Talks

Abstract Motivated by the discovery of hard-to-find social networks (such as MSM or A natural and well-known way to dPWIDs) or by finding contact-tracing strategies, we consider the question of exploring the topology of random structures (such as a random graph G) by random walks. The usual random walk jumps from a vertex of G to a neighboring vertex, providing information on the connected components of the graph G. The number of these connected components is the Betti number beta0. To gather further information on the higher Betti numbers that describe the topology of the graph, we can consider the simplicial complex C associated to the graph G: a k-simplex (edge for k=1, triangle for k=2, tetrahedron for k=3 etc.) belongs to C if all the lower (k-1)-simplices that constitute it also belong to the C. For example, a triangle belongs to C if its three edges are in the graph G. Several random walks have already been propose recently to explore these structures, mostly in Informatics Theory. We propose a new random walk, whose generator is related to a Laplacian of higher order of the graph, and to the Betti number betak. A rescaling of the walk for k=2 (cycle-valued random walk) is also detailed when the random walk visits a regular triangulation of the torus. We embed the space of chains into spaces of currents to establish the limiting theorem. Joint work with T. Bonis, L. Decreusefond and Z. Zhang.

Abstract We study a non-Markovian individual-based stochastic spatial epidemic model where the number of locations and the number of individuals at each location both grow to infinity while satisfying certain growth condition. Each individual is associated with a random infectivity function, which depends on the age of infection. The rate of infection at each location takes an averaging effect of infectivity from all the locations. The epidemic dynamics in each location is described by the total force of infection, the number of susceptible individuals, the number of infected individuals that are infected at each time and have been infected for a certain amount of time, as well as the number of recovered individuals. The processes can be described using a time-space representation. We prove a functional law of large numbers for these time-space processes, and in the limit, we obtain a set of time-space integral equations together with the limit of the number of infected individuals tracking the age of infection as a time-age-space integral equation. Joint work with G. Pang (Rice Univ)

Abstract People's interaction networks play a critical role in epidemics. However, precise mapping of the network structure is often expensive or even impossible. I will show that it is unnecessary to map the entire network. Instead, contact tracing a few samples from the population is enough to estimate an outbreak's likelihood and size. More precisely, I start by studying a simple epidemic model where one node is initially infected, and an infected node transmits the disease to its neighbors independently with probability p. In this model, I will present a nonparametric estimator on the likelihood of an outbreak based on local graph features and give theoretical guarantees on the estimator's accuracy for a large class of networks. Finally, I will extend the result to the general SIR model with random recovery time: Local graph features are enough to predict the time evolution of epidemics on a large class of networks.  

Abstract   Contact tracing, either manual by questioning diagnosed individuals for recent contacts or by an App keeping track of close contacts, is one out of many measures to reduce spreading. Mathematically this is hard to analyse because future infections of an individual are no longer independent of earlier contacts. In the talk I will describe a stochastic model for a simplified situation, allowing for both manual and digital contact tracing, for which it is possible to obtain results for the initial phase of the epidemic, with focus on the effective reproduction number $R_E$ which determines if contact tracing will prevent an outbreak or not. (Joint work with Dongni Zhang)