Epidemic dynamics on networks

Epidemic dynamics on networks

Kieran SharkeyUniversity of Liverpool

NeST workshop, June 2014

Overview

• Introduction to epidemics on networks• Description of moment-closure

representation• Description of “Message-passing”

representation• Comparison of methods

Some example network slides removed here due to potential data confidentiality issues.

Route 2: Water flow (down stream)Modelling aquatic infectious disease

Jonkers et al. (2010) Epidemics

Route 2: Water flow (down stream)

Jonkers et al. (2010) Epidemics

SusceptibleInfectiousRemoved

States of individual nodes could be:

The SIR compartmental model

SI

R

Infection

Removal

All processes Poisson

SusceptibleInfectiousRemoved

States of individual nodes could be:

Contact Networks

1

4

2

3

1 2 3 4 0 0 0 00 0 1 00 1 0 01 1 0 0

1

2

3

4

G

Transmission Networks

1

4

2

3

1 2 3 4 0 0 0 00 0 T23 00 T32 0 0T41T42 0 0

1

2

3

4

TT41

T42

T32

T23

Moment closure & BBGKY hierarchy

iS Probability that node i is Susceptible

N

jjiiji ISTS

1

i

N

jjiiji IgISTI

1

Sharkey, K.J. (2008) J. Math Biol., Sharkey, K.J. (2011) Theor. Popul. Biol.

i j𝑇 𝑖𝑗

N

jjiiji ISTS

1

i

N

jjiiji IgISTI

1

jijk

jiijjikikik

kjijkji ISgISTISITISSTIS

i j i ji j

k

i j

k



N

jjiiji ISTS

1

i

N

jjiiji IgISTI

1

jijk

jiijjikikik


ik

kjijk

jikikji ISSSSITSS


Hierarchy provably exact at all orders

j

kjji

kjiB

CBBACBA iA

jB

kC

To close at second order can assume:



Random Network of 100 nodes


Random Network of 100 nodes

Random K-Regular Network


Locally connected Network


Example: Tree graph

For any tree, these equations are exact

Sharkey, Kiss, Wilkinson, Simon. B. Math. Biol . (2013)

Extensions to Networks with Clustering

1 2

3

4

1 2

3

5 4

⟨ 𝐼 1 𝐼 2𝑆3 𝐼 4 ⟩=⟨ 𝐼 1 𝐼 2𝑆3 ⟩ ⟨𝑆3 𝐼 4 ⟩

⟨𝑆3 ⟩

⟨ 𝐼 1 𝐼 2𝑆3 𝐼 4 𝐼5 ⟩=⟨ 𝐼 1 𝐼 2𝑆3 ⟩ ⟨𝑆3 𝐼 4 𝐼 5 ⟩

⟨𝑆3 ⟩

Kiss, Morris, Selley, Simon, Wilkinson (2013) arXiv preprint arXiv:1307.7737

Application to SIS dynamics

i

N

jjiiji IgISTS

1

i

N

jjiiji IgISTI

1

jijijk

jiijjikikik

kjijkji IIgISgISTISITISSTIS

jijiik

kjijk

jikikji SIgISgISSSSITSS

j

kjji

kjiB

CBBACBA Closure:

Nagy, Simon Cent. Eur. J. Math. 11(4) (2013)

Exact on tree networks

Can be extended to exact models on clustered networks

Can be extended to other dynamics (e.g. SIS)

Problem: Limited to Poisson processes

Moment-closure model

Karrer and Newman Message-Passing

Karrer B and Newman MEJ, Phys Rev E 84, 036106 (2010)


Cavity state

i

Fundamental quantity: : Probability that i has not received an infectious contact from j when i is in the cavity state.

j



Cavity state


i

j

is the probability that j has not received an infectious contact by time t from any of its neighbours when i and j are in the cavity state.


Karrer and Newman Message Passing


is the probability that j has not received an infectious contact by time t from any of its neighbours when i and j are in the cavity state.

Define: of being infected is: (Combination of infection process and removal ).

