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Estimation of the Malthusian parameter using martingale methods
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
(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.

Place, publisher, year, edition, pages
, 17 p.
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-71712OAI: oai:DiVA.org:su-71712DiVA: diva2:485551
Available from: 2012-01-31 Created: 2012-01-29 Last updated: 2012-01-31Bibliographically approved
In thesis
1. Stochastic epidemic models in heterogeneous communities
Open this publication in new window or tab >>Stochastic epidemic models in heterogeneous communities
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:nbn:se:su:diva-70825 (URN)978-91-7447-436-7 (ISBN)
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

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CiteExportLink to record
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Citation style
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