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the R Package sr Sebastian Meyer Epidemic phenomena Examples: - - PowerPoint PPT Presentation

Sep 26, 2022 455 likes •762 views

Epidemiology, Biostatistics and Prevention Institute Spatio-Temporal Analysis of Epidemic Phenomena Using the R Package sr Sebastian Meyer Epidemic phenomena Examples: Earth quakes Riots / crimes

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Epidemiology, Biostatistics and Prevention Institute

Spatio-Temporal Analysis of Epidemic Phenomena Using the R Package s✉r✈❡✐❧❧❛♥❝❡

Sebastian Meyer

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Epidemic phenomena

Examples: – Earth quakes – Riots / crimes – Infectious diseases Data: Surveillance systems routinely collect – time-stamped – geo-referenced case reports

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 2

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Case study I: Invasive meningococcal disease

❧✐❜r❛r②✭✧s✉r✈❡✐❧❧❛♥❝❡✧✮❀ ❞❛t❛✭✧✐♠❞❡♣✐✧✮

♣❧♦t✭✐♠❞❡♣✐✱ ✧s♣❛❝❡✧✮

  • ●
  • type

B C

  • 1

2 4 8 16

Dot size proportional to the number of cases (residence postcode) ♣❧♦t✭✐♠❞❡♣✐✱ ✧t✐♠❡✧✮

2002 2004 2006 2008 5 10 15 20 Time (months) Number of cases 84 168 252 336 Cumulative number of cases

B C

Monthly and cumulative number of cases (by date of specimen sampling)

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 3

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❛♥✐♠❛t❡✭s✉❜s❡t✭✐♠❞❡♣✐✱ t②♣❡❂❂✧❇✧✮✱ t✐♠❡✳s♣❛❝✐♥❣ ❂ ✼✮ ❛♥✐♠❛t❡✭s✉❜s❡t✭✐♠❞❡♣✐✱ t②♣❡❂❂✧❈✧✮✱ t✐♠❡✳s♣❛❝✐♥❣ ❂ ✼✮

Does the force of infection depend on the bacterial finetype?

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 4

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Case study II: Measles

❧✐❜r❛r②✭✧s✉r✈❡✐❧❧❛♥❝❡✧✮❀ ❞❛t❛✭✧♠❡❛s❧❡s❲❡s❡r❊♠s✧✮

Publically available surveillance data: time series of counts of newly reported infections by district

♣❧♦t✭♠❡❛s❧❡s❲❡s❡r❊♠s✱ t②♣❡ ❂ ♦❜s❡r✈❡❞ ⑦ ✉♥✐t✮

2001/1 − 2002/52

036 100 196 324 400 484 576 676

♣❧♦t✭♠❡❛s❧❡s❲❡s❡r❊♠s✱ t②♣❡ ❂ ♦❜s❡r✈❡❞ ⑦ t✐♠❡✮

time

  • No. infected

2001 II 2001 IV 2002 II 2002 IV 10 20 30 40 50 60 useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 5

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❛♥✐♠❛t❡✭♠❡❛s❧❡s❲❡s❡r❊♠s✮

Is local vaccination coverage related to disease dynamics?

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 6

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Characteristics of epidemic-type data

– Low number of cases – Seasonality – Occassional outbreaks (“self-exciting” process) – Dependence between areas, age groups, etc. – Underreporting, reporting delays

s✉r✈❡✐❧❧❛♥❝❡

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 7

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Characteristics of epidemic-type data

– Low number of cases – Seasonality – Occassional outbreaks (“self-exciting” process) – Dependence between areas, age groups, etc. – Underreporting, reporting delays

Aims of s✉r✈❡✐❧❧❛♥❝❡

Monitoring (prospective): Outbreak prediction and detection (→ “Zombie Preparedness” talk by Michael Höhle) Modelling (retrospective): Quantify epidemicity and effects of external covariates on disease dynamics

