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An Epidemic on a Weighted Partially Directed Network
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0003-0233-0022
(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.

Keyword [en]
Epidemics, Reproduction number, Final size, Large outbreak, Weighted network, Partially directed, Configuration model
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-154929OAI: oai:DiVA.org:su-154929DiVA, id: diva2:1195905
Available from: 2018-04-06 Created: 2018-04-06 Last updated: 2018-04-12Bibliographically approved
In thesis
1. Random networks with weights and directions, and epidemics thereon
Open this publication in new window or tab >>Random networks with weights and directions, and epidemics thereon
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
Keyword
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:nbn:se:su:diva-154930 (URN)978-91-7797-278-5 (ISBN)978-91-7797-279-2 (ISBN)
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

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