### DIMACS Working Group on Spatio-Temporal and Network Modeling of Diseases

#### April 22 - 26, 2003 DIMACS Center, CoRE Building, Rutgers University

Organizers:
Valerie Isham, University College, London, valerie@stats.ucl.ac.uk
Alun Lloyd, Institute for Advanced Study, alun@alunlloyd.com
Bryan Grenfell, Cambridge University
Matthew Keeling, Cambridge
Denis Mollison, Heriot-Watt University
Presented under the auspices of the Special Focus on Computational and Mathematical Epidemiology.

### Abstracts

Frank Ball, University of Nottingham

Title: Stochastic modelling and statistical inference for multitype SIR epidemics amoung a population of households

Standard deterministic models of epidemics implicitly assume that the population among which the disease is spreading is locally as well as globally large. The same assumption is made when analysing the asymptotic behaviour of many stochastic epidemic models. However, this assumption seems inappropriate for many human and animal epidemics, since such populations are usually partitioned into small groups or households, and there has been recent interest in models which explicitly incorporate such household structure. Most of this work has been concerned with homogeneous populations (e.g. Becker and Dietz (1995) and Ball et al. (1997)). However, heterogeneities, such as those due to age, sex and response to vaccine, are known to have an important impact on disease spread.

This talk is concerned with a stochastic model (Ball and Lyne (2001)) for the spread of a an SIR (susceptible $\to$ infective $\to$ removed) epidemic among a closed, finite population that contains several types of individuals and is partitioned into households. The infection rate between two individuals depends on the types of the transmitting and receiving individuals and also on whether the infection is within- or between-household. The threshold behaviour of the model is briefly outlined and an associated threshold parameter $R_*$ is determined. A pseudo- likelihood framework is presented for making statistical inferences about the parameters governing such epidemics from final outcome data, when only some of the households in the population are observed. It turns out that the between-household infection rates (and hence $R_*$) are not identifiable from final outcome data, unless some constraints are placed on them. Thus a Perron-Frobenius argument is used to derive sharp lower and upper bounds for $R_*$, which are identifiable. Determining the allocation of vaccines that reduces the upper bound for $R_*$ to its threshold value of one with minimum vaccine coverage is shown to be a linear programming problem. The methodology is illustrated by applications to data on influenza epidemics in Tecumseh, Michigan.

References
• Ball F G, Mollison D and Scalia-Tomba G (1997) Epidemics with two levels of mixing. Ann Appl Prob 7, 46-89.
• Ball F G and Lyne O D (2001) Stochastic multitype SIR epidemics among a population partitioned into households. Adv Appl Prob 33, 99-123.
• Becker N G and Dietz K (1995) The effect of the household distribution on transmission and control of highly infectious diseases. Math Biosci 127, 207-219.
* Based on joint work with Tom Britton (Stockholm University) and Owen Lyne (University of Nottingham).

Ben Bolker, Zoology Department, University of Florida

Title: Correlation equations for spatial epidemics: dynamics and estimation

I will discuss a general framework developed for tracking the spatial correlations that develop over time in an epidemic among hosts distributed in a continuous landscape, and the feedbacks between spatial pattern and the progression of the epidemic. The hosts may be regularly spaced, randomly distributed or clustered; the "infection kernel", the probability of infection as a function of distance, may have a variety of forms. After discussing the derivation of the equations and presenting some basic results, I will consider briefly how estimates of transient or equilibrium spatial correlations can be used to try to estimate dispersal kernels and the relative importance of environmental heterogeneity and epidemic spread processes in determining overall spatial pattern.

Stephen Eubank, Los Alamos National Lab

Title: Modeling Disease Outbreak and Interventions on a Realistic Social Network

I will describe the process of using activity surveys to estimate contact patterns in a large urban region,. The resulting time-dependent social network in combination with models of person-to-person disease transmission enables highly resolved -- individual people at specific locations within the region -- simulation of an epidemic's path through the region. The high resolution also facilitates modeling proposed response strategies, taking into account associated resource constraints. I will compare simulated outcomes of several response strategies in a smallpox scenario. I will also discuss work in progress using graph-theoretic characteristics of the social network directly to optimize responses.

