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Stochastic epidemics on random networks
Stockholm University, Faculty of Science, Department of Mathematics.
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 [en]
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: urn:nbn:se:su:diva-167373ISBN: 978-91-7797-661-5 (print)ISBN: 978-91-7797-662-2 (electronic)OAI: oai:DiVA.org:su-167373DiVA, id: diva2:1299570
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
List of papers
1. Branching process approach for epidemics in dynamic partnership network
Open this publication in new window or tab >>Branching process approach for epidemics in dynamic partnership network
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.

Keywords
SI epidemic, Branching process, Basic reproduction number, Dynamic network, Stochastic epidemic model
National Category
Mathematics Biological Sciences
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-143930 (URN)10.1007/s00285-017-1147-0 (DOI)000419452400008 ()
Available from: 2017-06-05 Created: 2017-06-05 Last updated: 2019-03-27Bibliographically approved
2. The duration of an SIR epidemic on a configuration model
Open this publication in new window or tab >>The duration of an SIR epidemic on a configuration model
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider the spread of a stochastic SIR (Susceptible, Infec-tious, Recovered) epidemic on a configuration model random graph.We focus especially on the final stages of the outbreak and providelimit results for the duration of the entire epidemic, while we allowfor non-exponential distributions of the infectious period and for bothfinite and infinite variance of the asymptotic degree distribution in thegraph.

Our analysis relies on the analysis of some subcritical continuoustime branching processes and on ideas from first-passage percolation.

Keywords
SIR epidemics, Time to extinction, Branching process approximation, First passage percolation
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-167360 (URN)
Funder
Swedish Research Council
Available from: 2019-03-27 Created: 2019-03-27 Last updated: 2019-03-29
3. Effect of vaccination on the duration of an SIR epidemic in homogeneously mixing and structured populations
Open this publication in new window or tab >>Effect of vaccination on the duration of an SIR epidemic in homogeneously mixing and structured populations
(English)Manuscript (preprint) (Other academic)
Abstract [en]

This paper is concerned with the effects of vaccination on the properly scaled dura-tion of the stochastic SIR (Susceptible → Infected → Recovered/Removed) epidemicboth in homogeneous mixing populations and in populations structured through config-uration model random graphs. We examine uniform vaccination and leaky vaccinationin both homogeneous and structured populations. Furthermore, we consider acquain-tance vaccination on the configuration model. For these vaccination schemes, we studythe asymptotic time until the end of the epidemic and study the effect of the vaccinationon this duration.

We show that, depending on the degree distribution, uniform vaccination witha perfect vaccine may lead to both an increase and decrease in the duration of anepidemic in structured populations, whereas in homogeneously mixing populations,vaccination with a perfect vaccine either prevents or prolongs an epidemic in the largepopulation limit. In homogeneously mixing populations, the leaky vaccine has a similareffect as uniform vaccination on the duration of the epidemic, whereas in structuredpopulations, we conjecture that the leaky vaccine always increases the duration of anepidemic. For the acquaintance vaccination scheme we give, through the derivationof the effective degree distribution of unvaccinated individuals, a recipe to obtain theasymptotic duration of an epidemic and show that acquaintance vaccination may bothdecrease and increase the duration of an epidemic.

Keywords
Epidemic model, duration of an SIR epidemic, branching process approximation, vaccination
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-167361 (URN)
Available from: 2019-03-27 Created: 2019-03-27 Last updated: 2019-03-29Bibliographically approved
4. Modeling the spread of two successive SIR epidemics on a configuration model network
Open this publication in new window or tab >>Modeling the spread of two successive SIR epidemics on a configuration model network
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We present a stochastic model for two successive SIR (Susceptible, Infectious, Recov-ered) epidemics in the same network structured population. Individuals infected duringthe first epidemic might have (partial) immunity for the second one. The first epidemic isanalysed through a bond percolation model, while the second epidemic is approximated bya three-type branching process in which the types of individuals depend on their position inthe percolation clusters used for the first epidemic. This branching process approximationenables us to calculate a threshold parameter and the probability of a large outbreak for thesecond epidemic.

We illustrate our results through two examples. In the first example individuals infectedby the first epidemic are independently either completely susceptible or completely immuneto the second epidemic. The probability of being completely immune is the same for allindividuals infected in the first epidemic. In the second example the recovered individual inthe first epidemic have reduced susceptibility and infectivity for the second epidemic.

Keywords
Subsequent SIR epidemics, Reed-Frost model, Percolation, Multi-type branching processes
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-167372 (URN)
Available from: 2019-03-27 Created: 2019-03-27 Last updated: 2019-03-29Bibliographically approved

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