# Epidemic models

Alex Reinhart – Updated July 20, 2016

Written from the perspective of someone trying to do spatiotemporal crime modeling, not disease models.

Typical epidemic models focus on individuals, not locations – they look at patients, model them as Susceptible, Infected, or Recovered, and model how people transition between states. This is the SIR model, and sets up differential equations describing transitions. Natural extensions are things like network models for the interactions between individuals, so the structure of contacts between infected and susceptible people is captured.

Metapopulation models do analyze space, but only by breaking the population into sub-populations which have connectivity dependent on spatial factors. We could, for example, have sub-populations for different neighborhoods and adjust their connectivity based on how difficult it is to get from one to the next. But that doesn’t directly model risk in space – we’re still looking at the people, not the locations. The quantity of interest for all of these models is the number of infected people as a function of time, not the spatial risk of future infections.

An overview: Lawson, A. B. (2006). Statistical Methods in Spatial Epidemiology (2nd ed.). Wiley. doi:10.1002/9780470035771

• Brix, A., & Diggle, P. J. (2001). Spatiotemporal Prediction for Log-Gaussian Cox Processes. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63(4), 823–841. doi:10.1111/1467-9868.00315

Introduces a Cox model with a conditional intensity function \Lambda(x, t) = \lambda(x) \exp(Y(x, t)), where Y(x, t) is a stationary Gaussian process. In their version, \lambda(x) is the spatial variation in the population (say, its density) and Y(x, t) is the risk at a point and time.

This doesn’t seem amenable to self-excitation, since Y(x, t) is a Gaussian process and not some arbitrary function of previous events and covariates.

## Conditional intensity model

• Meyer, S., Elias, J., & Höhle, M. (2011). A Space-Time Conditional Intensity Model for Invasive Meningococcal Disease Occurrence. Biometrics, 68(2), 607–616. doi:10.1111/j.1541-0420.2011.01684.x

A self-exciting point process model with covariates, with data aggregated into small political units. See also Meyer’s MSc thesis (despite what the citation says): Meyer, S. (2010). Spatio-temporal infectious disease epidemiology based on point processes (PhD thesis). Ludwig-Maximilians-Universität München. https://epub.ub.uni-muenchen.de/11703/1/MA_Meyer.pdf

Run on a small dataset of just over 600 cases. Of dubious practicality for larger datasets, unless you abandon the Gaussian spatial kernel and go for something similar: the spatial covariates require numerical integration to get the log-likelihood, and preclude EM, so the model is fit via ordinary numerical maximization.

Method is thoroughly implemented in the R package surveillance, which is pure R.

• Meyer, S., & Held, L. (2014). Power-law models for infectious disease spread. Annals of Applied Statistics, 8(3), 1612–1639. doi:10.1214/14-AOAS743

Switches to a power-law kernel for spatial decay instead of the previous Gaussian kernel, and finds a better fit for their data.

• Meyer, S., Warnke, I., Rössler, W., & Held, L. (2016). Model-based testing for space-time interaction using point processes: An application to psychiatric hospital admissions in an urban area. Spatial and Spatio-Temporal Epidemiology, 17(C), 15–25. doi:10.1016/j.sste.2016.03.002

Applying the model (with simplified spatial kernel, since the dataset is larger) to test for epidemic behavior, by testing the coefficients on the self-exciting portion of the model. This doesn’t indicate where epidemic behavior is most likely, but merely tests for its presence.