The Future of Modelling Networks

Matt KeelingWarwick Infectious Disease Epidemiological Research

(WIDER) Centre

The Future of Modelling Networks1) 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 ModelsdSdt

= B – b S I – dS

dIdt

= b S I – gI – dI

dRdt

= gI – dR

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:

Formulating ModelsdSdt

= B – t [S I] – dS

dIdt

= t [S I] – gI – dI

dRdt

= gI – dR

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:

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].

[SI] PAIRS

S I

Creation

Destruction

d[SI]dt

=

S

I

I

S

Infection of an SS pairby an external source.Requires an SSI triple.

[SSI]

S R

g [SI]

Recovery of the infectedmember of the SI pair.

I IInfection of susceptible from within the pair

[SI]

I

IS I

Infection of susceptiblefrom outside the pair

[ISI]

Formulating Pair-wise Models

Pair-wise: Formulation of TriplesTHE TRIPLE APPROXIMATION

1) In the simplest scenario, we can conceptualise a triple as two over-lapping pairs, which share the middle individual.

S I R

thus [SIR] = when all individuals have n connections.n-1 [SI][IR] n [I]

[SI] [IR]

Pair-wise: Formulation of TriplesTHE TRIPLE APPROXIMATION

2) This enables us to close the equations and simulate an epidemic.

1) In the simplest scenario, we can conceptualise a triple as two over-lapping pairs, which share the middle individual.

S I R

thus [SIR] = when all individuals have n connections.n-1 [SI][IR] n [I]

[SI] [IR]

3) This has been shown to be exact for the SIR model on a regular unclustered network.

Results for a network of ten thousand, and

exactly 4 contacts per individual.

Pair-wise: Formulation of Triples

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 modelsStudying sexually transmitted infections (STIs) is an ideal use of networks – given that the network is often well-known and easier to define. However the SIS-model formulation used for STIs does not approximate well:

SIS modelsStudying sexually transmitted infections (STIs) is an ideal use of networks – given that the network is often well-known and easier to define. However the SIS-model formulation used for STIs does not approximate well:

Results for a network of ten thousand, and exactly 3 contacts per individual.

SIS modelsStudying sexually transmitted infections (STIs) is an ideal use of networks – given that the network is often well-known and easier to define. However the SIS-model formulation used for STIs does not approximate well:

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.

There is a need for more theoretical development in this area – but potential for impact is huge.

ClusteringUnderstanding 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

ClusteringUnderstanding 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.

ClusteringUnderstanding 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 HeterogeneityMultiple 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 HeterogeneityA 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 HeterogeneityA 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.

Networks are dynamic at a range of temporal scales:

• Our patterns of contacts change during each day, reflecting social context.

Dynamics Networks

1 2

7-9 am

3

5

4

9am-12

3

6

7

48

12-4 pm 1 2

5-8 pm

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

Its down-hill from 50

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.

Dynamics 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.