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Interacting simple contagions are complex contagions

Interacting simple contagions are complex contagions

Simple and complex contagions are typically distinguished based on whether their transmission mechanism is linear or nonlinear. Is the probability of transmission the same whether one has a single infected neighbour for 2 days or 2 infected neighbours for a single day? The latter scenario tends to be more contagious in social situations because of effects such as social reinforcement. These nonlinear effects lead to contagions with radically different phenomenology: discontinuous phase transitions, hysteresis loops, and speed up due to social clustering. However, similar phenomenology can be shown to occur in the case of simple contagions with synergistic interactions. We use this parallel to show how interacting simple contagions spreading on a clustered network look like a complex contagion if one is unaware of the interaction or not tracking the co-infections. In a social context, our results highlight the difficulties of identifying and quantifying mechanisms such as social reinforcement when innumerable amount of ideas, memes and behaviours interact on complex social networks. In the biological context, this allows us to use the toolbox of complex contagions to detect novel biological interactions between pathogens.

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Laurent Hébert-Dufresne

June 11, 2018
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  1. Interacting simple contagions are complex contagions (and why that matters)

    Laurent H´ ebert-Dufresne laurent.hebert-dufresne@uvm.edu :: @LHDnets Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  2. Simple contagions An edge is enough information. • All edges

    between susceptible and infectious nodes are identical. • Known rate of transmission βdt. • Includes more complicated models, such as non-Markovian dynamics. Example: Most models of disease transmission. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  3. Interacting simple contagions An edge is enough information. • All

    edges between susceptible and infectious nodes are not all identical. • The rate of transmission βdt depends on the state of another process. • The interaction alone depends on a lot of parameters. Example: Most real diseases interact, like influenza and pneumonia. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  4. Complex contagions An edge is not enough information. • All

    edges between susceptible and infectious nodes are not identical. • The rate of transmission βdt depends on the neighbourhood of the susceptible node. • The transmission rate per edge is now a function of mesoscopic information. Example: Social contagions with social reinforcement. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  5. Important properties of contagion dynamics Let’s focus on ease of

    spread in clustered networks. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  6. LHD et al., Phys. Rev. E (2010) Interacting simple contagions

    are complex contagions Laurent H´ ebert-Dufresne
  7. Important properties of contagion dynamics Let’s focus on ease of

    spread in clustered networks. 1 Simple contagions emerge continuously. 2 Clustering slows down simple contagions. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  8. Dodds & Watts, PRL (2004) Interacting simple contagions are complex

    contagions Laurent H´ ebert-Dufresne
  9. Centola, Science (2010) Interacting simple contagions are complex contagions Laurent

    H´ ebert-Dufresne
  10. O’Sullivan et al., Frontiers in Physics (2015) Interacting simple contagions

    are complex contagions Laurent H´ ebert-Dufresne
  11. Important properties of contagion dynamics Let’s focus on ease of

    spread in clustered networks. 1 Simple contagions emerge continuously. 2 Clustering slows down simple contagions. 3 Complex contagions can emerge discontinuously. 4 Clustering can speed up complex contagions. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  12. LHD & Althouse, Proc. Natl. Acad. Sci. U.S.A. (2015) Interacting

    simple contagions are complex contagions Laurent H´ ebert-Dufresne
  13. Important properties of contagion dynamics Let’s focus on ease of

    spread in clustered networks. 1 Simple contagions emerge continuously. 2 Clustering slows down simple contagions. 3 Complex contagions can emerge discontinuously. 4 Clustering can speed up complex contagions. 5 Interacting simple contagions can emerge discontinuously. 6 Clustering can speed up interacting simple contagions. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  14. Important properties of contagion dynamics Let’s focus on ease of

    spread in clustered networks. 1 Simple contagions emerge continuously. 2 Clustering slows down simple contagions. 3 Complex contagions can emerge discontinuously. 4 Clustering can speed up complex contagions. 5 Interacting simple contagions can emerge discontinuously. 6 Clustering can speed up interacting simple contagions. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  15. Can we tell complex contagions apart from interacting simple contagions?

    Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  16. Two types of approaches in the literature 1 Phenomenological: Based

    on structural properties 2 Statistical: Based on model selection Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  17. Phenomenological and statistical experiments Study properties of all SIS contagions

    on a cliquish graph Fit complex/simple contagions to interacting SI contagions on a clique graph Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  18. Results 1: Phenomenology of SIS contagions Study properties of all

    SIS contagions on a cliquish graph Simple SIS: transmission rate β and recovery rate α. Complex SIS: transmission rate β(k) = βmin + βmax − βmin 1 + exp (−w(k − k0)) and recovery rate α. Interacting SIS: transmission rates β1 and β2, recovery rates α1 and α2, and 10 interaction parameters! Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  19. Results 1: Phenomenology of SIS contagions Study properties of all

    SIS contagions on a cliquish graph 10−3 10−2 10−1 Fraction I of infected 0 2 4 6 8 10 I relative to rand. Sim. Com. Inter. 1.0 1.5 2.0 2.5 # inf. neigh. on inf. Sim. Com. Inter. 1.0 1.5 2.0 2.5 # inf. neigh. on rec. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  20. Results 2: Fitting to interacting simple contagions Fit complex/simple dynamics

    to interacting SI contagions on a clique graph d dt Si,j = − β(i + j)Si,j − (n − 1 − i)β(i + Θi)Si,j − (n − 1 − j)β(j + Θj)Si,j + (n − i)β(i − 1 + Θi−1)Si−1,j + (n − j)β(j − 1 + Θj−1)Si,j−1 Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  21. Results 2: Fitting to interacting simple contagions Fit complex/simple dynamics

    to interacting SI contagions on a clique graph 0 5 10 15 20 25 30 Interaction factor −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 normalized ∆ BIC Simple contagion Complex contagion Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  22. Results 2: Fitting to interacting simple contagions Fit complex/simple dynamics

    to interacting SI contagions on a clique graph 0 5 10 15 20 25 30 Interaction factor 0.00 0.05 0.10 0.15 0.20 0.25 0.30 βmax Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  23. Results 2: Fitting to interacting simple contagions Fit complex/simple dynamics

    to interacting SI contagions on a clique graph 0 5 10 15 20 25 30 Interaction factor 0 2 4 6 8 10 k0 Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  24. Interacting contagions are complex contagions An edge is enough information.

    • All edges between susceptible and infectious nodes are not all identical. • The rate of transmission βdt depends on the state of another process. • The interaction alone depends on a lot of parameters. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  25. Interacting contagions are complex contagions An edge is enough information,

    if you are aware of the interaction. • Edges that look identical are not truly identical. • The rate of transmission βdt depends on some hidden process. • The interactions are potentially impossible to parametrize. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  26. Interacting contagions are complex contagions An edge is enough information,

    if you are aware of the interaction. • Edges that look identical are not truly identical. • The rate of transmission βdt depends on some hidden process. • The interactions are potentially impossible to parametrize. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  27. Interacting contagions are complex contagions An edge is enough information,

    if you are aware of the interaction. • Edges that look identical are not truly identical. • The rate of transmission βdt depends on some hidden process. • The interactions are potentially impossible to parametrize. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  28. Conclusions (“and why that matters”) 1 Interacting contagions look like

    complex contagions, and they are complex contagions when using hidden Markov models. → Similar macro properties: discontinuous emergence, speed up with clustering. → Similar micro properties: heterogeneous spreading rates, more clustered cases. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  29. Conclusions (“and why that matters”) 2 Bad news: Phenomenology and

    model fitting do not distinguish spreading mechanisms. → Measurements of social reinforcement “in the wild” are confounded by interactions with billions of spreading memes. → Very controlled experiments (` a la Centola) can help, but results are hard to generalize. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  30. Conclusions (“and why that matters”) 3 Good news: Complex contagion

    models can detect novel pathogen interactions. → Many diseases interact significantly, and model complexity scales superlinearly with the number of pathogens. → Complex contagion offer an effective model where “social reinforcement” correlates with interaction strength. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne
  31. If you have any questions Preprint will be available eventually.

    Until then: laurent.hebert-dufresne@uvm.edu or @LHDnets Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne