Undergoes Hopf Bifurcation Research Articles

Dynamical Analysis of a Melanoma Model with Immune Response

In this paper, we study the stability and bifurcation behavior of a three-dimensional melanoma model with immune response. The system has at least one and at most three positive equilibria. It is proved that the system undergoes Hopf bifurcation and saddle-node bifurcation at the positive equilibrium. We investigate the direction of Hopf bifurcation and stability of the bifurcating periodic solution by center manifold theorem and normal form theory. Moreover, codimension two bifurcations of the system are analyzed. We demonstrate the existence of Bautin bifurcation and Bogdanov–Takens bifurcation of the system. The normal form of Bautin bifurcation and Bogdanov–Takens bifurcation are given. Finally, some numerical simulations are demonstrated to support our theoretical results, and the importance of some parameters of the system is discussed, in particular the activation rate of CD8[Formula: see text]T cells.

Dynamics of an HTLV-I infection model with delayed CTLs immune response

This paper deals with a four dimensional mathematical model for Human T-cell leukemia virus type-I (HTLV-I) infection that includes delayed CD8+ cytotoxic T-cells (CTLs) immune response. The proposed system has three biologically feasible steady states, namely disease-free steady state, CTL-inactive steady state and an interior steady state. Our theoretical analysis demonstrates that local and global stability analysis are established by the two critical parameters R0 and R1, basic reproduction numbers due to viral infection and due to CTLs immune response, respectively. The disease-free steady state E0 is globally stable if R0≤1, and the HTLV-I infections are eliminated. The asymptotic-carrier steady state E1 is globally stable if R1≤1<R0, which indicates HTLV-I infection is chronic without persistence of CTLs immune response. The interior steady state E2 is globally asymptotic stable if R1>1, which implies that the HTLV-I infection is choric in persistence of CTLs immune response. Due to immune response delay, our proposed model undergoes a destabilization of the interior steady state leading to Hopf bifurcation and periodic oscillations. We estimate the length of time delay that preserve the stability of period-1 limit cycle. We also derived the direction and stability of Hopf bifurcation around the interior steady state by center manifold theory and normal form method. To determine the robustness of the model, we performed normalized forward sensitivity analysis with reference to R0 and R1. Our proposed model undergoes Hopf bifurcation with respect to the production rate of uninfected CD4+T cells h, removal rate of virus-specific CTLs d4, spontaneous infected CD4+T cell activation d2 and transmissibility coefficient β. Implications of our numerical illustrations to the pathogenesis of HTLV-I infection and the development of HTLV-I related HAM/TSP are explored.

Stability and Numerical Simulations of a New SVIR Model with Two Delays on COVID-19 Booster Vaccination

As COVID-19 continues to threaten public health around the world, research on specific vaccines has been underway. In this paper, we establish an SVIR model on booster vaccination with two time delays. The time delays represent the time of booster vaccination and the time of booster vaccine invalidation, respectively. Second, we investigate the impact of delay on the stability of non-negative equilibria for the model by considering the duration of the vaccine, and the system undergoes Hopf bifurcation when the duration of the vaccine passes through some critical values. We obtain the normal form of Hopf bifurcation by applying the multiple time scales method. Then, we study the model with two delays and show the conditions under which the nontrivial equilibria are locally asymptotically stable. Finally, through analysis of official data, we select two groups of parameters to simulate the actual epidemic situation of countries with low vaccination rates and countries with high vaccination rates. On this basis, we select the third group of parameters to simulate the ideal situation in which the epidemic can be well controlled. Through comparative analysis of the numerical simulations, we concluded that the most appropriate time for vaccination is to vaccinate with the booster shot 6 months after the basic vaccine. The priority for countries with low vaccination rates is to increase vaccination rates; otherwise, outbreaks will continue. Countries with high vaccination rates need to develop more effective vaccines while maintaining their coverage rates. When the vaccine lasts longer and the failure rate is lower, the epidemic can be well controlled within 20 years.

Hopf bifurcation without parameters in deterministic and stochastic modeling of cancer virotherapy, part I

In this research, we propose deterministic and stochastic models to explain the complexity of interactions in cancer virotherapy and outcomes of current preclinical and clinical trials of oncolytic viral treatments. In Part I, we analyze the deterministic model. The model incorporates both innate and adaptive immune responses which have opposite effects on the outcome of the therapy. According to relative immune clearance rates, the model can be reduced to two subsystems, one with only innate immunity and one with only adaptive immunity, which provide detailed dynamical properties for the full model. The full system shows many different asymptotic behaviors which correspond to outcomes of the therapy. It undergoes classical Hopf bifurcation when the infectivity constant passes through a particular value and, interestingly, it also undergoes Hopf bifurcation without parameters when the tumor cell number passes through some particular value. We conduct numerical simulations to demonstrate our analytical results and provide detailed medical interpretations.

