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**Bill Amos**:
`Population genetics is dead, long live population genetics: new ways to
infer population history and population structure from genetics'

A cornerstone of traditional population genetic theory is the concept that rate of evolution of a DNA sequence is independent of population size. After all, how can a chromosome 'know' the size of the population in which it exists? However, there are now good grounds for believing that this assumption is false. During meiosis, homologous chromosomes pair and are subject to mismatch repair. Consequently, heterozygous sites have an opportunity to mutate which is not available to equivalent homozygous sites. Consequently, evolution may progress faster in circumstances where heterozygosity is high, namely large populations and during hybridisation events. I present empirical data to show that evolution really does progress faster in larger and hybrid populations, at least for markers such as minisatellites and microsatellites. Heterozygote instability has major and exciting implications for genome evolution and may generate a whole new range of tools with which to infer past and present population processes.

**Hakan Andersson**: `Epidemics and Random graphs'

Imagine a large population where each individual has a random number
of acquaintances. Let the *i*th individual have *D_i* friends, the
variables *D_i* being independent and identically distributed, and
suppose that two friends rarely have other friends in common. Now
introduce an infectious disease into the population by infecting one
individual. Assuming that a given infective individual infects a given
susceptible friend with probability *p*, under what conditions on *(D,p)*
is a large outbreak possible?

We model the situation above using random graphs, thereby discussing

- branching approximations (giving the probability of a large outbreak);
- deterministic approximations (giving the final epidemic size);
- constant vs random number of acquaintances;
- multitype extensions.

**Victor Apanius**:
Lymphocytes talk about modelling immunity

[Survey of biological aspects of immuno-genetics -- full text available]

**Frank Ball**:
`Models for the spread of epidemics with two levels of mixing'

*[Based on joint work with Denis Mollison and Gianpaolo Scalia-Tomba]*

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 also implicitly made when analysing the threshold behaviour of many stochastic epidemic models. However, this assumption is clearly inappropriate for most human and animal epidemics, since such populations are usually partitioned into small groups or households. In this talk, models for the spread of an epidemic among populations that mix at two levels, `local' and `global', will be considered.

The simplest model considers the spread of an SIR (susceptible ->
infective -> removed) epidemic among a population consisting of
*m* households, each of size *n*, with different infection rates for
local (within-household) and global (between-household) infections. The asymptotic situation
in which *m* tends to infinity with *n* fixed will be analysed, and the threshold behaviour of the model determined. The final outcome in the event
of a `major' epidemic will be determined and the problem of estimating
the parameters of the model from hosehold final size data will be discussed
briefly.

Another model considers the spread of an SIR epidemic among a population equally spaced on a circle, in which infectious individuals can make local (nearest-neighbour) and global (chosen at random from the whole population) infections. The threshold behaviour of this model and its final outcome in the event of a major epidemic will be determined.

Several extensions of the basic household model will be described, including unequal household sizes, heterogeneous populations (e.g. incorporating differences in infectiousness and susceptibility owing to age and sex) and more complex population structures, such as a population partitioned into towns consisting of households, with three levels of mixing. SIS (susceptible -> infective -> susceptible) models will also be considered briefly.

**Niels Becker and Tom Britton**:
`Statistical Studies of Disease Transmission: An Overview'

*[full text available as postscript file]*

Infectious disease data have two features that distinguish it from other data. They are highly dependent and the infection process is only partially observable. A consequence of these features is that the analysis of data is usually most effective when it is based on a model that describes aspects of the infection process, i.e. on a transmission model. Therefore modelling is clearly an integral part of statistical work in this area. While deterministic models can serve as a guide towards parameter estimates, the need to quantify the precision of estimates and the variation in data imply that stochastic models are the natural basis for the analysis of infectious disease data.

Statistical analysis, embracing modelling, parameter estimation, hypothesis testing and the design of studies, plays an essential role in bridging the gap between the mathematical theory and public health practice, and it is this aspect that motivates the present discussion. In other words, we attempt to promote the use of statistical analyses that provide practical insight and guidance for disease control, with emphasis on identifying issues that have not been addressed adequately.

