Subexponential Growth Research Articles

Finitary codings for the random-cluster model and other infinite-range monotone models

A random field X=(Xv)v∈G on a quasi-transitive graph G is a factor of an i.i.d. process if it can be written as X=φ(Y) for some i.i.d. process Y=(Yv)v∈G and equivariant map φ. Such a map, also called a coding, is finitary if, for every vertex v∈G, there exists a finite (but random) set U⊂G such that Xv is determined by {Yu}u∈U. We construct a coding for the random-cluster model on G, and show that the coding is finitary whenever the free and wired measures coincide. This strengthens a result of Häggström–Jonasson–Lyons [18]. We also prove that the coding radius has exponential tails in the subcritical regime. As a corollary, we obtain a similar coding for the subcritical Potts model. Our methods are probabilistic in nature, and at their heart lies the use of coupling-from-the-past for the Glauber dynamics. These methods apply to any monotone model satisfying mild technical (but natural) requirements. Beyond the random-cluster and Potts models, we describe two further applications – the loop O(n) model and long-range Ising models. In the case of G= Zd, we also construct finitary, translation-equivariant codings using a finite-valued i.i.d. process Y. To do this, we extend a mixing-time result of Martinelli–Olivieri [22] to infinite-range monotone models on quasi-transitive graphs of sub-exponential growth.

Transmission Dynamics and Short-Term Forecasts of COVID-19: Nepal 2020/2021.

Nepal was hard hit by a second wave of COVID-19 from April-May 2021. We investigated the transmission dynamics of COVID-19 at the national and provincial levels by using data on laboratory-confirmed RT-PCR positive cases from the official national situation reports. We performed 8 week-to-week sequential forecasts of 10-days and 20-days at national level using three dynamic phenomenological growth models from 5 March 2021-22 May 2021. We also estimated effective and instantaneous reproduction numbers at national and provincial levels using established methods and evaluated the mobility trends using Google's mobility data. Our forecast estimates indicated a declining trend of COVID-19 cases in Nepal as of June 2021. Sub-epidemic and Richards models provided reasonable short-term projections of COVID-19 cases based on standard performance metrics. There was a linear pattern in the trajectory of COVID-19 incidence during the first wave (deceleration of growth parameter (p) = 0.41-0.43, reproduction number (Rt) at 1.1 (95% CI: 1.1, 1.2)), and a sub-exponential growth pattern in the second wave (p = 0.61 (95% CI: 0.58, 0.64)) and Rt at 1.3 (95% CI: 1.3, 1.3)). Across provinces, Rt ranged from 1.2 to 1.5 during the early growth phase of the second wave. The instantaneous Rt fluctuated around 1.0 since January 2021 indicating well sustained transmission. The peak in mobility across different areas coincided with an increasing incidence trend of COVID-19. In conclusion, we found that the sub-epidemic and Richards models yielded reasonable short-terms projections of the COVID-19 trajectory in Nepal, which are useful for healthcare utilization planning.

Eigenfunctions growth of R-limits on graphs

A characterization of the essential spectrum of Schrödinger operators on infinite graphs is derived involving the concept of \mathcal{R} -limits. This concept, which was introduced previously for operators on \mathbb{N} and \mathbb{Z}^d as “right-limits,” captures the behaviour of the operator at infinity. For graphs with sub-exponential growth rate, we show that each point in \sigma_{\textup{ess}}(H) corresponds to a bounded generalized eigenfunction of a corresponding \mathcal{R} -limit of H . If, additionally, the graph is of uniform sub-exponential growth, also the converse inclusion holds.

