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- 1. How the information content of your contact pattern representation affects predictability of epidemics Petter Holme Sungkyunkwan University Umeå University HONS workshop, NetSci 2015 Zaragoza, Spain June 2, 2015 Title Presenter Affiliation Occasion Place Date
- 2. P Holme Information content of contact-pattern representations and the predictability of epidemic outbreaks arxiv:1503.06583
- 3. Compartmental models Contact structure To start with: use canonical compartmental models. SIR with fixed disease duration (and discrete time). probability λ time δ
- 4. Compartmental models Contact structure Fully mixed Network Temporal network information
- 5. Background / Motivation
- 6. Background / Motivation
- 7. … conditional on a large outbreak, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time.
- 8. “Weather is hard to predict because it is chaotic”
- 9. “Weather is hard to predict because it is modeled by equations that show chaotic behavior”
- 10. “Disease outbreaks are hard to predict because human contact structure has this-or-that structure”
- 11. predictability … in what sense? Assume we know the present, and can predict future contacts, then how well can we predict the ﬁnal outbreak size? … so it’s about the uncertainty of the SIR model rather than the contacts.
- 12. Datasets Human proximity data: who is close to whom at what time From the Sociopatterns project (RFID sensors, ~1.5m range, N = 75~250), T = 10h~5days From the Reality mining project N = 64, T = 9 hrs From Brazilian online prostitution N = 16,730, T = 6 hrs
- 13. 0.2 0.3 0.1 0 0.2 0.6 0.4 0 0.8 Temporal network Static network P(Ω)P(Ω) δ/T λ 0.001 0.01 0.1 1 0.001 0.01 0.1 1 δ/T λ 0.001 0.01 0.1 1 0.001 0.01 0.1 1 theknownstuff: differenceinoutbreaksize
- 14. 0 0.2 0.6 0.4 0 0.8 0.5 0 1 Static network Fully mixed P(Ω)P(Ω) λ 0.001 0.01 0.1 1 0.001 δ/T λ 0.001 0.01 0.1 1 0.001 0.01 0.1 1 δ/T λ 0.001 0.01 0.1 1 0.001 0.01 0.1 1 theknownstuff: differenceinoutbreaksize
- 15. Time Numberofinfected + time of infection + time of infection + time of infection
- 16. Time Numberofinfected
- 17. Time Numberofinfected Time s.d.
- 18. Results
- 19. Example Temporal networks Sociopatterns’ hospital data δ = 0.6, λ = 0.1 0 10 20 30 40 50 60 70 0 1 2 3 4 Time (days) Numberofinfected breaking time: 1h
- 20. Example Temporal networks Sociopatterns’ hospital data δ = 0.6, λ = 0.1 0 10 20 30 40 50 60 70 0 1 2 3 4 Time (days) Numberofinfected breaking time: 2h
- 21. Example Temporal networks Sociopatterns’ hospital data δ = 0.6, λ = 0.1 0 10 20 30 40 50 60 70 0 1 2 3 4 Time (days) Numberofinfected breaking time: 3h
- 22. Example Temporal networks Sociopatterns’ hospital data δ = 0.6, λ = 0.1 0 10 20 30 40 50 60 70 0 1 2 3 4 Time (days) Numberofinfected breaking time: 4h
