Combining social contact data with spatio-temporal models for infectious diseases Leonhard Held ISCB 2015, Utrecht, 26 August 2015 Joint work with Sebastian Meyer Financial support by the Swiss National Science Foundation Social Contact Data POLYMOD study estimated contact matrices in eight EU countries. Mossong et al. (2008) Leonhard Held (University of Zurich) Modelling infectious diseases 26 August 2015 2 / 24 Statistical Modelling of Infectious Disease Spread I Spatio-temporal models have been proposed for counts of infectious diseases Schrödle et al. (2011) Meyer and Held (2014) I Contact data has been used to analyse infectious disease spread between age groups Goeyvaerts et al. (2015) → Combine social contact data with spatio-temporal time series models for infectious disease counts Ygrt : I I I Age Group g Region r Time t Leonhard Held (University of Zurich) Modelling infectious diseases 26 August 2015 3 / 24 Case Study: Noroviral Gastroenteritis I Generation time similar to seasonal influenza: 3 − 4 days I Highly infectious via droplets I No vaccination available Weekly counts downloaded from https://survstat.rki.de/ I Age Group: Region: Time: Stratification in 5-year intervals Berlin Week 2011/27 to 2014/26 → 15 age groups → 12 city districts → 156 weeks I Latest revision of reporting scheme in 2011 → Only laboratory-confirmed cases are notifiable I Lower reporting rates during Christmas break Leonhard Held (University of Zurich) Modelling infectious diseases 26 August 2015 4 / 24 Weekly Counts (All Districts and Age Groups) | Christmas break 200 No. infected 150 100 50 0 2011 2011 III IV Leonhard Held (University of Zurich) 2012 2012 2012 II III 2013 2013 2013 IV II Time [week] Modelling infectious diseases III IV 2014 II 26 August 2015 5 / 24 Mean Incidence by District 52.65°N 52.6°N Pankow Reinickendorf 52.55°N Spandau Charlottenburg− Wilmersdorf 52.5°N 52.45°N Steglitz− Zehlendorf Mitte Lichtenberg Marzahn− Hellersdorf Friedrichshain− Kreuzberg Tempelhof− Neukoelln Schoeneberg Treptow− Koepenick 52.4°N 52.35°N 13.1°E 13.2°E 13.3°E 13.4°E 13.5°E 13.6°E 13.7°E 49.00 64.00 81.00 100.00 121.00 144.00 169.00 Mean yearly incidence [per 100 000 inhabitants] Leonhard Held (University of Zurich) Modelling infectious diseases 26 August 2015 6 / 24 300 200 100 + 70 −6 9 −6 4 65 −5 9 60 55 −5 4 −4 9 Modelling infectious diseases 50 −4 4 45 40 −3 9 −3 4 35 −2 9 30 25 −2 4 −1 9 Leonhard Held (University of Zurich) 20 −1 4 15 10 −0 9 00 −0 4 0 05 Mean yearly incidence [per 100 000 inhabitants] Mean Incidence by Age Group 26 August 2015 7 / 24 Aggregate into Larger Age Groups Aggregate into 5 age groups: 00-04, 05-14, 15-49, 50-64, 65+ 70+ 65−69 60−64 55−59 50−54 45−49 40−44 35−39 30−34 25−29 20−24 15−19 10−14 05−09 00−04 4.0 6 65+ 3.0 2.5 2.0 1.5 1.0 age group of participant 3.5 5 50−64 4 15−49 3 2 05−14 0.5 1 00−04 0.0 00 − 05 04 − 10 09 − 15 14 − 20 19 − 25 24 − 30 29 − 35 34 − 40 39 − 45 44 − 50 49 − 55 54 − 60 59 − 65 