Stability and bifurcation analyses of p53 gene regulatory network with time delay

&lt;abstract&gt;&lt;p&gt;In this paper, based on a p53 gene regulatory network regulated by Programmed Cell Death 5(PDCD5), a time delay in transcription and translation of Mdm2 gene expression is introduced into the network, the effects of the time delay on oscillation dynamics of p53 are investigated through stability and bifurcation analyses. The local stability of the positive equilibrium in the network is proved through analyzing the characteristic values of the corresponding linearized systems, which give the conditions on undergoing Hopf bifurcation without and with time delay, respectively. The theoretical results are verified through numerical simulations of time series, characteristic values and potential landscapes. Furthermore, combined effect of time delay and several typical parameters in the network on oscillation dynamics of p53 are explored through two-parameter bifurcation diagrams. The results show p53 reaches a lower stable steady state under smaller PDCD5 level, the production rates of p53 and Mdm2 while reaches a higher stable steady state under these larger ones. But the case is the opposite for the degradation rate of p53. Specially, p53 oscillates at a smaller Mdm2 degradation rate, but a larger one makes p53 reach a low stable steady state. Besides, moderate time delay can make the steady state switch from stable to unstable and induce p53 oscillation for moderate value of these parameters. Theses results reveal that time delay has a significant impact on p53 oscillation and may provide a useful insight into developing anti-cancer therapy.&lt;/p&gt;&lt;/abstract&gt;

Bifurcation analysis of a vaccination mathematical model with application to COVID-19 pandemic

In this research, we propose a delayed vaccination model with the application for predicting the evolution of infectious cases related to COVID-19 disease. The main purpose of this paper is to show the existence of Hopf bifurcation that can explain the multiple waves that the world witnessed this recent times. Therefore, it can be used the length between the doses for the vaccine that considered for different vaccines and its effect on the evolution of the infectious cases. It has been shown that the investigated model can undergo Hopf bifurcation in presence of delay time lags to the vaccine against a COVID-19, and can lead to the persistence of the disease. The obtained mathematical findings are checked using graphical representations with proper interpretations on the manner of controlling the outbreak of COVID-19 disease.

Dynamics and Hopf Bifurcation of a Chaotic Chemical Reaction Model

Taking into account an ideal mixture and a well-stirred reactor, some dynamical aspects for a 3-dimensional chaotic system are carried out. Positivity and boundedness of solutions are discussed. Equilibria are investigated and method of linearization is implemented for asymptotic behavior of system about these equilibria. Lyapunov function is constructed to prove the global stability of positive equilibrium point. Moreover, it is proved that system undergoes Hopf bifurcation about its interior (positive) equilibrium. An explicit criterion of Hopf bifurcation without finding the eigenvalues is used for the existence of Hopf bifurcation. Numerical simulation is presented for the illustration of theoretical discussion. Lyapunov dimension is approximated and maximum Lyapunov characteristic exponents are plotted to ensure the chaotic behavior of the model.

Global dynamics of a diffusive phytoplankton-zooplankton model with toxic substances effect and delay.

This paper examines a diffusive toxic-producing plankton system with delay. We first show the global attractivity of the positive equilibrium of the system without time-delay. We further consider the effect of delay on asymptotic behavior of the positive equilibrium: when the system undergoes Hopf bifurcation at some points of delay by the normal form and center manifold theory for partial functional differential equations. Global existence of periodic solutions is established by applying the global Hopf bifurcation theory.

Stability and bifurcation control analysis of a delayed fractional-order eco-epidemiological system.

Considering the factor of artificial intervention in biological control, a delayed fractional eco-epidemiological system with an extended feedback controller is proposed. By using the digestion delay as bifurcation parameter, the stability and Hopf bifurcation are investigated, and the branching conditions are given. The system undergoes Hopf bifurcation, when the parameter tau passes through the critical value. In addition, it can be pointed out that the negative feedback gain and the feedback delay could affect the bifurcation critical value of the system. Therefore, the Hopf bifurcation can also be induced by taking the feedback delay as a bifurcation parameter. Finally, by plotting the solution curve of the system, the significance of the controller to the stability of the eco-epidemiological system is verified.

