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 

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) 

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 

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) 

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 

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 

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. 

Many-to-one Many mechanistic models lead to outcomes 

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

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 , ( )] 

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 , ( )] 

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) 

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

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) 

I 

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. 

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 . 

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). 

I 

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 

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. 

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

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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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)