We provide a necessary and sufficient condition for the metastability of a Markov chain, expressed in terms of a property of the solutions of the resolvent equation. As an application of this result, we prove the metastability of reversible, critical zero-range processes starting from a configuration.

Sortition, i.e. random appointment for public duty, has been employed by societies throughout the years as a firewall designated to prevent illegitimate interference between parties in a legal case and agents of the legal system. In judicial systems of modern western countries, random procedures are mainly employed to select the jury, the court and/or the judge in charge of judging a legal case. Therefore, these random procedures play an important role in the course of a case, and should comply with some principles, such as transparency and complete auditability. Nevertheless, these principles are neglected by random procedures in some judicial systems, which are performed in secrecy and are not auditable by the involved parties. The assignment of cases in the Brazilian Supreme Court is an example of such a procedure, for it is performed using procedures unknown to the parties involved in the judicial cases. This article presents a review of how sortition has been historically employed by societies and discusses how Mathematical Statistics may be applied to random procedures of the judicial system, as it has been applied for almost a century on clinical trials, for example. A statistical model for assessing randomness in case assignment is proposed and applied to the Brazilian Supreme Court. As final remarks, guidelines for the development of good randomization procedures are outlined.

Mobile geolocation data is a valuable asset in the assessment of movement patterns of a population. Once a highly contagious disease takes place in a location the movement patterns aid in predicting the potential spatial spreading of the disease, hence mobile data becomes a crucial tool to epidemic models. In this work, based on millions of anonymized mobile visits data in Brazil, we investigate the most probable spreading patterns of the COVID-19 within states of Brazil. The study is intended to help public administrators in action plans and resources allocation, whilst studying how mobile geolocation data may be employed as a measure of population mobility during an epidemic. The first part of the study focus on the states of São Paulo and Rio de Janeiro during the period of March 2020, when the disease first started to spread in these states. Metapopulation models for the disease spread were simulated in order to evaluate the risk of infection of each city within the states, by ranking them according to the time the disease will take to infect each city. We observed that, although the high risk regions are those closer to the capital cities, where the outbreak has started, there are also cities in the countryside with great risk.

In response to the coronavirus pandemic, governments implemented social distancing, attempting to block the virus spread within territories. While it is well accepted that social isolation plays a role in epidemic control, the precise connections between mobility data indicators and epidemic dynamics are still a challenge. In this work, we investigate the dependency between a social isolation index and epidemiological metrics for several Brazilian cities. Classic statistical methods are employed to support the findings. As a first, initially surprising, result, we illustrate how there seems to be no apparent functional relationship between social isolation data and later effects on disease incidence. However, further investigations identified two regimes of successful employment of social isolation: as a preventive measure or as a remedy, albeit remedy measures require greater social isolation and bring higher burden to health systems. Additionally, we exhibit cases of successful strategies involving lockdowns and an indicator-based mobility restriction plan.