Bifurcations in an epidemic model with constant removal rate of the infectives

An epidemic model with a constant removal rate of infective individuals is proposed to understand the effect of limited resources for treatment of infectives on the disease spread. It is found that it is unnecessary to take such a large treatment capacity that endemic equilibria disappear to eradicate the disease. It is shown that the outcome of disease spread may depend on the position of the initial states for certain range of parameters. It is also shown that the model undergoes a sequence of bifurcations including saddle-node bifurcation, subcritical Hopf bifurcation, and homoclinic bifurcation.     

Backward bifurcation of an epidemic model with saturated treatment function

An epidemic model with saturated incidence rate and saturated treatment function is studied. Here the treatment function adopts a continuous and differentiable function which can describe the effect of delayed treatment when the number of infected individuals is getting larger and the medical condition is limited. The global dynamics of the model indicate that the basic reproduction number being the unity is a strict threshold for disease eradication when such effect is weak. However, it is shown that a backward bifurcation will take place when this delayed effect for treatment is strong. Therefore, driving the basic reproduction number below the unity is not enough to eradicate the disease. And a critical value at the turning point is deduced as a new threshold. Some sufficient conditions for the disease-free equilibrium and the endemic equilibrium being globally asymptotically stable are also obtained. Mathematical results in this paper suggest that giving the patients timely treatment, improving the cure efficiency and decreasing the infective coefficient are all valid methods for the control of disease.     

Bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action

In this paper, we study the bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action, which describes the psychological effects of the community on certain serious diseases when the number of infective is getting larger. By carrying out the bifurcation analysis of the model, we show that there exist some values of the model parameters such that numerous kinds of bifurcation occur for the model, such as Hopf bifurcation, Bogdanov–Takens bifurcation.     

Dynamical behavior of an epidemic model with a nonlinear incidence rate

In this paper, we study the global dynamics of an epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action. By carrying out global qualitative and bifurcation analyses, it is shown that either the number of infective individuals tends to zero as time evolves or there is a region such that the disease will be persistent if the initial position lies in the region and the disease will disappear if the initial position lies outside this region. When such a region exists, it is shown that the model undergoes a Bogdanov-Takens bifurcation, i.e., it exhibits a saddle-node bifurcation, Hopf bifurcations, and a homoclinic bifurcation. Existence of none, one or two limit cycles is also discussed.     

Hopf Bifurcation for a Susceptible-Infective Model with Infection-Age Structure

An SIS model is investigated in which the infective individuals are assumed to have an infection-age structure. The model is formulated as an abstract non-densely defined Cauchy problem. We study some dynamical properties of the model by using the theory of integrated semigroups, the Hopf bifurcation theory and the normal form theory for semilinear equations with non-dense domain. Qualitative analysis indicates that there exist some parameter values such that this SIS model has a non-trivial periodic solution which bifurcates from the positive equilibrium. Furthermore, the explicit formulae are given to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Numerical simulations are also carried out to support our theoretical results.     

Backward bifurcation of an epidemic model with standard incidence rate and treatment rate

An epidemic model with standard incidence rate and treatment rate of infectious individuals is proposed to understand the effect of the capacity for treatment of infectives on the disease spread. It is assumed that treatment rate is proportional to the numbers of infectives below the capacity and is a constant when the number of infectives is greater than the capacity. It is proved that the existence and stability of equilibria for the model is not only related to the basic reproduction number but also the capacity for treatment of infectives. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low.     

一类具有非线性发生率和治疗函数的传染病模型研究

传染病动力学系统的数学建模中,合理的使用非线性发生率往往更能使模型与实际相吻合.并且在实际的疾病防治过程中,由于受到空间人力物力资源的影响一般存在最大治疗容量的限制.结合这两种情况建立了一类含非线性发生率和最大治疗容量限制的传染病模型.通过分析这个模型,得到无病平衡点和正平衡点的存在性、稳定性.进一步取发生率和治疗系统达到最大容量时的感染者人数作为分支参数,得到了Hopf分支和Bogdanov-Takens分支的存在条件,并进行了数值模拟.     

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.     

An Infectious Disease Model With Media Coverage and Limited Medical Resources北大核心CSCD

该文建立和分析了一类具有媒体报道效应和有限医疗资源的传染病动力学模型,定义了疾病的基本再生数,分析了平衡点的存在性和稳定性,给出了系统发生前向分支、后向分支和Hopf分支的条件.通过数值模拟发现：提高媒体报道的信息覆盖率或医院对病人的最大容纳量,可以显著降低疾病流行的峰值或稳态时的感染人数;随着参数变化,系统不仅可能会产生后向分支或前向分支,还可能会出现鞍结点分支、Hopf分支以及地方病平衡点稳定性随参数变化而变化等动力学行为.     

