Pedro Gajardo

The present paper proposes a novel methodology for evaluating the impact of a vaccination plan against a transmissible disease. The methodology has two distinct stages. The initial stage comprises a compartmental model that describes the transmission of the disease within the population. This model is composed of fundamental parts representing the vaccination status and is used in this initial stage to estimate the number of cases and deaths averted by the vaccination plan. The case dynamics generated by the compartmental model serve as the input for a data-driven model in the second part of the methodology. The second model is statistical in nature and provides additional information regarding the number of hospitalizations and ICU admissions averted. This methodology is applied to assess the impact of booster SARS-CoV-2 vaccines in Chile, resulting in an estimation of 84 % (95 % confidence interval (CI): 74 %–92 %) of cases, 73 % (95 % CI: 57 %, 80 %) of hospital admissions, 77 % (95 % CI: 61 %, 84 %) of intensive care unit admissions, and 78 % (95 % CI: 62 %–85 %) of deaths averted between August 16 and December 31, 2021.

Let K K be a closed convex cone in the state space R n \mathbb {R}^n . This note characterizes the K K -monotonicity of a non-autonomous dynamical system x ˙ ( t ) = f ( t , x ( t ) ) \dot x (t) = f(t,x(t)) governed by a locally Lipschitz velocity field. We deviate from the classical literature in two important ways. Firstly, the velocity field f f is not required to be differentiable with respect to the state variables. And, secondly, the closed convex cone K K is allowed to be absolutely general. In particular, we impose neither pointedness, nor solidity.

In natural resource management, or more generally in the study of sustainability issues, the objective often consists of maintaining the state of a given system within a desirable configuration, typically established in terms of standards or thresholds. For instance, in fisheries management, the procedure for designing policies may include maintaining the spawning stock biomass over a precautionary threshold and ensuring minimal catches. With the evolution of some natural resources, under the action of controls and uncertainties, being represented by a dynamical system in discrete time, the aim of this paper is to characterize the set of robust sustainable thresholds. That is, the thresholds for which there exists a trajectory satisfying, for all possible uncertainty scenarios, prescribed constraints parametrized by such thresholds. This set provides useful information to users and decision‐makers, illustrating the tradeoffs between constraints. Using optimal control, maximin and level‐set approaches, we characterize the weak Pareto front of the set of robust sustainable thresholds and derive a numerical method for computing the entire set, as we show with a numerical example relying on renewable resource management. Recommandation for Resource Managers For biological, ecological or social systems, identifying robust sustainable thresholds that it is possible not to exceed over time regardless of uncertain future scenarios, is a fundamental challenge to operate in a sustainable way. The set of robust sustainable thresholds provides useful information to users and decision‐makers, illustrating the tradeoffs between constraints. The computation of the weak‐Pareto front of the set of sustainable thresholds provides a tool for the management and visualization of multiple objectives related to sustainability that can be accomplished in a robust manner, allowing to observe the (in)compatibility of multiple objectives and their tradeoffs.

Nonpharmaceutical interventions (NPI) such as banning public events or instituting lockdowns have been widely applied around the world to control the current COVID-19 pandemic. Typically, this type of intervention is imposed when an epidemiological indicator in a given population exceeds a certain threshold. Then, the nonpharmaceutical intervention is lifted when the levels of the indicator used have decreased sufficiently. What is the best indicator to use? In this paper, we propose a mathematical framework to try to answer this question. More specifically, the proposed framework permits to assess and compare different event-triggered controls based on epidemiological indicators. Our methodology consists of considering some outcomes that are consequences of the nonpharmaceutical interventions that a decision maker aims to make as low as possible. The peak demand for intensive care units (ICU) and the total number of days in lockdown are examples of such outcomes. If an epidemiological indicator is used to trigger the interventions, there is naturally a trade-off between the outcomes that can be seen as a curve parameterized by the trigger threshold to be used. The computation of these curves for a group of indicators then allows the selection of the best indicator the curve of which dominates the curves of the other indicators. This methodology is illustrated with indicators in the context of COVID-19 using deterministic compartmental models in discrete-time, although the framework can be adapted for a larger class of models.

In this work, we develop an inmate population model with a sentencing length structure. The sentence length structure of new inmates represents the problem data and can usually be estimated from the histograms corresponding to the conviction times that are sentenced in a given population. We obtain a transport equation, typically known as the McKendrick equation, the homogenous version of which is included in population models with age structures. Using this equation, we compute the inmate population and entry/exit rates in equilibrium, which are the values to consider in the design of a penitentiary system. With data from the Chilean penitentiary system, we illustrate how to perform these computations. In classifying the inmate population into two groups of sentence lengths (short and long), we incorporate the SIS (susceptible-infected-susceptible) epidemiological model, which considers the entry of infective individuals. We show that a failure to consider the structure of the sentence lengths -- as is common in epidemiological models developed for inmate populations -- for prevalences of new inmates below a certain threshold induces an underestimation of the prevalence in the prison population at steady state. The threshold depends on the basic reproduction number associated with the nonstructured SIS model with no entry of new inmates. We illustrate our findings with analytical and numerical examples for different distributions of sentencing lengths.

In this paper, we study an optimal control problem of a communicable disease in a prison population. In order to control the spread of the disease inside a prison, we consider an active case-finding strategy, consisting on screening a proportion of new inmates at the entry point, followed by a treatment depending on the results of this procedure. The control variable consists then in the coverage of the screening applied to new inmates. The disease dynamics is modeled by a SIS (susceptible-infected-susceptible) model, typically used to represent diseases that do not confer immunity after infection. We determine the optimal strategy that minimizes a combination between the cost of the screening/treatment at the entrance and the cost of maintaining infected individuals inside the prison, in a given time horizon. Using the Pontryagin Maximum Principle and Hamilton-Jacobi-Bellman equation, among other tools, we provide the complete synthesis of an optimal feedback control, consisting in a bang-bang strategy with at most two switching times and no singular arc trajectory, characterizing different profiles depending on model parameters.

Nonpharmaceutical interventions (NPI) such as banning public events or instituting lockdowns have been widely applied around the world to control the current COVID-19 pandemic. Typically, this type of intervention is imposed when an epidemiological indicator in a given population exceeds a certain threshold. Then, the nonpharmaceutical intervention is lifted when the levels of the indicator used have decreased sufficiently. What is the best indicator to use? In this paper, we propose a mathematical framework to try to answer this question. More specifically, the proposed framework permits to assess and compare different event-triggered controls based on epidemiological indicators. Our methodology consists of considering some outcomes that are consequences of the nonpharmaceutical interventions that a decision maker aims to make as low as possible. The peak demand for intensive care units (ICU) and the total number of days in lockdown are examples of such outcomes. If an epidemiological indicator is used to trigger the interventions, there is naturally a trade-off between the outcomes that can be seen as a curve parameterized by the trigger threshold to be used. The computation of these curves for a group of indicators then allows the selection of the best indicator the curve of which dominates the curves of the other indicators. This methodology is illustrated using indicators in the context of COVID-19 using deterministic compartmental models in discrete-time, although the framework can be adapted for a larger class of models.

The present paper is a continuation of our previous article: Various Lipschitz-like properties of functions and sets. I: Directional derivative and tangential characterizations [SIAM J. Optim. 20(4) (2010) 1766–1785]. Here we provide diverse subdifferential and normal characterizations of K-directionally Lipschitzian functions and sets for bounded sets K of a Banach space.