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In this work, effects of constant and time-dependent vaccination rates on the Susceptible-Exposed-Infected-Recovered-Susceptible (SEIRS) seasonal model are studied. Computing the Lyapunov exponent, we show that typical complex structures, such as shrimps, emerge for given combinations of constant vaccination rate and another model parameter. In some specific cases, the constant vaccination does not act as a chaotic suppressor and chaotic bands can exist for high levels of vaccination (e.g., $> 0.95$). Moreover, we obtain linear and non-linear relationships between one control parameter and constant vaccination to establish a disease-free solution. We also verify that the total infected number does not change whether the dynamics is chaotic or periodic. The introduction of a time-dependent vaccine is made by the inclusion of a periodic function with a defined amplitude and frequency. For this case, we investigate the effects of different amplitudes and frequencies on chaotic attractors, yielding low, medium, and high seasonality degrees of contacts. Depending on the parameters of the time-dependent vaccination function, chaotic structures can be controlled and become periodic structures. For a given set of parameters, these structures are accessed mostly via crisis and in some cases via period-doubling. After that, we investigate how the time-dependent vaccine acts in bi-stable dynamics when chaotic and periodic attractors coexist. We identify that this kind of vaccination acts as a control by destroying almost all the periodic basins. We explain this by the fact that chaotic attractors exhibit more desirable characteristics for epidemics than periodic ones in a bi-stable state.

Dengue is a viral vector-borne infectious disease that affects many countries worldwide, infecting around 390 million people per year. The main outbreaks occur in subtropical and tropical countries. We study here the influence of climate on dengue in Natal (2016-2019), Brazil, Iquitos (2001-2012), Peru, and Barranquilla (2011-2016), Colombia. For the analysis and simulations, we apply Machine Learning (ML) techniques, especially the Random Forest (RF) algorithm. In addition, regarding a feature in the ML technique, we analyze three possibilities: only dengue cases (D); climate and dengue cases (CD); humidity and dengue cases (HD). Depending on the city, our results show that the climate data can improve or not the forecast. For instance, for Natal, D induces a better forecast. For Iquitos, it is better to use CD. Nonetheless, for Barranquilla, the forecast is better, when we include cases and humidity data. For Natal, when we use more than 64\% and less than 80\% of the time series for training, we obtain results with correlation coefficients ($r$) among 0.917 and 0.949 and mean absolute errors (MAE) among 57.783 and 71.768 for the D case in forecasting. The optimal range for Iquitos is obtained when 79\% up to 88\% of the time series is considered for training. For this case, the best case is CD, having a minimum $r$ equal to 0.850 and maximum 0.887, while values of MAE oscillate among 2.780 and 4.156. For Barranquilla, the optimal range occurs between 72\% until 82\% of length training. In this case, the better approach is HD, where the measures exhibit a minimum $r$ equal to 0.942 and a maximum 0.953, while the minimum and maximum MAE vary between 6.085 and 6.669. We show that the forecast of dengue cases is a challenging problem and climate variables do not always help. However, when we include the mentioned climate variables, the most important one is humidity.

Spiral waves are spatial-temporal patterns that can emerge in different systems as heart tissues, chemical oscillators, ecological networks and the brain. These waves have been identified in the neocortex of turtles, rats, and humans, particularly during sleep-like states. Although their functions in cognitive activities remain until now poorly understood, these patterns are related to cortical activity modulation and contribute to cortical processing. In this work, we construct a neuronal network layer based on the spatial distribution of pyramidal neurons. Our main goal is to investigate how local connectivity and coupling strength are associated with the emergence of spiral waves. Therefore, we propose a trustworthy method capable of detecting different wave patterns, based on local and global phase order parameters. As a result, we find that the range of connection radius (R) plays a crucial role in the appearance of spiral waves. For R < 20 {\mu}m, only asynchronous activity is observed due to small number of connections. The coupling strength (gsyn ) greatly influences the pattern transitions for higher R, where spikes and bursts firing patterns can be observed in spiral and non-spiral waves. Finally, we show that for some values of R and gsyn bistable states of wave patterns are obtained.

The description of neuronal activity has been of great importance in neuroscience. In this field, mathematical models are useful to describe the electrophysical behaviour of neurons. One successful model used for this purpose is the Adaptive Exponential Integrate-and-Fire (Adex), which is composed of two ordinary differential equations. Usually, this model is considered in his standard formulation, i.e., with integer order derivatives. In this work, we propose and study the fractal extension of Adex model, which in simple terms corresponds to replacing the integer derivative by non-integer. As non-integer operators, we choose the fractal derivatives. We explore the effects of equal and different orders of fractal derivatives in the firing patterns and mean frequency of the neuron described by the Adex model. Previous results suggest that fractal derivatives can provide a more realistic representation due to the fact that the standard operators are generalized. Our findings show that the fractal order influences the inter-spike intervals and changes the mean firing frequency. In addition, the firing patterns depend not only on the neuronal parameters but also on the order of respective fractal operators. As our main conclusion, the fractal order below the unit value increases the influence of the adaptation mechanism in the spike firing patterns.