A single-agent extension of the SIR model describes the impact of mobility restrictions on the COVID-19 epidemic

Mobility restrictions are successfully used to contain the diffusion of epidemics. In this work we explore their effect on the epidemic growth by investigating an extension of the Susceptible-Infected-Removed (SIR) model in which individual mobility is taken into account. In the model individual agents move on a chessboard with a Lévy walk and, within each square, epidemic spreading follows the standard SIR model. These simple rules allow to reproduce the sub-exponential growth of the epidemic evolution observed during the Covid-19 epidemic waves in several countries and which cannot be captured by the standard SIR model. We show that we can tune the slowing-down of the epidemic spreading by changing the dynamics of the agents from Lévy to Brownian and we investigate how the interplay among different containment strategies mitigate the epidemic spreading. Finally we demonstrate that we can reproduce the epidemic evolution of the first and second COVID-19 waves in Italy using only 3 parameters, i.e , the infection rate, the removing rate, and the mobility in the country. We provide an estimate of the peak reduction due to imposed mobility restrictions, i. e., the so-called flattening the curve effect. Although based on few ingredients, the model captures the kinetic of the epidemic waves, returning mobility values that are consistent with a lock-down intervention during the first wave and milder limitations, associated to a weaker peak reduction, during the second wave.

Periodic points and measures for a class of skew-products

We consider the C1-open set \(\cal{V}\) of partially hyperbolic diffeomorphisms on the space \(\mathbb{T}^{2}\times\mathbb{T}^{2}\) whose non-wandering set is not stable, introduced by M. Shub in [57]. Firstly, we show that the non-wandering set of each diffeormorphism in \(\cal{V}\) is a limit of horseshoes in the sense of entropy. Afterwards, we establish the existence of a C2-open set \(\cal{U}\) of C2-diffeomorphisms in \(\cal{V}\) and of a C2-residual subset ℜ of \(\cal{U}\) such that any diffeomorphism in ℜ has equal topological and periodic entropies, is asymptotic per-expansive, has a sub-exponential growth rate of the periodic orbits and admits a principal strongly faithful symbolic extension with embedding. Besides, such a diffeomorphism has a unique probability measure with maximal entropy describing the distribution of periodic orbits. Under an additional assumption, we prove that the skew-products in \(\cal{U}\) preserve a unique ergodic SRB measure, which is physical, whose basin has full Lebesgue measure and which coincides with the measure with maximal entropy.

Splitting Algorithms for Rare Events of Semimartingale Reflecting Brownian Motions

We study rare event simulations of semimartingale reflecting Brownian motions (SRBMs) in an orthant. The rare event of interest is that a d-dimensional positive recurrent SRBM enters the set [Formula: see text] before hitting a small neighborhood of the origin [Formula: see text] as [Formula: see text] with a starting point outside the two sets and of order o(n). We show that, under two regularity conditions (the Dupuis–Williams stability condition of the SRBM and the Lipschitz continuity assumption of the associated Skorokhod problem), the probability of the rare event satisfies a large deviation principle. To study the variational problem (VP) for the rare event in two dimensions, we adapt its exact solution from developed by Avram, Dai, and Hasenbein in 2001. In three and higher dimensions, we construct a novel subsolution to the VP under a further assumption that the reflection matrix of the SRBM is a nonsingular [Formula: see text]-matrix. Based on the solution/subsolution, particle-based simulation algorithms are constructed to estimate the probability of the rare event. Our estimator is asymptotically optimal for the discretized problem in two dimensions and has exponentially superior performance over standard Monte Carlo in three and higher dimensions. In addition, we establish that the growth rate of the relative bias term arising from discretization is subexponential in all dimensions. Therefore, we can estimate the probability of interest with subexponential complexity growth in two dimensions. In three and higher dimensions, the computational complexity of our estimators has a strictly smaller exponential growth rate than the standard Monte Carlo estimators.

Universal singular exponents in catalytic variable equations

Catalytic equations appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that the asymptotic estimate for the coefficients of the solutions of (so-called) positive catalytic equations has a universal asymptotic behavior. In particular, this provides a rationale why the number of maps of size n in various planar map classes grows asymptotically like c⋅n−5/2γn, for suitable positive constants c and γ. Essentially we have to distinguish between linear catalytic equations (where the subexponential growth is n−3/2) and non-linear catalytic equations (where we have n−5/2 as in planar maps). Furthermore we provide a quite general central limit theorem for parameters that can be encoded by catalytic functional equations, even when they are not positive.

