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Stochastic epidemic models in heterogeneous communities
Stockholm University, Faculty of Science, Department of Mathematics.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The aim of Paper I is to explain where randomness should be taken into account when modelling epidemic spread, i.e. when a stochastic model is preferable to a deterministic counterpart. Two examples are used to show that the probability of a large outbreak and the initial growth rate of the epidemic are affected by randomness in infectious period and latent period. It follows that the basic reproduction number is sensitive to assumptions about the distributions of the infectious and latent periods when using data from the early stages of an outbreak, which we illustrate with data from the H1N1 influenza A pandemic. In paper II we analyse an open population stochastic epidemic S-I-S model.  That is, individuals in the population move between the states of infectiousness and susceptibility, and enter of leave the population through birth and death. An approximate expression for the outbreak probability is derived using a coupling argument. It is proved that the number of infectives and susceptibles close to quasi-stationarity behaves like an Ornstein-Uhlenbeck process, for an exponentially distributed time before going extinct. In Paper III we analyse an estimator, based on martingale methods, of the Malthusian parameter, which determines the speed of epidemic spread. Asymptotic properties of the estimator are obtained, and compared to the results from simulations. The advantage of the estimator is that it may use any proportion of the information contained in the epidemic curve, in contrast to the more common simpler estimators. In paper IV a social (sexual) network is modeled by an extension of the configuration model to the situation where edges have weights. The aim is to analyse how individual heterogeneity in susceptibility and infectivity affects the basic reproduction number, but also the size and probability of a major outbreak. The main qualitative conclusion is that the basic reproduction number gets larger as the community becomes more heterogeneous.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2012. , 14 p.
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-70825ISBN: 978-91-7447-436-7 (print)OAI: oai:DiVA.org:su-70825DiVA: diva2:482854
Public defence
2012-02-24, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 10:00 (English)
Opponent
Supervisors
Note
At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Submitted. Paper 4: Submitted.Available from: 2012-02-02 Created: 2012-01-24 Last updated: 2012-01-31Bibliographically approved
List of papers
1. Epidemic modelling: aspects where stochasticity matters
Open this publication in new window or tab >>Epidemic modelling: aspects where stochasticity matters
2009 (English)In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 22, no 2, 109-116 p.Article in journal (Refereed) Published
Abstract [en]

Epidemic models are always simplifications of real world epidemics. Which real world features to include, and which simplifications to make, depend both on the disease of interest and on the purpose of the modelling. In the present paper we discuss some such purposes for which a stochastic model is preferable to a deterministic counterpart. The two main examples illustrate the importance of allowing the infectious and latent periods to be random when focus lies on the probability of a large epidemic outbreak and/or on the initial speed, or growth rate, of the epidemic. A consequence of the latter is that estimation of the basic reproduction number R0 is sensitive to assumptions about the distributions of the infectious and latent periods when using data from the early stages of an outbreak, which we illustrate with data from the H1N1 influenza A pandemic. Some further examples are also discussed as are some practical consequences related to these stochastic aspects.

Keyword
stochastic epidemic model, major outbreak probability, infectious period, latency period, exponential growth rate
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-35206 (URN)10.1016/j.mbs.2009.10.001 (DOI)000273101900005 ()
Available from: 2010-01-15 Created: 2010-01-15 Last updated: 2017-12-12Bibliographically approved
2. A stochastic SIS epidemic with demography: initial stages and time to extinction
Open this publication in new window or tab >>A stochastic SIS epidemic with demography: initial stages and time to extinction
2011 (English)In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 62, no 3, 333-348 p.Article in journal (Refereed) Published
Abstract [en]

We study an open population stochastic epidemic model from the time of introduction of the disease, through a possible outbreak and to extinction. The model describes an SIS (susceptible–infective–susceptible) epidemic where all individuals, including infectious ones, reproduce at a given rate. An approximate expression for the outbreak probability is derived using a coupling argument. Further, we analyse the behaviour of the model close to quasi-stationarity, and the time to disease extinction, with the aid of a diffusion approximation. In this situation the number of susceptibles and infectives behaves as an Ornstein–Uhlenbeck process, centred around the stationary point, for an exponentially distributed time before going extinct.

Keyword
Stochastic epidemic model, Quasi stationarity, SIS model, Coupling, Ornstein–Uhlenbeck, Diffusion approximation, Outbreak probability
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-57948 (URN)10.1007/s00285-010-0336-x (DOI)000287250300002 ()
Available from: 2011-05-24 Created: 2011-05-24 Last updated: 2017-12-11Bibliographically approved
3. Estimation of the Malthusian parameter using martingale methods
Open this publication in new window or tab >>Estimation of the Malthusian parameter using martingale methods
(English)Article in journal (Refereed) Submitted
Abstract [en]

Data gathered from a large epidemic outbreak can be used to estimate disease related parameters. We analyse an estimate, based on martingale methods, of the Malthusian parameter, which determines the growth rate of the epidemic. This is done using a simple epidemic SIR model, with deterministic infectious period. We derive asymptotic properties of the estimate and compare them to the results from simulations of the epidemic. The advantage of our estimate is that is uses all the information contained in the epidemic curve, in contrast to the more common simpler estimates which uses only data from the start of the outbreak. The theoretical and numerical results show this in that the variance of the estimate decreases the more data we use.

Publisher
17 p.
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-71712 (URN)
Available from: 2012-01-31 Created: 2012-01-29 Last updated: 2012-01-31Bibliographically approved
4. Inhomogeneous epidemics on weighted networks
Open this publication in new window or tab >>Inhomogeneous epidemics on weighted networks
(English)Article in journal (Refereed) Submitted
Abstract [en]

A social (sexual) network is modeled by an extension of the configuration model to the situation where edges have weights, e.g. reflecting the number of sex-contacts between the individuals. An epidemic model is defined on the network such that individuals are heterogeneous in terms of how susceptible and infectious they are. The basic reproduction number R_0 is derived and studied for various examples, but also the size and probability of a major outbreak. The qualitative conclusion is that R_0 gets larger as the community becomes more heterogeneous but that different heterogeneities (degree distribution, weight, susceptibility and infectivity can sometimes have the cumulative effect of homogenizing the community, thus making R_0 smaller. The effect on the probability and final size of an outbreak is more complicated.

National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-71714 (URN)
Available from: 2012-01-31 Created: 2012-01-29 Last updated: 2012-01-31Bibliographically approved

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