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., Comment: 10 pages, 9 figures

In this work, we study the unpredictability of seasonal infectious diseases considering a SEIRS model with seasonal forcing. To investigate the dynamical behaviour, we compute bifurcation diagrams type hysteresis and their respective Lyapunov exponents. Our results from bifurcations and the largest Lyapunov exponent show bistable dynamics for all the parameters of the model. Choosing the inverse of latent period as control parameter, over 70% of the interval comprises the coexistence of periodic and chaotic attractors, bistable dynamics. Despite the competition between these attractors, the chaotic ones are preferred. The bistability occurs in two wide regions. One of these regions is limited by periodic attractors, while periodic and chaotic attractors bound the other. As the boundary of the second bistable region is composed of periodic and chaotic attractors, it is possible to interpret these critical points as tipping points. In other words, depending on the latent period, a periodic attractor (predictability) can evolve to a chaotic attractor (unpredictability). Therefore, we show that unpredictability is associated with bistable dynamics preferably chaotic, and, furthermore, there is a tipping point associated with unpredictable dynamics.

In this work, to support decision making of immunisation strategies, we propose the inclusion of two vaccination doses in the SEIR model considering a stochastic cellular automaton. We analyse three different scenarios of vaccination: $i) unlimited doses, (ii) limited doses into susceptible individuals, and (iii) limited doses randomly distributed overall individuals. Our results suggest that the number of vaccinations and time to start the vaccination is more relevant than the vaccine efficacy, delay between the first and second doses, and delay between vaccinated groups. The scenario (i) shows that the solution can converge early to a disease-free equilibrium for a fraction of individuals vaccinated with the first dose. In the scenario (ii), few two vaccination doses divided into a small number of applications reduce the number of infected people more than into many applications. In addition, there is a low waste of doses for the first application and an increase of the waste in the second dose. The scenario (iii) presents an increase in the waste of doses from the first to second applications more than the scenario $(ii)$. In the scenario (iii), the total of wasted doses increases linearly with the number of applications. Furthermore, the number of effective doses in the application of consecutive groups decays exponentially overtime.

We study the time delay in the synaptic conductance for suppression of spike synchronisation in a random network of Hodgkin Huxley neurons coupled by means of chemical synapses. In the first part, we examine in detail how the time delay acts over the network during the synchronised and desynchronised neuronal activities. We observe a relation between the neuronal dynamics and the syaptic conductance distributions. We find parameter values in which the time delay has high effectiveness in promoting the suppression of spike synchronisation. In the second part, we analyse how the delayed neuronal networks react when pulsed inputs with different profiles (periodic, random, and mixed) are applied on the neurons. We show the main parameters responsible for inducing or not synchronous neuronal oscillations in delayed networks.

Cancer is a term used to refer to a large set of diseases. The cancerous cells grow and divide and, as a result, they form tumours that grow in size. The immune system recognise the cancerous cells and attack them, though, it can be weakened by the cancer. One type of cancer treatment is chemotherapy, which uses drugs to kill cancer cells. Clinical, experimental, and theoretical research has been developed to understand the dynamics of cancerous cells with chemotherapy treatment, as well as the interaction between tumour growth and immune system. We study a mathematical model that describes the cancer growth, immune system response, and chemotherapeutic agents. The immune system is composed of resting cells that are converted to hunting cells to combat the cancer. In this work, we consider drug sensitive and resistant cancer cells. We show that the tumour growth can be controlled not only by means of different chemotherapy protocols, but also by the immune system that attacks both sensitive and resistant cancer cells. Furthermore, for all considered protocols, we demonstrate that the time delay from resting to hunting cells plays a crucial role in the combat against cancer cells.

The brain has the phenomenal ability to reorganize itself by forming new connections among neurons and by pruning others. The so-called neural or brain plasticity facilitates the modification of brain structure and function over different time scales. Plasticity might occur due to external stimuli received from the environment, during recovery from brain injury, or due to modifications within the body and brain itself. In this paper, we study the combined effect of short-term (STP) and spike-timing-dependent plasticities (STDP) on the synaptic strength of excitatory coupled Hodgkin-Huxley neurons and show that plasticity can facilitate the formation of modular neural networks with complex topologies that resemble those of networks with preferential attachment properties. In particular, we use an STDP rule that alters the synaptic coupling intensity based on time intervals between spikes of postsynaptic and presynaptic neurons. Previous works have shown that STDP may induce the appearance of directed connections from high to low-frequency spiking neurons. On the other hand, STP is attributed to the release of neurotransmitters in the synaptic cleft of neurons that alter its synaptic efficiency. Our results suggest that the combined effect of STP and STDP with high recovery time facilitates the formation of connections among neurons with similar spike frequencies only, a kind of preferential attachment. We then pursue this further and show that, when starting with all-to-all neural configurations, depending on the STP recovery time and distribution of neural frequencies, modular neural networks can emerge as a direct result of the combined effect of STP and STDP., Comment: 18 pages, 11 figures

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 spikes and bursts activities. Furthermore, those desynchronised 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.

Brain plasticity, also known as neuroplasticity, is a fundamental mechanism of neuronal adaptation in response to changes in the environment or due to brain injury. In this review, we show our results about the effects of synaptic plasticity on neuronal networks composed by Hodgkin-Huxley neurons. We show that the final topology of the evolved network depends crucially on the ratio between the strengths of the inhibitory and excitatory synapses. Excitation of the same order of inhibition revels an evolved network that presents the rich-club phenomenon, well known to exist in the brain. For initial networks with considerably larger inhibitory strengths, we observe the emergence of a complex evolved topology, where neurons sparsely connected to other neurons, also a typical topology of the brain. The presence of noise enhances the strength of both types of synapses, but if the initial network has synapses of both natures with similar strengths. Finally, we show how the synchronous behaviour of the evolved network will reflect its evolved topology.