Download Random graphs: a useful tool in epidemic modelling

Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Random graphs: a useful tool in epidemic
modelling
Tom Britton
June, 2008
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Infectious disease spread
Infectious diseases: worlds most common death cause
Historical: plague, cholera, small-pox, ...
Major present: malaria, TB, HIV, ...
New: SARS, Ebola, foot & mouth, avian influenza
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Infectious disease spread
Infectious diseases: worlds most common death cause
Historical: plague, cholera, small-pox, ...
Major present: malaria, TB, HIV, ...
New: SARS, Ebola, foot & mouth, avian influenza
=⇒ Need for understanding spreading dynamics
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Infectious disease spread
Infectious diseases: worlds most common death cause
Historical: plague, cholera, small-pox, ...
Major present: malaria, TB, HIV, ...
New: SARS, Ebola, foot & mouth, avian influenza
=⇒ Need for understanding spreading dynamics
Two features have equal importance: disease agent
(transmissability) and social structure
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Infectious disease spread
New disease: could either take off or die out
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Infectious disease spread
New disease: could either take off or die out
“Take off” possible if: R0 > 1, where
R0 = average number of new infections caused by an infected
during initial phase
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Infectious disease spread
New disease: could either take off or die out
“Take off” possible if: R0 > 1, where
R0 = average number of new infections caused by an infected
during initial phase
Initial phase of epidemic ≈ branching process
If R0 > 1 (super-critical) disease may become endemic
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Intervention – Control
Aim: to stop or reduce spreading (either during early stages or
when endemic)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Intervention – Control
Aim: to stop or reduce spreading (either during early stages or
when endemic)
How: Reduce R below 1 =⇒ epidemic will not take off, or
endemic disease will go extinct
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Intervention – Control
Aim: to stop or reduce spreading (either during early stages or
when endemic)
How: Reduce R below 1 =⇒ epidemic will not take off, or
endemic disease will go extinct
Means:
Vaccination, use of condoms, ...: changes transmittability
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Intervention – Control
Aim: to stop or reduce spreading (either during early stages or
when endemic)
How: Reduce R below 1 =⇒ epidemic will not take off, or
endemic disease will go extinct
Means:
Vaccination, use of condoms, ...: changes transmittability
isolation, travelling restrictions, reduce prostitution, sauna
clubs, ...: changes social structure
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Stochastic network model
Social structure only partly known: modelled using random
graph/network with structure
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Stochastic network model
Social structure only partly known: modelled using random
graph/network with structure
Graph/Network: nodes (individuals) and edges (“friendship”)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Stochastic network model
Social structure only partly known: modelled using random
graph/network with structure
Graph/Network: nodes (individuals) and edges (“friendship”)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Stochastic epidemic model
Also spreading is uncertain =⇒ stochastic model
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Stochastic epidemic model
Also spreading is uncertain =⇒ stochastic model
Simplest model: an infected person infects each friend
independently with prob p (and then recovers)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Stochastic epidemic model
Also spreading is uncertain =⇒ stochastic model
Simplest model: an infected person infects each friend
independently with prob p (and then recovers)
Effect on graph: thinning – each edge is removed with prob 1 − p
Interpretation: remaining edges reflect “potential spreading”
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Graph and its thinned version
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Graph and its thinned version
The thinned graph is also a random graph!
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Graph and its thinned version
The thinned graph is also a random graph!
Those connected to index case make upp final outbreak!
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Scientific questions
Given social structure (random network) + epidemic model (p):
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Scientific questions
Given social structure (random network) + epidemic model (p):
Can a big outbreak occur? (R0 > 1?)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Scientific questions
Given social structure (random network) + epidemic model (p):
Can a big outbreak occur? (R0 > 1?)
If so, how many will get infected?
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Scientific questions
Given social structure (random network) + epidemic model (p):
Can a big outbreak occur? (R0 > 1?)
If so, how many will get infected?
How about when control measures are put into place?
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Motivation
Stochastic models
Scientific questions
Given social structure (random network) + epidemic model (p):
Can a big outbreak occur? (R0 > 1?)
If so, how many will get infected?
How about when control measures are put into place?
