### An extended SIR model for the spread of COVID-19 in different communities

In early 2020 the novel coronavirus Covid-19 began to spread rapidly throughout the world. To understand the impact of the virus, and how to control infections, scientists looked towards data modelling to provide examples of the infection rate and the impact on communities.

In this seminar, Dr Chris Antonopoulos looked at the effectiveness of the modelling approach on the pandemic due to the spreading of the novel COVID-19 disease and develop an extended-susceptible-infected-removed (eSIR) model that provides a theoretical framework to investigate its spread within a community. A particular focus of this research has been the time evolution of different populations, and how to monitor diverse significant parameters for the spread of the disease in various communities, represented by countries and the state of Texas in the USA.

Dr Antonopoulos explained that the eSIR model can provide us with insights and predictions of the spread of the virus in communities that recorded data alone cannot. It was interesting to hear that the spread of COVID-19 can be under control in all communities considered, if proper restrictions and strong policies are implemented to control the infection rates early from the spread of the disease.

#### Related papers

- Cooper, I., Mondal, A. and Antonopoulos, CG., (2020). A SIR model assumption for the spread of COVID-19 in different communities. Chaos, Solitons and Fractals. 139, 110057-110057
- Cooper, I., Mondal, A. and Antonopoulos, CG., (2020). Dynamic tracking with model-based forecasting for the spread of the COVID-19 pandemic. Chaos, Solitons and Fractals. 139, 110298-110298

### K-stability of Fano 3-folds

Dr Anne-Sophie Kaloghiros, a Senior Lecturer at Brunel University, joined us to discuss her work on K-stability of Fano 3-folds.

Fano varieties are geometric shapes which are positively curved. They arise in a wide array of fields from theoretical physics to phylogenetic trees. There are rich interactions between differential geometric and algebro-geometric properties of Fano manifolds (and more generally of Kahler manifolds).

An instance of this phenomenon was conjectured by Yau Tian and Donaldson (and proved by Donaldson, Chen and Sun): they proved that on Fano manifolds the existence of special canonical metrics is equivalent to a stability property. This is an equivalence between properties that are subtle, and still little understood.

In her talk, Dr Kaloghiros explained the algebro-geometric approaches to this problem, as well as recent developments in this area of research and their applications to our understanding of Fano surfaces and 3-folds.

#### Related papers

- Ahmadinezhad, H. and Kaloghiros, A-S. (2015) 'Non-rigid quartic 3-folds'. Compositio Mathematica, 152 (5). pp. 955 - 983. ISSN: 1570-5846
- Kaloghiros, AS. (2011) 'The defect of Fano 3-folds'. Journal of Algebraic Geometry, 20 (1). pp. 127 - 149. ISSN: 1056-3911

### Deep tensor decompositions for sampling from high-dimensional distributions

Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation, for example, in the solution of Bayesian inverse problems. The recent surge of transport maps offers a mathematical foundation and new insights for tackling this challenge by coupling intractable random variables with tractable reference random variables.

In this talk Dr Sergey Dolgov, from the University of Bath, presented a nested coordinate transformation framework inspired by deep neural networks but driven by functional tensor-train approximation of tempered probability density functions instead. This bypasses slow gradient descent optimisation by a direct inverse Rosenblatt transformation. The resulting deep inverse Rosenblatt transport significantly expands the capability of tensor approximations and transport maps to random variables with complicated nonlinear interactions and concentrated density functions.

Sergey demonstrated the efficiency of the proposed approach on a range of applications in uncertainty quantification, including parameter estimation for dynamical systems and inverse problems constrained by partial differential equations.

#### Related papers

Sergey Dolgov, Dante Kalise, and Karl K. Kunisch, "Tensor Decomposition Methods for High-dimensional Hamilton--Jacobi--Bellman Equations", SIAM Journal on Scientific Computing, 2021, Vol. 43, No. 3.

Dolgov, S., Anaya-Izquierdo, K., Fox, C., Scheichl, R., "Approximation and sampling of multivariate probability distributions in the tensor train decomposition", Statistics and Computing volume 30, 603–625 (2020).