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Modelling power-law spread of infectious diseases Sebastian Meyer - - PowerPoint PPT Presentation

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ISPM, Division of Biostatistics Modelling power-law spread of infectious diseases Sebastian Meyer and Leonhard Held Financially supported by the Swiss National Science Foundation (project 137919: Statistical methods for spatio-temporal modelling

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ISPM, Division of Biostatistics

Modelling power-law spread

  • f infectious diseases

Sebastian Meyer and Leonhard Held

Financially supported by the Swiss National Science Foundation (project 137919: Statistical methods for spatio-temporal modelling and prediction of infectious diseases)

SMIDDY 2013, 13 September 2013 Page 1

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ISPM, Division of Biostatistics

Epidemic Modelling

– Prospective surveillance: outbreak detection (Farrington). – This talk is concerned with retrospective surveillance:

– Explain the spread of epidemics through statistical modelling – Assess influential factors, e.g., seasonality, climate, concurrent epidemics of related pathogens, contact networks

– Data basis: routine public health surveillance including temporal as well as spatial information – This talk deals with two types of surveillance data:

– individual case reports – aggregated counts by week and administrative district

Meyer & Held: Modelling power-law spread of infectious diseases Page 2

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ISPM, Division of Biostatistics

Mobility networks determine the spread of epidemics

Source: Max Planck Institute for Dynamics and Self-Organization (http://www.mpg.de/4406928/)

How to quantify spatial interaction between regions or individuals in the absence of network data?

Meyer & Held: Modelling power-law spread of infectious diseases Page 3

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ISPM, Division of Biostatistics

Power law! Why?

Brockmann et al., 2006: – Analysed trajectories of 464 670 dollar bills in the USA – Short-time travel behaviour follows a power law wrt distance

  • Fig. 1c: Histogram of the

distance r traversed within 4 days. Dashed line: P(r) ∝ r−1.59

– “Starting point for the development of a new class of models for the spread of infectious diseases”

Meyer & Held: Modelling power-law spread of infectious diseases Page 4

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ISPM, Division of Biostatistics

Power law! Why?

Brockmann et al., 2006: – Analysed trajectories of 464 670 dollar bills in the USA – Short-time travel behaviour follows a power law wrt distance

  • Fig. 1c: Histogram of the

distance r traversed within 4 days. Dashed line: P(r) ∝ r−1.59

– “Starting point for the development of a new class of models for the spread of infectious diseases” Let’s do it! We use this finding to improve upon two previously established model frameworks for infectious disease spread.

Meyer & Held: Modelling power-law spread of infectious diseases Page 4

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ISPM, Division of Biostatistics

Two additive components (Held et al., 2005) ⊕

Endemic: seasonality, population, socio-demography, climate, . . . Epidemic: dependency on previously infected individuals

Space-time point process model for individual case reports

λ∗(t, s) = ν[t][s] ρ[t][s] +

  • j:tj <t

ηj · g(t − tj) · f (s − sj) log(ν[t][s]) = β0 + β⊤z[t][s] , log(ηj) = γ0 + γ⊤mj (Meyer, Elias, and H¨

  • hle,

2012)

Multivariate time-series model for counts

Yit|Y·,t−1 ∼ NegBin(µit, ψ) µit = νit eit + λit Yi,t−1 + φit

  • j=i

wjiYj,t−1 log(·it) = β(·) + b(·)

i

+ β(·)⊤z(·)

it

· ∈ {ν, λ, φ} (Held and Paul, 2012, and previous work)

Meyer & Held: Modelling power-law spread of infectious diseases Page 5

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ISPM, Division of Biostatistics

Two additive components (Held et al., 2005) ⊕

Endemic: seasonality, population, socio-demography, climate, . . . Epidemic: dependency on previously infected individuals

Space-time point process model for individual case reports

λ∗(t, s) = ν[t][s] ρ[t][s] +

  • j:tj <t

ηj · g(t − tj) · f (s − sj) log(ν[t][s]) = β0 + β⊤z[t][s] , log(ηj) = γ0 + γ⊤mj (Meyer, Elias, and H¨

  • hle,

2012)