64 −6 9 70 + age group of participant I 0 00−04 15−49 50−64 65+ age group of contact age group of contact Leonhard Held (University of Zurich) 05−14 Modelling infectious diseases 26 August 2015 8 / 24 Weekly Incidence by Age Group Aggregated across districts, per 100 000 inhabitants 30 00−04 30 05−14 30 25 25 25 20 20 20 15 15 15 10 10 10 5 5 5 0 0 0 30 50−64 30 25 25 20 20 15 15 10 10 5 5 0 0 Leonhard Held (University of Zurich) 15−49 65+ | Christmas break Modelling infectious diseases 26 August 2015 9 / 24 Regression Model for Infectious Disease Counts I Additive endemic-epidemic decomposition of disease incidence Held et al. (2005) Paul et al. (2008) Endemic ⊕ Epidemic I seasonality, population, socio-demography, climate, . . . force of previously infected individuals ⇒ spatio-temporal interaction Multivariate branching process formulation → epidemic proportion λ Leonhard Held (University of Zurich) Modelling infectious diseases 26 August 2015 10 / 24 Spatio-temporal Model for Infectious Disease Counts I Time series model for weekly counts Yrt in region r and week t Meyer and Held (2014) I Negative binomial likelihood with mean X bwr 0 r c Yr 0 ,t−1 µrt = ert νrt + φrt r0 ert νrt , φrt wr 0 r bwr 0 r c known population fractions log-linear predictors weights for r 0 to r transmission, e. g . power law: wr 0 r = (or 0 r + 1)−d with adjacency order or 0 r and unknown decay parameterPd normalized weights such that r bwr 0 r c = 1 Leonhard Held (University of Zurich) Modelling infectious diseases 26 August 2015 11 / 24 Specific Model Formulation µrt = er exp{α + βxt + γ sin(ωt) + δ cos(ωt)} X b(or 0 r + 1)−d cYr 0 ,t−1 + φr r0 I I I I xt : Christmas break indicator Sinusoidal log-rate with frequency ω = 2π/52 Power-law distance decay Model selection: Model endemic-only + power-law Leonhard Held (University of Zurich) dim 5 18 ∆ AIC 0.0 −537.3 Modelling infectious diseases d̂ – 2.5 λ̂ 0 0.64 26 August 2015 12 / 24 Fitted Mean by District Charlottenburg−Wilmersdorf 30 epidemic endemic 25 20 ● 15 ● ● ● ● ● 25 10 0 ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ●● ● ●● ● ● ●● ● ●● ● ● ● ●●● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ●● ●● ● ●● ● ● ● ● ●●● ● ●● ●● ●●● ●●● ●● ● ● ● ● ●●● ● ● ●● ● ●● ●● ● ● 0 ● ● 10 ● ● ● ●● ● ● ● ●●● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ●● ● ● ●● ● ● ● ●●● ● ● ●●●● ● ● ● ● ● ● ● ●●●● ● ● ● ● ●●● ●● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● 5 ●● ● ● ● ●● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● 15 ● ● ● 0 20 ● ● ●● ● ● ● ● 5 ● ● ● 15 ● 25 ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●● ●● ● ● ● ● ●● ● ●● ●● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ●● ●● ●● ● ● ● 30 ● 20 ● ● ● Marzahn−Hellersdorf ● 25 20 ● 5 Lichtenberg 30 15 ● ● 10 Friedrichshain−Kreuzberg 30 ● ●● ● ● ● ●● ● ● ● ●● ● ● 5 0 ● ● ● ●● 10 ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ●● ● ●● ● ● ● ●● ● ● ●● ●● ● ●● ●● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● Mitte 