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Stochastic Epidemic Models: Different Aspects of Heterogeneity
Stockholm University, Faculty of Science, Department of Mathematics.
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: urn:nbn:se:su:diva-8335ISBN: 978-91-7155-784-1 (print)OAI: oai:DiVA.org:su-8335DiVA: diva2:200110
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
List of papers
1. Endemic persistence or disease extinction: The effect of separation into sub-communities
Open this publication in new window or tab >>Endemic persistence or disease extinction: The effect of separation into sub-communities
2007 In: Theoretical Population Biology, ISSN 0040-5809, Vol. 72, no 2, 253-263 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-25636 (URN)000249308900007 ()
Note
Part of urn:nbn:se:su:diva-8335Available from: 2008-11-27 Created: 2008-11-20Bibliographically approved
2. On the time to extinction for a two-type version of Bartlett's epidemic model
Open this publication in new window or tab >>On the time to extinction for a two-type version of Bartlett's epidemic model
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.

Keyword
stochastic SIR epidemic model, quasi-stationary distribution, diffusion approximation, endemic diseases
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-25637 (URN)10.1016/j.mbs.2008.01.005 (DOI)000254730600006 ()
Available from: 2008-11-27 Created: 2008-11-20 Last updated: 2017-12-13Bibliographically approved
3. Epidemics on random graphs with tunable clustering
Open this publication in new window or tab >>Epidemics on random graphs with tunable clustering
2008 (English)In: Journal of Applied Probability, ISSN 0021-9002, Vol. 45, no 3, 743-756 p.Article in journal (Refereed) Published
Abstract [en]

In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.

Keyword
Epidemics; random graph; clustering; branching process; epidemic threshold
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-25638 (URN)10.1239/jap/1222441827 (DOI)000260171800014 ()
Note
Part of urn:nbn:se:su:diva-8335Available from: 2008-11-27 Created: 2008-11-20 Last updated: 2012-03-07Bibliographically approved
4. A note on the component structure in random intersection graphs with tunable clustering
Open this publication in new window or tab >>A note on the component structure in random intersection graphs with tunable clustering
2008 (English)In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 15, no 1, N10- p.Article in journal (Refereed) Published
Abstract [en]

We study the component structure in random intersection graphs with tunable clustering, and show that the average degree works as a threshold for a phase transition for the size of the largest component. That is, if the expected degree is less than one, the size of the largest component is a.a.s. of logarithmic order, but if the average degree is greater than one, a.a.s. a single large component of linear order emerges, and the size of the second largest component is at most of logarithmic order.

National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-25639 (URN)000254892000003 ()
Available from: 2008-11-27 Created: 2008-11-20 Last updated: 2017-12-13Bibliographically approved
5. The early stage behaviour of a stochastic SIR epidemic with term-time forcing
Open this publication in new window or tab >>The early stage behaviour of a stochastic SIR epidemic with term-time forcing
2009 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 46, no 4, 975-992 p.Article in journal (Refereed) Published
Abstract [en]

The general stochastic SIR epidemic in a closed population under the influence of a term-time forced environment is considered. An 'environment' in this context is any external factor that influences the contact rate between individuals in the population, but is itself unaffected by the population. Here 'term-time forcing' refers to discontinuous but cyclic changes in the contact rate. The inclusion of such an environment into the model is done by replacing a single contact rate λ with a cyclically alternating renewal process with k different states denoted {A(t)}<sub>t≥0</sub>. Threshold conditions in terms of R<sub>⋆</sub> are obtained, such that R<sub>⋆</sub> > 1 implies that π, the probability of a large outbreak, is strictly positive. Examples are given where π is evaluated numerically from which the impact of the distribution of the time periods that Λ(t) spends in its different states is clearly seen.

Keyword
branching process in a seasonal environment, seasonal forcing, Stochastic epidemic, term-time forcing, threshold conditions
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-35205 (URN)10.1239/jap/1261670683 (DOI)000273995700004 ()
Available from: 2010-01-15 Created: 2010-01-15 Last updated: 2017-12-12Bibliographically approved

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