Qualitative analysis of a SIR epidemic model with saturated treatment rate

On account of the effect of limited treatment resources on the control of epidemic disease, a saturated removal rate is incorporated into Hethcote’s SIR epidemiological model (Hethcote, SIAM Rev. 42:599–653, 2000). Unlike the original model, the model has two endemic equilibria when R ₀<1. Furthermore, under some conditions, both the disease free equilibrium and one of the two endemic equilibria are asymptotically stable, i.e., the model has bistable equilibria. Therefore, disease eradication not only depends on R ₀ but also on the initial sizes of all sub-populations. By the Poincaré-Bendixson theorem, Poincaré index, center manifold theorem, Hopf bifurcation theorem and Lyapunov-Lasalle theorem, etc., the existence and asymptotical stability of the equilibria, the existence, stability and direction of Hopf bifurcation are discussed, respectively.     

Mathematical Analysis of SIR Epidemic Model with Piecewise Infection Rate and Control Strategies

The limitation of contact between susceptible and infected individuals plays an important role in decreasing the transmission of infectious diseases. Prevention and control strategies contribute to minimizing the transmission rate. In this paper, we propose SIR epidemic model with delayed control strategies, in which delay describes the response and effect time. We study the dynamic properties of the epidemic model from three aspects: steady states, stability and bifurcation. By eliminating the existence of limit cycles, we establish the global stability of the endemic equilibrium, when the delay is ignored. Further, we find that the delayed effect on the infection rate does not affect the stability of the disease-free equilibrium, but it can destabilize the endemic equilibrium and bring Hopf bifurcation. Theoretical results show that the prevention and control strategies can effectively reduce the final number of infected individuals in the population. Numerical results corroborate the theoretical ones.     

Necessary and sufficient conditions for Hopf bifurcation in exponential RED algorithm with communication delay

In this paper, we investigate a novel congestion control algorithm, i.e., exponential RED algorithm, with communication delay. We derive some necessary and sufficient conditions ensuring Hopf bifurcation to occur for this model. By choosing the delay as a bifurcation parameter, we demonstrated that Hopf bifurcation would occur when the delay exceeds a critical value. A formula for determining the bifurcation direction and stability of bifurcation periodic solutions is given by applying the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical results are also provided.     

Global Dynamics of a Tuberculosis Epidemic Model and the Influence of Backward Bifurcation

In this paper, we propose and analyze a tuberculosis (TB) model with exogenous re-infection. We assume that treated individual may be again infected by infectious individual. The model exhibits two bifurcations viz. transcritical bifurcation when the basic reproductive number R ₀?=?1 and backward bifurcation where the disease transmission rate β plays as control parameter. The persistent of the model and, the local and global stability criteria of disease-free and endemic equilibria are discussed. By carrying out bifurcation analysis, it is shown that the model exhibits the bistability and undergoes the Hopf bifurcation when immunological memory is everlasting i.e. when σ?=?0. Lastly, some simulations are given to verify our analytical results.     

Nonlinear oscillations in business cycle model with time lags

In this paper we analyse the dynamics of the Kaldor–Kalecki business cycle model. This model is based on the classical Kaldor model in which capital stock changes are caused by past investment decisions. This lag is connected with time delay needed for new capital to be installed. The dynamics of the model is reduced to the form of damped oscillator with negative feedback connected with lag parameter and next it is analysed in terms of bifurcation theory. We find conditions for existence and persistence of oscillatory behaviour which is represented by limit cycle on some central manifold in phase space, i.e., single Hopf bifurcation. We demonstrate that the Hopf cycles may be exhibited for nonzero measure set of the parameter space. The conditions for bifurcation of co-dimension two connected with interaction of bifurcations as well as bifurcation diagrams are also given. Finally, we obtain numerical values describing an amplitude and a period of oscillation for different parameter of the system. It is also proved that while the investment function is not nonlinear a quasi-periodic solution (a 1:2 resonant double Hopf point) can appear. The source of such a behaviour is rather a consequence of time lag than nonlinearity of the investment function. Our results confirm the existence of asymmetric (two periodic) cycles in the Kaldor–Kalecki model with time-to-build.     

Complicated endemics of an SIRS model with a generalized incidence under preventive vaccination and treatment controls

In this paper, we propose and study an SIRS epidemic model that incorporates: a generalized incidence rate function describing mechanisms of the disease transmission; a preventive vaccination in the susceptible individuals; and different treatment control strategies depending on the infective population. We provide rigorous mathematical results combined with numerical simulations of the proposed model including: treatment control strategies can determine whether there is an endemic outbreak or not and the number of endemic equilibrium during endemic outbreaks, in addition to the effects of the basic reproduction number; the large value of the preventive vaccination rate can reduce or control the spread of disease; and the large value of the psychological or inhibitory effects in the incidence rate function can decrease the infective population. Some of our interesting findings are that the treatment strategies incorporated in our SIRS model are responsible for backward or forward bifurcations and multiple endemic equilibria; and the infective population decreases with respect to the maximal capacity of treatment. Our results may provide us useful biological insights on population managements for disease that can be modeled through SIRS compartments.     