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 7

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Place in the world of R packages

s✉r✈❡✐❧❧❛♥❝❡ is the first and only software package dedicated to the space-time modelling and monitoring of epidemic phenomena Related packages:

s♣❛❝❡t✐♠❡: Basic classes and methods for spatio-temporal data s♣❛tst❛t: THE package for purely spatial point patterns ts❝♦✉♥t, ❊♣✐❊st✐♠, ♦✉t❜r❡❛❦❡r, ❛♠❡✐: Several packages dealing with purely temporal epidemic data st♣♣: Simulation & visualization of space-time point patterns

For a more complete picture: → CRAN task view “Handling and Analyzing Spatio-Temporal Data”

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 8

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Three modelling frameworks in s✉r✈❡✐❧❧❛♥❝❡

Data Resolution Example Model Function individual events in continuous space-time cases of invasive meningococcal disease (IMD) Meyer et al., 2012 spatio-temporal point process t✇✐♥st✐♠✭✮ event counts aggregated in space & time week×district counts of measles Meyer et al., 2014 multivariate NegBin time series ❤❤❤✹✭✮ individual SIR event history of a fixed population spread of classi- cal swine fever among domes- tic pig farms Höhle, 2009 multivariate temporal point process t✇✐♥❙■❘✭✮

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 9

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Basic modelling concept

Stochastic branching process with immigration

– Decomposed disease risk:

Endemic: seasonality, population, socio-demography, . . .

⊕

Epidemic: force of previously infected individuals

– Ebola: R0 of about 1.5 – 2.5 – Force of infection may depend on age and spatial/temporal distance to infective

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 10

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Spatial interaction

Tobler’s First Law of Geography:

Everything is related to everything else, but near things are more related than distant things.

Brockmann et al., 2006 (dollar bill tracking):

The distribution of travelling distances decays as a power law.

500 1000 2000 3000

f(x) = x−1.6 Distance x

1 5 50 500

log(f(x)) = −1.6 ⋅ log(x) Distance x

  • −1.6

2 3 4 3 1 3 2 3 2 4 2 1 3 2 1 3

0.0 0.2 0.4 0.6 0.8 1.0

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 11

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Case study I: Invasive meningococcal disease

Regression framework for the conditional intensity function

λ(s, t) = ρ[s][t] ν[s][t]

Endemic component – Piecewise constant on a suitable space-time grid – Explanatory variables in a log-linear predictor ν[s][t] – Equivalent to Poisson-GLM for aggregated counts ♥❧♠✐♥❜✭✮ ♣♦❧②❈✉❜

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 12

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Case study I: Invasive meningococcal disease

Regression framework for the conditional intensity function

λ(s, t) = ρ[s][t] ν[s][t] +

  • j:tj<t

ηj f(s − sj) g(t − tj)

Endemic component – Piecewise constant on a suitable space-time grid – Explanatory variables in a log-linear predictor ν[s][t] – Equivalent to Poisson-GLM for aggregated counts Force of infection – Depends on event-specific characteristics mj via log(ηj) = γ0 + γ⊤mj – Decays over space/time according to parametric interaction function f(·)/g(·) ♥❧♠✐♥❜✭✮ ♣♦❧②❈✉❜

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 12

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Case study I: Invasive meningococcal disease

Regression framework for the conditional intensity function

λ(s, t) = ρ[s][t] ν[s][t] +

  • j:tj<t

ηj f(s − sj) g(t − tj)

Endemic component – Piecewise constant on a suitable space-time grid – Explanatory variables in a log-linear predictor ν[s][t] – Equivalent to Poisson-GLM for aggregated counts Force of infection – Depends on event-specific characteristics mj via log(ηj) = γ0 + γ⊤mj – Decays over space/time according to parametric interaction function f(·)/g(·) Likelihood inference – ♥❧♠✐♥❜✭✮ with analytical score function and Fisher info – R package ♣♦❧②❈✉❜ for cubature of f(s) over polygons

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 12

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Case study I: Invasive meningococcal disease

Model estimation

✐♠❞❢✐t ❁✲ t✇✐♥st✐♠✭ ❡♥❞❡♠✐❝ ❂ ⑦♦❢❢s❡t✭❧♦❣✭♣♦♣❞❡♥s✐t②✮✮ ✰ ■✭st❛rt✴✸✻✺ ✲ ✸✳✺✮ ✰ s✐♥✭✷ ✯ ♣✐ ✯ st❛rt✴✸✻✺✮ ✰ ❝♦s✭✷ ✯ ♣✐ ✯ st❛rt✴✸✻✺✮✱ ❡♣✐❞❡♠✐❝ ❂ ⑦t②♣❡ ✰ ❛❣❡❣r♣✱ s✐❛❢ ❂ s✐❛❢✳♣♦✇❡r❧❛✇✭✮✱ t✐❛❢ ❂ t✐❛❢✳❝♦♥st❛♥t✭✮✱ ❞❛t❛ ❂ ✐♠❞❡♣✐✱ s✉❜s❡t ❂ ✦✐s✳♥❛✭❛❣❡❣r♣✮✱ st❛rt ❂ ❝✭✧❡✳✭■♥t❡r❝❡♣t✮✧❂✲✻✳✺✱ ✧❡✳s✐❛❢✳✶✧❂✶✳✺✱ ✧❡✳s✐❛❢✳✷✧❂✵✳✾✮✱ ♦♣t✐♠✳❛r❣s ❂ ❧✐st✭❢✐①❡❞ ❂ ✧❡✳s✐❛❢✳✶✧✮✱ ♠♦❞❡❧ ❂ ❚❘❯❊✱ ❝♦r❡s ❂ ✹✮ ①t❛❜❧❡✭✐♠❞❢✐t✮

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 13

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Case study I: Invasive meningococcal disease

Model estimation

✐♠❞❢✐t ❁✲ t✇✐♥st✐♠✭ ❡♥❞❡♠✐❝ ❂ ⑦♦❢❢s❡t✭❧♦❣✭♣♦♣❞❡♥s✐t②✮✮ ✰ ■✭st❛rt✴✸✻✺ ✲ ✸✳✺✮ ✰ s✐♥✭✷ ✯ ♣✐ ✯ st❛rt✴✸✻✺✮ ✰ ❝♦s✭✷ ✯ ♣✐ ✯ st❛rt✴✸✻✺✮✱ ❡♣✐❞❡♠✐❝ ❂ ⑦t②♣❡ ✰ ❛❣❡❣r♣✱ s✐❛❢ ❂ s✐❛❢✳♣♦✇❡r❧❛✇✭✮✱ t✐❛❢ ❂ t✐❛❢✳❝♦♥st❛♥t✭✮✱ ❞❛t❛ ❂ ✐♠❞❡♣✐✱ s✉❜s❡t ❂ ✦✐s✳♥❛✭❛❣❡❣r♣✮✱ st❛rt ❂ ❝✭✧❡✳✭■♥t❡r❝❡♣t✮✧❂✲✻✳✺✱ ✧❡✳s✐❛❢✳✶✧❂✶✳✺✱ ✧❡✳s✐❛❢✳✷✧❂✵✳✾✮✱ ♦♣t✐♠✳❛r❣s ❂ ❧✐st✭❢✐①❡❞ ❂ ✧❡✳s✐❛❢✳✶✧✮✱ ♠♦❞❡❧ ❂ ❚❘❯❊✱ ❝♦r❡s ❂ ✹✮ ①t❛❜❧❡✭✐♠❞❢✐t✮

RR 95% CI p-value h.I(start/365 - 3.5) 0.959 0.92–1.00 0.071 h.sin(2 * pi * start/365) 1.231 1.08–1.41 0.0022 h.cos(2 * pi * start/365) 1.379 1.21–1.57 <0.0001 e.typeC 0.450 0.27–0.74 0.0017 e.agegrp[3,19) 2.133 1.10–4.12 0.024 e.agegrp[19,Inf) 0.824 0.33–2.05 0.68

