Spread of a disease and its effect on population dynamics in an eco-epidemiological system

In this paper, an eco-epidemiological model with simple law of mass action and modified Holling type II functional response has been proposed and analyzed to understand how a disease may spread among natural populations. The proposed model is a modification of the model presented by Upadhyay et al. (2008) [1]. Existence of the equilibria and their stability analysis (linear and nonlinear) has been studied. The dynamical transitions in the model have been studied by identifying the existence of backward Hopf-bifurcations and demonstrated the period-doubling route to chaos when the death rate of predator (μ₁) and the growth rate of susceptible prey population (r) are treated as bifurcation parameters. Our studies show that the system exhibits deterministic chaos when some control parameters attain their critical values. Chaotic dynamics is depicted using the 2D parameter scans and bifurcation analysis. Possible implications of the results for disease eradication or its control are discussed.     

Dynamical behavior of computer virus on Internet

In this paper, we presented a computer virus model using an SIRS model and the threshold value R₀ determining whether the disease dies out is obtained. If R₀ is less than one, the disease-free equilibrium is globally asymptotically stable. By using the time delay as a bifurcation parameter, the local stability and Hopf bifurcation for the endemic state is investigated. Numerical results demonstrate that the system has periodic solution when time delay is larger than a critical values. The obtained results may provide some new insight to prevent the computer virus.     

Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate

In this paper, an SVEIS epidemic model for an infectious disease that spreads in the host population through horizontal transmission is investigated. The role that temporary immunity (natural, disease induced, vaccination induced) plays in the spread of disease, is incorporated in the model. The total host population is bounded and the incidence term is of the Holling-type II form. It is shown that the model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. The global dynamics are completely determined by the basic reproduction number R₀. If R₀<1, the disease-free equilibrium is globally stable which leads to the eradication of disease from population. If R₀>1, a unique endemic equilibrium exists and is globally stable in the feasible region under certain conditions. Further, the transcritical bifurcation at R₀=1 is explored by projecting the flow onto the extended center manifold. We use the geometric approach for ordinary differential equations which is based on the use of higher-order generalization of Bendixson’s criterion. Further, we obtain the threshold vaccination coverage required to eradicate the disease. Finally, taking biologically relevant parametric values, numerical simulations are performed to illustrate and verify the analytical results.     

Bifurcation and asymptotic bifurcation for equations involving A-proper mappings with applications to differential equations

Let X, Y be real Banach spaces, T: X → YA-proper, and C: X → Y compact. Section 1 of this paper is devoted to the study of bifurcation and asymptotic bifurcation problems for Eq. $\text{(1): Tx ? λCx = 0}$. In Theorem 1 it is shown that if T(0) = C(0) = 0 and T and C have F-derivatives T₀ and C₀ at 0 with T₀A-proper and injective, then each eigenvalue of T₀x ? λC₀x = 0 of odd multiplicity is a bifurcation point for Eq. (1). Theorem 2 shows that if T and C have asymptotic derivatives T_∞ and C_∞, then each eigenvalue of T_∞x ? λC_∞x = 0 of odd multiplicity is an asymptotic bifurcation point for Eq. (1). Special cases are treated when Y = X and T = I ? F with Fk-ball-contractive or when Y ≠ X and T is either of type (S) or of strongly accretive type. Section 2 is devoted to applications of Theorems 1 and 2 to bifurcation problems involving elliptic operators. The usefulness of Theorems 1 and 2 stems from the fact that they are directly applicable to differential eigenvalue problems without the preliminary reduction of Eq. (1) to equivalent problems involving compact operators. Moreover, in some cases they are applicable in situations to which the known bifurcation results are not applicable.     

Mathematical Analysis of West Nile Virus Model with Discrete Delays

The paper presents the basic model for the transmission dynamics of West Nile virus (WNV). The model, which consists of seven mutually-exclusive compartments representing the birds and vector dynamics, has a locally-asymptotically stable disease-free equilibrium whenever the associated reproduction number (?₀) is less than unity. As reveal in [3, 20], the analyses of the model show the existence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity). It is shown, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. Analysis of the reproduction number of the model shows that, the disease will persist, whenever ?₀ > 1, and increase in the length of incubation period can help reduce WNV burden in the community if a certain threshold quantities, denoted by Δ_b and Δ_v are negative. On the other hand, increasing the length of the incubation period increases disease burden if Δ_b > 0 and Δ_v > 0. Furthermore, it is shown that adding time delay to the corresponding autonomous model with standard incidence (considered in [2]) does not alter the qualitative dynamics of the autonomous system (with respect to the elimination or persistence of the disease).     

