Matheus Hansen

The pursuit of strategies that minimize the number of individuals needing vaccination to control an outbreak is a well-established area in mathematical epidemiology. However, for certain diseases, public policy tends to prioritize immunizing vulnerable individuals over epidemic control. As a result, optimal vaccination strategies may not always be effective in supporting real-world public policies. A similar situation happens when a new vaccine is introduced and is in short supply, as target priority groups for vaccination have to be defined. In this work, we focus on a disease that results in long-term immunity and spreads through a heterogeneous population, represented by a contact network. We study four well-known group centrality measures and show that the GED-Walk offers a reliable means of estimating the impact of vaccinating specific groups of individuals, even in suboptimal cases. Additionally, we depart from the search for target individuals to be vaccinated and provide proxies for identifying optimal groups. While the GED-Walk is the most useful centrality measure for suboptimal cases, the betweenness (a related, but different centrality measure) stands out when looking for optimal groups. This indicates that optimal vaccination is not concerned with breaking the largest number of transmission routes, but interrupting geodesic ones.

In this study, we identify an intricate class of periodic behavior, coined as "fishbone," within the chaotic dynamics induced by external periodic disturbances in a neuronal system. Alongside this structure, we observe well-known patterns such as shrimps and cockroaches, revealing the system's sensitivity to external perturbations. Notably, our work highlights the significant impact of altering biophysical parameters, showing that changes in the ionic conductance of neurons can lead to the cohesion or detachment of complex periodic structures from the primary fishbone-shaped structure. This finding underscores the role of ionic channels in shaping neuronal responses, which is relevant given their involvement in various neuronal disorders. Furthermore, our study provides insights into how external perturbations influence neuron signal processing.

The study of billiards investigates the trajectories of particles that move freely in a region and reflect elastically at boundaries. Although there is already considerable understanding about invariant spanning curves, also known as whispering gallery orbits in the context of billiards, their determination in the phase space of the system, in addition to the analysis of their existence is still an open question. Our proposal is to present a numerical method based on Slater’s theorem, capable of determining the location of these curves in phase space, as well as finding the critical parameter at which these curves are no longer observed. In this work, we apply this method to determine the location of a set of invariant spanning curves in an oval billiard for different parameter values. Furthermore, we identified the critical parameter at which the phase space no longer presents these curves and local chaos becomes global. We compared our numerical results with analytical results present in the literature, proving the effectiveness of the proposed method. By studying the rotation number, we obtain additional information about the behavior of these curves and also of the systems.

This paper presents a recursive method for identifying extreme and superstable curves in the parameter space of dissipative one-dimensional maps. The method begins by constructing an Archimedean spiral with a constant arc length. Subsequently, it identifies extreme and superstable curves by calculating an observable ψ. The spiral is used to locate a region where ψ changes sign. When this occurs, a bisection method is applied to determine the first point on the desired superstable or extreme curve. Once the initial direction is established, the recursive method identifies subsequent points using an additional bisection method, iterating the process until the stopping conditions are met. The logistic-Gauss map demonstrates each step of the method, as it exhibits a wide variety of periodicity structures in the parameter space, including cyclic extreme and superstable curves, which contribute to the formation of period-adding structures. Examples of extreme and superstable curves obtained by the recursive method are presented. It is important to note that the proposed method is generalizable and can be adapted to any one-dimensional map.

This study focuses on the analysis of a unique composition between two well-established models, known as the Logistic–Gauss map. The investigation cohesively transitions to an exploration of parameter space, essential for unraveling the complexity of dissipative mappings and understanding the intricate relationships between periodic structures and chaotic regions. By manipulating control parameters, our approach reveals intriguing patterns, with findings enriched by extreme orbits, trajectories that connect local maximum and minimum values of one-dimensional maps. This theory enhances our perception of structural organization and offers valuable perceptions of the system behaviors, contributing to an expanded understanding of chaos and periodicity in dynamic systems. The analysis reveals Complex Sets of Periodicity (CSP) in the parameter space, characterized by superstable curves that traverse their main bodies. The exploration of different combinations of parameters shows cascades of CSP structures with added periods and are organized based on extreme curves. This investigation offers valuable discoveries of the dynamics of dissipative mappings, opening avenues for future explorations in chaotic systems.

Nearly half of the bee species can perform a fascinating stereotyped behavior to collect pollen grains by vibrating flowers, known as buzz pollination. During the floral visit, these bees mechanically transfer the vibrations produced by their thoracic indirect flight muscles to the flower anther, inducing the movement of the pollen grains and leading them to be released through a small pore or slit placed at the tip of the anther in poricidal flowers. In such flowers, pollen release is affected by the vibrational behavior of buzzing bees, primarily their duration and velocity amplitude. However, we know little about how poricidal anther morphology may influence it. In this work, we investigated through a theoretical and numerical point of view the buzz pollination process considering a typical poricidal anther of a tomato flower (Solanum lycopersicum), which in our work will be approached by a rectangular billiard, experiencing vibrations applied by a bumblebee (Bombus terrestris). Our primary goals in this paper are (i) to understand the mechanism behind the pollen release in this model, (ii) to observe some scale effects associated with morphological variations of the anther (as pore size and anther shape), and (iii) analyze how these results are related to natural buzz pollination systems.

This article investigates the emergence of phase synchronization in a network of randomly connected neurons by chemical synapses. The study uses the classic Hodgkin–Huxley model to simulate the neuronal dynamics under the action of a train of Poissonian spikes. In such a scenario, we observed the emergence of irregular spikes for a specific range of conductances and also that the phase synchronization of the neurons is reached when the external current is strong enough to induce spiking activity but without overcoming the coupling current. Conversely, if the external current assumes very high values, then an opposite effect is observed, i.e., the prevention of the network synchronization. We explain such behaviors considering different mechanisms involved in the system, such as incoherence, minimization of currents, and stochastic effects from the Poissonian spikes. Furthermore, we present some numerical simulations where the stimulation of only a fraction of neurons, for instance, can induce phase synchronization in the non-stimulated fraction of the network, besides cases in which for larger coupling values, it is possible to propagate the spiking activity in the network when considering stimulation over only one neuron.

In this paper, we consider the evolution of an ensemble of noninteracting classical particles confined in a closed region. Each one of these particles experiences several elastic collisions with a rigid-smooth oval-like boundary. As known in the literature, numerical simulations for this kind of system demand too much time, mostly because of the time spent in solving transcendental equations. To deal with that, we highlight an alternative approach to speed up the search of solutions for the transcendental equations via the tangent method, which has proved to be a faster way to solve this problem when compared with other methods, like the bisection one. The methods applied here are generic and can be used or adapted for other billiard boundaries.

Buzz pollination is described using a mathematical model considering a billiard approach. Applications to a rough morphology of a typical poricidal anther of a tomato flower (Solanum lycopersicum) experiencing vibrations applied by a bumblebee (Bombus terrestris) are made. The anther is described by a rectangular billiard with a pore on its tip while the borders are perturbed by specific oscillations according to the vibrational properties of the bumblebee. Pollen grains are considered as noninteracting particles that can escape through the pore. Our results not only recover some observed data but also provide a possible answer to an open problem involving buzz pollination.