| Asikainen, Tommi |
| Presentation of ECDC, modelling networks and data availability |
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| Ball, Frank |
| Epidemics on random networks with household structure |
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There has been a growing interest in
models for epidemics among structured populations, which incorporate
realistic departures from homogeneous mixing whilst maintaining
mathematical tractability. Two classes of structured population epidemic
models that have attracted considerable recent attention are network
models (in which there is a random graph describing possible infectious
contacts) and household models (in which the population is partitioned
into households with different contact rates for within- and
between-household infection). In this talk a model for the spread of an
SIR (susceptible-infective-removed) epidemic that includes both of these
features is described and analysed. The analysis includes deriving a
threshold parameter which determines whether or not an epidemic with few
initial infectives can become established and lead to a major outbreak,
and determining the probability and expected relative final size of a
major outbreak. Extensions of the model, including some unsolved
problems, are briefly discussed. Frank Ball (University of Nottingham),
David Sirl (Loughborough University), Pieter Trapman (Stockholm
University).
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| Barrass, Iain |
| Interpretation of large-scale stochastic epidemic models |
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Modern computational systems have allowed
the development of highly detailed models of disease dynamics.
Stochastic agent-based and metapopulation models usually give rise to an
ensemble of results which must be interpreted in some way. Within the
large ensembles that can routinely be obtained quite varied behaviour
can be observed. For a confident statement of results one must consider
the predictive power of the models as well as how best to understand
and present the variability seen. This talk presents recent work within
HPA examining the issues around model selection, and predictability and
interpretation of a contingency planning model in use. We examine how
considerations of the model and its output depend on a study’s aims and
how results can be visualised and then presented in a manner appropriate
to the target audience.
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| Britton, Tom |
| Network modelling: future directions |
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| A presentation for discussion |
| Cannings, Chris |
| Genetics and games on graphs |
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| A presentation for discussion |
Clancy, Damian |
| The effects of population heterogeneities on spread of infection |
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Heterogeneities in individual
infectivities, individual susceptibilities, and population mixing
(assortative or dissortative) may all have effects upon infectious
spread, compared to a corresponding homogeneous model. Specifically, the
corresponding homogeneous model here is chosen to have the same value
of the basic reproduction number R_0 as the heterogeneous model of
interest. Effects of heterogeneity are considered in terms of (i) the
probability of a major outbreak; (ii) the equilibrium prevalence level,
given that infection becomes established; and (iii) the distribution of
time until eventual fade-out of infection.
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| Deijfen, Maria |
| Scale-free percolation |
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I will describe a model for inhomogeneous
long-range percolation on Z^d with potential applications in network
modelling. Each vertex is independently assigned a non-negative random
weight and the probability that there is an edge between two given
vertices is then determined by a certain function of their weights and
of the distance between them. The results concern the degree
distribution in the resulting graph, the percolation properties of the
graph and the graph distance between remote pairs of vertices. The model
interpolates between long-range percolation and inhomogeneous random
graphs, and is shown to inherit the interesting features of both these
model classes.
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| Ford, Ashley |
| Indian Buffet Epidemics: a
non-parametric Bayesian approach to modelling heterogeneity |
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Bipartite graphs provide a natural representation of the way in which
people meet in
groups at different locations such as households, schools,
workplaces, buses etcetera.
The relation between this representation and the widely studied graph
(or network) models will be described.
Ways in which standard epidemic models such as household models fit
into this framework will be described.
The Indian Buffet Epidemic model has been developed as an approach to
a non-parametric model, within this bipartite network framework,
which does not require a priori defining a meaningful and useful set
of locations.
The Indian Buffet Epidemic combines the bipartite network model with
the Indian Buffet process to provide a model which is simple to define
and simulate from and on which Bayesian inference is possible.
