Modelling HIV dynamics with cell-to-cell transmission and CTL response

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell-to-cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so-called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov-Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle-node bifurcation, Bogdanov-Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell-to-cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so-called new type bifurcation persist. Finally, some conclusions and discussions are also given.

Chaotic Dynamics by Some Quadratic Jerk Systems

This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.

Multiple Outbreaks Resulting from Asymptomatic Infection in Spreading of COVID-19

In this paper, we establish a compartmental model for the transmission of COVID-19 with both symptomatic and asymptomatic infections. The difference is that in our model, we consider the infection caused by multiple contacts with asymptomatic infected population. Through the dynamic analysis of the model, we found that when there is only one contact with an asymptomatic infected people, the system has a unique threshold to control the disease. When there is two contact with asymptomatic infected population, the system will undergo forward bifurcation, backward bifurcation, saddle-node bifurcation, supercritical Hopf bifurcation, subcritical Hopf bifurcation, and Bogdanov-Takens bifurcation with codimension 2. Finally, we give The complete bifurcation diagram and global phase diagram of the system, and the biological significance of our results are also given.

Control of Hopf Bifurcation Type of a Neuron Model Using Washout Filter

A quantitative mathematical model of neurons should not only include enough details to consider the dynamics of single neurons but also minimize the complexity of the model so that the model calculation is convenient. The two-dimensional Prescott model provides a good compromise between the authenticity and computational efficiency of a neuron. The dynamic characteristics of the Prescott model under external electrical stimulation are studied by combining analytical and numerical methods in this paper. Through the analysis of the equilibrium point distribution, the influence of model parameters and external stimulus on the dynamic characteristics is described. The occurrence conditions and the type of Hopf bifurcation in the Prescott model are analyzed, and the analytical determination formula of the Hopf bifurcation type in the neuron model is obtained. Washout filter control is used to change the Hopf bifurcation type, so that the subcritical Hopf bifurcation transforms to supercritical Hopf bifurcation, so as to realize the change of the dynamic characteristics of the model.

Spatiotemporal Dynamics and Pattern Formations of an Activator-Substrate Model with Double Saturation Terms

By using center manifold theory, Poincaré–Bendixson theorem, spatiotemporal spectrum and dispersion relation of linear operators, the spatiotemporal dynamics of an activator-substrate model with double saturation terms under the homogeneous Neumann boundary condition are considered in the present paper. It is surprising to find that the system can induce new dynamics, such as subcritical Hopf bifurcation and the coexistence of two limit cycles. Moreover, Turing instability in equilibrium mainly generates stripe patterns, while homogeneous periodic solutions mainly generate spot patterns or spot-stripe patterns, where the pattern formations are enormously consistent with the theoretical results. Interestingly, Turing instability can create equilibrium and periodic solution simultaneously in the subcritical Hopf bifurcation, which is the new finding of the diffusion-driven instability. In fact, those theoretical methods are also valid for finding the patterns of other models in one-dimensional space.

Bogdanov–Takens bifurcation of a Holling IV prey–predator model with constant-effort harvesting

A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation.

Nonlinear Vibration Analysis of a Flexible Rotor Supported by a Flexure Pivot Tilting Pad Journal Bearing in Comparison to a Fixed Profile Journal Bearing with Experimental Verification

In this paper, an efficient numerical method consisting of the real mode component mode synthesis (CMS) model reduction, shooting method with parallel computing, and Floquet analysis was developed for nonlinear rotordynamics analysis of a flexible rotor supported by a 4-lobe flexure pivot tilting pad journal bearing (FPTPJB) in load-on-pad (LOP) and load-between-pad (LBP) orientations in comparison to a fixed profile journal bearing (JB) of the same pad geometry. The method used the rotor's finite elements and bearing forces obtained from directly solving the Reynolds equation to determine the limit cycles and Hopf bifurcation types. For the investigated rotor and bearing parameters, the numerical results indicated that the onset speed of instability (OSI) of FPTPJB is considerably higher than that of JB of the same orientation. Also, FPTPJB in LOP orientation yielded higher OSI than the LBP one, whereas the OSI of JB in LOP orientation was substantially higher than the LBP counterpart. Nonlinear calculation results indicated that all bearing types and orientations gave subcritical Hopf bifurcation. The FPTPJB in LOP orientation produced the largest stable operating region, whereas the JB in LBP configuration yield the smallest one. The experiment showed subcritical Hopf bifurcation occurred at speed close to the calculated OSI in all cases except FPTPJB in LOP orientation that the OSI is higher than the maximum test rig speed. The whirling orbit had the same frequency as the first critical speed and precessed in the direction of shaft rotation.

Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate

<p style='text-indent:20px;'>In this paper, we analyze a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. The nonmonotone incidence rate describes the "psychological effect": when the number of the infected individuals (denoted by <inline-formula><tex-math id="M1">\begin{document}$ I $\end{document}</tex-math></inline-formula>) exceeds a certain level, the incidence rate is a decreasing function with respect to <inline-formula><tex-math id="M2">\begin{document}$ I $\end{document}</tex-math></inline-formula>. The piecewise-smooth treatment rate describes the situation where the community has limited medical resources, treatment rises linearly with <inline-formula><tex-math id="M3">\begin{document}$ I $\end{document}</tex-math></inline-formula> until the treatment capacity is reached, after which constant treatment (i.e., the maximum treatment) is taken.Our analysis indicates that there exists a critical value <inline-formula><tex-math id="M4">\begin{document}$ \widetilde{I_0} $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M5">\begin{document}$ ( = \frac{b}{d}) $\end{document}</tex-math></inline-formula> for the infective level <inline-formula><tex-math id="M6">\begin{document}$ I_0 $\end{document}</tex-math></inline-formula> at which the health care system reaches its capacity such that:<b>(i)</b> When <inline-formula><tex-math id="M7">\begin{document}$ I_0 \geq \widetilde{I_0} $\end{document}</tex-math></inline-formula>, the transmission dynamics of the model is determined by the basic reproduction number <inline-formula><tex-math id="M8">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula>: <inline-formula><tex-math id="M9">\begin{document}$ R_0 = 1 $\end{document}</tex-math></inline-formula> separates disease persistence from disease eradication. <b>(ii)</b> When <inline-formula><tex-math id="M10">\begin{document}$ I_0 &lt; \widetilde{I_0} $\end{document}</tex-math></inline-formula>, the model exhibits very rich dynamics and bifurcations, such as multiple endemic equilibria, periodic orbits, homoclinic orbits, Bogdanov-Takens bifurcations, and subcritical Hopf bifurcation.</p>

Dynamic expression of a HR neuron model under an electric field

The movement of large amounts of ions (e.g., potassium, sodium and calcium) in the nervous system triggers time-varying electromagnetic fields that further regulate the firing activity of neurons. Accordingly, the discharge states of a modified Hindmarsh–Rose (HR) neuron model under an electric field are studied by numerical simulation. By using the Matcont software package and its programming, the global basins of attraction for the model are analyzed, and it is found that the model has a coexistence oscillation pattern and hidden discharge behavior caused by subcritical Hopf bifurcation. Furthermore, the model’s unstable branches are effectively controlled based on the Washout controller and eliminating the hidden discharge states. Interestingly, by analyzing the two-parametric bifurcation analysis, we also find that the model generally has a comb-shaped chaotic structure and a periodic-adding bifurcation pattern. Additionally, considering that the electric field is inevitably disturbed periodically, the discharge states of this model are more complex and have abundant coexisting oscillation modes. The research results will provide a useful reference for understanding the complex dynamic characteristics of neurons under an electric field.