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Bayesian inference of effective contagion models from population level data

Bayesian inference of effective contagion models from population level data

Flash talk at SINM 2019 (http://danlarremore.com/sinm2019/)

Preprint: https://arxiv.org/abs/1906.01147
Software: https://github.com/jg-you/complex-coinfection-inference/

See also: "Interacting simple contagions are complex contagions," by Laurent Hébert-Dufresne: https://speakerdeck.com/laurenthebert/interacting-simple-contagions-are-complex-contagions

Extended abstract
==============
Contagions never occur in a vacuum. Instead, diseases and ideas interact with each other and with externalities such as host connectivity, behaviour, and mobility. Several recent studies have shown that many of these non-linear mechanism lead to rich dynamics that can exhibit, for example, a non-monotonous relation between the expected epidemic size and their average transmission rate and discontinuous phase transitions. Surprisingly, many of these features arise from minor alterations to the mechanistic rules of the models. In other words, innocuous looking modeling choices can produce drastically different outcomes that would lead to very different conclusions about intervention strategies or risk. Understanding how to properly generalize contagion models is, as a result, perhaps one of the most pressing challenge of network epidemiology.

Recent work shows that so-called “complex contagions models” can induce many of the defining features of the new wave of non-linear mechanistic models. Complex contagion models achieve this by modifying the transmission rate β(I), as to let it depend on the density of infected individuals in the neighbourhood of the susceptible individuals. In this work, we show that some complex contagion models are in fact indistinguishable from a number of non-linear mechanistic models at the population level. This motivates us to think of complex contagion (on Erdős-Rényi graphs) as a useful effective model of contagion on networks. The complex contagion function β(I) that appears in this model captures arbitrary non-linear effects, allowing us to the contagions without making any mechanistic assumptions. By understanding how mechanistic models map unto this complex contagion model, we can interpret the function β(I) and draw tentative inference about the population under scrutiny.

We develop a fully Bayesian method to fit our effective contagion model to population-level data. This allows us to infer posterior distributions of: Potential epidemiological trajectory, complex contagion functions β(I), and noise components. We avoid overfitting—a problem that arises in related approaches—by parameterizing the complex contagion functions as low degree polynomials. The net results is a flexible, efficient, yet expressive model that can be easily fitted to real and synthetic data alike.

Jean-Gabriel Young

May 27, 2019
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  1. B
    SINM : F
    Jean-Gabriel Young
    Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI, USA
    jgyoung.ca @_jgyou
    May th,
    Based on joint work with Laurent Hébert-Dufresne and Samuel V. Scarpino

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  2. 1998
    1999
    2000
    2001
    2002
    2003
    2004
    2005
    2006
    2007
    2008
    2009
    2010
    2011
    2012
    2013
    2014
    0.0
    2.5
    5.0
    7.5
    New cases (% national)
    National Influenza data, CDC FluView Portal (WHO & ILINet)

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  3. At the population level : a longstanding approach
    Compartmental modeling, e.g. :
    Susceptible node Infected node Recovered node
    Infection
    (rate β)
    Recovery
    (rate γ)
    d
    dt
    S −βIS
    d
    dt
    I βIS − γI
    d
    dt
    r γI

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  4. At the population level : a longstanding approach
    Well-mixed
    0 10 20
    t
    10−2
    10−1
    100
    Densities
    I(t)
    R(t)
    Heterogeneous
    networks
    0 10 20
    t
    10−2
    10−1
    100
    Densities
    I(t)
    R(t)

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  5. Challenges to conventional : Two examples ( of )
    M : Prudent behavior
    [ S. V. Scarpino, A. Allard, and L. Hébert-Dufresne, Nature Physics ( )]
    Replace infected nodes by susceptible nodes

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  6. Challenges to conventional : Two examples ( of )
    M : Prudent behavior
    [ S. V. Scarpino, A. Allard, and L. Hébert-Dufresne, Nature Physics ( )]
    Replace infected nodes by susceptible nodes

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  7. Challenges to conventional wisdom : Two examples ( of )
    M : Interaction in clustered networks
    [ L. Hébert-Dufresne, S. V. Scarpino, and JGY (in preparation)]
    T diseases in well-mixed populations. Nodes with both become superspreaders.

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  8. Many-to-one
    Many mechanistic models lead to
    outcomes

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  9. Many-to-one
    Many mechanistic models lead to
    outcomes
    At the population level : We need an effective, mechanism agnostic model

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  10. Complex SIR model
    In a well-mixed population
    Susceptible node Infected node Recovered node
    Infection
    (rate β)
    Recovery
    (rate γ)
    d
    dt
    S −βIS
    d
    dt
    I βIS − γI
    d
    dt
    r γI
    Early reference [P. S. Dodds and D. J. Watts, Physical Review Letters , ( )]

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  11. Complex SIR model
    In a well-mixed population
    Susceptible node Infected node Recovered node
    Infection
    (rate β)
    Recovery
    (rate γ)
    d
    dt
    S −β(I)IS
    d
    dt
    I β(I)IS − γI
    d
    dt
    r γI
    Early reference [P. S. Dodds and D. J. Watts, Physical Review Letters , ( )]

