Dia 9 de Fevereiro (quinta-feira), às 16h30, sala 6.2.38

Vector borne diseases on an urban environment—a multi-group model approach

Abstract: Starting from a class of meta-population models for the  dynamics of a vector-borne disease, we will show how different mathematical tools lead to very complete analysis of the system. The underlying dynamics will be reduced to a coupled SIR (human)/SI (mosquito) system; notification districts are taken for patches. We focus on the role of human movement in sustaining the epidemics. It turns out that considering different aspects of urban districts leads to very heterogeneous networks, which might lead to very distinctive dynamics.

In a worst case scenario, one might have local basic reproduction numbers all less than unity, but with the network basic reproduction number ($R_0$) larger than one. In particular, we can obtain a correction to the uniform $R_0$ (aggregating data as a single region) which is given by the principal singular value of a certain interaction matrix. We also completely analyse the model with respect to global stability. This is joint work with Abderrahman Iggidr, Jair Koiller, Maria Lúcia Penna, Gauthier Sallet and Moacyr Silva.

This seminar is supported by National Funding from FCT - Fundação para a Ciência e a Tecnologia, under the project: UID/MAT/04561/2013

## 2015

Sala B3-01 do Instituto para a Investigação Interdisciplinar, 30 de junho 2015, 16.00

Peter Schwertfeger

Centre for Theoretical Chemistry and Physics, The New Zealand
Institute for Advanced Study, Massey University Auckland, New Zealand.

"The Topology of Fullerenes"

Abstract:

Fullerenes are 3-connected cubic planar graphs consisting of pentagons and hexagons only. There has been great progress over the last twodecades describing the topological and graph theoretical properties of ullerenes, but leaving still many unsolved and interesting mathematical (and chemical) problems open in this field. A few example are, i) how to generate all possible non-isomorphic graphs for a fixed vertex count, ii) are fullerenes Hamiltonian (Barnette’s conjecture) and what is the number of distinct Hamiltonian cycles, iii) the Pauling bond order and the number of perfect matchings, iv) the search for suitable topological indices to find the most stable fullerene structure out of the many (N9) possibilities, or how to pack fullerene
cages in 3D space (Hilbert problem)? Our research group in Albany isndeveloping a general-purpose program (Program Fullerene) [1,2] that creates 2D graphs and accurate 3D structures for any fullerene isomer through various different graph-theoretical methods and algorithms, and subsequently performs a topological analysis. A general overview on topological and graph theoretical aspects of fullerenes is presented, and illustrated for many different fullerenes ranging from N = 20 to 20,000 vertices, and some new conjectures and theorems are presented [3].

[1] P. Schwerdtfeger, L. Wirz, J. Avery, J. Comput. Chem. 34, 1508-1526 (2013).

[2] O. Ori, M. V. Putz, I. Gutman, P. Schwerdtfeger, “Generalized
Stone-Wales Transformations for Fullerene Graphs Derived from Berge’s
Switching Theorem”, submitted.

[3] P. Schwerdtfeger, L. Wirz, J. Avery, “The Topology of Fullerenes”,
Wiley Interdisciplinary Reviews: Computational Molecular Science 5,
96-145 (2015).

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