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Random networks with weights and directions, and epidemics thereon
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0003-0233-0022
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Networks, consisting of nodes and of edges, can be used to model numerous phenomena, e.g, web pages linking to each other or interactions between people in a population. Edges can be directed, such as a one way link from one web page to another, or undirected (bi-directional), such as physical contacts between pairs of people, which potentially could spread an infection either way between them. Edges can also have weights associated with them, in this thesis corresponding to the probability that an infection is transmitted on the edge.

Empirical networks are often only partially known, in the form of ego-centric network data where only a subset of the nodes and the number of adjacent edges of each node have been observed. This situation lends itself well to analysis through the undirected or partially directed configuration model - a random network model where the number of edges of each node (the degree) is given but where the way these edges are connected is random.

The four papers in this thesis are concerned with the properties of the configuration model and with the usefulness of it with respect to its ability to model the spread of epidemics on empirical networks. Paper I proves the asymptotic convergence to a given degree distribution for the partially directed configuration model. In Paper II it is shown that epidemics on some empirical and theoretically constructed networks grow exponentially, similarly to what can be seen on the corresponding configuration models. Finally, in Papers III and IV, large population analytical results for the reproduction number, the probability of a large epidemic outbreak and the final size of such an outbreak are derived assuming a configuration model network with weighted and/or partially directed edges. These results are then evaluated on several large empirical networks upon which epidemics are simulated. We find that on some of these networks the analytical expressions are compatible with the results of the simulations. This makes the model useful as a tool for analyzing such networks.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2018. , p. 28
Keywords [en]
Epidemics, Reproduction number, Final size, Large outbreak, Weighted network, Undirected, Partially directed, Configuration model, Copula
National Category
Probability Theory and Statistics Mathematics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-154930ISBN: 978-91-7797-278-5 (print)ISBN: 978-91-7797-279-2 (electronic)OAI: oai:DiVA.org:su-154930DiVA, id: diva2:1197294
Public defence
2018-05-31, Sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 14:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2009-5759Riksbankens Jubileumsfond, P12-0705:1Swedish Research Council, 2016-04566Swedish Research Council, 2015-05015
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: 2018-05-07 Created: 2018-04-12 Last updated: 2018-05-04Bibliographically approved
List of papers
1. The Configuration Model for Partially Directed Graphs
Open this publication in new window or tab >>The Configuration Model for Partially Directed Graphs
2015 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 161, no 4, p. 965-985Article in journal (Refereed) Published
Abstract [en]

The configuration model was originally defined for undirected networks and has recently been extended to directed networks. Many empirical networks are however neither undirected nor completely directed, but instead usually partially directed meaning that certain edges are directed and others are undirected. In the paper we define a configuration model for such networks where vertices have in-, out-, and undirected degrees that may be dependent. We prove conditions under which the resulting degree distributions converge to the intended degree distributions. The new model is shown to better approximate several empirical networks compared to undirected and completely directed networks.

Keywords
Configuration model, Partially directed, Semi-directed, Degree distribution, Asymptotic convergence
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-123511 (URN)10.1007/s10955-015-1360-4 (DOI)000363257600010 ()
Funder
Riksbankens Jubileumsfond, P12-0705:1Swedish Research Council, 2009-5759
Available from: 2015-11-30 Created: 2015-11-27 Last updated: 2018-04-12Bibliographically approved
2. Characterizing the Initial Phase of Epidemic Growth on some Empirical Networks
Open this publication in new window or tab >>Characterizing the Initial Phase of Epidemic Growth on some Empirical Networks
2018 (English)In: Stochastic Processes and Applications / [ed] Sergei Silvestrov, Anatoliy Malyarenko, Milica Rančić, Springer, 2018, p. 315-334Conference paper, Published paper (Refereed)
Abstract [en]

A key parameter in models for the spread of infectious diseases is the basic reproduction number R0">R 0  R0 , which is the expected number of secondary cases a typical infected primary case infects during its infectious period in a large mostly susceptible population. In order for this quantity to be meaningful, the initial expected growth of the number of infectious individuals in the large-population limit should be exponential. We investigate to what extent this assumption is valid by simulating epidemics on empirical networks and by fitting the initial phase of each epidemic to a generalised growth model, allowing for estimating the shape of the growth. For reference, this is repeated on some elementary graphs, for which the early epidemic behaviour is known. We find that for the empirical networks tested in this paper, exponential growth characterizes the early stages of the epidemic, except when the network is restricted by a strong low-dimensional spacial constraint.

Place, publisher, year, edition, pages
Springer, 2018
Series
Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009, E-ISSN 2194-1017 ; 271
Keywords
Epidemics, Exponential growth, Generalized growth model, Reproduction number, Stochastic processes
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-154927 (URN)10.1007/978-3-030-02825-1_13 (DOI)978-3-030-02824-4 (ISBN)978-3-030-02825-1 (ISBN)
Conference
SPAS2017, International Conference on Stochastic Processes and Algebraic Structures – From Theory Towards Applications, Västerås and Stockholm, Sweden, October 4-6, 2017
Funder
Swedish Research Council, 2016-04566
Available from: 2018-04-06 Created: 2018-04-06 Last updated: 2018-12-13Bibliographically approved
3. An Epidemic on a Weighted Network
Open this publication in new window or tab >>An Epidemic on a Weighted Network
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We introduce a weighted configuration model graph, where edge weights correspond to the probability of infection in an epidemic on the graph, focusing on two different weights. We study the basic reproduction number R0, the probability of a large outbreak and the relative final size of a large outbreak, using discrete time and Markovian continuous time settings. Results are compared with those for a calibrated unweighted graph. The degree distributions are based both on empirical network data and on theoretical constructs. Using copulas to model the dependence between the degrees of the different edge types allows for modeling the correlation over a wide range. The weighted model produces much richer results than the unweighted model. Also, while R0 always increases with increasing correlation between the two degrees, this is not necessarily true for the probability of an epidemic nor for the relative final size of it. The copula model can produce results that are similar to those of the empirical degree distributions, indicating that itis a viable alternative to using the full empirical data.

Keywords
Epidemics, Basic reproduction number, Weighted graph, Configuration model, Final size, Large outbreak, Copula
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-154928 (URN)
Funder
Swedish Research Council, 2015-05015
Available from: 2018-04-06 Created: 2018-04-06 Last updated: 2018-04-16Bibliographically approved
4. An Epidemic on a Weighted Partially Directed Network
Open this publication in new window or tab >>An Epidemic on a Weighted Partially Directed Network
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We introduce a weighted partially directed configuration model graph, where the weights on undirected and directed edges correspond to the probability of transmitting an infection. The model is especially useful when the degree distribution is known, but not the exact structure of the network. On such graphs, we study SIR epidemics in discrete time and determine the basic reproduction number R0, the probability of a large outbreak and the relative final size of the outbreak. The analytical results of the model are compared with simulated epidemics on three empirical networks. While the networks are large and have a very high maximum degree, thanks to the sparse nature of the degree distribution for many empirical networks, the analysis is still computationally feasible on standard computers. While results are different on the three networks, we see that for two of them the proposed model is compatible with the simulated epidemics on the empirical networks. This makes the model useful for analyzing such networks.

Keywords
Epidemics, Reproduction number, Final size, Large outbreak, Weighted network, Partially directed, Configuration model
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-154929 (URN)
Available from: 2018-04-06 Created: 2018-04-06 Last updated: 2018-04-12Bibliographically approved

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