Title: The construction and analysis of epidemic trees with reference to the 2001 UK foot-and-mouth outbreak

Reliable estimation of R0 is a problem central to epidemiology, most often accomplished by fitting dynamic models to data and estimating combinations of parameters that equate to the case reproduction ratio. I will describe an alternative non-parametric method that permits direct reconstruction of the branching process by which the epidemic arose. From these 'epidemic trees' case reproduction ratios can be estimated directly. Bootstrap algorithms are used to generate confidence intervals for these estimates that show the estimation procedure to be encouragingly robust. Identifying and 'pruning' branches from these trees whose occurrence might have been prevented by implementation of more stringent control measures permits estimation of the possible efficacy of these alternative measures. Examination of the cladistic structure of these trees as a function of the distance of each case from its infection source reveals some insights about the relationship between long distance transmission events and epidemic size. The methods are demonstrated using data from the 2001 foot-and-mouth-disease outbreak in the United Kingdom.

Alun L. Lloyd, Institute for Advanced Study

Title: Spatiotemporal dynamics of childhood diseases in the US, 1950-present

Using the historical incidence records of measles, mumps, rubella and chickenpox, I shall discuss aspects of the spatiotemporal dynamics of childhood diseases in the United States. For some of these diseases, the incidence records cover the period when mass-vaccination programs were introduced, and the impact of such programs on both spatial and temporal dynamics will be described.

The spatial coherence of epidemics across the country is of particular interest, partly because of the important role played by spatial effects in enhancing disease persistence. Both global and local mechanisms can give rise to synchrony between outbreaks, and these will be discussed both in terms of the data and with the aid of simple epidemiological models. These patterns of synchrony changed with the onset of vaccination, and I shall discuss the ability-- or inability-- of simple models to reproduce these patterns.

James Lloyd-Smith, Biophysics, University of California at Berkeley

Title: Pair formation, disease-induced changes in behaviour, and frequency-dependent transmission in sexually-transmitted disease models

We explore the relationship between partnership dynamics and transmission of a sexually-transmitted disease (STD), and consider situations where disease status influences pairing behaviour. Starting from a pair-formation model, we apply a steady-state approximation to derive analytic expressions for the incidence rate, basic reproductive ratio (R0), and endemic prevalence (i%) of an epidemic, for four cases of increasing disease-induced behavioural change. When disease does not alter behaviour, the pair-formation model is precisely equivalent to the classical frequency-dependent transmission model; when behaviour is more complex, we derive generalized frequency-dependent results. The expression for R0 is identical for all cases, giving refined insights into determinants of invasion ability for STDs. Effects of disease-induced changes of behaviour are illustrated by simulating epidemics of bacterial and viral STDs, and the robustness of the steady-state approximation is explored. Our mechanistic derivation clarifies conditions required for simple frequency-dependent formulations to accurately represent STD incidence, which depend strongly on the disease being studied: such formulations are more suitable for chronic, less-transmissible infections than for transient, highly-transmissible infections. Our results thus support earlier proposals to divide STDs into these two functional classes. We discuss application of our results to STDs (in humans and animals) and other infectious diseases.
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James will also briefly discuss some of his current work on the effects of host movement on disease spread.

Mark Newman, Center for the Study of Complex Systems, University of Michigan

Title: How the structure of contact networks affects disease propagation

I will discuss observational results on the statistical properties of contact networks, including degree distributions, correlation properties, transitivity, assortativity, and so forth. Then I will talk about models of disease propagation on networks with these properties and describe how the properties in question affect the spread of the disease. Some results are already well known, such as the fact that highly degree-assortative networks support epidemics at lower density, but others are not, such as the fact that high levels of transitivity cause diseases to saturate populations in regimes only slightly above the epidemic threshold.