The blood-stage dynamics of malaria infection with immune response

This article is concerned with the dynamics of malaria infection model with diffusion and delay. The disease free threshold and the immune response threshold value of the malaria infection are obtained, which characterize the stability of the disease free equilibrium and infection equilibrium (with or without immune response). In addition, fluctuations occur when the model undergoes Hopf bifurcation as the delay passes through a certain critical value . In this case, periodic oscillation appears near the positive steady state, which implies the recurrent attacks of disease. Finally, numerical simulations are provided to illustrate the theoretical results.

Impact of the fear and Allee effect on a Holling type II prey\u2013predator model

In this paper, we propose and investigate a prey–predator model with Holling type II response function incorporating Allee and fear effect in the prey. First of all, we obtain all possible equilibria of the model and discuss their stability by analyzing the eigenvalues of Jacobian matrix around the equilibria. Secondly, it can be observed that the model undergoes Hopf bifurcation at the positive equilibrium by taking the level of fear as bifurcation parameter. Moreover, through the analysis of Allee and fear effect, we find that: (i) the fear effect can enhance the stability of the positive equilibrium of the system by excluding periodic solutions; (ii) increasing the level of fear and Allee can reduce the final number of predators; (iii) the Allee effect also has important influence on the permanence of the predator. Finally, numerical simulations are provided to check the validity of the theoretical results.

Complex oscillatory motion of multiple spikes in a three-component Schnakenberg system

In this paper, we introduce a three-component Schnakenberg model, whose key feature is that it has a solution consisting of N spikes that undergoes Hopf bifurcations with respect to N distinct modes nearly simultaneously. This results in complex oscillatory dynamics of the spikes, not seen in typical two-component models. For parameter values beyond the Hopf bifurcations, we derive reduced equations of motion which consist of coupled ordinary differential equations (ODEs) of dimension 2N for spike positions and their velocities. These ODEs fully describe the slow-time evolution of the spikes near the Hopf bifurcations. We then apply the method of multiple scales to the resulting ODEs to derive the long-time dynamics. For a single spike, we find that its long-time motion consists of oscillations near the steady state whose amplitude can be computed explicitly. For two spikes, the long-time behavior can be either in-phase or out-of-phase oscillations. Both in-phase and out-of-phase oscillations are stable, coexist for the same parameter values, and the fate of motion depends solely on the initial conditions. Further away from the Hopf bifurcation points, we offer numerical experiments indicating the existence of highly complex oscillations.

Bifurcation analysis and global dynamics in a predator–prey system of Leslie type with an increasing functional response

The dynamical behaviors of a Leslie type predator–prey system are explored when the functional response is increasing for both predator and prey. Qualitative and quantitative analysis methods based on stability theory, bifurcation theory and numerical simulation are adopted. It is showed that the system is dissipative and permanent, and its solutions are bounded. Global stability of the unique positive equilibrium is investigated by constructing Dulac function and applying Poincaré–Bendixson theorem. The bifurcation behaviors are further explored and the number of limit cycles is determined. By calculating the first Lyapunov number and the first two focus values, it is proved that the positive equilibrium is not a center but a weak focus of multiplicity at most two, so the system undergoes Hopf bifurcation and Bautin bifurcation. The normal form of Bautin bifurcation is also obtained by introducing the complex system. Moreover, numerical simulations are run to demonstrate the validity of theoretical results.

Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease Model with Two Delays and Reinfection

In this paper, a vector-borne disease model with two delays and reinfection is established and considered. First of all, the existence of the equilibrium of the system, under different cases of two delays, is discussed through analyzing the corresponding characteristic equation of the linear system. Some conditions that the system undergoes Hopf bifurcation at the endemic equilibrium are obtained. Furthermore, by employing the normal form method and the center manifold theorem for delay differential equations, some explicit formulas used to describe the properties of bifurcating periodic solutions are derived. Finally, the numerical examples and simulations are presented to verify our theoretical conclusions. Meanwhile, the influences of the degree of partial protection for recovered people acquired by a primary infection on the endemic equilibrium and the critical values of the two delays are analyzed.

