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Risk Assessment from Past Temporal Contacts Modelling Disease Spreading with Message Passing Andreas Koher, Hartmut Lentz, Philipp Hövel
Risk Assessment from Past Temporal ContactsModelling Disease Spreading with Message Passing
Andreas Koher, Hartmut Lentz, Philipp Hövel
SIR – Type Outbreaks
SIR – Type Outbreaks
Temporal Contact Networkcatle trade in Germany 2010 – 2011 weighted, directed, daily resolution
Temporal Contact Networkcatle trade in Germany 2010 – 2011 weighted, directed, daily resolution
SIR – Model
Infection:
Recovery:
Temporal Contact Networkcatle trade in Germany 2010 – 2011 weighted, directed, daily resolution
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: 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.
Example 1: Quenched Mean Field
: Prob. b is susceptible
: Prob. b is infected
: Prob. b is recovered
: transmission prob.
: recovery prob.
Example 1: Quenched Mean Field
: Prob. b is susceptible
: Prob. b is infected
: Prob. b is recovered
: transmission prob.
: recovery prob.
Example 1: Quenched Mean Field
: Prob. b is susceptible
: Prob. b is infected
: Prob. b is recovered
: transmission prob.
: recovery prob.
Example 1: Quenched Mean Field
A B
Example 1: Quenched Mean Field
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
...
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)
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 “
...
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)
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
Example 1: Message Passing
A B
asdfasdfSpectral methods for risk estimation
Low prevalence limit
Linearisation around
disease free solution:
Low prevalence limit
Linearisation around
disease free solution:
A B
C
Low prevalence limit
Linearisation around
disease free solution:
Vectorisation with non-backtracking matrix:
Low prevalence limit
Linearisation around
disease free solution:
Propagator matrix
Vectorisation with non-backtracking matrix:
Spectral Condition for Global Oubreaks
Propagator matrix
Spectral Condition for Global Oubreaks
Propagator matrix
Global Outbreak Condition
: Non-Backtracking Matrix
Spectral Condition for Global Oubreaks
Propagator matrix
Global Outbreak Condition
: 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: 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
Tree Network: Message PassingAverage Number of Infected and Recovered
SIR – Type OutbreaksTemporal Contact Network
catle trade in Germany 2010 – 2011 weighted, directed, daily resolution
SIR – Type OutbreaksTemporal Contact Network
catle trade in Germany 2010 – 2011 weighted, directed, daily resolution
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
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
Global Outbreak Condition
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: Quenched Mean Field
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
Example 3: Quenched Mean Field
Example 3: Quenched Mean Field
