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On the time to extinction for a two-type version of Bartlett's epidemic model
Stockholm University, Faculty of Science, Department of Mathematics.
2008 (English)In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 212, no 1, 99-108 p.Article in journal (Refereed) Published
Abstract [en]

We are interested in how the addition of type heterogeneities affects the long time behaviour of models for endemic diseases. We do this by analysing a two-type version of a model introduced by Bartlett under the restriction of proportionate mixing. This model is used to describe diseases for which individuals switch states according to susceptible infectious recovered and immune, where the immunity is life-long. We describe an approximation of the distribution of the time to extinction given that the process is started in the quasi-stationary distribution, and we analyse how the variance and the coefficient of variation of the number of infectious individuals depends on the degree of heterogeneity between the two types of individuals. These are then used to derive an approximation of the time to extinction. From this approximation we conclude that if we increase the difference in infectivity between the two types the expected time to extinction decreases, and if we instead increase the difference in susceptibility the effect on the expected time to extinction depends on which part of the parameter space we are in, and we can also obtain non-monotonic behaviour. These results are supported by simulations.

Place, publisher, year, edition, pages
2008. Vol. 212, no 1, 99-108 p.
Keyword [en]
stochastic SIR epidemic model, quasi-stationary distribution, diffusion approximation, endemic diseases
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-25637DOI: 10.1016/j.mbs.2008.01.005ISI: 000254730600006OAI: oai:DiVA.org:su-25637DiVA: diva2:200106
Available from: 2008-11-27 Created: 2008-11-20 Last updated: 2017-12-13Bibliographically approved
In thesis
1. Stochastic Epidemic Models: Different Aspects of Heterogeneity
Open this publication in new window or tab >>Stochastic Epidemic Models: Different Aspects of Heterogeneity
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is concerned with the study of stochastic epidemic models for infectious diseases in heterogeneous populations. All diseases treated are of SIR type, i.e. individuals are either Susceptible, Infectious or Recovered (and immune). The transitions between these states are according to S to I to R.

The thesis consists of five papers. Papers I and II treat approximations for the distribution of the time to extinction. In Paper I, a sub-community version of the SIR model with demography is considered. The interest is in how the distribution of the time to extinction is affected by varying the degree of interaction between the sub-communities. Paper II is concerned with a two-type version of Bartlett's model. The distribution of the time to extinction is studied when the difference in susceptibility/infectivity between the types of individuals is varied.

Papers III and IV treat random intersection graphs with tunable clustering. In Paper III a Reed-Frost epidemic is run on such a random intersection graph. The critical parameter R_0 and the probability of a large outbreak are derived and it is investigated how these quantities are affected by the clustering in the graph. In Paper IV the interest is in the component structure of such a graph, i.e. the size and the emergence of a giant component is studied.

The last paper, Paper V, treats the situation when a simple epidemic is running in a varying environment. A varying environment is in this context any external factor that affects the contact rate in the population, but is itself unaffected by the population. The model treated is a term-time forced version of the stochastic general epidemic where the contact rate is modelled by an alternating renewal process. A threshold parameter R_* and the probability of a large outbreak are derived and studied.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2008. 21 p.
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-8335 (URN)978-91-7155-784-1 (ISBN)
Public defence
2008-12-19, sal 14, hus 5, Kräftriket, Stockholm, 13:00 (English)
Opponent
Supervisors
Available from: 2008-11-27 Created: 2008-11-20 Last updated: 2012-07-02Bibliographically approved

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