Section 2 provides an overview of some standard epidemic models and methods for making statistical inference about their parameters. Its aims are both to help make the paper self-contained for readers unfamiliar with epidemic models and to act as a foundation for the discussion in the rest of the paper. Later sections are directed at more specific objectives. Methodology that can help to determine the way the disease is transmitted is reviewed in Section 3. In Section 4 the focus is on identifying sources of heterogeneity in disease transmission, as well as on estimation problems when heterogeneity is present and assessing the consequences of ignoring heterogeneity. The particular heterogeneity arising from having a structured community is investigated in greater detail in Section 5. A major motivation for the study of epidemic models is the insight they provide about the control of disease transmission. Section 6 looks at the estimation of parameters that are needed to determine control specifications, such as the vaccination coverage required to prevent epidemics and the estimation of vaccine efficacy. A relatively new area of work is that concerned with the transmission of HIV, the virus that leads to AIDS. The analysis of AIDS data has attracted a wide range of statisticians who have developed methodology for the study of this, somewhat unique, epidemic. In Section 7 we focus on some of this methodology with a view to seeing if we can utilize these methods for the analysis of data on other infectious diseases. In the final section we identify some statistical problems that are worthy of further attention.

**Niels Becker**:
`Assessing the consequences of ignoring heterogeneity when planning vaccination strategies'

Estimation of the immunity coverage required to effectively control disease transmission is an important public health problem. We compare estimates based on the simplifying assumption that the community consists of uniformly mixing individuals with estimates obtained when the more complex community structure is acknowledged. The alternative community structures considered include households and localities that are quite separate. Several inequalities are established for estimates of the critical immunity coverage. For several settings it is found that the coverage estimated by assuming an oversimplified community structure is actually an underestimate. A serious consequence of this is that we may be misled into believing that we have estimated an immunity coverage that can prevent epidemics when it in fact cannot. The conclusion is that the heterogeneity in the community must be taken into account when estimating the critical immunity coverage.

**Ben Bolker**:
`Heterogeneity of mixing and spatio-temporal models:
advances and open problems'

*[full text available as postscript file]*

Despite optimistic predictions {Garrett95},
many infectious diseases of humans, other animals,
and plants persist in the face of
modern medicine and technology.
Epidemiological theory has a vital role to play in disease
control; even if medical and molecular science could provide a `magic
bullet' - a cheap, 100% effective treatment with no side effects - we
would still need practical advice about how to target it effectively
in the population. In the absence of perfect
cures and unlimited resources, theory has to provide practical
advice on how to predict, control, and (eventually) eradicate
various infectious diseases.
These practical challenges connect
with particular issues in theoretical epidemiology.
To *predict* the course of an epidemic
requires us to understand the mechanisms of disease spread;
to *control* disease,
we need to know
how much effort will reduce disease incidence
to a certain level, which in turn requires us to understand what
determines endemic levels of disease; and
to *eradicate* disease, we must understand how
disease persists in a population.
These problems are not new; the classical foundations
of theoretical epidemiology are based on answering
these questions in a hypothetical, homogeneous population
{KermMcK32,KermMcK33}.
Much of the subsequent development of theoretical (dynamical)
epidemiology has extended these results to populations
with various forms of heterogeneity,
particularly
*heterogeneity of mixing*:
variability in the probability or rate of
epidemiological contact between individuals.
For example, epidemiological modellers have studied differences
in contact rates among and between age groups
{AndeMay85,AndeMay85c,DietSche85,Sche84}, and
among and between different sexes, racial groups and
sexual activity groups {MorrDean94,Hadeler+CCC95,LajmYork76,DietHade88}.
Spatial structure is one of the most important
forms of mixing heterogeneity.
Space matters in epidemics because transmission
depends on the distance between individuals.
`Distance' can be defined either narrowly, as
Euclidean distance between two points on the plane
or geographic distance between two cities, or broadly,
as the social distance between individuals in
different families, neighborhoods, and so forth.
In fact, we can really turn the definition around and
define space in terms of epidemics, so that all distances
are proportional to transmission probabilities.
All we really mean by space is a particular
kind of structured contact between individuals;
spatial models use spatial structure to narrow
the impossibly general variety of
possible contact structures in populations.

Human, non-human animal, and plant epidemics differ both in their spatial dynamics and in treatment and control options.