Global and Bifurcation Analysis of an HIV Pathogenesis Model with Saturated Reverse Function

In this paper, an HIV dynamics model with the proliferation of CD4 T cells is proposed. The authors consider nonnegativity, boundedness, global asymptotic stability of the solutions and bifurcation properties of the steady states. It is proved that the virus is cleared from the host under some conditions if the basic reproduction number R_0 is less than unity. Meanwhile, the model exhibits the phenomenon of backward bifurcation. We also obtain one equilibrium is semi-stable by using center manifold theory. It is proved that the endemic equilibrium is globally asymptotically stable under some conditions if R_0 is greater than unity. It also is proved that the model undergoes Hopf bifurcation from the endemic equilibrium under some conditions. It is novelty that the model exhibits two famous bifurcations,backward bifurcation and Hopf bifurcation. The model is extended to incorporate the specific Cytotoxic T Lymphocytes(CTLs) immune response. Stabilities of equilibria and Hopf bifurcation are considered accordingly. In addition, some numerical simulations for justifying the theoretical analysis results are also given in paper.     

Bifurcation analysis of a delayed epidemic model

In this paper, Hopf bifurcation for a delayed SIS epidemic model with stage structure and nonlinear incidence rate is investigated. Through theoretical analysis, we show the positive equilibrium stability and the conditions that Hopf bifurcation occurs. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. In addition, we also study the effect of the inhibition effect on the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.     

Backward bifurcation in simple SIS model

We describe and analyze a simple SIS model with treatment. In particular, we give a completely qualitative analysis by means of the theory of asymptotically autonomous system. It is found that a backward bifurcation occurs if the adequate contact rate or the capacity is small. It is also found that there exists bistable endemic equilibria. In the case of disease-induced death, it is shown that the backward bifurcation also occurs. Moreover, there is no limit cycle under some conditions, and the subcritical Hopf bifurcation occurs under another conditions. Supported by the National Natural Science Foundation of China (No. 10571143, 30770555)     

Dynamics of an intraguild predation food web model with strong Allee effect in the basal prey

Since intraguild predation (IGP) is a ubiquitous and important community module in nature and Allee effect has strong impact on population dynamics, in this paper we propose a three-species IGP food web model consisted of the IG predator, IG prey and basal prey, in which the basal prey follows a logistic growth with strong Allee effect. We investigate the local and global dynamics of the model with emphasis on the impact of strong Allee effect. First, positivity and boundedness of solutions are studied. Then existence and stability of the boundary and interior equilibria are presented and the Hopf bifurcation curve at an interior equilibrium is given. The existence of a Hopf bifurcation curve indicates that if competition between the IG prey and IG predator for the basal resource lies below the curve then the interior equilibrium remains stable, while if it lies above the curve then the interior equilibrium loses its stability. In order to explore the impact of Allee effect, the parameter space is classified into sixteen different regions and, in each region, the number of interior equilibria is determined and the corresponding bifurcation diagrams on the Allee threshold are given. The extinction parameter regions of at least one species and the necessary coexistence parameter regions of all three species are provided. In addition, we explore possible dynamical patterns, i.e., the existence of multiple attractors. By theoretical analysis and numerical simulations, we show that the model can have one (i.e. extinction of all species), two (i.e. bi-stability) or three (i.e. tri-stability) attractors. It is also found by simulations that when there exists a unique stable interior equilibrium, the model may generate multiple attracting periodic orbits and the coexistence of all three species is enhanced as the competition between the IG prey and IG predator for the basal resource is close to the Hopf bifurcation curve from below. Our results indicate that the intraguild predation food web model exhibits rich and complex dynamic behaviors and strong Allee effect in the basal prey increases the extinction risk of not only the basal prey but also the IG prey or/and IG predator.     

Delayed feedback on the dynamical model of a financial system

We investigate the effect of delayed feedbacks on the financial model, which describes the time variation of the interest rate, the investment demand, and the price index, for establishing the fiscal policy. By local stability analysis, we theoretically prove the occurrences of Hopf bifurcation. Through numerical bifurcation analysis, we obtain the supercritical and subcritical Hopf bifurcation curves which support the theoretical predictions. Moreover, the fold limit cycle and Neimark–Sacker bifurcation curves are detected. We also confirm that the double Hopf and generalized Hopf codimension-2 bifurcation points exist.