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Risk Assessment from Past Temporal ContactsModelling Disease Spreading with Message Passing

Andreas Koher, Hartmut Lentz, Philipp Hövel

SIR – Type Outbreaks

Temporal Contact Networkcatle trade in Germany 2010 – 2011 weighted, directed, daily resolution


SIR – Model

Infection:

Recovery:


SIR – Model

Goal1. Find a dynamical model

2. Linearize around disease free state2. Apply spectral properties to risk assessment

Infection:

Recovery:

asdfasdfModelling Disease Spreading

Example 1

A B

Two connected nodes

Undirected, unweighted, static

Example 1

A B

Two connected nodes

Undirected, unweighted, static

SI – Model

Discrete node state: Uniform transmission prob.Node A is infected with prob. 0.5 Node B is susceptible

α

S , I

Example 1: Monte – Carlo Simulation

A B

Example 1: Quenched Mean Field

: Prob. b is susceptible

: Prob. b is infected

: Prob. b is recovered

: transmission prob.

: recovery prob.


A B

Message Passing

...

Inferring the origin of an epidemic with a dynamic message-passing algorithmA. Y. Lokhov, M. Mézard, H. Ohta, and L. ZdeborováPhys. Rev. E 90.1, 012801 (2014)

Message passing approach for general epidemic modelsB. Karrer and M. E. J. NewmanPhys. Rev. E 80, 016101 (2010)

A B

C

Message Passing

...



Features

1. Edge-based dynamics

2. Accounts for dynamical

correlations - echo chamber

effect

3. Exact on tree–topologies

A B

C

Message Passing

Key idea

„If B infects A, then B has been previously infected by

some neighbor “

...



Features

1. Edge-based dynamics

2. Accounts for dynamical

correlations - echo chamber

effect

3. Exact on tree–topologies

A B

C

C≠A

Quenched Mean Field

Quenched Mean Field Message Passing

: b is susceptible given a is susceptible

: b is infected given a is susceptible

: No disease transmission from b to a

Example 1: Message Passing

A B

asdfasdfSpectral methods for risk estimation

Low prevalence limit

Linearisation around

disease free solution:




A B

C




Vectorisation with non-backtracking matrix:




Propagator matrix

Vectorisation with non-backtracking matrix:

Spectral Condition for Global Oubreaks

Propagator matrix


Propagator matrix

Global Outbreak Condition

: Non-Backtracking Matrix


Propagator matrix


: Non-Backtracking Matrix

Previous Result

: Adjacency Matrix

Valdano et al. Phys. Rev. X 5, 021005 (2015)

Example 2

α

Temporal Tree - Network

Construction Priciple:

1. Static backbone: undirected, unweighted tree2. One undirected edge → two directed edges3. Edges appears with a fixed prob. per time step

SIR – Model

Discrete node state: Uniform transmission prob.Uniform recovery prob.One initially infected node: Center

S , I , Rα

Tree Network: Quenched Mean FieldAverage Number of Infected and Recovered

Tree Network: Monte - Carlo SimulationAverage Number of Infected and Recovered

Tree Network: Message PassingAverage Number of Infected and Recovered

SIR – Type OutbreaksTemporal Contact Network

catle trade in Germany 2010 – 2011 weighted, directed, daily resolution

Outbreak Condition

Critical transmission prob. for a

given recovery prob. Cattle trade in Germany from 2010 to 2011

Summary

Message Passing for epidemic modelling

Epidemic threshold based on the non-backtracking matrix

Summary

Message Passing for epidemic modelling

Epidemic threshold based on the non-backtracking matrix

Thank you… and

Philipp Hövel

Hartmut Lorenz

Eugenio Valdano

Vittoria Colizza

asdfasdfBackup

Comparison


Critical transmission prob. for a given recovery prob.

0 0 0 0,01 0,01 0,01 0,01 0,01 0,02 0,02

Quenched Mean Field

Message Passing

Cattle trade in Germany from 2010 to 2011

Example 2

α

Temporal Tree - Network

Construction Priciple:

1. Static backbone: undirected, unweighted tree2. One undirected edge → two directed edges3. Edges appears with a fixed prob. per time step

SIR – Model

Discrete node state: Uniform transmission prob.Uniform recovery prob.One initially infected node: Center

S , I , Rα

β

Example 2: Monte - Carlo Simulation

Example 2: Message Passing

Example 2: Message Passing (QMF)

Example 3

α

Complex temporal network

Hypertext Conference 2009 (Sociopaterns.org)Nodes:

SIR – Model

Discrete node state: Uniform transmission prob.Uniform recovery prob.One initially infected node

S , I , Rα

β

Example 3: Monte - Carlo Simulation

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