- Plant epidemics are clearly spatial, because of the fixed spatial positions of plants; spatial transmission occurs by random movement of wind or water, or by more directed movement of vectors, usually insect herbivores or pollinators. Plant epidemics can be controlled by chemical treatments, or perhaps by producing resistant types of plants which replace or mix with susceptible varieties.
- Animal epidemics are most clearly spatial when they spread among distinct populations, or when they involve territorial animals. Epidemics of livestock can sometimes be treated, but with many agricultural and most wildlife epidemics the only choices are vaccination or destruction of infected animals and susceptible animals, the latter in hopes of reducing susceptible density below a threshold level.
- Human epidemics have the most complicated form of spatial dynamics; whether we consider countries, cities and towns, or neighborhoods, humans interact along a complicated hierarchical social network. Human epidemics can be controlled by vaccines or behavioral changes, or by quarantining or treating infectious individuals.

*Persistence.*Many diseases persist surprisingly well in populations where non-spatial (deterministic and stochastic) models suggest they should go extinct {Grif73}; spatial models suggest that the nonlinear interactions of weakly-coupled stochastic populations may lead to `rescue effects' {Brown77}, where disease is reintroduced from neighboring populations. (Note that as always we can also generate explanations for disease persistence at the level of individual hosts, for example if asymptomatic carriers exist. The distinction between individual- and population-level explanations must be empirical.)*Endemic levels.*Any time that infectives come in contact with each other or with recovered (immune) individuals more often than with susceptible individuals (relative to some rate expected from homogeneous-mixing), infective contacts are `wasted' and speeds of spread and endemic levels of disease are reduced. Wasted contacts occur in any non-homogeneous model, but can be particularly important in stochastic spatial models where infectives tend to infect nearby susceptibles, and as a result end up eventually (at equilibrium) in neighborhoods with more infectives and recovered individuals than average. Wasted contacts will lower the endemic incidence of disease - as infectiousness and contact rates go up, so will the proportion of wasted contacts. Conversely, `macroscopic' estimates of epidemic parameters based on endemic incidence levels will be biased downward, leading to overoptimistic control predictions {JDthesis}.*Spread.*If we want to know about spread of a disease, we almost always care about space. Questions about when a spatially-spreading disease will reach a particular community are obviously spatial; less obviously, spatial structure and spatial effects of a population will affect even average rates of spread because contacts will be wasted, as described above. Although epidemiologists have studied some forms of spreading waves of disease for a long time, new studies of the patchiness of disease spread are suggesting that the wasted-contact phenomenon may be important.

**Tom Britton**: `Estimation of individual heterogeneities'

Due to recent progress in epidemic modelling several types of individual heterogeneities can be incorporated into an epidemic model, for example susceptibility, infectivity, distribution of the infectious period, non-uniform mixing, etcetera. From a statistical point of view such generalizations are mainly useful if the heterogeneities are estimable. This will in turn depend on the detail of data, and whether one or several epidemic outbreaks (as in household outbreaks) are observed.

We will study this matter for a specific example: one epidemic is observed and variation in susceptibility and infectivity is considered. If only the final state of the epidemic is observed (the most common data!) it turns out that only the susceptibilities may be estimated consistently. In fact, we cannot even estimate the basic reproduction number R_0 consistently! If the epidemic process is observed continuously then the infectivities can also be estimated consistently, except if the corresponding susceptibilities are equal.

**Mart de Jong**:
`Experiments to test measures that stop
transmission: statistical analysis and power calculations'

*[Work with B Kroese]*

For veterinary problems and in animal models of human diseases the effect of certain interventions, as for example vaccination, can be tested experimentally. Such experiments are a useful extension to observational studies, because definition difficulties as which individuals are infectious and which are susceptible can be resolved. For the statistical analysis of these experiments detailed assumptions on the transmission process are made but it can be argued that the resulting test procedure is conservative. Even with this conservative test the power with the optimal experimental design is very high for the relevant comparisons. The same statistical methods can also be used in field trails where treatments are given to different groups. As there is no control over challenge in these trials the design is less optimal but the high number of animals involved makes these trials normally even more powerful then the experiments.