- 23. Example Temporal networks Sociopatterns’ hospital data δ = 0.6, λ = 0.1 0 10 20 30 40 50 60 70 0 1 2 3 4 Time (days) Numberofinfected breaking time: 6h
- 24. Example Temporal networks Sociopatterns’ hospital data δ = 0.6, λ = 0.1 0 10 20 30 40 50 60 70 0 1 2 3 4 Time (days) Numberofinfected breaking time: 12h
- 25. Example Temporal networks Sociopatterns’ hospital data δ = 0.6, λ = 0.1 0 10 20 30 40 50 60 70 0 1 2 3 4 Time (days) Numberofinfected breaking time: 24h
- 26. Example Temporal networks Sociopatterns’ hospital data δ = 0.6, λ = 0.1 0 10 20 30 40 50 60 70 0 1 2 3 4 Time (days) Numberofinfected breaking time: 36h
- 27. Example Temporal networks Sociopatterns’ hospital data δ = 0.6, λ = 0.1 breaking time: 48h 0 10 20 30 40 50 60 70 0 1 2 3 4 Time (days) Numberofinfected
- 28. A λ = 0.00428, δ = 0.695 Temporal network 0 0.25 0.5 0.75 1 t 0 0.25 0.5 0.75 1∆ B λ = 0.0127, δ = 0.233 Temporal network 0 0.25 0.5 0.75 1 t 0 0.25 0.5 0.75 1 ∆ C λ = 0.233, δ = 0.112 Temporal network 0 0.25 0.5 0.75 1 t 0 0.25 0.5 0.75 1 ∆ D λ = 0.233, δ = 0.162 Temporal network 0 0.25 0.5 0.75 1 t 0 0.25 0.5 0.75 1 ∆ E λ = 0.00428, δ = 0.695 Static network 0 0.25 0.5 0.75 1 t 0 0.25 0.5 0.75 1 ∆ F λ = 0.0263, δ = 0.112 Static network 0 0.25 0.5 0.75 1 t 0 0.25 0.5 0.75 1 ∆ 00.050.10.150.2 P(∆)
- 29. 0 2 4 6 8 10 D Prostitution 0 0.2 0.4 0.6 0.8 1 t / T –5 ×10 ∆Ω 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Static network Temporal network Fully mixed A Conference 0 0.2 0.4 0.6 0.8 1 t / T ∆Ω 0 0.01 0.02 0.03 B Gallery 0 0.2 0.4 0.6 0.8 1 t / T ∆Ω 0 0.02 0.04 0.06 0.08 C Hospital 0 0.2 0.4 0.6 0.8 1 t / T ∆Ω 0 0.01 0.02 0.03 0.04 0.05 0.06 F School 0 0.2 0.4 0.6 0.8 1 t / T ∆Ω 0 0.02 0.04 0.06 0.08 E Reality 0 0.2 0.4 0.6 0.8 1 t / T ∆Ω
- 30. 0 0.1 0.2 0.3 0.4 max∆Ω 0 0.2 0.4 0.6 0.8 1 t / T B Gallery E Reality 0 0.1 0.2 0.3 0.4 max∆Ω 0 0.2 0.4 0.6 0.8 1 t / T F School 0 0.2 0.4 0.6 0.8 1 t / T 0 0.1 0.2 0.3 0.4 max∆Ω Static network Temporal network Fully mixed A Conference 0 0.2 0.4 0.6 0.8 1 t / T 0 0.1 0.2 0.3max∆Ω 0 1 2 3 D Prostitution 0 0.2 0.4 0.6 0.8 1 t / T ×10 max∆Ω –4 0.1 0.2 0.3 0 0.2 0.4 0.6 0.8 1 t / T 0 max∆Ω C Hospital
- 31. Temporalnetwork,Sociopatterns’hospitaldata λ 0.001 0.01 0.1 1 0.001 0.01 0.1 1 A Temporal network B Static network 0.5 0 1 tp/T 0 tp/T δ/T 0.001 0.01 0.1 1 0.001 0.01 0.1 1 λ δ/T 0.8 0.6 0.4 0.2
- 32. Temporalnetwork,Sociopatterns’hospitaldata δ/T λ λ C Fully mixed 0.001 0.01 0.1 1 0.001 0.01 0.1 1 0.001 0.01 0.1 1 0.001 0.01 0.1 1 B Static network 0.5 0 tp/T 0.5 0 1 tp/T δ/T 1
- 33. P Holme, N Masuda The basic reproduction number as a predictor for epidemic outbreaks in temporal networks PLOS ONE 10: e0120567 (2015)
- 34. R₀ — basic reproductive number, reproduction ratio, reproductive ratio, ... The expected number of secondary infections of an infectious individual in a population of susceptible individuals.