In what follows we will study these questions for some examples
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Heavy tail degree distribution: Motivation
Example 1: Heavy tailed degree distribution
Social structure described by degree distribution
Degree distribution {pk ; k ≥ 0}:
D ∼ {pk } is the number of friends of randomly selected individual
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Heavy tail degree distribution: Motivation
Example 1: Heavy tailed degree distribution
Social structure described by degree distribution
Degree distribution {pk ; k ≥ 0}:
D ∼ {pk } is the number of friends of randomly selected individual
Most empirical networks have heavy tail degree distribution
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Random graph & Epidemic model
Model
Social structure: Individuals have degree distribution
D ∼ {pk } and friends are chosen completely at random
Epidemic model: each susc. friend is infected with prob p
initially one randomly selected infectious (n − 1 susceptibles)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Random graph & Epidemic model
Model
Social structure: Individuals have degree distribution
D ∼ {pk } and friends are chosen completely at random
Epidemic model: each susc. friend is infected with prob p
initially one randomly selected infectious (n − 1 susceptibles)
What is R0 ?
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Random graph & Epidemic model
Model
Social structure: Individuals have degree distribution
D ∼ {pk } and friends are chosen completely at random
Epidemic model: each susc. friend is infected with prob p
initially one randomly selected infectious (n − 1 susceptibles)
What is R0 ?
R0 = pE (D)?
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Random graph & Epidemic model
Model
Social structure: Individuals have degree distribution
D ∼ {pk } and friends are chosen completely at random
Epidemic model: each susc. friend is infected with prob p
initially one randomly selected infectious (n − 1 susceptibles)
What is R0 ?
R0 = pE (D)?– Wrong!
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Random graph & Epidemic model
Model
Social structure: Individuals have degree distribution
D ∼ {pk } and friends are chosen completely at random
Epidemic model: each susc. friend is infected with prob p
initially one randomly selected infectious (n − 1 susceptibles)
What is R0 ?
R0 = pE (D)?– Wrong!
R0 = p(E (D) − 1)?
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Random graph & Epidemic model
Model
Social structure: Individuals have degree distribution
D ∼ {pk } and friends are chosen completely at random
Epidemic model: each susc. friend is infected with prob p
initially one randomly selected infectious (n − 1 susceptibles)
What is R0 ?
R0 = pE (D)?– Wrong!
R0 = p(E (D) − 1)?– Wrong!
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
The basic reproduction number
What is the degree distribution of infectives (during early stages)?
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
The basic reproduction number
What is the degree distribution of infectives (during early stages)?
Answer: {p̃k ; k ≥ 1}, where p̃k = const · kpk = kpk /E (D)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
The basic reproduction number
What is the degree distribution of infectives (during early stages)?
Answer: {p̃k ; k ≥ 1}, where p̃k = const · kpk = kpk /E (D)
V (D) − E (D)
=⇒ R0 = p(E (D̃) − 1) = · · · = p E (D) +
E (D)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
The probability of an outbreak
The initial phase of epidemic ≈ branching process
=⇒ π = π(p, {pk }) := P(big outbreak) can be computed
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Size of outbreak
A big class of random graphs: one giant component
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Size of outbreak
A big class of random graphs: one giant component
π = P(big outbreak) = P(belong to giant) = rel size of giant
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Size of outbreak
A big class of random graphs: one giant component
π = P(big outbreak) = P(belong to giant) = rel size of giant
=⇒ outbreak size can also be derived
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination
Suppose a fraction v are vaccinated prior to outbreak
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination
Suppose a fraction v are vaccinated prior to outbreak
Effect on graph: thinning – vaccinated nodes (and their
surrounding edges) are removed
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination
Suppose a fraction v are vaccinated prior to outbreak
Effect on graph: thinning – vaccinated nodes (and their
surrounding edges) are removed
Who?
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination
Suppose a fraction v are vaccinated prior to outbreak
Effect on graph: thinning – vaccinated nodes (and their
surrounding edges) are removed
Who?
a) Randomly chosen individuals
=⇒ Rv = p(1 − v )(E (D̃) − 1) = (1 − v )R0
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination
Suppose a fraction v are vaccinated prior to outbreak
Effect on graph: thinning – vaccinated nodes (and their
surrounding edges) are removed
Who?
a) Randomly chosen individuals
=⇒ Rv = p(1 − v )(E (D̃) − 1) = (1 − v )R0
=⇒ if v ≥ 1 − 1/R0 then Rv ≤ 1 =⇒ no outbreak!
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination
Suppose a fraction v are vaccinated prior to outbreak
Effect on graph: thinning – vaccinated nodes (and their
surrounding edges) are removed
Who?
a) Randomly chosen individuals
=⇒ Rv = p(1 − v )(E (D̃) − 1) = (1 − v )R0
=⇒ if v ≥ 1 − 1/R0 then Rv ≤ 1 =⇒ no outbreak!
Critical vaccination coverage: vC = 1 − 1/R0
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination
Suppose a fraction v are vaccinated prior to outbreak
Effect on graph: thinning – vaccinated nodes (and their
surrounding edges) are removed
Who?
a) Randomly chosen individuals
=⇒ Rv = p(1 − v )(E (D̃) − 1) = (1 − v )R0
=⇒ if v ≥ 1 − 1/R0 then Rv ≤ 1 =⇒ no outbreak!
Critical vaccination coverage: vC = 1 − 1/R0
Problem: If R0 large, vC ≈ 1
Tom Britton
=⇒ impossible
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination, cont’d
Can we do better?
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination, cont’d
Can we do better? Yes! Vaccinate social people
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination, cont’d
Can we do better? Yes! Vaccinate social people
But social structure usually unknown ...
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination, cont’d
Can we do better? Yes! Vaccinate social people
But social structure usually unknown ...
b) Acquaintance vaccination strategy
Choose individuals at random
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination, cont’d
Can we do better? Yes! Vaccinate social people
But social structure usually unknown ...
b) Acquaintance vaccination strategy
Choose individuals at random
vaccinate one of their friends
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Vaccination, cont’d
Can we do better? Yes! Vaccinate social people
But social structure usually unknown ...
b) Acquaintance vaccination strategy
Choose individuals at random
vaccinate one of their friends
Vaccinees will have degree distribution {p̃k } rather than {pk }
=⇒ much more efficient
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Uniform vaccination: Proportion infected as function of v
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Acquaintance vaccination: Proportion infected as function
of v
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Second example: An STI in heterosexual community
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
An application to Sexually Transmitted Infections
A model for sexually transmitted infections
D = # sex-partners (e.g. during a year)
p = P(transmission in a relationship)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
An application to Sexually Transmitted Infections
A model for sexually transmitted infections
D = # sex-partners (e.g. during a year)
p = P(transmission in a relationship)
Heterosexual community: Df , Dm , pf , pm
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
An application to Sexually Transmitted Infections
A model for sexually transmitted infections
D = # sex-partners (e.g. during a year)
p = P(transmission in a relationship)
Heterosexual community: Df , Dm , pf , pm
=⇒ bipartite graph
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
An application to Sexually Transmitted Infections
A model for sexually transmitted infections
D = # sex-partners (e.g. during a year)
p = P(transmission in a relationship)
Heterosexual community: Df , Dm , pf , pm
=⇒ bipartite graph
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
It can be shown that
R0
r (Df )
= pf E (Df ) + V (DEf )−E
(Df )
r )−E (Dm )
× pm E (Dm ) + V (DEm (D
m)
Similar to before
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Improved analysis
However: P(transmission) depends on # sex-acts in
relationship
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Improved analysis
However: P(transmission) depends on # sex-acts in
relationship
Promisquous individuals tend to have fewer sex-acts per
partner
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Improved analysis
However: P(transmission) depends on # sex-acts in
relationship
Promisquous individuals tend to have fewer sex-acts per
partner
This should reduce R0 !