[Work with Gareth Roberts]
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| Hall, Ian |
| Building behavioural response effects into stochastic models for contingency planning |
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Potential bioterrorist activity demands
careful contingency planning to mitigate the likely impacts of a release
and present communication challenges about uncertainty. The Health
Protection Agency has been working on developing plans assuming releases
of a number of possible agents (smallpox, plague and anthrax). However,
often these models are left making assumptions about the public
perceptions and responses to public health interventions. To provide a
more evidence based approach to this problem we have performed
population surveys to assess these factors for pneumonic plague and
recently for pandemic influenza. Here we discuss the results of the
surveys, the incorporation into a range of stochastic models (individual
based and meta-population) and lessons learnt.
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| Holroyd, Alexander |
| Invariant matching |
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Suppose that red and blue points occur as independent point processes
in R^d, and consider translation-invariant schemes for perfectly
matching the red points to the blue points. (Translation-invariance
can be interpreted as meaning that the matching is constructed in a
way that does not favour one spatial location over another). What is
best possible cost of such a matching, measured in terms of the edge
lengths? What happens if we insist that the matching is
non-randomized, or if we forbid edge crossings, or if the points act
as selfish agents? I will review recent progress and open problems on
this topic, as well as on the related topic of fair allocation. In
particular I will address some surprising new discoveries on
multi-colour matching and multi-edge matching.
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| Isham, Valerie |
| Spread of information/infection on networks |
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The basic (SIR) epidemic model for the spread of infection in a homogeneously-mixing population is a special case of a more general stochastic model used for the spread of information (a `rumour’). In both cases it is well known that there is a threshold for widespread transmission.
More generally, for both epidemics and rumours, there is particular
interest in using a network to represent population structure. This
ensures that some pairs of individuals are never in contact, and
direct spread between them cannot occur. Natural applications are to
the spread of infection or information on social networks.
In this talk, I review simple epidemic and rumour models, and describe networks generated by a number of random mechanisms. The effect of different network structures on the transmission dynamics of infection or information on networks, including some insights gained by use of approximate models, is discussed.
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| Jordan, Jonathan |
| Randomised reproducing graphs |
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We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random element, and there are three parameters, a, b and c, which are the probabilities of edges appearing between different types of vertices. We show that as the probabilities associated with the model vary there are a number of phase transitions, in particular concerning the degree sequence. If (1+a)(1+c)<1 then the degree distribution converges to a stationary distribution, which in most cases has an approximately power law tail with an index which depends on a and c. If (1+a)(1+c)>1 then the degree of a typical vertex grows to infinity, and the proportion of vertices having any fixed degree d tends to zero. We also give some results on the number of edges and on the spectral gap.
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| Kretzschmar, Mirjam |
| Dynamic pair formulation models |
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I would like to discuss how to extend
dynamic pair formation models to include some features of networks, for
example describing models with dynamics stars instead of only dyads.
This is a deterministic approach, but could of course also be formulated
in a stochastic framework.
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| Kypraios, Theo (with Philip O'Neill) |
| Statistical inference for epidemics on networks (joint talk with Philip O'Neill) |
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We review different approaches to the
problem of inferring model parameters for stochastic epidemic models
defined on networks and highlight possible future research directions.
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| Limic, Vlada |
| Exchangeable coalescents |
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We consider a model of genealogy corresponding to a Lambda-coalescent or more generally a regular exchangeable coalescent (also known as Xi-coalescent) started from a large finite configuration, and undergoing neutral mutations. Asymptotic expressions for the number of active lineages were obtained using a martingale-based technique. Analogous results for the number of active mutation-free lineages and the combined lineage lengths are derived using a variation of the same technique. They are given in terms of convergence in probability, while extensions to convergence in moments and convergence almost surely (already derived in the Lambda setting) are discussed in the Xi setting. The above mentioned results have direct consequences on the sampling theory in the Xi-coalescent setting. In particular, the regular Xi-coalescents that come down from infinity (i.e., with locally finite genealogies), have an asymptotically equal number of families under the corresponding infinite alleles and infinite sites models. In special cases, quantitative asymptotic formulae for the number of families that contain a fixed number of individuals can be given. |
| Miller, Joel |
| Edge-based compartmental models for epidemics on networks |
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The potentially infectious contacts in a
population can be represented by a network which controls how an
infectious disease spreads. A number of results have been found for the
final size of epidemics in random static networks with given degree
distributions. However, relatively little has been known about their
dynamics or what impact the changing structure of a contact network has
on disease spread. We have recently developed an "edge-based
compartmental modelling" approach that focuses on the status of edges
rather than individuals. This approach allows us to calculate the
dynamics of an epidemic spreading on a network in a straightforward way.