- 35. One of few concepts that went from mathematical to medical epidemiology
- 36. Disease R₀ Measles 12–18 Pertussis 12–17 Diphtheria 6–7 Smallpox 5–7 Polio 5–7 Rubella 5–7 Mumps 4–7 SARS 2–5 Inﬂuenza 2–4 Ebola 1–2
- 37. SIR model ds dt = –βsi— di dt = βsi – νi— = νi dr dt — S I I I I R Ω = r(∞) = 1 – exp[–R₀ Ω] where R₀ = β/ν Ω > 0 if and only if R₀ > 1 The epidemic threshold
- 38. Problems with R₀ Hard to estimate Can be hard for models & even harder for outbreak data and many datasets lack the important early period The threshold isn’t R₀ = 1 in practice The meaning of a threshold in a ﬁnite population. In temporal networks, the outbreak size needn’t be a monotonous function of R₀
- 39. Plan Use empirical contact data Simulate the entire parameter space of the SIR model Plot Ω vs R₀ Figure out what temporal network structure that creates the deviations
- 40. 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Averageoutbreaksize,Ω Basic reproductive number, R0 Conference
- 41. 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Averageoutbreaksize,Ω Basic reproductive number, R0 Conference 0.001 0.01 0.1 1 1 0.1 0.01 0.001 transmission probability diseaseduration
- 42. 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Averageoutbreaksize,Ω Basic reproductive number, R0 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Averageoutbreaksize,Ω Basic reproductive number, R0 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Averageoutbreaksize,Ω Basic reproductive number, R0 Conference Hospital Forum 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Averageoutbreaksize,Ω Basic reproductive number, R0 School, day 2
- 43. Shape index (example)— discordant pair separation in Ω 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Basic reproductive number, R0 Averageoutbreaksize,Ω μΩ=0.304 ρΩ = 2.663
- 44. avg. fraction of nodes present when 50% of contact happened avg. fraction of links present when 50% of contact happened avg. fraction of nodes present at 50% of the sampling time avg. fraction of links present at 50% of the sampling time frac. of nodes present 1st and last 10% of the contacts frac. of links present 1st and last 10% of the contacts frac. of nodes present 1st and last 10% of the sampling time frac. of links present 1st and last 10% of the sampling time Time evolution degree distribution, mean degree distribution, s.d. degree distribution, coefficient of variation degree distribution, skew Degree distribution link duration, mean link duration, s.d. link duration, coefficient of variation link duration, skew link interevent time, mean link interevent time, s.d. link interevent time, coefficient of variation link interevent time, skew Link activity Node activity node duration, mean node duration, s.d. node duration, coefficient of variation node duration, skew node interevent time, mean node interevent time, s.d. node interevent time, coefficient of variation node interevent time, skew Other network structure number of nodes clustering coefficient assortativity Temporal network structure
- 45. Correlation between point-cloud shape & temporal network structure * * ** ** ** ** ** * ** ** ** * ∆R0 0 0.2 0.4 0.6 0.8 1 R² Time evolution Node activity Link activity Degree distribution Network structure fLT fNT fLC fNC FLT FNT FLC FNC γNt σNt cNt µNt γNτ σNτ cNτ µNτ γLt σLt cLt µLt γLτ σLτ cLτ µLτ γk σk ck µk N C r
- 46. *** ** ∆Ω 0 0.2 0.4 0.6 0.8 1 R² Time evolution Node activity Link activity Network structure fLT fNT fLC fNC FLT FNT FLC FNC γNt σNt cNt µNt γNτ σNτ cNτ µNτ γLt σLt cLt µLt γLτ σLτ cLτ µLτ γk σk ck µk N C r Degreedistribution Correlation between point-cloud shape & temporal network structure Holme & Masuda, 2015, PLoS ONE 10:e0120567.
- 47. P Holme, T Takaguchi Time evolution of predictability of epidemics on networks Phys. Rev. E 91: 042811 (2015)
- 48. Only static networks Constant recovery rate SIR Diﬀerent topologies (RR, SW, LW, SF w expo 2, 2.5, 3) Two diﬀerent assumptions of what is known about the outbreak. Standard deviation as measure of outbreak diversity or non-predictability
- 49. s,t (a) (b) (d) (e) 0 t 0 t 0 t 0 t –7 –9 –7 –7 –7 s,t s,t s,t s,t s,t
- 50. (b) (c) (f)(e) 0 t 0 t 0 t 0 t –7 –7 –7 –7 s,t s,t s,t s,t
- 51. 0 1 2 3 4 5 6 0 1 2 3 1 4 161/16 1/4 0 1 2 0 1 2 3 0 1 2 3 0 1 2 3 1 4 161/16 1/4 1 4 161/16 1/4 1 4 161/16 1/41 4 161/16 1/41 4 161/16 1/4 T T T T T T (a) Large world (b) Small world (c) Random regular (d) (f)(e) 4 5
- 52. 0 50 70 90 t 0 50 70 t 0 5 t 0 5 t (a) (b) (d) (e) |R |R |R |R
- 53. 0 50 70 t 0 5 t 0 5 t (b) (c) (f)(e) 0 50 70 90 t |R |R |R |R
- 54. 0 2 4 6 8 10 0 1 2 3 1 4 161/16 1/4 0 0 1 2 3 0 1 2 0 1 2 1 4 161/16 1/4 1 4 161/16 1/4 1 4 161/16 1/41 4 161/16 1/41 4 161/16 1/4 T T T T T T (a)Largeworld (b)Smallworld (c)Randomregular (d) (f) (e) 4 6 2 4 6 8
- 55. Thank you!