30 Neukoelln 25 ● ● ● ● Pankow 30 ● ● 25 ● ● 20 ● Reinickendorf ● 30 ● 25 ● ● ● 30 ● ● ● ● ● 20 ● ● ● ● 10 ● ● ● ● 0 ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ●● ●●● ●● ● ●● ● ●●● ● ●● ● ● ● ● ● 5 ● ●● ● ● ● ●● ● ● ● ●● ●● ● ● ●●● ● ● ● ● 15 ● ● ●●●● ● ● ● ● ● ● ● 10 5 0 ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ●●● ● ● ●●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ●● ●● ●● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ●●● ●● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● 10 20 ● 15 ● ● ● 20 ● ● ● ● ● ● 15 ● 25 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ●● ● ● ● ●● ●● ● ●● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● 15 ● ● ● ● ● ● 0 ●● ● ● ● ● ●● ● ●●● ●● ● ● 0 ● ● ● ●● ●● 10 5 ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●● ● ●● ●● ● ● ●● ● ● ● ●● ● 5 ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●●● ●● ● ● ● ● ● ●● ●● ●● ●● ● ●● ● ● ● ● ●● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● Spandau Steglitz−Zehlendorf ● Tempelhof−Schoeneberg ● ● 30 30 ● 30 ● ● ● Treptow−Koepenick 30 ● 25 25 ● ● ● ● ● ● ●● 25 ● ● ● ●● ●● 20 ● ● ● ● 15 ● ● ● ● ● 10 ● ● ● ● ● 5 0 ● ● ● ● ●● ● ●● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ●● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ●● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● 10 ● ● ● ● 5 0 Leonhard Held (University of Zurich) ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ●● ● ●● ● ● ●● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● 15 ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ●● ●● ● ●● ● ● ●● 10 5 0 ● 20 ● ● ● ● ● 15 ● ● 20 ● 25 ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ●● ●● ● ● ● ● ● 15 ● ●● ● ●● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ●●● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ●●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● Modelling infectious diseases ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ●● ● ●● ● ● ● ● ● ● ●● ●● ●● ● ●●● ● ● ● ● ●● ● ●● ● ● ●●● ● ●● ● ●● ● ● 10 ● ● ●● ● ● ● ● ●● ●● 5 0 ●● ● 26 August 2015 13 / 24 Power Law Weights 1.0 ● ● weight 0.8 power law unconstrained 0.6 0.4 0.2 ● ● 0.0 0 1 2 ● ● 3 4 adjacency order o Leonhard Held (University of Zurich) Modelling infectious diseases 26 August 2015 14 / 24 Normalized Power Law Weights Pankow 46.8% 8.4% 1.5% 3.0% 1.4% Leonhard Held (University of Zurich) 7.9% 7.9% 3.0% 3.0% 2.9% 44.4% 2.9% 3.0% 8.4% 3.0% 7.9% 7.9% 8.4% 8.4% 3.0% Mitte 2.9% Modelling infectious diseases 7.9% 2.9% 2.9% 26 August 2015 15 / 24 Age-Stratified Spatio-temporal Model µgrt = egr exp{αgG + αrR + βxt + γg sin(ωt) + δg cos(ωt)} X R + φG φ bCg 0 g (or 0 r + 1)−d c Yg 0 ,r 0 ,t−1 g r (g 0 ,r 0 ) I Age group and region-specific effects αgG and αrR I Age group-specific seasonality γg , δg I R Parsimonious “main effects” decomposition: φgr = φG g φr I Cg 0 g : Number of social contacts of participant in age group g 0 with age group g I Power law