**Klaus Dietz**: `Demography, immunology and transmission dynamics'

**Andy Dobson**: `Allometric scaling of mathematical models for macroparasites and microparasites'

**Mervi Eerola**:
*(INFEMAT Project)*

**Neil Ferguson**: `Outstanding issues in BSE epidemiology'

**Bryan Grenfell** *et al*:
`Four years on: recent progress in
parasite ecology and evolutionary biology'

[Full text available]

This brief review takes as its starting point the symposium volume by Grenfell and Dobson (G&D) from the Newton Epidemics meeting in 1993. We survey theoretical (and relevant empirical) progress since then in understanding the ecology and evolution of infectious diseases. In particular, this leads us to suggest open problems and fruitful directions for future theoretical work.

The review follows G&D by focusing mainly on the ecological and evolutionary dynamics of infectious diseases in natural animal and plant populations. However, we also use human diseases examples, where these illuminate ecological and evolutionary issues.

**Hans Heesterbeek**:
`Population dynamics of immunity'

**Barbara Hellriegel**:
`Immunoepidemiology: a problem-oriented approach'

[Full text available]

**Peter Hudson**:
`Tick-borne diseases'

**Matt Keeling**:
`The Use of Dyad Models for Measles Epidemics'

It has long been known that heterogeneities at both the local and large scale are important to the spread of epidemics. Although much attention has been focused upon large-scale heterogeneity using meta-population models, behaviour at the local scale has been largely ignored. I will discuss the formation of a dyad version of the standard SEIR model and show that in many cases this model can be simplified. This leads to non mass-action behaviour between S and I. The addition of age structure (cf RAS model) produces more realistic dynamics, where irregular cycles and low critical community sizes are possible. The stochastic version of this model has been shown to display power-law behaviour, where different scaling due to transmission within families and schools can be seen.

**Niels Keiding**:
`Statistical themes in infectious disease epidemiology'

The Becker and Britton survey provides an excellent overview of statistical issues in various contexts in infectious disease epidemiology. In my presentation my aim is to outline some perspectives within general biostatistical methodology. I list below some commented keywords for this discussion:

- Incomplete data have been a general source of development of
biostatistical methodology; just look at the whole field of survival
analysis which developed because we cannot wait for everybody to die.
In the present area there are similar obvious examples in analysis of
current (immunization) status data and the estimation of the AIDS
incubation time distribution by e.g. back projection.

In survival analysis and its generalization the 'classical' approach is the conditional, where the modelling goes on in the underlying structure that we had hoped to observe but which is disturbed. A recent counter-movement (the 'marginal' approach) prefers to model directly what you observe.

Are there parallels here? - The models that one can realistically fit here are often (always?) over-simplistic. How sensitive is our inference (e.g. for public health purposes, as stressed by B&B) to this fact? Do we need to become more interested in a 'forward selection' or elaboration approach, where the fit and inference of obviously simplistic models are evaluated by various kinds of sensitivity analysis?
- Put another way: our stochastic models often have a dual role in BOTH describing the stochasticity in the epidemic development AND the noise in the relevant measurement process. Might a heretical thought such as adding noise to a deterministic epidemic model sometimes be more relevant than an inherent stochastic model? A third use of models, emphasized by B&B, comes from their necessity when data are sparse.
- Most modelling and consequent statistical theory in this area are concerned with steady state situations. But authorities are often rather more interested in initial transient stages of e.g. immunization programmes.

I intend to combine the general discussion of some of these issues with our own experience with modelling heterogeneity in diarrhea epidemics and with surveying the early stage of measles, mumps and rubella vaccination.

**Margaret Mackinnon**:
`Host selection against parasite recombination'
*[with Andrew Read]*

The structuring of parasite populations into discrete hosts has two profound effects on the way parasites evolve. First, the strength of host selection on the parasite population depends on how this selection is structured across the host population. For example, frequency dependent immune selection, or directional selection by drugs is fragmented because hosts impose heterogeneous selection pressure. The way parasites are distributed throughout the host population is therefore important, especially when selection is for multi-locus gene combinations. Second, the opportunity for recombination breakdown is limited because of the small number of available sexual partners within a single host or vector. This means that the force of recombination breaking down selected gene combinations is weak. We have used population genetics models to predict the outcome of selection versus recombination in two pertinent examples. In the case of multiple drug selection against multiply resistant parasites, selection almost always beats recombination. In the case of immune selection against multiple immunogenic loci, selection and recombination strike a balance which determines the degree of strain and population structure which will be maintained. The key influence of transmission rate on these outcomes is emphasized.

