Unpredictability in seasonal infectious diseases spread

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

Fractional dynamics and recurrence analysis in cancer model

In this work, we analyze the effects of fractional derivatives in the chaotic dynamics of a cancer model. We begin by studying the dynamics of a standard model, {\it i.e.}, with integer derivatives. We study the dynamical behavior by means of the bifurcation diagram, Lyapunov exponents, and recurrence quantification analysis (RQA), such as the recurrence rate (RR), the determinism (DET), and the recurrence time entropy (RTE). We find a high correlation coefficient between the Lyapunov exponents and RTE. Our simulations suggest that the tumor growth parameter ($\rho_1$) is associated with a chaotic regime. Our results suggest a high correlation between the largest Lyapunov exponents and RTE. After understanding the dynamics of the model in the standard formulation, we extend our results by considering fractional operators. We fix the parameters in the chaotic regime and investigate the effects of the fractional order. We demonstrate how fractional dynamics can be properly characterized using RQA measures, which offer the advantage of not requiring knowledge of the fractional Jacobian matrix. We find that the chaotic motion is suppressed as $\alpha$ decreases, and the system becomes periodic for $\alpha \lessapprox 0.9966$. We observe limit cycles for $\alpha \in (0.9966,0.899)$ and fixed points for $\alpha<0.899$. The fixed point is determined analytically for the considered parameters. Finally, we discover that these dynamics are separated by an exponential relationship between $\alpha$ and $\rho_1$. Also, the transition depends on a supper transient which obeys the same relationship

Effects of drug resistance in the tumour-immune system with chemotherapy treatment

Acknowledgement This study was possible by partial financial support from the following Brazilian government agencies: Fundaao Araucaria, National Council for Scientific and Technological Development, Coordination for the Improvement of Higher Education Personnel, and Sao Paulo Research Foundation (2015/07311-7, 2017/18977- 1, 2018/03211-6, 2020/04624-2)Peer reviewedPostprin

Anomalous Relaxation and Three-Level System: A Fractional Schrödinger Equation Approach

We investigate a three-level system in the context of the fractional Schrödinger equation by considering fractional differential operators in time and space, which promote anomalous relaxations and spreading of the wave packet. We first consider the three-level system omitting the kinetic term, i.e., taking into account only the transition among the levels, to analyze the effect of the fractional time derivative. Afterward, we incorporate a kinetic term and the fractional derivative in space to analyze simultaneous wave packet transition and spreading among the levels. For these cases, we obtain analytical and numerical solutions. Our results show a wide variety of behaviors connected to the fractional operators, such as the non-conservation of probability and the anomalous spread of the wave packet

Fractional Diffusion Equation under Singular and Non-Singular Kernel and Its Stability

The fractional reaction–diffusion equation has been used in many real-world applications in fields such as physics, biology, and chemistry. Motivated by the huge application of fractional reaction–diffusion, we propose a numerical scheme to solve the fractional reaction–diffusion equation under different kernels. Our method can be particularly employed for singular and non-singular kernels, such as the Riemann–Liouville, Caputo, Fabrizio–Caputo, and Atangana–Baleanu operators. Moreover, we obtained general inequalities that guarantee that the stability condition depends explicitly on the kernel. As an implementation of the method, we numerically solved the diffusion equation under the power-law and exponential kernels. For the power-law kernel, we solved by considering fractional time, space, and both operators. In another example, we considered the exponential kernel acting on the time derivative and compared the numerical results with the analytical ones. Our results showed that the numerical procedure developed in this work can be employed to solve fractional differential equations considering different kernels

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