Forecasting COVID-19 infections with the semi-unrestricted Generalized Growth Model

Recently, the Generalized Growth Model (GGM) has played a prominent role as an effective tool to predict the spread of pandemics exhibiting subexponential growth. A key feature of this model is a damping parameter p that is bounded to the [0,1] interval. By allowing this parameter to take negative values, we show that the GGM can also be useful to predict the spread of COVID-19 in countries that are at middle stages of the pandemic. Using both in-sample and out-of-sample evaluations, we show that a semi-unrestricted version of the model outperforms the traditional GGM in a number of countries when predicting the number of infected people at short horizons. Reductions in Root Mean Squared Prediction Errors (RMSPE) are shown to be substantial. Our results indicate that our semi-unrestricted version of the GGM should be added to the traditional set of phenomenological models used to generate forecasts during early to middle stages of epidemic outbreaks.

Inhomogeneous Transmission and Asynchronic Mixing in the Spread of COVID-19 Epidemics

The ongoing epidemic of COVID-19 first found in China has reinforced the need to develop epidemiological models capable of describing the progression of the disease to be of use in the formulation of mitigation policies. Here, this problem is addressed using a metapopulation approach to consider the inhomogeneous transmission of the spread arising from a variety of reasons, like the distribution of local epidemic onset times or of the transmission rates. We show that these contributions can be incorporated into a susceptible-infected-recovered framework through a time-dependent transmission rate. Thus, the reproduction number decreases with time despite the population dynamics remaining uniform and the depletion of susceptible individuals is small. The obtained results are consistent with the early subexponential growth observed in the cumulated number of confirmed cases even in the absence of containment measures. We validate our model by describing the evolution of COVID-19 using real data from different countries, with an emphasis in the case of Mexico, and show that it also correctly describes the longtime dynamics of the spread. The proposed model yet simple is successful at describing the onset and progression of the outbreak, and considerably improves the accuracy of predictions over traditional compartmental models. The insights given here may prove to be useful to forecast the extent of the public health risks of the epidemics, thus improving public policy-making aimed at reducing such risks.

Confined subgroups of Thompson's group $F$ and its embeddings into wobbling groups

We obtain a characterisation of confined subgroups of Thompson's group F . As a result, we deduce that the orbital graph of a point under an action of F has uniformly subexponential growth if and only if this point is fixed by the commutator subgroup. This allows us to prove non-embeddability of F into wobbling groups of graphs with uniformly subexponential growth.

Power law behaviour in the saturation regime of fatality curves of the COVID-19 pandemic

We apply a versatile growth model, whose growth rate is given by a generalised beta distribution, to describe the complex behaviour of the fatality curves of the COVID-19 disease for several countries in Europe and North America. We show that the COVID-19 epidemic curves not only may present a subexponential early growth but can also exhibit a similar subexponential (power-law) behaviour in the saturation regime. We argue that the power-law exponent of the latter regime, which measures how quickly the curve approaches the plateau, is directly related to control measures, in the sense that the less strict the control, the smaller the exponent and hence the slower the diseases progresses to its end. The power-law saturation uncovered here is an important result, because it signals to policymakers and health authorities that it is important to keep control measures for as long as possible, so as to avoid a slow, power-law ending of the disease. The slower the approach to the plateau, the longer the virus lingers on in the population, and the greater not only the final death toll but also the risk of a resurgence of infections.

Epidemiological model for the inhomogeneous spatial spreading of COVID-19 and other diseases.