The approach allows us to consider the impact of changes in the network
structure as well as heterogeneity in contact levels among individuals.
The resulting equations are comparable in complexity to the usual mass
action SIR equations. I will demonstrate the approach with a simple
network structure, and then discuss what features are needed for the
approach to work more generally.
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| Mollison, Denis |
| Network models for epidemics |
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A review of the history of network models
for epidemics, discussing their relation to other models, and their
advantages and disadvantages. At best, the network approach extends
the situations and features that we can model, and facilitates new
insights into epidemic dynamics and new data analysis techniques. But,
as with all stochastic mathematics, sometimes we can be diverted into
what is easy or mathematically attractive to model, into what is elegant
rather than practically appropriate.
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| Neal, Peter |
| Variance of the giant component of a random graph |
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Provided that a random graph is supercritical, it is well known that there exists a unique giant
component, Rn , of size O(n), where n is the total number of vertices in the random graph. Moreover,
there exists 0 < \rho <= 1 such that Rn/n -> \rho (in probability) as
n -> \infty. In this talk we show that for the configuration
model random graph with an independent and identically distributed (iid) vertex degree distribution D
and E[D^12] < \infty, there exists \sigma^2 > 0, such that
var(\sqrt{n}(Rn/n - \rho)) -> \sigma^2 as n -> \infty.
An explicit, easy to compute, formula is given for \sigma^2. Moreover, the
results are extended to the Molloy-Reed
random graph where the iid vertex degree distribution is replaced by a deterministic degree sequence with
var(\sqrt{n}(Rn/n - \rho)) -> \sigma^2_MR as n -> \infty, where \sigma^2_MR < \sigma^2.
[Work with Frank Ball]
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| O'Neill, Philip (with Theo Kypraios) |
| Statistical inference for epidemics on networks (joint with Theo Kypraios) |
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We review different approaches to the
problem of inferring model parameters for stochastic epidemic models
defined on networks and highlight possible future research directions.
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| Pardoux, Etienne |
| Continuous limit of probabilistic models of population with interaction |
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We describe a continuous time Galton-Watson branching diffusion, with
superimposed
quadratic competition (i. e. we add a death rate which is proportional
to the square of the
population size - the resulting process is no longer a branching
process, due to the interactions
between branches). We take the limit of the renormalized population
process as the size of the
initial population tends to infinity. The limiting population size
process solves a Feller diffusion
SDE with logistic term. Our main result is the description of the
genealogy of the limiting
population in terms of a continuum random tree in the sense of Aldous.
This is joint work with Anton Wakolbinger, and in part with my student
Vi Le.
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| Pellis, Lorenzo |
| $R_0$ and other reproduction numbers for epidemic models with household structure. |
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The basic reproduction number $R_0$ is one
of the most important quantities in epidemiology. However, for epidemic
models with explicit social structure involving small mixing units such
as households, its definition is not straightforward and a wealth of
other surrogate threshold parameters has appeared in the literature. I
will show how to use branching processes to define $R_0$, apply this
definition to models with households (or other more complex social
structures) and provide methods for calculating it. Extending previous
work, I will provide a complete overview of the inequalities holding
among existing threshold parameters, including $R_0$. I will show how
random vaccination of a fraction $1-1/R_0$ of the population is not
enough to prevent large epidemics and I will provide sharper bounds than
the existing ones for bracketing the critical vaccination coverage
associated with a perfect vaccine between analytically tractable
quantities.