Multivariate time-series model for counts

Yit|Y·,t−1 ∼ NegBin(µit, ψ) µit = νit eit + λit Yi,t−1 + φit

  • j=i

wji Yj,t−1 log(·it) = β(·) + b(·)

i

+ β(·)⊤z(·)

it

· ∈ {ν, λ, φ} (Held and Paul, 2012, and previous work)

Meyer & Held: Modelling power-law spread of infectious diseases Page 5

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ISPM, Division of Biostatistics

Two additive components (Held et al., 2005) ⊕

Endemic: seasonality, population, socio-demography, climate, . . . Epidemic: dependency on previously infected individuals

Space-time point process model for individual case reports

λ∗(t, s) = ν[t][s] ρ[t][s] +

  • j:tj <t

ηj · g(t − tj) · f (s − sj) log(ν[t][s]) = β0 + β⊤z[t][s] , log(ηj) = γ0 + γ⊤mj

“twinstim”

Multivariate time-series model for counts

Yit|Y·,t−1 ∼ NegBin(µit, ψ) µit = νit eit + λit Yi,t−1 + φit

  • j=i

wji Yj,t−1 log(·it) = β(·) + b(·)

i

+ β(·)⊤z(·)

it

· ∈ {ν, λ, φ}

“hhh4”

Meyer & Held: Modelling power-law spread of infectious diseases Page 5

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ISPM, Division of Biostatistics

Power-law distance decay in twinstim

f (x) = x−d not suitable: pole at x = 0 ⇒ not integrable.

Meyer & Held: Modelling power-law spread of infectious diseases Page 6

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ISPM, Division of Biostatistics

Power-law distance decay in twinstim

f (x) = x−d not suitable: pole at x = 0 ⇒ not integrable. “Lagged”power law with uniform short-range dispersal: fL(x) =

  • 1

for x < σ, x

σ

−d

  • therwise.

20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Distance x fL(x)

σ = 10 d=0.5 d=1 d=1.59 d=3 Meyer & Held: Modelling power-law spread of infectious diseases Page 6

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ISPM, Division of Biostatistics

Power-law distance decay in twinstim

f (x) = x−d not suitable: pole at x = 0 ⇒ not integrable. Kernel of the density of the shifted Pareto distribution: f (x) = (x + σ)−d

20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Distance x f(x)

σ = 1 d=0.5 d=1 d=1.59 d=3 Meyer & Held: Modelling power-law spread of infectious diseases Page 6

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ISPM, Division of Biostatistics

Power-law distance decay in twinstim

f (x) = x−d not suitable: pole at x = 0 ⇒ not integrable. Kernel of the density of the shifted Pareto distribution: f (x) = (x + σ)−d

20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Distance x f(x)

σ = 1 d=0.5 d=1 d=1.59 d=3

– Joint ML-inference for all model parameters – Numerical cubature of f2D(s) = f (s) over polygonal domains in likelihood via product-Gauss cubature (Sommariva and Vianello, 2007)

Meyer & Held: Modelling power-law spread of infectious diseases Page 6

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ISPM, Division of Biostatistics

Power-law weights in hhh4

– On which distance measure between regions should the power law act? − → Order of neighbourhood oji!

1 3 2 1 3 1 2 1 2 2 3

Meyer & Held: Modelling power-law spread of infectious diseases Page 7

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ISPM, Division of Biostatistics

Power-law weights in hhh4

– On which distance measure between regions should the power law act? − → Order of neighbourhood oji!

1 3 2 1 3 1 2 1 2 2 3

– Generalisation of previously used first-order weights wji:

first-order power law 1(j ∼ i)

  • −d

ji

– Normalisation: wji/

k wjk

2 4 6 8 10 12 14 0.0 0.2 0.4 0.6 0.8 1.0 Neighbourhood order o

  • −d
  • d

0.5 1 1.59 ∞ Meyer & Held: Modelling power-law spread of infectious diseases Page 7

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ISPM, Division of Biostatistics

Power-law weights in hhh4

– On which distance measure between regions should the power law act? − → Order of neighbourhood oji!

