transmission Rate, t
1
0.03
0.02
0.01
0
0
10
0
0
10
D Growth Rate
D Mean Prevelance
Simulation
ODE
Pairwise
New Pairwise
Star Model
Growth Rate, r
Mean Prevelance
1
0.8
0.6
0.4
0.2
0
2
4
6
Transmission Rate, t
8
0
10
2
Transmission Rate, t
There is a need for
more theoretical
development in this
area – but potential for
impact is huge.
Relative Error (%)
Relative Error (%)
100
10
5
0
0
2
4
6
Transmission Rate, t
8
10
80
60
40
20
0
0
2
4
6
Transmission Rate, t
These preliminary
results suggest that
counting infection
episodes (cyan) allows
growth rates to be
estimated; while the
prevalence is only
captured when entire
neighbourhoods are
models.
8
10
Clustering
Understanding the implications of triangles in a network has long been a goal of
network modelling. Some recent work is highlighting the important factors:
The impact of clustering
depends what we hold
constant as clustering is
added:
• Transmission rate
• Early growth
• R0
Clustering
Understanding the implications of triangles in a network has long been a goal of
network modelling. Some recent work is highlighting the important factors:
Studying the simplest clustered network
(a single triangle) in extreme detail has
highlighted the problems and potential
solutions.
We find it is the random
recovery of individuals that
breaks any of the
approximation
models.
Maximum Entropy
appears to offer the
smallest errors.
Clustering
Understanding the implications of triangles in a network has long been a goal of
network modelling. Some recent work is highlighting the important factors:
Household models as formulated by Ball
and co-workers represent highly clustered
networks where analytical traction is
possible.
Can these insights be used to underpin
novel approximation models.
In general, new theoretical
approaches are required to
create the next generation
of models.
Extreme Heterogeneity
Extreme Heterogeneity
Multiple solutions to the problem of heterogeneous degree (number of contacts).
• The pairwise models can be expanded to include degree:
[SnIm] is the number of susceptibles with n contacts linked to infecteds with m.
• Sharkey’s methods capture the correlation between individuals in a network,
although it requires the full network to be known or approximated.
• Volz’s PGF models automatically incorporates
degree heterogeneity, but implicitly assume
a configuration network – ie random
connections.
Extreme Heterogeneity
A bigger problem is the heterogeneity in local structure:
• some links are strong, long and frequent, others are weak and rare.
• some contacts are interconnected (clustered) others are not.
• there will be correlations between the demographics of connected individuals –
eg age, occupation, gender etc.
Extreme Heterogeneity
A bigger problem is the heterogeneity in local structure:
• some links are strong, long and frequent, others are weak and rare.
• some contacts are interconnected (clustered) others are not.
• there will be correlations between the demographics of connected individuals –
eg age, occupation, gender etc.
In principle all of these can be accommodated
by increasing the number of classes /
compartments within our models.
But it is not an ideal solution as the number of
classes grows exponentially.
Work is needed to develop models where
these correlated network structures arise
naturally.
Dynamics Networks
Networks are dynamic at a range of temporal scales:
• Our patterns of contacts change during each day, reflecting social context.
12-4 pm
2
1
2
1
9am-12
5
7-9 am
3
4
5-8 pm
6
3
4
8
7
Dynamics Networks
Networks are dynamic at a range of temporal scales:
• Our patterns of contacts change during each day, reflecting social context.
• Our contacts change during each week, especially at weekends.
Dynamics Networks
Networks are dynamic at a range of temporal scales:
• Our patterns of contacts change during each day, reflecting social context.
• Our patterns of contact change during each week, especially at weekends.
• Our patterns of contacts change as we grow older.
Dynamics Networks
Networks are dynamic at a range of temporal scales:
•
•
•
•
Our patterns of contacts change during each day, reflecting social context.
Our patterns of contact change during each week, especially at weekends.
Our patterns of contacts change as we grow older.
Contact patterns change as a consequence of infection.
Dynamics Networks
Networks are dynamic at a range of temporal scales:
•
•
•
•
Our patterns of contacts change during each day, reflecting social context.
Our patterns of contact change during each week, especially at weekends.
Our patterns of contacts change as we grow older.
Contact patterns change as a consequence of infection.
Again new modelling approaches are required to incorporate these changes. These
need to be under-pinned by detailed data collected over time and during illnesses.
In addition, we need a new ‘vocabulary’ to deal with dynamic networks.
The Future of
Modelling Networks
New insights are only likely from a blend of approaches:
• Theoretical analysis of exact cases
• Richer data sets on which to test ideas – networks and infection
• Analysis of simulation results
The interplay between networks and infections has many open
problems for future generations of researchers.