Impact of periodic vaccination in SEIRS seasonal model.

We study three different strategies of vaccination in an SEIRS (Susceptible-Exposed-Infected-Recovered-Susceptible) seasonal forced model, which are (i) continuous vaccination; (ii) periodic short-time localized vaccination, and (iii) periodic pulsed width campaign. Considering the first strategy, we obtain an expression for the basic reproduction number and infer a minimum vaccination rate necessary to ensure the stability of the disease-free equilibrium (DFE) solution. In the second strategy, short duration pulses are added to a constant baseline vaccination rate. The pulse is applied according to the seasonal forcing phases. The best outcome is obtained by locating intensive immunization at inflection of the transmissivity curve. Therefore, a vaccination rate of 44.4% of susceptible individuals is enough to ensure DFE. For the third vaccination proposal, additionally to the amplitude, the pulses have a prolonged time width. We obtain a non-linear relationship between vaccination rates and the duration of the campaign. Our simulations show that the baseline rates, as well as the pulse duration, can substantially improve the vaccination campaign effectiveness. These findings are in agreement with our analytical expression. We show a relationship between the vaccination parameters and the accumulated number of infected individuals, over the years, and show the relevance of the immunization campaign annual reaching for controlling the infection spreading. Regarding the dynamical behavior of the model, our simulations show that chaotic and periodic solutions as well as bi-stable regions depend on the vaccination parameters range.

Multistability and chaos in SEIRS epidemic model with a periodic time-dependent transmission rate.

In this work, we study the dynamics of a susceptible-exposed-infectious-recovered-susceptible epidemic model with a periodic time-dependent transmission rate. Emphasizing the influence of the seasonality frequency on the system dynamics, we analyze the largest Lyapunov exponent along parameter planes finding large chaotic regions. Furthermore, in some ranges, there are shrimp-like periodic structures. We highlight the system multistability, identifying the coexistence of periodic orbits for the same parameter values, with the infections maximum distinguishing by up one order of magnitude, depending only on the initial conditions. In this case, the basins of attraction have self-similarity. Parametric configurations, for which both periodic and non-periodic orbits occur, cover 13.20% of the evaluated range. We also identified the coexistence of periodic and chaotic attractors with different maxima of infectious cases, where the periodic scenario peak reaches approximately 50% higher than the chaotic one.

Nontwist field line mapping in a tokamak with ergodic magnetic limiter.

For tokamaks with uniform magnetic shear, Martin and Taylor have proposed a symplectic map which has been used to describe the magnetic field lines at the plasma edge perturbed by an ergodic magnetic limiter. We propose an analytical magnetic field line map, based on the Martin-Taylor map, for a tokamak with arbitrary safety factor profile. With the inclusion of a nonmonotonic profile, we obtain a nontwist map which presents the characteristic properties of degenerate systems, such as the twin islands scenario, shearless curve, and separatrix reconnection. We estimate the width of the islands and describe their changes of shape for large values of the limiter current. From our numerical simulations about the shearless curve, we show that its position and aspect depend on the control parameters.

The Roles of Potassium and Calcium Currents in the Bistable Firing Transition.

Healthy brains display a wide range of firing patterns, from synchronized oscillations during slow-wave sleep to desynchronized firing during movement. These physiological activities coexist with periods of pathological hyperactivity in the epileptic brain, where neurons can fire in synchronized bursts. Most cortical neurons are pyramidal regular spiking (RS) cells with frequency adaptation and do not exhibit bursts in current-clamp experiments (in vitro). In this work, we investigate the transition mechanism of spike-to-burst patterns due to slow potassium and calcium currents, considering a conductance-based model of a cortical RS cell. The joint influence of potassium and calcium ion channels on high synchronous patterns is investigated for different synaptic couplings (gsyn) and external current inputs (I). Our results suggest that slow potassium currents play an important role in the emergence of high-synchronous activities, as well as in the spike-to-burst firing pattern transitions. This transition is related to the bistable dynamics of the neuronal network, where physiological asynchronous states coexist with pathological burst synchronization. The hysteresis curve of the coefficient of variation of the inter-spike interval demonstrates that a burst can be initiated by firing states with neuronal synchronization. Furthermore, we notice that high-threshold (IL) and low-threshold (IT) ion channels play a role in increasing and decreasing the parameter conditions (gsyn and I) in which bistable dynamics occur, respectively. For high values of IL conductance, a synchronous burst appears when neurons are weakly coupled and receive more external input. On the other hand, when the conductance IT increases, higher coupling and lower I are necessary to produce burst synchronization. In light of our results, we suggest that channel subtype-specific pharmacological interactions can be useful to induce transitions from pathological high bursting states to healthy states.

The Role of Potassium and Calcium Currents in the Bistable Firing Transition.

