On March 23rd and March 30th, 2020, the Mexican Federal government implemented social distancing measures to mitigate the COVID-19 epidemic. In this work a mathematical model is used to explore atypical transmission events within the confinement period, triggered by the timing and strength of short time perturbations of social distancing. Is shown that social distancing measures were successful in achieving a significant reduction of the epidemic curve growth rate in the early weeks of the intervention. However, “flattening the curve” had an undesirable effect, since the epidemic peak was delayed too far, almost to the government preset day for lifting restrictions (June 1st, 2020). If the peak indeed occurs in late May or early June, then the events of children's day and Mother’s Day may either generate a later peak (worst case scenario), a long plateau with relatively constant but high incidence (middle case scenario) or the same peak date as in the original baseline epidemic curve, but with a post-peak interval of slower decay.
A SEIRS model was developed to describe the spread of COVID-19 in Mexico, assuming different quarantine scenarios as a function of the conditions of hospital shortage. The presented model takes into account the heterogeneity of the state of infection, that is, the groups of clinical variants that can occur when the disease is contracted. Finally, the model allows different policy options to be implemented in different sectors of population.
This is the prototype of an agent based model for a closed universe of a population experiencing a contagion-based epidemic, in which risk factors, movement, time of incubation and asymptomatic infection are all parameters. The model allows the operator to intervene at any step and change parameters, thus analytically visualizing the effect of policies like more testing, contract tracing, and shelter in place. Under current development, CovidSimMV is an ABM that supports a Multiverse of different environments, in which agents move from one to another according to ticket with stops. Each universe has its own characteristic mix of residents, transients and attached staff, and persons are able to adopt different roles and characteristics in different universes. The fundamental disease characteristics of incubation, asymptomatic infection, confirmed cases will be preserved. The Multiverse model will support a rich diversity of environments and interpersonal dynamics. These are JavaScript programs that can be run in a browser as HTML files. The code is open source, and available on github.com/ecsendmail.
Given a discrete subset V in the plane, how many points would you expect there to be in a ball of radius 100? What if the radius is 10,000? Due to the results of Fairchild and forthcoming work with Burrin, when V arises as orbits of non-uniform lattice subgroups of SL(2,R), we can understand asymptotic growth rate with error terms of the number of points in V for a broad family of sets. A crucial aspect of these arguments and similar arguments is understanding how to count pairs of saddle connections with certain properties determining the interactions between them, like having a fixed determinant or having another point in V nearby. We will spend the first 40 minutes discussing how these sets arise and counting results arise from the study of concrete translation surfaces. The following 40 minutes will be spent highlighting the proof strategy used to obtain these results, and advertising the generality and strength of this argument that arises from the computation of all higher moments of the Siegel--Veech transform over quotients of SL(2,R) by non-uniform lattices.
Given a probability preserving system (X, \mu, T) and a set U of positive measure contained in X we denote by N(U,U) the set of integers n such that the measure of U intersected with T^n(U) is positive. These sets are called return-time sets and are of very special nature. For instance, Poincaré recurrence theorem tells us that the set must have bounded gaps while Sarkozy-Furstenberg theorem tells us that it must have a square. The subject of this talk is a very old question (going back to Følner-1954 if not earlier) whether they give rise to the same family of the sets as when we restrict ourselves to compact group rotations. This was answered negatively by Kříž in 1987 and recently it was proved by Griesmer that a return-time set need not contain any translate of a return-time set arising from compact group rotations. In this talk, I will try to sketch some of these proofs and give a flavour of results and questions in this direction.
There are simple formulas defining "naive entropy" for continuous/measure preserving actions of a countable group G on a compact metric/probability space. It turns out that if G is amenable, then this naive entropy coincides with topological/Kolmogoro-Sinai entropy of the action, while for non-amenable groups both naive entropies take only two values: 0 or infinity. During my talk, I will try to sketch the proofs of these facts. I will follow: T. Downarowicz, B. Frej, P.-P. Romagnoli, Shearer's inequality and infimum rule for Shannon entropy and topological entropy. Dynamics and numbers, 63-75, Contemp. Math., 669, Amer. Math. Soc., Providence, RI, 2016. MR3546663 and P. Burton, Naive entropy of dynamical systems. Israel J. Math. 219 (2017), no. 2, 637-659. MR3649602.
Diffusion is the enemy of life. This is because diffusion is a ubiquitous feature of molecular motion that is constantly spreading things out, destroying molecular aggregates. However, all living organisms, whether single cell or multicellular have ways to use the reality of molecular diffusion to their advantage. That is, they expend energy to concentrate molecules and then use the fact that molecules move down their concentration gradient to do useful things. In this talk, I will show some of the ways that cells use diffusion to their advantage, to signal, to form structures and aggregates, and to make measurements of length and size of populations. Among the examples I will describe are signalling by nerves, cell polarization, bacterial quorum sensing, and regulation of flagellar molecular motors. In this way, I hope to convince you that living organisms have made diffusion their friend, not their enemy.
This main result of this talk is that there exists a billiard flow in a polygon that is weakly mixing with respect to Lebesgue measure on the unit tangent bundle to the billiard. This strengthens Kerckhoff, Masur and Smillie's result that there exists ergodic billiard flows in polygons. The existence of a weakly mixing billiard follows, via a Baire category argument, from showing that for any translation surface the product of the flows in almost every pair of directions is ergodic with respect to Lebesgue measure. This in turn is proven by showing that for every translation surface the flows in almost every pair of directions do not share non-trivial common eigenvalues. This talk will explain the problem, related results, and approach. The talk will not assume familiarity with translation surfaces. This is joint work with Giovanni Forni.
Furstenberg's conjecture on the dimension of the intersection of x2,x3-invariant Cantor sets can be restated as a bound on the dimension of linear slices of the product of x2,x3-Cantor sets, which is a self-affine set in the plane. I will discuss some older and newer variants of this, where the self-affine set is replaced by a self-similar set such as the Sierpinski triangle, Sierpinski carpet or (support of) a complex Bernoulli convolution. Among other things, I will show that the intersection of the Sierpinski carpet with circles has small dimension, but on the other hand the Sierpinski carpet can be covered very efficiently by linear tubes (neighborhoods of lines). The latter fact is a recent result joint with A. Pyörälä, V. Suomala and M. Wu.
There is a rich connection between homogeneous dynamics and number theory. Often in such applications it is desirable for dynamical results to be effective (i.e. the rate of convergence for dynamical phenomena are known). In the first part of this talk, I will provide the necessary background and relevant history to state an effective equidistribution result for horospherical flows on the space of unimodular lattices in Rn. I will then describe an application to studying the distribution of almost-prime times (integer times having fewer than a fixed number of prime factors) in horospherical orbits and discuss connections of this work to Sarnak’s Mobius disjointness conjecture. In the second part of the talk I will describe some of the ingredients and key steps that go into proving these results.