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Improved analysis: continued
Data: (Anonymous) study of sexual habits in Gotland
≈ 1000 people (17-28 yrs)
Among other things: How many sex-partners during last year
and how many sex-acts in each relationship
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Improved analysis: continued
Data: (Anonymous) study of sexual habits in Gotland
≈ 1000 people (17-28 yrs)
Among other things: How many sex-partners during last year
and how many sex-acts in each relationship
=⇒ Include in model: short and long term relationships
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Improved analysis: continued
Data: (Anonymous) study of sexual habits in Gotland
≈ 1000 people (17-28 yrs)
Among other things: How many sex-partners during last year
and how many sex-acts in each relationship
=⇒ Include in model: short and long term relationships
=⇒ two types of edges in graph (diff transm prob)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Improved analysis, cont’d
P(transmission in relationship) estimated from
P(transmission) = 1 − (1 − p)# sex-acts
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Improved analysis, cont’d
P(transmission in relationship) estimated from
P(transmission) = 1 − (1 − p)# sex-acts
R0 fitted to data and computed as a function of p
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
R0 as function of p (fitted to Gotland data)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Conclusions:
1. Acknowledgeing short and longterm relationships reduces R0
2. Endemicity not possible (for realistic p’s)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Conclusions:
1. Acknowledgeing short and longterm relationships reduces R0
2. Endemicity not possible (for realistic p’s)but maybe in
subgroups ...
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Graphs with clustering
Third example: Social structures with clustering
In many social networks (perhaps not sexual networks!)
friends of an individual are quite often friends themselves
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Graphs with clustering
Third example: Social structures with clustering
In many social networks (perhaps not sexual networks!)
friends of an individual are quite often friends themselves
c := P(two friends of an individual are friends)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Graphs with clustering
Third example: Social structures with clustering
In many social networks (perhaps not sexual networks!)
friends of an individual are quite often friends themselves
c := P(two friends of an individual are friends)
How construct a random network with predefined clustering c?
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
One solution: bipartite graphs
One solution: bipartite graphs
Specific construction:
1. Type 1: ”true individuals” (n), Type 2: ”groups” (βn)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
One solution: bipartite graphs
One solution: bipartite graphs
Specific construction:
1. Type 1: ”true individuals” (n), Type 2: ”groups” (βn)
2. An individual is attached to a group, independently, with prob
γ/n
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
One solution: bipartite graphs
One solution: bipartite graphs
Specific construction:
1. Type 1: ”true individuals” (n), Type 2: ”groups” (βn)
2. An individual is attached to a group, independently, with prob
γ/n
3. Project the graph on ”true individuals”: individuals that share
a common group are connected
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
One solution: bipartite graphs
One solution: bipartite graphs
Specific construction:
1. Type 1: ”true individuals” (n), Type 2: ”groups” (βn)
2. An individual is attached to a group, independently, with prob
γ/n
3. Project the graph on ”true individuals”: individuals that share
a common group are connected
4. An infected individual infects each not yet infected ”friend”
with prob p and then recovers.
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Illustration of bipartite graph
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Resulting graph:
Conclusions from analysis
Positive clustering: c =
1
1+βγ
E (D) = βγ 2
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Resulting graph:
Conclusions from analysis
Positive clustering: c =
1
1+βγ
E (D) = βγ 2
How is the epidemic affected by c?
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Resulting graph:
Conclusions from analysis
Positive clustering: c =
1
1+βγ
E (D) = βγ 2
How is the epidemic affected by c?
Next slide: R0 and P(major outbreak) and studied as functions of
c, (and E (D) and p)
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
Summary
Random graphs are useful
Social structure is very important in spreading of diseases
Tom Britton
Random graphs: a useful tool in epidemic modelling
Introduction
Heavy tail degree distribution
An STI in heterosexual community
Social structures with clustering
References
Britton, T., Janson, S., Martin-Löf A. (2007): Graphs with specified
degree distributions, simple epidemics and local vacination
strategies. Adv. Appl. Prob., 39, 922-948.
Britton T., Nordvik, M.K., and Liljeros, F. (2007): Modelling
sexually transmitted infections: the effect of partnership activity and
number of partners on R0 . Theor Pop Biol., 72, 389-399.
Britton T., Deijfen, M., Lindholm, M. and Nordvall Lagerås, A.:
Epidemics on random graphs with tunable clustering. To appear in
J. App. Prob.
Tom Britton
Random graphs: a useful tool in epidemic modelling