Healthy brains display a wide range of firing patterns, from synchronized oscillations during slowwave sleep to desynchronized firing during movement. These physiological activities coexist with periods of pathological hyperactivity in the epileptic brain, where neurons can fire in synchronized bursts. Most cortical neurons are pyramidal regular spiking cells (RS) with frequency adaptation and do not exhibit bursts in current-clamp experiments ( in vitro ). In this work, we investigate the transition mechanism of spike-to-burst patterns due to slow potassium and calcium currents, considering a conductance-based model of a cortical RS cell. The joint influence of potassium and calcium ion channels on high synchronous patterns is investigated for different synaptic couplings ( g syn ) and external current inputs ( I ). Our results suggest that slow potassium currents play an important role in the emergence of high-synchronous activities, as well as in the spike-to-burst firing pattern transitions. This transition is related to bistable dynamics of the neuronal network, where physiological asynchronous states coexist with pathological burst synchronization. The hysteresis curve of the coefficient of variation of the inter-spike interval demonstrates that a burst can be initiated by firing states with neuronal synchronization. Furthermore, we notice that high-threshold ( I L ) and low-threshold ( I T ) ion channels play a role in increasing and decreasing the parameter conditions ( g syn and I ) in which bistable dynamics occur, respectively. For high values of I L conductance, a synchronous burst appears when neurons are weakly coupled and receive more external input. On the other hand, when the conductance I T increases, higher coupling and lower I are necessary to produce burst synchronization. In light of our results, we suggest that channel subtype-specific pharmacological interactions can be useful to induce transitions from pathological high bursting states to healthy states.

Analyzing bursting synchronization in structural connectivity matrix of a human brain under external pulsed currents.

Cognitive tasks in the human brain are performed by various cortical areas located in the cerebral cortex. The cerebral cortex is separated into different areas in the right and left hemispheres. We consider one human cerebral cortex according to a network composed of coupled subnetworks with small-world properties. We study the burst synchronization and desynchronization in a human neuronal network under external periodic and random pulsed currents. With and without external perturbations, the emergence of bursting synchronization is observed. Synchronization can contribute to the processing of information, however, there are evidences that it can be related to some neurological disorders. Our results show that synchronous behavior can be suppressed by means of external pulsed currents.

Stickiness and recurrence plots: An entropy-based approach.

The stickiness effect is a fundamental feature of quasi-integrable Hamiltonian systems. We propose the use of an entropy-based measure of the recurrence plots (RPs), namely, the entropy of the distribution of the recurrence times (estimated from the RP), to characterize the dynamics of a typical quasi-integrable Hamiltonian system with coexisting regular and chaotic regions. We show that the recurrence time entropy (RTE) is positively correlated to the largest Lyapunov exponent, with a high correlation coefficient. We obtain a multi-modal distribution of the finite-time RTE and find that each mode corresponds to the motion around islands of different hierarchical levels.

Curry-Yorke route to shearless attractors and coexistence of attractors in dissipative nontwist systems.

The routes to chaos play an important role in predictions about the transitions from regular to irregular behavior in nonlinear dynamical systems, such as electrical oscillators, chemical reactions, biomedical rhythms, and nonlinear wave coupling. Of special interest are dissipative systems obtained by adding a dissipation term in a given Hamiltonian system. If the latter satisfies the so-called twist property, the corresponding dissipative version can be called a "dissipative twist system." Transitions to chaos in these systems are well established; for instance, the Curry-Yorke route describes the transition from a quasiperiodic attractor on torus to chaos passing by a chaotic banded attractor. In this paper, we study the transitions from an attractor on torus to chaotic motion in dissipative nontwist systems. We choose the dissipative standard nontwist map, which is a non-conservative version of the standard nontwist map. In our simulations, we observe the same transition to chaos that happens in twist systems, known as a soft one, where the quasiperiodic attractor becomes wrinkled and then chaotic through the Curry-Yorke route. By the Lyapunov exponent, we study the nature of the orbits for a different set of parameters, and we observe that quasiperiodic motion and periodic and chaotic behavior are possible in the system. We observe that they can coexist in the phase space, implying in multistability. The different coexistence scenarios were studied by the basin entropy and by the boundary basin entropy.

Dynamics of epidemics: Impact of easing restrictions and control of infection spread.

During an infectious disease outbreak, mathematical models and computational simulations are essential tools to characterize the epidemic dynamics and aid in design public health policies. Using these tools, we provide an overview of the possible scenarios for the COVID-19 pandemic in the phase of easing restrictions used to reopen the economy and society. To investigate the dynamics of this outbreak, we consider a deterministic compartmental model (SEIR model) with an additional parameter to simulate the restrictions. In general, as a consequence of easing restrictions, we obtain scenarios characterized by high spikes of infections indicating significant acceleration of the spreading disease. Finally, we show how such undesirable scenarios could be avoided by a control strategy of successive partial easing restrictions, namely, we tailor a successive sequence of the additional parameter to prevent spikes in phases of low rate of transmissibility.

Ratchet current in nontwist Hamiltonian systems.

