## The Annals of Applied Statistics

- Ann. Appl. Stat.
- Volume 8, Number 3 (2014), 1612-1639.

### Power-law models for infectious disease spread

Sebastian Meyer and Leonhard Held

#### Abstract

Short-time human travel behaviour can be described by a power law with respect to distance. We incorporate this information in space–time models for infectious disease surveillance data to better capture the dynamics of disease spread. Two previously established model classes are extended, which both decompose disease risk additively into endemic and epidemic components: a spatio-temporal point process model for individual-level data and a multivariate time-series model for aggregated count data. In both frameworks, a power-law decay of spatial interaction is embedded into the epidemic component and estimated jointly with all other unknown parameters using (penalised) likelihood inference. Whereas the power law can be based on Euclidean distance in the point process model, a novel formulation is proposed for count data where the power law depends on the order of the neighbourhood of discrete spatial units. The performance of the new approach is investigated by a reanalysis of individual cases of invasive meningococcal disease in Germany (2002–2008) and count data on influenza in 140 administrative districts of Southern Germany (2001–2008). In both applications, the power law substantially improves model fit and predictions, and is reasonably close to alternative qualitative formulations, where distance and order of neighbourhood, respectively, are treated as a factor. Implementation in the R package *surveillance* allows the approach to be applied in other settings.

#### Article information

**Source**

Ann. Appl. Stat. Volume 8, Number 3 (2014), 1612-1639.

**Dates**

First available in Project Euclid: 23 October 2014

**Permanent link to this document**

http://projecteuclid.org/euclid.aoas/1414091227

**Digital Object Identifier**

doi:10.1214/14-AOAS743

**Mathematical Reviews number (MathSciNet)**

MR3271346

**Zentralblatt MATH identifier**

1304.62135

**Keywords**

Power law spatial interaction function infectious disease surveillance stochastic epidemic modelling branching process with immigration multivariate time series of counts spatio-temporal point process

#### Citation

Meyer, Sebastian; Held, Leonhard. Power-law models for infectious disease spread. Ann. Appl. Stat. 8 (2014), no. 3, 1612--1639. doi:10.1214/14-AOAS743. http://projecteuclid.org/euclid.aoas/1414091227.

#### Supplemental materials

- Supplementary material A: Animations of the IMD and influenza epidemics. •Observed evolution of the IMD and influenza epidemics. •Simulated counts from various models for the 2008 influenza wave. •Weekly mean PIT histograms for these predictions.URL: Link to itemSupplemental files available for subscribers.
- Supplementary material B: Inference details, integration of isotropic functions over polygons, and additional figures and tables. •Details on likelihood inference for both models. •Integration of radially symmetric functions over polygonal domains. •Additional figures and tables of the power-law models for invasive meningococcal disease and influenza.Digital Object Identifier: doi:10.1214/14-AOAS743SUPPBSupplemental files available for subscribers.