Colloquium Mathematics - Dr. J. Komjáthy

Title: Epidemic spread on inhomogeneous networks

Abstract:

In the first part of the presentation, I will highlight some lines of research I have been doing in the last years. One line of my research focuses on the mathematical understanding of various random graphs models designed to mimic real life networks, and epidemic spread thereon. Studied graph properties include typical distances, degree distribution, size of the giant component, percolation, and describing the first peak of an epidemic spreading on the network: characterising explosion and proper time scaling. In this part I will also describe a recent (Covid-19 related) project that communicated our understanding on epidemic spread via a simulation study to the applied mathematical community, focusing on the dependence of the height of the first and second peak on the network topology (and interventions applied).

In the second part of the presentation, I will describe a recent result on epidemic spread on spatially embedded inhomogeneous random graphs. We take a scale-free spatial random graph, where the degree of a vertex follows a power law with exponent tau>1. Examples of such graphs include: Scale free percolation, Geometric Inhomogeneous Random Graphs, and Hyperbolic Random Graphs. Then we equip each edge with a random and iid transmission delay L, and study the ball-growth of the first-passage infected cluster around the source vertex as a function of time. For a second, more realistic spreading model, the iid random transmission delay L through an edge with expected degrees W and Z is multiplied by a factor that is a polynomial of W,Z, (the penalty factor).

We call the model outwards (inwards) explosive if it is possible to reach infinitely many vertices within finite time (if infinitely many vertices can reach a target vertex within finite time) in an infinite model. Finite versions will be also discussed. We will discuss the criterion for explosion in the original model (no penalty factor) and in the penalised model. In particular, we will discuss that asymmetric penalty functions can lead to `outwards' explosion but no `inwards’ explosion or the other way round.

Joint work with John Lapinskas and Johannes Lengler.

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