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A stochastic SIS epidemic with demography: initial stages and time to extinction
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
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.

Place, publisher, year, edition, pages
2011. Vol. 62, no 3, 333-348 p.
Keyword [en]
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: urn:nbn:se:su:diva-57948DOI: 10.1007/s00285-010-0336-xISI: 000287250300002OAI: oai:DiVA.org:su-57948DiVA: diva2:418779
Available from: 2011-05-24 Created: 2011-05-24 Last updated: 2017-12-11Bibliographically approved
In thesis
1. Four applications of stochastic processes: Contagious disease, credit risk, gambling and bond portfolios
Open this publication in new window or tab >>Four applications of stochastic processes: Contagious disease, credit risk, gambling and bond portfolios
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers on applications of stochastic processes.

In Paper I we study an open population SIS (Susceptible - Infective - Susceptible) stochastic epidemic model from the time of introduction of the disease, through a possible outbreak and to extinction. The analysis uses coupling arguments and diffusion approximations.

In Paper II we propose a model describing an economy where companies may default due to contagion. The features of the model are analyzed using diffusion approximations. We show that the model can reproduce oscillations in the default rates similar to what has been observed empirically.

In Paper III we consider the problem of finding an optimal betting strategy for a house-banked casino card game that is played for several coups before reshuffling. A limit result for the return process is found and the optimal card counting strategy is derived. This continuous time strategy is shown to be a natural generalization of the discrete time strategy where the so called effects of removals are replaced by the infinitesimal generator of the card process.

In Paper IV we study interest rate models where the term structure is given by an affine relation and in particular where the driving stochastic processes are so-called generalised Ornstein-Uhlenbeck processes. We show that the return and variance of a portfolio of bonds which are continuously rolled over, also called rolling horizon bonds, can be expressed using the cumulant generating functions of the background driving Lévy processes associated with the OU processes. We also show that if the short rate, in a risk-neutral setting, is given by a linear combination of generalised OU processes, the implied term structure can be expressed in terms of the cumulant generating functions.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2011. 27 p.
Keyword
Stochastic processes
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-57949 (URN)978-91-7447-318-6 (ISBN)
Public defence
2011-08-26, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Submitted. Paper 3: Submitted. Paper 4: Manuscript.Available from: 2011-06-06 Created: 2011-05-24 Last updated: 2011-06-14Bibliographically approved
2. 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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