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- Fransson, Carolina, 1992-
(författare)
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Stochastic epidemics on random networks and competition in growth
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The COVID-19 pandemic has dramatically demonstrated the importance of epidemic models in understanding and predicting disease spread and in assessing the effectiveness of interventions. The overarching topic of this thesis is stochastic epidemic modelling, with the main focus on the role of the underlying social structure in infectious disease spread.In Paper I we study the spread of stochastic SIR-epidemics on an extended version of the configuration model with group structure. We present expressions for the basic reproduction number R0, the probability of a major outbreak and the expected final size, and investigate random vaccination with a perfect vaccine. We weaken the assumptions of earlier results for epidemics on this type of graph by allowing for heterogeneous infectivity both in individual infectivity and between different kinds of edges. An important special case of this model is the spread of a disease with arbitrary infectious period distribution in continuous time. Paper II concerns multi-type competition in a variant of Pólya's urn model with interaction, where balls of different colours/types annihilate upon contact. The model dynamics are governed by the structure of an underlying graph. In the special case of a cycle graph, this urn model is equivalent to a planar growth model with competing pathogens. It has earlier been shown that in the two-type case, indefinite coexistence has probability 0 for any (finite and connected) underlying graph, while for K ≥ 3 types the possibility of coexistence depends on the structure of this graph. We show that for K ≥ 3 types competing on a cycle graph, there is with probability 1 eventually only one remaining type.In Paper III we study the real-time growth rate of SIR epidemics on random intersection graphs with mixed Poisson degree distribution. We show that during the early stage of the epidemic, the number of infected individuals grows exponentially and the Malthusian parameter is shown to satisfy a version of the Euler-Lotka equation. These results are obtained via an approximating embedded single-type Crump-Mode-Jagers branching process. In addition, we provide a lower bound on the cumulative number of individuals that get infected before the branching process approximation breaks down. In Paper IV we consider stochastic SIR epidemics on inhomogeneous random graphs with degree-dependent contact rates. In this model, the per-neighbour contact rate of an individual decrease but its overall expected contact rate increases with its expected number of neighbours. We provide the basic reproduction number R0, the probability of a large outbreak and the final size of an epidemic. We show that reducing heterogeneity in contact rates results in a higher value of the basic reproduction number R0, and demonstrate that this result does not generally extend to the probability of a major outbreak and the final size.
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| 2. |
- Spricer, Kristoffer, 1966-
(författare)
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Random networks with weights and directions, and epidemics thereon
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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