1 3 2 1 3 1 2 1 2 2 3

– Generalisation of previously used first-order weights wji:

first-order power law 1(j ∼ i)

  • −d

ji

– Normalisation: wji/

k wjk

– Estimate d within the penalised likelihood framework simultaneously with all other model parameters.

2 4 6 8 10 12 14 0.0 0.2 0.4 0.6 0.8 1.0 Neighbourhood order o

  • −d
  • d

0.5 1 1.59 ∞ Meyer & Held: Modelling power-law spread of infectious diseases Page 7

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ISPM, Division of Biostatistics

Example of individual-level surveillance data: Invasive meningococcal disease in Germany (2002–8)

48°N 50°N 52°N 54°N 6°E 8°E 10°E 12°E 14°E

  • ●
  • 100 km

50 100 200 500 1000 2000

  • 1

2 5 10 15

Type

  • B

C District population [1000 inhabitants] 5 10 15 20 Time (months) Number of cases 2002 2003 2004 2005 2006 2007 2008 Type B C 85 170 255 340 Cumulative number of cases

Meyer & Held: Modelling power-law spread of infectious diseases Page 8

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ISPM, Division of Biostatistics

⊲ Estimated power law

10 20 30 40 50

Distance x from host [km]

2 ⋅ 10−6 4 ⋅ 10−6 6 ⋅ 10−6 8 ⋅ 10−6 1 ⋅ 10−5

eγ0 ⋅ f(x) Gaussian Power law

Endemic: seasonality, trend, population density as offset Epidemic: type, age group Decay parameter: ˆ d = 2.3 (95% CI: [1.74, 3.03])

AIC ˆ R(B) ˆ R(C) Gaussian 18972.04 0.22 [0.17,0.31] 0.10 [0.06,0.15] Power law 18944.25 0.26 [0.14,0.35] 0.13 [0.06,0.19]

Meyer & Held: Modelling power-law spread of infectious diseases Page 9

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ISPM, Division of Biostatistics

Example of aggregated surveillance data: Influenza in Southern Germany (2001–8)

100 km

10 20 30 40 50 60 70

Mean yearly incidence per 100 000 inhabitants in the 140 districts of Baden-W¨ urttemberg and Bavaria

200 400 600 800 1000 1200 Time (416 weeks) Number of cases 2001 2002 2003 2004 2005 2006 2007 2008

Meyer & Held: Modelling power-law spread of infectious diseases Page 10

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ISPM, Division of Biostatistics

⊲ Estimated power law

2 4 6 8 10 12 14 0.0 0.2 0.4 0.6 0.8 1.0 Neighbourhood order o

  • −d
  • Decay parameter d

0 [uniform] 1.80 [power law] ∞ [first−order]

– population fractions as endemic offset – seasonality, region-specific random intercepts in all three components

Meyer & Held: Modelling power-law spread of infectious diseases Page 11

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ISPM, Division of Biostatistics

⊲ Predictive performance

– Use strictly proper scoring rules to evaluate consistency of predictive distribution with later observed value: logarithmic score (logS) and ranked probability score (RPS) (Czado et al., 2009) – Based on one-week-ahead predictions in the last two years – Calculate mean scores and p-values via permutation tests

logS RPS first order 0.5511 0.4194 power law 0.5448 0.4168 p-value 0.0001 0.19

Meyer & Held: Modelling power-law spread of infectious diseases Page 12

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ISPM, Division of Biostatistics

⊲ Long-term predictive performance

– Simulate the 2008 wave of influenza – Based on models fitted on 2001–2007 – Initialised by the 18 cases of the last week of 2007 – Run 1000 simulations for each model and evaluate by

– the final size distribution – proper scoring rules on the empirical distribution of the simulations compared to the later reported counts

– Additional benchmark against

– endemic-only model – model without neighbourhood effects – model with additional population effect in spatio-temporal component

Meyer & Held: Modelling power-law spread of infectious diseases Page 13

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ISPM, Division of Biostatistics

⊲ Long-term predictive performance final size

  • ●
  • ● ●
  • ●
  • ●
  • power law + pop.

power law first order endemic + AR endemic only Final size (sqrt−scale) 5000 10000 20000 30000 5781

Meyer & Held: Modelling power-law spread of infectious diseases Page 14

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ISPM, Division of Biostatistics

⊲ Long-term predictive performance time domain

200 400 600 800 1000 Week of 2008 Number of cases

  • endemic only

endemic + AR first order power law power law + pop.