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| Penrose, Mathew |
| Random geometric graphs |
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This is a survey talk on random geometric graphs
G(n,r) which are obtained by dropping n points
at random into the unit square and connecting
any two points less than distance r apart.
Items discussed include the distribution of
cluster sizes and small subgraphs, the giant component phenomenon,
central and local limit theorems for the
number of components, connectivity, and hamiltonian
paths.
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| Poletti, Piero |
| ECDC work on varicella vaccination modelling in EU/EEA/EFTA countries |
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Introduction. The introduction of mass immunization against varicella
in Europe is currently under paralysis because of uncertainty on
critical processes and parameters.
Methods. We use age-structured mathematical models for investigating
the transmission dynamics and control of varicella and zoster in
different European countries. Different assumptions on the process of
zoster development are considered. The model is parametrised in a
two-stage process: (a) transmission is estimated first, using mixing
matrices from a variety of recent contact studies and serological data
on varicella infection in European countries; (b) the process leading
to the development of zoster after first varicella infection is
estimated conditionally on transmission, by using available European
data on zoster age-specific incidence.
Results. We are able to accurately reproduce the observed incidence of
zoster. We show that the issue of the lack of identifiability of the
main intermediate parameters, i.e. rates of development of zoster
susceptibility, of boosting, and zoster disease, is critical. However
this problem can be partly coped with an extensive sensitivity
analysis of those outputs related to the impact of immunization
programmes which are of interest for the public health policy maker.
Discussion. The present work clearly identifies which are the sources
of major uncertainty in the predictions of the impact of immunization
programmes, and consequently which model predictions are robust, given
the current limited knowledge on mechanisms underlying zoster
development. Even though acquiring further and new data remains an
urgent need, we believe that this work has clearly identified which
are the priorities of future efforts in data collection.
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| Reinert, Gesine |
| The shortest distance in random multi-type intersection graphs |
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Using an associated branching process as the basis of our approximation, we show that
typical inter-point distances in a multi-type random intersection graph have a defective
distribution, which is well described by a mixture of translated and scaled Gumbel
distributions, the missing mass corresponding to the event that the vertices are not
in the same component of the graph.
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| Scalia-Tomba, Gianpaolo |
| ECDC work on generation times in epidemic modelling |
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There has been recent interest in the so
called generation time in epidemic models, i.e. the average time between
the infection of a primary case and one of its secondary cases. It is
related to the latent and infectious period distributions and is
involved in a useful relationship between initial speed of growth and
the basic reproductive number R0 in SIR models. The natural framework
for considerations about various times in epidemic models and analysis
of their statistical properties is stochastic. Various facts about the
generation time distribution will be presented and links to demography
and statistics will also be discussed.
References SCALIA TOMBA G., SVENSSON Å., ASIKAINEN T. and GIESECKE J.
(2010): Some model based considerations on observing generation times
for communicable diseases. Mathematical Biosciences, 223, 24-31.
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| Schmid, Boris |
| How does the sexual behaviour of individuals relate to commonly used sexual behaviour summary measures. |
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There are four major quantities that are measured in sexual behaviour
surveys that are thought to be especially relevant for the performance
of sexual network models in terms of disease transmission. These are (i)
the cumulative distribution of lifetime number of partners, (ii) the
distribution of partnership durations, (iii) the distribution of gap
lengths between partnerships, and (iv) the number of recent partners.
Fitting a network model to these quantities as measured in sexual
behaviour surveys is expected to result in a good description of
Chlamydia trachomatis (Ct) transmission in terms of the heterogeneity of
the distribution of infection in the population. Here we present work
on how these population level summary measures depend on four sexual
behaviours and their heterogeneity on the individual level; namely the
onset of sexual availability, partner age preference, partnership type
preference, and the ability to maintain concurrent partnerships. We use
an individual based simulation model that defines individuals in terms
of their personal sexual behaviour, and demonstrate how including sexual
behaviour on the individual level improved the performance of the model
in describing population level measures of sexual behaviour, compared
to the original Kretzschmar model from 1996.
