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Branching process approach for epidemics in dynamic partnership network
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
2018 (English)In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 76, no 1-2, p. 265-294Article in journal (Refereed) Published
Abstract [en]

We study the spread of sexually transmitted infections (STIs) and other infectious diseases on a dynamic network by using a branching process approach. The nodes in the network represent the sexually active individuals, while connections represent sexual partnerships. This network is dynamic as partnerships are formed and broken over time and individuals enter and leave the sexually active population due to demography. We assume that individuals enter the sexually active network with a random number of partners, chosen according to a suitable distribution and that the maximal number of partners that an individual can have at a time is finite. We discuss two different branching process approximations for the initial stages of an outbreak of the STI. In the first approximation we ignore some dependencies between infected individuals. We compute the offspring mean of this approximating branching process and discuss its relation to the basic reproduction number R0. The second branching process approximation is asymptotically exact, but only defined if individuals can have at most one partner at a time. For this model we compute the probability of a minor outbreak of the epidemic starting with one or few initial cases. We illustrate complications caused by dependencies in the epidemic model by showing that if individuals have at most one partner at a time, the probabilities of extinction of the two approximating branching processes are different. This implies that ignoring dependencies in the epidemic model leads to a wrong prediction of the probability of a large outbreak. Finally, we analyse the first branching process approximation if the number of partners an individual can have at a given time is unbounded. In this model we show that the branching process approximation is asymptomatically exact as the population size goes to infinity.

Place, publisher, year, edition, pages
2018. Vol. 76, no 1-2, p. 265-294
Keywords [en]
SI epidemic, Branching process, Basic reproduction number, Dynamic network, Stochastic epidemic model
National Category
Mathematics Biological Sciences
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-143930DOI: 10.1007/s00285-017-1147-0ISI: 000419452400008OAI: oai:DiVA.org:su-143930DiVA, id: diva2:1105655
Available from: 2017-06-05 Created: 2017-06-05 Last updated: 2019-03-27Bibliographically approved
In thesis
1. Stochastic epidemics on random networks
Open this publication in new window or tab >>Stochastic epidemics on random networks
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis considers stochastic epidemic models for the spread of epidemics in structured populations. The asymptotic behaviour of the models is analysed by using branching process approximations. The thesis contains four manuscripts.

Paper I is concerned with the study of the spread of sexually transmitted infections, or any other infectious diseases on a dynamic network. The model we investigate is about the spread of an SI (Susceptible → Infectious) type infectious disease in a population where partnerships are dynamic. We derive explicit formulas for the probability of extinction and the threshold parameter R0 using two branching process approximations for the model. In the first approximation some dependencies between infected individuals are ignored while the second branching process approximation is asymptotically exact and only defined if every individual in the population can have at most one partner at a time. By comparing the two approximations, we show that ignoring subtle dependencies in the dynamic epidemic model leads to wrong prediction of the probability of a large outbreak.

In paper II, we study a stochastic SIR (Susceptible → Infectious → Removed) epidemic model for the spread of an epidemic in populations structured through configuration model random graphs. We study the asymptotic (properly scaled) time until the end of an epidemic. This paper heavily relies on the theory of branching processes in continuous time.

In paper III, the effect of vaccination strategies on the duration of an epidemic in a large population is investigated. We consider three vaccination strategies: uniform vaccination, leaky vaccination and acquaintance vaccination.

In paper IV, we present a stochastic model for two successive SIR epidemics in the same network structured population. Individuals infected during the first epidemic might have (partial) immunity for the second one. The first epidemic is analysed through a bond percolation model, while the second epidemic is approximated by a three-type branching process in which the types of individuals depend on their status in the percolation clusters used for the analysis of the first epidemic. This branching process approximation enables us to calculate a threshold parameter and the probability of a large outbreak for the second epidemic. We use two special cases of acquired immunity for further evaluation.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2019
Keywords
Branching process, Configuration model, Random graph, Epidemic process, Final size, Threshold behaviour, Duration of an epidemic, Vaccination
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-167373 (URN)978-91-7797-661-5 (ISBN)978-91-7797-662-2 (ISBN)
Public defence
2019-05-16, 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 2: Manuscript. Paper 3: Manuscript. Paper 4: Manuscript.

Available from: 2019-04-23 Created: 2019-03-27 Last updated: 2019-04-09Bibliographically approved

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