Non-monotonic area-preserving maps violate the twist condition locally in phase space, giving rise to shearless invariant barriers surrounded by twin island chains in these regions of phase space. For the extended standard nontwist map, with two resonant perturbations with distinct wave numbers, we investigate the presence of such barriers and their associated island chains and compare our results with those that have been reported for the standard nontwist map with only one perturbation. Furthermore, we determine in the control parameter space the existence of the shearless barrier and the influence of the additional wave number on this condition. We show that only for odd second wave numbers are the twin island chains symmetrical. Moreover, for even wave numbers, the lack of symmetry between the chains of twin islands generates a ratchet effect that implies a directed transport in the phase space.

Influence of Delayed Conductance on Neuronal Synchronization.

In the brain, the excitation-inhibition balance prevents abnormal synchronous behavior. However, known synaptic conductance intensity can be insufficient to account for the undesired synchronization. Due to this fact, we consider time delay in excitatory and inhibitory conductances and study its effect on the neuronal synchronization. In this work, we build a neuronal network composed of adaptive integrate-and-fire neurons coupled by means of delayed conductances. We observe that the time delay in the excitatory and inhibitory conductivities can alter both the state of the collective behavior (synchronous or desynchronous) and its type (spike or burst). For the weak coupling regime, we find that synchronization appears associated with neurons behaving with extremes highest and lowest mean firing frequency, in contrast to when desynchronization is present when neurons do not exhibit extreme values for the firing frequency. Synchronization can also be characterized by neurons presenting either the highest or the lowest levels in the mean synaptic current. For the strong coupling, synchronous burst activities can occur for delays in the inhibitory conductivity. For approximately equal-length delays in the excitatory and inhibitory conductances, desynchronous spikes activities are identified for both weak and strong coupling regimes. Therefore, our results show that not only the conductance intensity, but also short delays in the inhibitory conductance are relevant to avoid abnormal neuronal synchronization.

Spike-burst chimera states in an adaptive exponential integrate-and-fire neuronal network.

Chimera states are spatiotemporal patterns in which coherence and incoherence coexist. We observe the coexistence of synchronous (coherent) and desynchronous (incoherent) domains in a neuronal network. The network is composed of coupled adaptive exponential integrate-and-fire neurons that are connected by means of chemical synapses. In our neuronal network, the chimera states exhibit spatial structures both with spike and burst activities. Furthermore, those desynchronized domains not only have either spike or burst activity, but we show that the structures switch between spikes and bursts as the time evolves. Moreover, we verify the existence of multicluster chimera states.

Using rotation number to detect sticky orbits in Hamiltonian systems.

In Hamiltonian systems, depending on the control parameter, orbits can stay for very long times around islands, the so-called stickiness effect caused by a temporary trapping mechanism. Different methods have been used to identify sticky orbits, such as recurrence analysis, recurrence time statistics, and finite-time Lyapunov exponent. However, these methods require a large number of map iterations and to know the island positions in the phase space. Here, we show how to use the small divergence of bursts in the rotation number calculation as a tool to identify stickiness without knowing the island positions. This new procedure is applied to the standard map, a map that has been used to describe the dynamic behavior of several nonlinear systems. Moreover, our procedure uses a small number of map iterations and is proper to identify the presence of stickiness phenomenon for different values of the control parameter.

Bistable Firing Pattern in a Neural Network Model.

Excessively high, neural synchronization has been associated with epileptic seizures, one of the most common brain diseases worldwide. A better understanding of neural synchronization mechanisms can thus help control or even treat epilepsy. In this paper, we study neural synchronization in a random network where nodes are neurons with excitatory and inhibitory synapses, and neural activity for each node is provided by the adaptive exponential integrate-and-fire model. In this framework, we verify that the decrease in the influence of inhibition can generate synchronization originating from a pattern of desynchronized spikes. The transition from desynchronous spikes to synchronous bursts of activity, induced by varying the synaptic coupling, emerges in a hysteresis loop due to bistability where abnormal (excessively high synchronous) regimes exist. We verify that, for parameters in the bistability regime, a square current pulse can trigger excessively high (abnormal) synchronization, a process that can reproduce features of epileptic seizures. Then, we show that it is possible to suppress such abnormal synchronization by applying a small-amplitude external current on > 10% of the neurons in the network. Our results demonstrate that external electrical stimulation not only can trigger synchronous behavior, but more importantly, it can be used as a means to reduce abnormal synchronization and thus, control or treat effectively epileptic seizures.

Recurrence-based analysis of barrier breakup in the standard nontwist map.

We study the standard nontwist map that describes the dynamic behaviour of magnetic field lines near a local minimum or maximum of frequency. The standard nontwist map has a shearless invariant curve that acts like a barrier in phase space. Critical parameters for the breakup of the shearless curve have been determined by procedures based on the indicator points and bifurcations of periodical orbits, a methodology that demands high computational cost. To determine the breakup critical parameters, we propose a new simpler and general procedure based on the determinism analysis performed on the recurrence plot of orbits near the critical transition. We also show that the coexistence of islands and chaotic sea in phase space can be analysed by using the recurrence plot. In particular, the measurement of determinism from the recurrence plot provides us with a simple procedure to distinguish periodic from chaotic structures in the parameter space. We identify an invariant shearless breakup scenario, and we also show that recurrence plots are useful tools to determine the presence of periodic orbit collisions and bifurcation curves.