[Work with Mirjam Kretzschmar]
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| Tran, Viet Chi |
| Large graph limit and Volz' equations for an SIR epidemic spreading on a configuration model graph |
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We consider an SIR epidemic model
propagating on a Configuration Model network, where the degree
distribution of the vertices is given and where the edges are randomly
matched. The evolution of the epidemic is summed up into three
measure-valued equations that describe the degrees of the susceptible
individuals and the number of edges from an infectious or removed
individual to the set of susceptibles. These three degree distributions
are sufficient to describe the course of the disease. The limit in large
population is investigated. This allows us to provide a rigorous proof
of the equations obtained by Volz (2008) where the spread of the
epidemics is summed up into 5 ordinary differential equations only.
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| Trapman, Pieter |
| SIR epidemics on random intersection graphs |
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We consider a model for the spread of a
stochastic SIR epidemic on a network of individuals described by a
random intersection graph. The number of cliques a typical individual
belongs to follows a mixed-Poisson distribution, as does the size of a
typical clique. Infection can be transmitted between two individuals if
and only if they belong to the same clique. An infinite-type branching
process approximation (with type being given by the length of an
individual's infectious period) for the early stages of an epidemic is
developed and made fully rigorous by proving an associated limit theorem
as the population size tends to infinity. This leads to a threshold
parameter R*. We also provide a law of large numbers for the size of
such a large outbreak. This talk is based on joint work with Frank
Ball and David Sirl.
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| Turner, Amanda |
| Scaling limits of planar random growth models |
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In 1998 Hastings and Levitov proposed a model for planar random growth, in
which clusters are represented as compositions of conformal mappings. We
introduce some natural versions of this model, and describe the scaling
limits that arise. This is based on joint work with James Norris, and with
Fredrik Johansson Viklund and Alan Sola.
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| van der Hofstad, Remco |
| Hypercube percolation |
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Consider bond percolation on the hypercube
{0,1}^n at the critical probability p_c defined such that the expected
cluster size equals 2^{n/3}, where 2^{n/3} acts as the cube root of the
number of vertices of the n-cube. Percolation on the Hamming cube was
proposed by Erd"os and Spencer (1979), and has proved to be
substantially harder than percolation on the complete graph.
In this talk, I will describe the phase transition for percolation on
the hypercube, and show that it shares many features with that on the
complete graph.
In previous work, we have identified the subcritical and critical
regimes of percolation on the hypercube. In particular, we know that for
p=p_c(1+O(2^{-n/3})), the largest connected component is of size
roughly 2^{2n/3} and that this quantity is non-concentrated. So far, we
were missing an analysis of the behavior of the largest connected
component above the critical value, to show that the critical value
really is critical. In this work, we identify the supercritical behavior
of percolation on the hypercube, by showing that, for any sequence
epsilon_n tending to zero, but epsilon_n being much larger than
2^{-n/3}, percolation at p_c(1+epsilon_n) has, with high probability, a
unique giant component of size (2+o(1))epsilon_n 2^n. (This is joint
work with Asaf Nachmias, building on previous work with Markus
Heydenreich, Gordon Slade, Christian Borgs, Jennifer Chayes and Joel
Spencer).
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| van Sighem, Ard |
| ECDC work on estimating HIV prevalence in EU/EEA/EFTA countries |
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Accurate estimates of the number of HIV-infected people in European
countries are necessary for understanding the true burden of HIV, the
corresponding need for treatment, and for intensifying testing for
HIV. Two classes of methods have been used to estimate the number
living with HIV, including methods based on prevalence surveys and
estimates of the size of groups most at risk for HIV like men who have
sex with men and injecting drug users, and methods based on case
report data, which are available in many European countries. A review
of both classes of methods is given and specific advantages and
disadvantages of each class will be discussed.
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