Title: Modelling and Bayesian inference for structured-population epidemics

We consider the problem of Bayesian statistical inference for stochastic epidemic models in structured populations. Two situations are described. In the first, temporal data from an outbreak of Gastroentiritis is considered. Here, Bayesian model choice can be used to address questions concerning transmission routes. In the second, a two-level mixing model is considered in which the likelihood required for inference is intractable. An approximation method is developed that relies on limiting results for large-population dynamics. The method is applied to influenza outbreak data, and the corresponding approximation for a homogeneous model is evaluated using exact results.

Lisa Sattenspiel, University of Missouri-Columbia

Title: Modeling the process of contact between subgroups in spatial epidemics

One of the most essential aspects of the spatial spread of disease is the nature of contact between members of different populations, since these patterns of contact determine where, when, and how fast an epidemic spreads throughout a region. A variety of approaches have been used to model spatial contact patterns, ranging from simple nearest-neighbor diffusion processes to completely stochastic individual-based models. The choice of an appropriate contact model is a difficult one, however, and is constrained not only by important modeling questions, such as the type of model being developed and the degree of complexity to include in the model, but also by the existing or potential availability of adequate data with which to estimate essential model parameters. In this paper I will review the most common approaches used to model contact among populations in spatial epidemic models, with an emphasis on human diseases. The goal will be to stimulate further discussions of how best to capture the essential nature of between population contact while maintaining a manageable model that contains potentially estimable parameters.

Title: Patterns of synchrony in influenza epidemics at the national and regional scale: 1968-1998.

KEYWORDS : influenza/epidemiology/mortality, comparative study, seasons, synchrony, United States, Australia, France.

Influenza epidemics occur once a year in wintertime in temperate areas of the world. The extent of similarity between epidemics in different locations is unknown. Here we analyzed pneumonia and influenza series from 1968 to 1998 in the United States, France and Australia to investigate the "synchrony", or similarities, of influenza dynamics at the national and regional level. Weekly number of deaths from pneumonia and influenza (P&I) were obtained from death certificates collected by national agencies for vital statistics over the period 1968-98. We used codes 470-474 and 480-486 from the International Classification of Diseases (ICD) 8th revision, before 1979, and codes 480-487 from ICD-9 from 1979 onwards. Demographic data were obtained from the same services.

At the national level, 3 measures of synchrony were derived from the raw P&I series (N=1410 weeks in 3 countries): the date of the peaks, phases, and excess mortality. We found that there was a high level of synchrony between France and the United States, as far as the time of the peak (median time lag = 0.5 week, P<0.001), the excess deaths (correlation coefficient \rho=0.76, P<0.001) and the phases (\rho=0.73, P<0.001). Because of the latitudes of the 3 countries, influenza epidemics in Australia are naturally out of phase from those in the US and France. Given a systematic half-year lead and systematic delay between the two hemispheres, there was no evidence of synchrony of influenza epidemics in Australia as compared with the US or France. These results support a high level of coherence at the hemisphere level and suggest that the global interhemispheric circulation of epidemics follows an irregular pathway with recurrent changes in the leading hemisphere.

At the regional level, P&I series in the US were aggregated by month and broken down by state and age (0-24 years old, 25-64, >= 65). Four measures of synchrony derived from the raw series were considered (N=372 months in 51 states): log transformed and first-differenced amplitudes, excess deaths, peak values and phases. Investigation of the between states cross-correlations revealed that for all 4 measures, a decline in synchrony was observed as the distance between the states increases; this relation was stronger in persons over 65 years old. The correlation coefficient between synchrony and distance ranged from -0.53 to -0.63 depending on the measure (randomization tests; P<0.001).

Synchrony of influenza epidemics might result from dispersal factors such as population fluxes, regionally correlated environmental forces, or both. Current knowledge points to population movements as a driving force of epidemic spread, and hence synchrony, while the role of environmental factors remains debated. Detailed studies on the functional form of synchrony at various geographical scales may shed light on the respective role of these mechanisms.