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  12. Complex contagion
    Complex contagion function β(I)
    0 1
    t
    0.0
    0.5
    1.0
    I(t)
    0 1
    I
    0
    10
    20
    β(I)

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  13. Complex contagion
    Complex contagion function β(I)
    0 25
    t
    0.0
    0.5
    I(t)
    0 1
    I
    0
    10
    20
    β(I)

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  14. Complex contagion
    Complex contagion function β(I)
    0 25
    t
    0.0
    0.5
    I(t)
    0 1
    I
    0
    10
    20
    β(I)
    0 20
    t
    0.0
    0.2
    I(t)
    0 1
    I
    0
    10
    β(I)
    0 25
    t
    0.0
    0.5
    I(t)
    0 1
    I
    50
    100
    β(I)
    0 25
    t
    0
    1
    I(t)
    0 1
    I
    5.0
    7.5
    β(I)
    0 25
    t
    0.0
    0.5
    I(t)
    0 1
    I
    8
    10
    β(I)

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  15. Bayesian inference : Likelihood
    Hidden dynamics : X1:T : Series of “micro-states” Xt
    ∈ X(who’s in what state?)
    Observed : Noisy macro states Y1:T. Ex. : number of infectious w/ Gaussian noise.

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  16. Bayesian inference : Likelihood
    Hidden dynamics : X1:T : Series of “micro-states” Xt
    ∈ X(who’s in what state?)
    Observed : Noisy macro states Y1:T. Ex. : number of infectious w/ Gaussian noise.
    L
    P(Y1:T
    |β, σ, γ)
    G

    XT
    P(Y1:T
    , x1:T
    , G|β, γ, σ)dx1:T
    .

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  17. Bayesian inference : Likelihood
    Hidden dynamics : X1:T : Series of “micro-states” Xt
    ∈ X(who’s in what state?)
    Observed : Noisy macro states Y1:T. Ex. : number of infectious w/ Gaussian noise.
    L
    P(Y1:T
    |β, σ, γ)
    G

    XT
    P(Y1:T
    , x1:T
    , G|β, γ, σ)dx1:T
    .
    A
    Maximal contribution under a mean-field (MF) approximation :
    P(Y1:T
    |β, γ, σ)
    t
    q(Yt
    | ˜
    yt
    (β, γ), σ)
    where q(Yt
    |y, σ) is the pdf of a Gaussian, and ˜
    yt is the MF solution of the dynamics
    on ensemble P(G).

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  18. Complex contagion on a well-mixed population
    Various level of sub-sampling :
    0.00 0.35 0.70
    I
    0
    10
    20
    30
    β(I)/γ
    t
    Prevalence
    0.00 0.35 0.70
    I
    0
    10
    20
    30
    β(I)/γ
    t
    Prevalence
    0.00 0.35 0.70
    I
    0
    10
    20
    30
    β(I)/γ
    t
    Prevalence
    Various level of noise :
    0.00 0.35 0.70
    I
    0
    10
    20
    30
    β(I)/γ
    t
    Prevalence
    0.00 0.35 0.70
    I
    0
    10
    20
    30
    β(I)/γ
    t
    Prevalence
    0.00 0.35 0.70
    I
    0
    10
    20
    30
    β(I)/γ
    t
    Prevalence

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  19. Interacting contagions on networks
    Structure Time-series Inference
    0 10000 20000
    t
    0.0
    0.2
    0.4
    Prevalence
    Indep.
    Coop.
    0.00 0.15 0.30 0.45
    I
    5
    10
    β(I)/γ
    1 3 5
    max β(I)/ min β(I)
    Prob.

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  20. Reference : arXiv : .xxxxx
    Code : github.com/jg-you/comp ex-coinfection-inference
    [email protected] jgyoung.ca @_jgyou

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  21. Take-home message
    Mechanistic models : Many-to-one at the population level
    Complex contagion : universal effective model for prevalence
    We infer β(I) with a Bayesian mean-field model
    Reference : arXiv : .xxxxx
    Software : github.com/jg-you/comp ex-coinfection-inference

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  22. Bayesian inference : Posterior
    P β(I)
    Bernstein polynomials of fixed degree N :
    β(I)
    N
    ν 0
    ξν
    N
    ν
    Iν(1 − I)N−ν
    Represent any function in the limit N → ∞. Nonnegative on [0, 1] if ξν > 0 ∀ν.
    Prior : ξν ∼ µξ + ∆ν where ∆ν is centered on .
    0 25
    t
    0.0
    0.5
    I(t)
    0 1
    I
    0
    10
    20
    β(I)
    0 20
    t
    0.0
    0.2
    I(t)
    0 1
    I
    0
    10
    β(I)
    0 25
    t
    0.0
    0.5
    I(t)
    0 1
    I
    50
    100
    β(I)
    0 25
    t
    0
    1
    I(t)
    0 1
    I
    5.0
    7.5
    β(I)
    0 25
    t
    0.0
    0.5
    I(t)
    0 1
    I
    8
    10
    β(I)

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