𝑓 𝑖𝑗 (𝜏 )𝑑𝜏=𝑠𝑖𝑗 (𝜏 )(1−∫0

𝜏

𝑟 𝑗 (𝜏 ′ )𝑑𝜏 ′)Message passing equation:

⟨𝑆 𝑖 ⟩=𝑧 𝑖 ∏𝑗∈𝑁 𝑖

𝐻 𝑖← 𝑗 (𝑡 ) ⟨𝑅𝑖 ⟩=∫0

𝑡

𝑟 (𝜏 ) [1− ⟨𝑆𝑖 ⟩𝑡 −𝜏 ]𝑑𝜏 ⟨ 𝐼 𝑖 ⟩=1− ⟨𝑆𝑖 ⟩− ⟨𝑅 𝑖 ⟩


1 if j initially susceptible¿ 𝑠𝑖𝑗 (𝜏 )∫

𝜏

∞

𝑟 𝑗 (𝜏 ′ )𝑑𝜏 ′

1) Applies to arbitrary transmission and removal processes

2) Not obvious to see how to extend it to other scenarios including generating exact models with clustering and dynamics such as SIS


Useful to relate the two formalisms to each other


Relationship to moment-closure equations

⟨𝑆 𝑖𝑆 𝑗 ⟩=𝑧 𝑖 𝑧 𝑗Φ 𝑗𝑖 (𝑡 )Φ𝑖

𝑗 (𝑡 )

⟨𝑆 𝑖 𝐼 𝑗 ⟩=𝑧 𝑖Φ 𝑗𝑖 (𝑡 )[1−𝑧 𝑗Φ𝑖

𝑗 (𝑡 )−∫0

𝑡

( 𝑓 𝑖𝑗 (𝜏 )+𝑔𝑖𝑗 (𝜏 ) )× (1−𝑧 𝑗 )Φ𝑖𝑗 (𝑡−𝜏 )𝑑𝜏 ]

𝐻 𝑖← 𝑗=1−∫0

𝑡

𝑓 𝑖𝑗 (𝜏 ) [1−𝑧 𝑗Φ𝑖𝑗 (𝑡−𝜏 ) ]𝑑𝜏

When the contact processes are Poisson, we have:

𝑑𝐻 𝑖← 𝑗 (𝑡 ) 𝑑𝑡

=−𝑇 𝑖𝑗 [𝐻 𝑖← 𝑗 (𝑡 )−𝑧 𝑗Φ𝑖𝑗 (𝑡 )−∫

0

𝑡

𝑟 𝑗 (𝜏 )𝑒−𝑇 𝑖𝑗𝜏 (1−𝑧𝑖Φ𝑖𝑗 (𝑡−𝜏 ) )𝑑𝜏 ]

so:

Wilkinson RR and Sharkey KJ, Phys Rev E 89, 022808 9 (2014)


When the removal processes are also Poisson:

jijk

jiijjikikik


jijk

jiijjikikik


⟨ 𝐴𝑖𝐵 𝑗𝐶𝑘 ⟩=⟨𝐴𝑖𝐵 𝑗 ⟩ ⟨𝐵 𝑗𝐶𝑘 ⟩

⟨𝐶𝑘 ⟩Wilkinson RR and Sharkey KJ, Phys Rev E 89, 022808 9 (2014)


When the removal process is fixed, Let


SIR with Delay


SIR with Delay


Summary part 1

Exact correspondence with stochastic simulation for tree networks.

Extensions to: • Exact models in networks with clustering• Non-SIR dynamics (eg SIS).

Pair-based moment closure:

Message passing:Exact on trees for arbitrary transmission and removal processes

Not clear how to extend to models with clustering or other dynamics

Limited to Poisson processes

Summary part 2

Extension of message passing models to include:a)Heterogeneous initial conditionsb)Heterogeneous transmission and removal processes

Extension of the pair-based moment-closure models to include arbitrary removal processes.

Linking the models enabled:

Proof that the pair-based SIR models provide a rigorous lower bound on the expected Susceptible time series.

Acknowledgements

• Robert Wilkinson (University of Liverpool, UK)

• Istvan Kiss (University of Sussex, UK)• Peter Simon (Eotvos Lorand University,

Hungary)

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Epidemic dynamics on networks

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