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 13

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Case study I: Invasive meningococcal disease

Estimated spatial interaction

♣❧♦t✭✐♠❞❢✐t✱ ✇❤✐❝❤ ❂ ✧s✐❛❢✧✱ ①❧✐♠ ❂ ❝✭✵✱ ✺✵✮✮ ✐♠❞❢✐t❴❢st❡♣ ❁✲ ✉♣❞❛t❡✭✐♠❞❢✐t✱ s✐❛❢ ❂ s✐❛❢✳st❡♣✭ ❦♥♦ts ❂ ❡①♣✭✭✶✿✹✮✯❧♦❣✭✶✵✵✮✴✺✮✱ ♠❛①❘❛♥❣❡ ❂ ✶✵✵✮✱ ♦♣t✐♠✳❛r❣s ❂ ❧✐st✭❢✐①❡❞ ❂ ◆❯▲▲✮✮ ♣❧♦t✭✐♠❞❢✐t❴❢st❡♣✱ ✇❤✐❝❤ ❂ ✧s✐❛❢✧✱ ❛❞❞ ❂ ❚❘❯❊✱ ❝♦❧✳❡st✐♠❛t❡ ❂ ✶✮

10 20 30 40 50 0e+00 2e−05 4e−05 Distance x from host eγ0 ⋅ f(x) Power law Step (df=4)

s✐❛❢✳✯ t✐❛❢✳✯ ❝♦♥st❛♥t ❝♦♥st❛♥t ❣❛✉ss✐❛♥ ❡①♣♦♥❡♥t✐❛❧ ♣♦✇❡r❧❛✇ st❡♣ ♣♦✇❡r❧❛✇▲ st❡♣ st✉❞❡♥t

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 14

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Case study I: Invasive meningococcal disease

Estimated spatial interaction

♣❧♦t✭✐♠❞❢✐t✱ ✇❤✐❝❤ ❂ ✧s✐❛❢✧✱ ①❧✐♠ ❂ ❝✭✵✱ ✺✵✮✮ ✐♠❞❢✐t❴❢st❡♣ ❁✲ ✉♣❞❛t❡✭✐♠❞❢✐t✱ s✐❛❢ ❂ s✐❛❢✳st❡♣✭ ❦♥♦ts ❂ ❡①♣✭✭✶✿✹✮✯❧♦❣✭✶✵✵✮✴✺✮✱ ♠❛①❘❛♥❣❡ ❂ ✶✵✵✮✱ ♦♣t✐♠✳❛r❣s ❂ ❧✐st✭❢✐①❡❞ ❂ ◆❯▲▲✮✮ ♣❧♦t✭✐♠❞❢✐t❴❢st❡♣✱ ✇❤✐❝❤ ❂ ✧s✐❛❢✧✱ ❛❞❞ ❂ ❚❘❯❊✱ ❝♦❧✳❡st✐♠❛t❡ ❂ ✶✮

10 20 30 40 50 0e+00 2e−05 4e−05 Distance x from host eγ0 ⋅ f(x) Power law Step (df=4)

Predefined interaction functions:

Spatial (s✐❛❢✳✯) Temporal (t✐❛❢✳✯) ❝♦♥st❛♥t ❝♦♥st❛♥t ❣❛✉ss✐❛♥ ❡①♣♦♥❡♥t✐❛❧ ♣♦✇❡r❧❛✇ st❡♣ ♣♦✇❡r❧❛✇▲ st❡♣ st✉❞❡♥t

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 14

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Case study I: Invasive meningococcal disease