Dynamical of Ratio-Dependent Eco-epidemical Model with Prey Refuge

This paper discusses the dynamic analysis of three species in the eco-epidemiology model by considering the ratio-dependent function and prey refuge. The prey refuge is applied under the fact that infected prey has protection instincts that allow it to reduce predation risk. Here, we get the boundedness and three equilibrium points where are existence under certain conditions. In the model, three equilibrium points are locally asymptotically stable and one of the equilibrium points is globally asymptotically stable. We find that the system undergoes Hopf bifurcation around the interior equilibrium point by choosing as a bifurcation parameter. We also find a condition for uniform persistence. Finally, several simulations of numerical are performed not only to illustrate the analytical results but also to illustrate the effect of the prey refuge.

Effect of Prey Refuge and Harvesting on Dynamics of Eco-epidemiological Model with Holling Type III

In this research, we formulate and analyze an eco-epidemiology model of the modified Leslie-Gower model with Holling type III by incorporating prey refuge and harvesting. In the model, we find at most six equilibrium where three equilibrium points are unstable and three equilibrium points are locally asymptotically stable. Furthermore, we find an interesting phenomenon, namely our model undergoes Hopf bifurcation at the interior equilibrium point by selecting refuge as the bifurcation parameter. Moreover, we also conclude that the stability of all populations occurs faster when the harvesting rate increases. In the end, several numerical solutions are presented to check the analytical results.

Effect of Prey Refuge and Harvesting on Dynamics of Eco-epidemiological Model with Holling Type III

In this research, we formulate and analyze an eco-epidemiology model of the modified Leslie-Gower model with Holling type III by incorporating prey refuge and harvesting. In the model, we find at most six equilibrium where three equilibrium points are unstable and three equilibrium points are locally asymptotically stable. Furthermore, we find an interesting phenomenon, namely our model undergoes Hopf bifurcation at the interior equilibrium point by selecting refuge as the bifurcation parameter. Moreover, we also conclude that the stability of all populations occurs faster when the harvesting rate increases. In the end, several numerical solutions are presented to check the analytical results.

Time-Delayed Sampled-Data Feedback Control of Differential Systems Undergoing Hopf Bifurcation

In this paper, time-delayed sampled-data feedback control technique is used to asymptotically stabilize a class of unstable delayed differential systems. Through the analysis for the distribution change of eigenvalues, an effective interval of the control parameter is obtained for a given sampling period. Here an indirect strategy is taken. Specifically, the system of continuous-time delayed feedback control is studied first by Hopf bifurcation theory. And then, the result and implicit function theorem are used to analyze the system of time-delayed sampled-data feedback control with a sufficiently small sampling period. Considering the practical criterion for the size of sampling period, the upper bound of sampling period is estimated. Finally, an application example, an unstable Mackey–Glass model, is asymptotically stabilized by introducing a blood transfusion item with time-delayed sampled-data feedback control. The blood transfusion speed and blood collection test period are derived from the main results. Some simulations and comparisons show the correctness and advantages of the main theoretical results.

Stability and bifurcation analysis of a tumor-immune system with two delays and diffusion.

A tumor-immune system with diffusion and delays is proposed in this paper. First, we investigate the impact of delay on the stability of nonnegative equilibrium for the model with a single delay, and the system undergoes Hopf bifurcation when delay passes through some critical values. We obtain the normal form of Hopf bifurcation by applying the multiple time scales method for determining the stability and direction of bifurcating periodic solutions. Then, we study the tumor-immune model with two delays, and show the conditions under which the nontrivial equilibria are locally asymptotically stable. Thus, we can restrain the diffusion of tumor cells by controlling the time delay associated with the time of tumor cell proliferation and the time of immune cells recognizing tumor cells. Finally, numerical simulations are presented to illustrate our analytic results.

Rich Dynamics of a Predator-Prey System with Different Kinds of Functional Responses

In this study, we investigate a mathematical model that describes the interactive dynamics of a predator-prey system with different kinds of response function. The positivity, boundedness, and uniform persistence of the system are established. We investigate the biologically feasible singular points and their stability analysis. We perform a comparative study by considering different kinds of functional responses, which suggest that the dynamical behavior of the system remains unaltered, but the position of the bifurcation points altered. Our model system undergoes Hopf bifurcation with respect to the growth rate of the prey population, which indicates that a periodic solution occurs around a fixed point. Also, we observed that our predator-prey system experiences transcritical bifurcation for the prey population growth rate. By using normal form theory and center manifold theorem, we investigate the direction and stability of Hopf bifurcation. The biological implications of the analytical and numerical findings are also discussed in this study.