1 5 9 13 17

DSS RPS 27.03 149.77 31.36 112.15 26.46 108.61 16.41 110.2 15.49 111.86

Meyer & Held: Modelling power-law spread of infectious diseases Page 15

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ISPM, Division of Biostatistics

⊲ Long-term predictive performance space domain

  • bserved

endemic only endemic + autoregressive first order power law power law + population

0 4 9 16 25 36 49 64 81 100 121 144 169 196 225

Incidence [per 100000 inhabitants]

DSS RPS

endemic only

7.85 15.39

endemic + AR

7.59 15.04

first order

7.51 15.63

power law

7.36 14.75

power law + pop.

7.24 14.3

Meyer & Held: Modelling power-law spread of infectious diseases Page 16

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ISPM, Division of Biostatistics

⊲ Long-term predictive performance space-time domain

DSS RPS endemic only 2.91 1.31 endemic + AR 2.58 1.26 first order 2.5 1.26 power law 2.29 1.25 power law + pop. 2.29 1.24

[animation]

Meyer & Held: Modelling power-law spread of infectious diseases Page 17

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ISPM, Division of Biostatistics

Discussion

– Human mobility

– is an important driver of epidemic spread – follows a power law with respect to distance

– Predictive performance improves when using a power law for spatial interaction of cases – Infectious imports increase with population size (Bartlett, 1957) – Edge effects:

– random intercepts account for unobserved heterogeneity – incorporate region-specific incoming traffic from abroad

Meyer & Held: Modelling power-law spread of infectious diseases Page 18

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ISPM, Division of Biostatistics

Outlook

– Semiparametric estimate of weight function to confirm power law – Estimate impact of traffic data on neighbourhood weights wji (Geilhufe et al., 2013)

Meyer & Held: Modelling power-law spread of infectious diseases Page 19

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ISPM, Division of Biostatistics

Outlook

– Semiparametric estimate of weight function to confirm power law – Estimate impact of traffic data on neighbourhood weights wji (Geilhufe et al., 2013) – Further reading: arXiv:1308.5115 – Further application: all methods are implemented in the

  • pen-source

package surveillance for visualisation, modelling and monitoring of epidemic phenomena.

Meyer & Held: Modelling power-law spread of infectious diseases Page 19

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References

◮ Bartlett, M. S. (1957). Measles periodicity and community size. Journal of the Royal Statistical Society. Series A (Statistics in Society), 120(1):48–70. ◮ Brockmann, D., Hufnagel, L., and Geisel, T. (2006). The scaling laws of human travel. Nature, 439(7075):462–465. ◮ Czado, C., Gneiting, T., and Held, L. (2009). Predictive model assessment for count data. Biometrics, 65(4):1254–1261. ◮ Geilhufe, M., Held, L., Skrøvseth, S. O., Simonsen, G. S., and Godtliebsen, F. (2013). Power law approximations of movement network data for modeling infectious disease

  • spread. Biometrical Journal. In press.

◮ Held, L., H¨

  • hle, M., and Hofmann, M. (2005). A statistical framework for the analysis of

multivariate infectious disease surveillance counts. Statistical Modelling, 5:187–199. ◮ Held, L. and Paul, M. (2012). Modeling seasonality in space-time infectious disease surveillance data. Biometrical Journal, 54(6):824–843. ◮ H¨

  • hle, M., Meyer, S., and Paul, M. (2013). surveillance: Temporal and

Spatio-Temporal Modeling and Monitoring of Epidemic Phenomena. ◮ Meyer, S., Elias, J., and H¨

  • hle, M. (2012). A space-time conditional intensity model for

invasive meningococcal disease occurrence. Biometrics, 68(2):607–616. ◮ Meyer, S. and Held, L. (2013). Modelling power-law spread of infectious diseases. Submitted to Annals of Applied Statistics. ◮ Sommariva, A. and Vianello, M. (2007). Product Gauss cubature over polygons based on Green’s integration formula. Bit Numerical Mathematics, 47(2):441–453.