Fitted ground intensity ˆ λ(s, t) ❞s

♣❧♦t✭✐♠❞❢✐t✱ ✇❤✐❝❤ ❂ ✧t♦t❛❧ ✐♥t❡♥s✐t②✧✱ ❛❣❣r❡❣❛t❡ ❂ ✧t✐♠❡✧✱ t②♣❡s ❂ ✶✱ ②❧✐♠ ❂ ❝✭✵✱✵✳✸✮✱ t❣r✐❞ ❂ ✷✺✵✵✮

500 1000 1500 2000 2500 0.00 0.10 0.20 0.30

B

Time [days] Fitted intensity process total intensity endemic intensity 500 1000 1500 2000 2500 0.00 0.10 0.20 0.30

C

Time [days] Fitted intensity process total intensity endemic intensity useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 15

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Case study I: Invasive meningococcal disease

Methods for ✧t✇✐♥st✐♠✧ Display Extract Modify Other ♣r✐♥t ♥♦❜s ✉♣❞❛t❡ s✐♠✉❧❛t❡ s✉♠♠❛r② ✈❝♦✈ ❛❞❞✶ ❝♦❡❢❧✐st ①t❛❜❧❡ ❧♦❣▲✐❦ ❞r♦♣✶ ♣❧♦t ❡①tr❛❝t❆■❈ st❡♣❈♦♠♣♦♥❡♥t ✐♥t❡♥s✐t②♣❧♦t ♣r♦❢✐❧❡ ✐❛❢♣❧♦t r❡s✐❞✉❛❧s ❝❤❡❝❦❘❡s✐❞✉❛❧Pr♦❝❡ss t❡r♠s ❘✵

useR! 2015, Spatial Session 1 July 2015 s✉r✈❡✐❧❧❛♥❝❡: Spatio-Temporal Analysis of Epidemic Phenomena Page 16

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Case study II: Measles (areal count time series)

Regression framework

Number of cases in region r at time t Yrt | Y ·,t−1 ∼ NegBin(µrt, ψ) Endemic-Epidemic decomposition of disease risk: µrt = ert νrt + λrtYr,t−1 + φrt

  • s=r

wsr Ys,t−1 ert population offset νrt, λrt, φrt log-linear predictors, e.g., vaccination coverage wsr weight for s to r transmission, e.g., wsr = o−d

sr

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Case study II: Measles (areal count time series)

Model estimation

✭❡♥❞❡♠✐❝ ❁✲ ❛❞❞❙❡❛s♦♥✷❢♦r♠✉❧❛✭⑦❧♦❣✭♣❙✉s❝❡♣t✐❜❧❡✮ ✰ t✮✮ ★★ ⑦❧♦❣✭♣❙✉s❝❡♣t✐❜❧❡✮ ✰ t ✰ s✐♥✭✷ ✯ ♣✐ ✯ t✴✺✷✮ ✰ ❝♦s✭✷ ✯ ♣✐ ✯ t✴✺✷✮ ♠❡❛s❧❡s▼♦❞❡❧ ❁✲ ❧✐st✭ ❡♥❞ ❂ ❧✐st✭❢ ❂ ❡♥❞❡♠✐❝✱ ♦❢❢s❡t ❂ ♣♦♣✉❧❛t✐♦♥✭♠❡❛s❧❡s❲❡s❡r❊♠s✮✮✱ ❛r ❂ ❧✐st✭❢ ❂ ⑦✶✮✱ ♥❡ ❂ ❧✐st✭❢ ❂ ⑦✶✱ ✇❡✐❣❤ts ❂ ❲❴♣♦✇❡r❧❛✇✭♠❛①❧❛❣ ❂ ✺✮✮✱ ❢❛♠✐❧② ❂ ✧◆❡❣❇✐♥✶✧✱ ❞❛t❛ ❂ ❧✐st✭♣❙✉s❝❡♣t✐❜❧❡ ❂ ✶ ✲ ♣❱❛❝❝✮✮ ♠❡❛s❧❡s❋✐t ❁✲ ❤❤❤✹✭♠❡❛s❧❡s❲❡s❡r❊♠s✱ ❝♦♥tr♦❧ ❂ ♠❡❛s❧❡s▼♦❞❡❧✮