weights (or 0 r + 1)−d Leonhard Held (University of Zurich) Modelling infectious diseases 26 August 2015 16 / 24 Model Selection Model endemic-only with age group and region-specific effects + age group-specific seasonality + power-law, homogeneous mixing between age groups + power-law with social contact data + power-law, no contact between age groups Leonhard Held (University of Zurich) Modelling infectious diseases dim 20 28 45 45 45 ∆ AIC 0.0 −604.8 −944.1 −1095.4 −1052.3 d̂ – – 2.8 2.5 2.2 26 August 2015 λ̂ – – 0.47 0.68 0.7 17 / 24 Fitted Mean By Age Group Aggregated across districts 00−04 140 05−14 15−49 140 140 120 120 100 100 80 80 80 60 60 60 from other groups within group endemic 120 100 ● 40 20 0 ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ●●● ● ● ●●● ● ●● ● ●● ● ●● ● ●● ● ● ● ● ● ●●●● ● ● ● ●●●● ● ●● ●●● ●● ● ●●●● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ●●●● ● ● ●●●● ● ●● ● ●● ● ● ● ●● ●●●●● ● ● ●●● ● ● ● ● ●● ● ● ● ● ●●●●●●●● ● 40 40 20 20 ● ● 0 ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 50−64 0 ● ● ● ● ● ●● ● ● ● ●● ● ●● ●● ●●● ●● ● ●● ● ●●●●● ●●●● ● ●● ● ● ●● ● ● ●● ●● ●●●●●● ● ● ●● ● ● ● ●● ●● ●●● ● ● ●● ●● ●● ● ● ● ●● ● ● ●● ●● ● ● ●● ● ●● ● ●● ● ●● ●●● ● ● ●● ● ● ● ● ●●●● ● ●●● ● ●●● ●●● ● ● ● ●● ● ●● ●● ● ● ● ● ●● ● ●● ● ● ● ●● ●● ●● ● ● 65+ 140 140 120 120 100 100 ● ●● ● ● ● ● ● ● 80 ● ● ● ● ●●● 80 ● ● ● ● ● ● ● ● ● ●● 60 60 40 40 ● 20 0 Leonhard Held (University of Zurich) 20 0 ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ●● ● ●●●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ●● ●●● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ●●●● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ● ●● ● ● ●● ●●● ●●● ● ●●●●●●● ● ●●● ●● ● ● ●●● ● ● ● ●●● ●●● ●● ●● ● ●● ● ● ● ● ●● ●●●● ● ●●● ●● ● ● ● ●● ●●●●●●●●● ●● ● ●●●●● ● ●● ●●●● ●●●●●●●●●● ● ● ●● ● ●●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ●●●●●●● Modelling infectious diseases ● ● ● ● ● ●● ●● ●●● 26 August 2015 18 / 24 Regional Effects Endemic Epidemic 0.6 0.4 Pankow Reinickendorf 0.6 0.4 Pankow Reinickendorf 0.2 Spandau Charlottenburg− Wilmersdorf Mitte Lichtenberg Marzahn− Friedrichshain− Hellersdorf Kreuzberg 0.0 0.2 Spandau Charlottenburg− Wilmersdorf Mitte Lichtenberg Marzahn− Friedrichshain− Hellersdorf Kreuzberg −0.2 Steglitz− Zehlendorf Tempelhof− Neukoelln Schoeneberg −0.2 Treptow− Steglitz− Koepenick Zehlendorf −0.4 Tempelhof− Neukoelln Schoeneberg Treptow− Koepenick −0.6 Leonhard Held (University of Zurich) 0.0 Modelling infectious diseases −0.4 −0.6 26 August 2015 19 / 24 Normalized Power-Law and Contact Weights Transmission pattern from a case in Mitte aged 15-49 years 00−04 0.1% 0.1% 0.1% 05−14 0.1% 0.5% 0.1% 0.8% 0.0% 0.1% 0.1% 0.1%0.1% 0.1% 0.2% 0.5% 1.0% 1.0%0.4% 5.8% 0.2% 0.5%0.2% 0.2% 2.1% 5.8% 2.1% 5.8% 2.1% 32.9% 1.0% 5.8% 5.8%2.1% 2.1% 65+ 0.4% 0.4% 5.8% 0.4% 0.2% 1.0% 1.0% 0.4% 0.5% 0.2% 3.1% 0.1% 0.5% 50−64 1.0% 15−49 0.4% Leonhard Held (University of Zurich) 0.4% 0.1% 2.1% 