**Martina Morris**: `Spatial network information available from the Ugandan sexual behavior study'

**Ingemar Nasell**:
`On endemic SIR models'

The phenomenon of stochastic fade-out is important for models that account for both epidemic and demographic forces. It is related to the concept of critical community size and also to the concepts of invasion and persistence thresholds. Models where these phenomena appear are studied with regard to quasi-stationary distributions and time to extinction. Numerical studies are made and the results are compared with approximations that have been suggested in the literature. Open problems are identified.

**Philip O'Neill**:
`Bayesian inference for partially observed stochastic epidemics'

Performing inference based on real-life epidemic data sets is usually hampered by a lack of complete information. Typically, data concerning the infection process are absent. We describe a framework for performing Bayesian inference by treating the missing data as extra model parameters, and using MCMC methods.

**David Rand**:
`Correlations & fluctuations in ecology and infection'

I will discuss a correlation dynamics formalism for stochastic spatial population dynamics. I will consider a number of examples where it allows analysis of phenomena due to spatial correlations and fluctuations that are missed by mean-field models. It also allows analysis of the spatial structure of invading populations and the effect of this on the invasion exponent (and R_0).

**Mick Roberts**:
`The immunoepidemiology of parasites of farmed animals'

*[full text available as postscript file
(figures not yet included)]*

The population dynamics of farmed animals are controlled by humans, and often encourage higher parasite burdens than would be usual in wild animals. As a result, the immunity to reinfection acquired by the host is an important determinant of parasite population dynamics. For example, lambs are highly susceptible to gastrointestinal nematodes as they begin to graze, but develop an immunity that accounts for observed within-year variation in parasite load and pasture contamination. In the longer term, control measures are compromised by the development of parasite strains resistant to chemotherapy, focusing attention on the development of ''natural'' measures including antiparasite vaccines. The immunoepidemiology of parasites of farmed animals must therefore be considered on three levels: the interaction between the parasite and the host's immune system determining the individual's level of protection; the development of acquired immunity determining the within-year parasite population dynamics; and the long-term effects of control measures on the between-year parasite population dynamics. The population dynamics of farmed animals are controlled by humans, and often encourage higher parasite burdens than would be usual in wild animals. As a result, the immunity to reinfection acquired by the host is an important determinant of parasite population dynamics. For example, lambs are highly susceptible to gastrointestinal nematodes as they begin to graze, but develop an immunity that accounts for observed within-year variation in parasite load and pasture contamination. In the longer term, control measures are compromised by the development of parasite strains resistant to chemotherapy, focusing attention on the development of ''natural'' measures including antiparasite vaccines. The immunoepidemiology of parasites of farmed animals must therefore be considered on three levels: the interaction between the parasite and the host's immune system determining the individual's level of protection; the development of acquired immunity determining the within-year parasite population dynamics; and the long-term effects of control measures on the between-year parasite population dynamics.

**Mick Roberts**:
`The dynamics of measles in New Zealand'

An SIR model has been used to explore the dynamics of measles in
New Zealand. The model predicts that the current vaccination programme has
increased the inter-epidemic period from two to six years, which is
consistent with observations. The model also demonstrates that the
vaccination programme has changed the age structure of the susceptible
population, and hence the threshold at which an epidemic is triggered; with
children under 5 yr. now playing a smaller part in epidemics, and those over
16 yr. having an increasing role. The predicted timing and sizes of measles
epidemics are very sensitive to assumed values of the basic reproduction
ratio, *R_0*, and the assumed mixing pattern between age classes.

**Ake Svensson**:
`Statistical analysis based on incomplete and distorted observations of
epidemic processes'

**Rodney Wolff**:
`Non-parametric detection of outbreaks'

**Woolhouse**: `Heterogeneous transmission and the 20-80 rule'

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Please send any comments or corrections to
* Denis Mollison*

*22nd March 1997*