We suggest a novel mathematical framework for the in-homogeneous spatial spreading of an infectious disease in human population, with particular attention to COVID-19. Common epidemiological models, e.g., the well-known susceptible-exposed-infectious-recovered (SEIR) model, implicitly assume uniform (random) encounters between the infectious and susceptible sub-populations, resulting in homogeneous spatial distributions. However, in human population, especially under different levels of mobility restrictions, this assumption is likely to fail. Splitting the geographic region under study into areal nodes, and assuming infection kinetics within nodes and between nearest-neighbor nodes, we arrive into a continuous, "reaction-diffusion", spatial model. To account for COVID-19, the model includes five different sub-populations, in which the infectious sub-population is split into pre-symptomatic and symptomatic. Our model accounts for the spreading evolution of infectious population domains from initial epicenters, leading to different regimes of sub-exponential (e.g., power-law) growth. Importantly, we also account for the variable geographic density of the population, that can strongly enhance or suppress infection spreading. For instance, we show how weakly infected regions surrounding a densely populated area can cause rapid migration of the infection towards the populated area. Predicted infection "heat-maps" show remarkable similarity to publicly available heat-maps, e.g., from South Carolina. We further demonstrate how localized lockdown/quarantine conditions can slow down the spreading of disease from epicenters. Application of our model in different countries can provide a useful predictive tool for the authorities, in particular, for planning strong lockdown measures in localized areas-such as those underway in a few countries.

The relationship between asymptotic decomposition properties

In this paper, we study the relationship between complementary-finite asymptotic dimension and transfinite asymptotic dimension. We prove that for every metric space X, coasdim(X)≤γ implies trasdim(X)≤γ for every countable ordinal number γ. Therefore, the metric space with complementary-finite asymptotic dimension has asymptotic property C. Furthermore, we construct an example of a metric space satisfying asymptotic property C and finite decomposition complexity with arbitrary high asymptotic dimension growth, which shows that asymptotic property C and finite decomposition complexity need not imply sub-exponential dimension growth in general.

Transmission dynamics and control of COVID-19 in Chile, March-October, 2020.

Since the detection of the first case of COVID-19 in Chile on March 3rd, 2020, a total of 513,188 cases, including ~14,302 deaths have been reported in Chile as of November 2nd, 2020. Here, we estimate the reproduction number throughout the epidemic in Chile and study the effectiveness of control interventions especially the effectiveness of lockdowns by conducting short-term forecasts based on the early transmission dynamics of COVID-19. Chile’s incidence curve displays early sub-exponential growth dynamics with the deceleration of growth parameter, p, estimated at 0.8 (95% CI: 0.7, 0.8) and the reproduction number, R, estimated at 1.8 (95% CI: 1.6, 1.9). Our findings indicate that the control measures at the start of the epidemic significantly slowed down the spread of the virus. However, the relaxation of restrictions and spread of the virus in low-income neighborhoods in May led to a new surge of infections, followed by the reimposition of lockdowns in Greater Santiago and other municipalities. These measures have decelerated the virus spread with R estimated at ~0.96 (95% CI: 0.95, 0.98) as of November 2nd, 2020. The early sub-exponential growth trend (p ~0.8) of the COVID-19 epidemic transformed into a linear growth trend (p ~0.5) as of July 7th, 2020, after the reimposition of lockdowns. While the broad scale social distancing interventions have slowed the virus spread, the number of new COVID-19 cases continue to accrue, underscoring the need for persistent social distancing and active case detection and isolation efforts to maintain the epidemic under control.

α1-Antitrypsin deficiency and the risk of COVID-19: an urgent call to action

α1-Antitrypsin deficiency and the risk of COVID-19: an urgent call to action

Epidemic growth and Griffiths effects on an emergent network of excited atoms

Whether it be physical, biological or social processes, complex systems exhibit dynamics that are exceedingly difficult to understand or predict from underlying principles. Here we report a striking correspondence between the excitation dynamics of a laser driven gas of Rydberg atoms and the spreading of diseases, which in turn opens up a controllable platform for studying non-equilibrium dynamics on complex networks. The competition between facilitated excitation and spontaneous decay results in sub-exponential growth of the excitation number, which is empirically observed in real epidemics. Based on this we develop a quantitative microscopic susceptible-infected-susceptible model which links the growth and final excitation density to the dynamics of an emergent heterogeneous network and rare active region effects associated to an extended Griffiths phase. This provides physical insights into the nature of non-equilibrium criticality in driven many-body systems and the mechanisms leading to non-universal power-laws in the dynamics of complex systems.

Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra

We analyze properties of semigroups generated by Schrödinger operators Δ−V or polyharmonic operators −(−Δ)m, on metric graphs both on Lp-spaces and spaces of continuous functions. In the case of spatially constant potentials, we provide a semi-explicit formula for their kernel. Under an additional sub-exponential growth condition on the graph, we prove analyticity, ultracontractivity, and pointwise kernel estimates for these semigroups; we also show that their generators' spectra coincide on all relevant function spaces and present a Kreĭn-type dimension reduction, showing that their spectral values are determined by the spectra of generalized discrete Laplacians acting on various spaces of functions supported on combinatorial graphs.

Early Phasic Containment of COVID-19 in Substantially Affected States of India

Introduction: India is experiencing the global COVID-19 pandemic caused with the infection of SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2). To explore the early epidemic course and the effectiveness of lockdowns on COVID-19 pandemic in some worst-affected Indian states. Methods: Using publicly available real data and model-based prediction, the growth rate, case fatality rate, serial interval, and time-varying reproduction number (R) of COVID-19 were estimated, before and after lockdown implementation in India. Results: The spread of COVID-19 epidemic in some highly-impacted Indian states displayed a characteristic sub-exponential growth projected up to 3 May 2020, as a consequence of lockdown strategies, in addition to improvement of reproduction number (R), serial interval, and daily growth rate, but not case fatality rate (CFR). The effect of COVID-19 containment was more prominent in second phase of lockdown with declining R, which was still >1. Conclusion: The current findings suggest the requirement of sustained interventions for effective containment of COVID-19 pandemic in Indian context. Keywords: COVID-19, SARS-CoV-2, Indian states, epidemiological parameters, lockdown effect.

Understanding transmission and intervention for the COVID-19 pandemic in the United States

The outbreak of the novel coronavirus disease (COVID-19) severely threatens the public health worldwide, but the transmission mechanism and the effectiveness of mitigation measures remain uncertain. Here we assess the role of airborne transmission in spreading the disease and the effectiveness of face covering in preventing inter-human transmission for the top-fifteen infected U.S. states during March 1 and May 18, 2020. For all fifteen states, the curve of total confirmed infections exhibits an initial sub-exponential growth and a subsequent linear growth after implementing social distancing/stay-at-home orders. The linearity extends one to two months for the six states without mandated face covering and to the onset of mandated face covering for the other nine states with this measure, reflecting a dynamic equilibrium between first-order transmission kinetics and intervention. For the states with mandated face covering, significant deviation from this linearity and curve flattening occur after the onset of this measure for seven states, with exceptions for two states. Most states exhibit persistent upward trends in the daily new infections after social distancing/stay-at-home orders, while reversed downward or slowing trends occur for eight states after implementing mandated face covering. The inadequacy of social distancing and stay-at-home measures alone in preventing inter-human transmission is reflected by the continuous linear growth in the total infection curve after implementing these measures, which is mainly driven by airborne transmission. We estimate that the number of the total infections prevented by face covering reaches ~252,000 on May 18 in seven states, which is equivalent to ~17% of the total infections in the nation. We conclude that airborne transmission and face covering play the dominant role in spreading the disease and flattening the total infection curve, respectively. Our findings provide policymakers and the public with compelling evidence that universal face covering, in conjunction with social distancing and hand hygiene, represents the maximal protection against inter-human transmission and the combination of these intervention measures with rapid and extensive testing as well as contact tracing is crucial in containing the COVID-19 pandemic.

Bilinear Pseudo-differential Operators with Gevrey–Hörmander Symbols

We consider bilinear pseudo-differential operators whose symbols may have a sub-exponential growth at infinity, together with all their derivatives. It is proved that those symbol classes can be described by the means of the short-time Fourier transform and modulation spaces. Our first main result is the invariance property of the corresponding bilinear operators. Furthermore, we prove the continuity of such operators when acting on modulation spaces. As a consequence, we derive their continuity on anisotropic Gelfand–Shilov type spaces.