0.1% 0.1% 0.4% 0.4% 0.1% 0.4%0.1% 0.1% Modelling infectious diseases 26 August 2015 20 / 24 Normalized Power-Law and Contact Weights Transmission pattern from a case in Mitte aged 65+ years 00−04 0.1% 0.0% 0.1% 05−14 0.1% 0.6% 0.0% 0.4% 0.0% 0.1% 0.0% 0.1%0.0% 0.0% 0.2% 0.6% 1.8% 1.8%0.7% 3.2% 0.2% 0.6%0.2% 0.2% 1.2% 3.2% 1.2% 3.2% 1.2% 18.2% 0.6% 3.2% 3.2%1.2% 1.2% 65+ 2.3% 0.7% 10.1% 0.7% 0.3% 1.8% 1.8% 0.7% 0.6% 0.2% 3.1% 0.1% 0.6% 50−64 1.8% 15−49 0.7% Leonhard Held (University of Zurich) 2.3% 0.8% 12.8% 0.8% 0.4% 2.3% 2.3% 0.8% 2.3%0.8% 0.8% Modelling infectious diseases 26 August 2015 21 / 24 Normalized Power-Law and Contact Weights Transmission pattern from a case in Mitte aged 00-04 years 00−04 1.7% 0.6% 1.7% 05−14 1.7% 1.0% 0.6% 9.4% 0.3% 1.7% 0.6% 1.7%0.6% 0.6% 0.4% 1.0% 0.8% 0.8%0.3% 4.0% 0.4% 1.0%0.4% 0.4% 1.4% 4.0% 1.4% 4.0% 1.4% 22.4% 0.7% 4.0% 4.0%1.4% 1.4% 65+ 0.5% 0.3% 4.5% 0.3% 0.1% 0.8% 0.8% 0.3% 1.0% 0.4% 5.7% 0.2% 1.0% 50−64 0.8% 15−49 0.3% Leonhard Held (University of Zurich) 0.5% 0.2% 2.8% 0.2% 0.1% 0.5% 0.5% 0.2% 0.5%0.2% 0.2% Modelling infectious diseases 26 August 2015 22 / 24 Summary and Outlook I Endemic-epidemic time series modelling is implemented in the open package surveillance source Meyer et al. (2014) I 3-dim. model describes disease spread by borrowing strength from I I different regions and different age groups Outlook: √ Age group-specific overdispersion → Increases model fit by ≈ 130 AIC units, model order remains the same I Comparison with age group-specific modelling → Improved predictive performance? I Estimation of parameters within the contact matrix? Leonhard Held (University of Zurich) Modelling infectious diseases 26 August 2015 23 / 24 References I Goeyvaerts, N., Willem, L., Kerckhove, K. V., Vandendijck, Y., Hanquet, G., Beutels, P., and Hens, N. (2015). Estimating dynamic transmission model parameters for seasonal influenza by fitting to age and season-specific influenza-like illness incidence. Epidemics, 13:1–9. Held, L., Höhle, M., and Hofmann, M. (2005). A statistical framework for the analysis of multivariate infectious disease surveillance counts. Statistical Modelling, 5(3):187–199. 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 surveillance. arXiv, 1411.0416. Mossong, J., Hens, N., Jit, M., Beutels, P., Auranen, K., Mikolajczyk, R., Massari, M., Salmaso, S., Tomba, G. S., Wallinga, J., Heijne, J., Sadkowska-Todys, M., Rosinska, M., and Edmunds, W. J. (2008). Social contacts and mixing patterns relevant to the spread of infectious diseases. PLoS Medicine, 5(3):e74. Paul, M., Held, L., and Toschke, A. (2008). Multivariate modelling of infectious disease surveillance data. Statistics in Medicine, 27(29):6250–6267. Schrödle, B., Held, L., and Rue, H. (2011). Assessing the impact of a movement network on the spatiotemporal spread of infectious diseases. Biometrics, 68:736–744. Leonhard Held (University of Zurich) Modelling infectious diseases 26 August 2015 24 / 24