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Spatial Marriage Problems and Epidemics
Stockholm University, Faculty of Science, Department of Mathematics.
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This thesis consists of four papers covering three different topics on the modeling of large real networks and phenomena thereon. In Papers I and II, we propose and study the properties of a bipartite version of the model introduced by Deijfen, Holroyd and Häggström for generating translation-invariant spatial random graphs with prescribed degree distribution. In particular, we focus our attention on spatial random graphs generated by a matching scheme based on the Gale-Shapley stable marriage problem. In paper III, we propose a random graph model for generating edge-weighted graphs with prescribed degree and weight distributions, and tunable degree-degree correlation. We then study a simple inhomogeneous epidemic model on such graphs, where the infection probabilities are functions of the edge-weights, and investigate how the epidemic threshold is affected by the degree-degree correlation. In paper IV, we study a simple stochastic model aimed at representing a competition between two virus strains in a population. A longstanding principle in ecology known as the competitive exclusion principle predicts that when one of the strains has even the slightest advantage over the other, the one with the advantage will either drive the competitor to extinction or lead to a transformation in the ecological niche. We investigate how long it will take for the strain to drive its competitor to extinction.

##### Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2014. , p. 25
Mathematics
##### Research subject
Mathematical Statistics
##### Identifiers
ISBN: 978-91-7447-970-6 (print)OAI: oai:DiVA.org:su-106796DiVA, id: diva2:739314
##### Public defence
2014-09-24, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
##### Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: In press. Paper 4: Manuscript.

Available from: 2014-09-02 Created: 2014-08-20 Last updated: 2015-03-27Bibliographically approved
##### List of papers
1. Invariant Bipartite random graphs on R
Open this publication in new window or tab >>Invariant Bipartite random graphs on R
2014 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 51, no 3, p. 769-779Article in journal (Refereed) Published
##### Abstract [en]

Suppose that red and blue points occur in $\mathbb{R}^d$ according to two simple point processes  with finite intensities $\lambda_{\mathcal{R}}$ and $\lambda_{\mathcal{B}}$, respectively. Furthermore, let $\nu$ and $\mu$ be two probability distributions on the strictly positive integers with means $\bar{\nu}$ and $\bar{\mu}$, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law $\nu$ ($\mu$).  We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a  prescribed degree distribution with law $\nu$ or $\mu$ depending on its color. For a large class of point processes we show that such translation-invariant schemes matching a.s. all stubs are possible if and only if$\lambda_{\mathcal{R}} \bar{\nu}= \lambda_{\mathcal{B}} \bar{\mu},$also including the case when $\bar{\nu}=\bar{\mu}=\infty$ so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme we give sufficient conditions on $\nu$ and $\mu$ for the presence and absence of infinite components. These results are  two-color versions of those obtained by Deijfen, Holroyd and H\"{a}ggstr\"{o}m.

##### Keyword
Poisson process, random graphs, bipartite, stable matching, percolation
Mathematics
##### Research subject
Mathematical Statistics
##### Identifiers
urn:nbn:se:su:diva-106729 (URN)000342035400013 ()
Available from: 2014-08-18 Created: 2014-08-18 Last updated: 2017-12-05Bibliographically approved
2. Bipartite Stable Poisson Graphs on R
Open this publication in new window or tab >>Bipartite Stable Poisson Graphs on R
2012 (English)In: Markov processes and related fields, ISSN 1024-2953, Vol. 18, no 4, p. 583-594Article in journal (Refereed) Published
##### Abstract [en]

Let red and blue points be distributed on R according to two independent Poisson processes R and B and let each red (blue) point independently be equipped with a random number of half-edges according to a probability distribution nu (mu). We consider translation-invariant bipartite random graphs with vertex classes defined by the point sets of R, and B, respectively, generated by a scheme based on the Gale - Shapley stable marriage for perfectly matching the half-edges. Our main result is that, when all vertices have degree 2, then the resulting graph almost surely does not contain an infinite component. The two-color model is hence qualitatively different from the one-color model, where Deijfen, Holroyd and Peres have given strong evidence that there is an infinite component. We also present simulation results for other degree distributions.

##### Keyword
Poisson process, random graph, degree distribution, percolation, matching
##### National Category
Probability Theory and Statistics
##### Identifiers
urn:nbn:se:su:diva-87670 (URN)000313221100001 ()
##### Note

AuthorCount:2;

Available from: 2013-02-18 Created: 2013-02-14 Last updated: 2014-08-21Bibliographically approved
3. Epidemics on a weighted network with tunable degree-degree correlation
Open this publication in new window or tab >>Epidemics on a weighted network with tunable degree-degree correlation
2014 (English)In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 253, p. 40-49Article in journal (Refereed) Published
##### Abstract [en]

We propose a weighted version of the standard configuration model which allows for a tunable degree-degree correlation. A social network is modeled by a weighted graph generated by this model, where the edge weights indicate the intensity or type of contact between the individuals. An inhomogeneous Reed-Frost epidemic model is then defined on the network, where the inhomogeneity refers to different disease transmission probabilities related to the edge weights. By tuning the model we study the impact of different correlation patterns on the network and epidemics therein. Our results suggest that the basic reproduction number R-0 of the epidemic increases (decreases) when the degree-degree correlation coefficient rho increases (decreases). Furthermore, we show that such effect can be amplified or mitigated depending on the relation between degree and weight distributions as well as the choice of the disease transmission probabilities. In addition, for a more general model allowing additional heterogeneity in the disease transmission probabilities we show that rho can have the opposite effect on R-0.

##### Keyword
Branching processes, Configuration model, Weighted graph, Epidemic threshold, Degree-degree correlation
Mathematics
##### Identifiers
urn:nbn:se:su:diva-106327 (URN)10.1016/j.mbs.2014.03.013 (DOI)000337874100006 ()
##### Note

AuthorCount:1;

Available from: 2014-08-07 Created: 2014-08-04 Last updated: 2017-12-05Bibliographically approved
4. A note on the coexistence times for two competing SIS epidemics
Open this publication in new window or tab >>A note on the coexistence times for two competing SIS epidemics
##### Abstract [en]

The stochastic SIS logistic epidemic process is a well-known continuous-time Markov chain with finite state space, describing the spread of an epidemic in a homogeneously mixing  population of size N. This process  eventually reaches an absorbing state (extinction'') and its extinction time is well-understood.  Namely, we can identify a phase transition depending on the infectious rate of the epidemic. There is a \textit{subcritical} phase where the process goes extinct in  time $O_{P}(\log N)$, and a \textit{supercritical} phase where the extinction time grows exponentially in the population size. In this work we consider two SIS epidemics with distinct supercritical infectious rates competing under cross-immunity, i.e. during its infectious period an infective individual is immune to the other infection. We show that with high probability the process with the lower infectious rate dies out first and the two epidemics coexist for a time that is $O_{P}(\log N)$. Furthermore, we conjecture the limiting distribution of the coexistence time.

##### Keyword
SIS epidemic model, competitive Lotka-Volterra systems, extinction times
Natural Sciences
##### Identifiers
urn:nbn:se:su:diva-106730 (URN)
Available from: 2014-08-18 Created: 2014-08-18 Last updated: 2014-08-22Bibliographically approved

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