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Case study II: Measles (areal count time series)

Fitted mean components

♣❧♦t✭♠❡❛s❧❡s❋✐t✱ t②♣❡ ❂ ✧❢✐tt❡❞✧✱ ✉♥✐ts ❂ ❝✭✼✱✶✷✮✱ ❤✐❞❡✵s ❂ ❚❘❯❊✮

2001.0 2002.0 2003.0 2 4 6 8

  • No. infected

LK Aurich

  • ●
  • ●
  • spatiotemporal

autoregressive endemic 2001.0 2002.0 2003.0 10 20 30 40 50

  • No. infected

LK Leer

  • ● ●● ● ●
  • ●
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Case study II: Measles (areal count time series)

Association with vaccination coverage vr

Endemic incidence is proportional to (1 − vr)βv:

s✉♠♠❛r②✭♠❡❛s❧❡s❋✐t✮✩❢✐①❡❢❬✧❡♥❞✳❧♦❣✭♣❙✉s❝❡♣t✐❜❧❡✮✧✱ ❪ ★★ ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ★★ ✷✳✵✺✹ ✵✳✸✼✾

✉♣❞❛t❡✭✮ s✐♠✉❧❛t❡✭✮ ♦♥❡❙t❡♣❆❤❡❛❞✭✮

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Case study II: Measles (areal count time series)

Association with vaccination coverage vr

Endemic incidence is proportional to (1 − vr)βv:

s✉♠♠❛r②✭♠❡❛s❧❡s❋✐t✮✩❢✐①❡❢❬✧❡♥❞✳❧♦❣✭♣❙✉s❝❡♣t✐❜❧❡✮✧✱ ❪ ★★ ❊st✐♠❛t❡ ❙t❞✳ ❊rr♦r ★★ ✷✳✵✺✹ ✵✳✸✼✾

Other methods: ✉♣❞❛t❡✭✮, s✐♠✉❧❛t❡✭✮, ♦♥❡❙t❡♣❆❤❡❛❞✭✮, . . .

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Conclusion

s✉r✈❡✐❧❧❛♥❝❡ offers a comprehensive framework for the spatio-temporal analysis of epidemic phenomena, including visualisation, modelling, inference and simulation of: – (multivariate) surveillance time series – spatio-temporal point patterns – geo-referenced SIR event histories Key references: – http://surveillance.r-forge.r-project.org/ – arXiv:1411.0416 (Meyer et al., 2014): a guide to the three presented endemic-epidemic model classes

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Acknowledgments

Joint work with: – Leonhard Held (University of Zurich) – Michael Höhle (University of Stockholm) Funding: – Munich Center of Health Sciences (2007–2010) – Swiss National Science Foundation (2012–2015)

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References

◮ Brockmann, D., Hufnagel, L., and Geisel, T. (2006). The Scaling Laws of Human Travel. Nature, 439(7075):462–465. ◮ Höhle, M. (2009). Additive-Multiplicative Regression Models for Spatio-Temporal Epidemics. Biometrical Journal, 51(6):961–978. ◮ Meyer, S., Elias, J., and Höhle, M. (2012). A Space-Time Conditional Intensity Model for Invasive Meningococcal Disease Occurrence. Biometrics, 68(2):607–616. ◮ Meyer, S. and Held, L. (2014). Power-Law Models for Infectious Disease

  • Spread. The Annals of Applied Statistics, 8(3):1612–1639.

◮ Meyer, S., Held, L., and Höhle, M. (2014). Spatio-Temporal Analysis of Epidemic Phenomena Using the R Package s✉r✈❡✐❧❧❛♥❝❡. arxiv:1411.0416. ◮ Tobler, W. R. (1970). A Computer Movie Simulating Urban Growth in the Detroit

  • Region. Economic Geography, 46:234–240.

Feedback? sebastian.meyer@uzh.ch