A delayed computer virus propagation model and its dynamics

In this paper, we propose a delayed computer virus propagation model and study its dynamic behaviors. First, we give the threshold value R₀ determining whether the virus dies out completely. Second, we study the local asymptotic stability of the equilibria of this model and it is found that, depending on the time delays, a Hopf bifurcation may occur in the model. Next, we prove that, if R₀ = 1, the virus-free equilibrium is globally attractive; and when R₀ < 1, it is globally asymptotically stable. Finally, a sufficient criterion for the global stability of the virus equilibrium is obtained.     

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.     

Hopf bifurcation and topological horseshoe of a novel finance chaotic system

This paper deals with the existence of both Hopf bifurcation and topological horseshoe for a novel finance chaotic system. First, through rigorous mathematical analysis, we show that a Hopf bifurcation occurs at systems’ three equilibriums S_0,1,2 and Hopf bifurcation at equilibrium S₀ is non-degenerate and supercritical. Second, the computer-assisted verifications for horseshoe chaos in the system are given. Simulation results are presented to support the analysis.     

Stable oscillations in a predator-prey model with time lag

The Lotka-Volterra model with carrying capacity at the prey and time delay in the equation concerning the predator is considered. The time delay is taken into consideration by an integral with the weight function a exp(?at). It is shown that under certain conditions imposed upon the parameters of the system a supercritical Hopf bifurcation takes place at a certain value a₀, of a and the bifurcating closed paths are orbitally asymptotically stable for values of a below a₀.     

Bifurcation of infinite Prandtl number rotating convection

We consider infinite Prandtl number convection with rotation which is the basic model in geophysical fluid dynamics. For the rotation free case, the rigorous analysis has been provided by Park (2005, 2007, revised for publication) [5], [6] and [25] under various boundary conditions. By thoroughly investigating we prove in this paper that the solutions bifurcate from the trivial solution u=0 to an attractor Σ_R which consists of only one cycle of steady state solutions and is homeomorphic to S¹. We also see how intensively the rotation inhibits the onset of convective motion. This bifurcation analysis is based on a new notion of bifurcation, called attractor bifurcation which was developed by Ma and Wang (2005); see [15].     

Bifurcation analysis in a model of cytotoxic T-lymphocyte response to viral infections

In this paper, we study the dynamics of a mathematical model on primary and secondary cytotoxic T-lymphocyte (CTL) response to viral infections by Wodarz et al. This model has three equilibria and their stability criteria are discussed. The system transitions from one equilibrium to the next as the basic reproductive number, R₀, increases. When R₀ increases even further, we analytically show that periodic solutions may arise from the third equilibrium via Hopf bifurcation. Numerical simulations of the model agree with the theoretical results and these dynamics occur within biologically realistic parameter range. The normal form theory is also applied to find the amplitude, phase and stability information on the limit cycles. Biological implications of the results are discussed.     

Existence of Infinitely Many Non-Trivial Bifurcation Points

We consider the non-linear two point boundary value problem where λ > 0,f ∈ C², f′ ≥ 0, f(0) < 0 and lim_{u → ∞} f(u) > 0. By considering the non-negative as well as all sign changing solutions, we establish the existence of infinitely many non-trivial bifurcation points. Further, when f is superlinear, we prove that there exists a constant λ* > 0, such that for each λ ∈ (0, λ*) there are exactly two solutions with m interior zeros for every m = 1,2, …We apply our results to the case when f(u) = u ³ - k; k > 0, and also discuss the evolution of the bifurcation diagram as k → 0.     

Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type

We consider a nonlinear Neumann logistic equation driven by the p-Laplacian with a general Carathéodory superdiffusive reaction. We are looking for positive solutions of such problems. Using minimax methods from critical point theory together with suitable truncation techniques, we show that the equation exhibits a bifurcation phenomenon with respect to the parameter λ > 0. Namely, we show that there is a λ_* > 0 such that for λ < λ_*, the problem has no positive solution; for λ = λ_*, it has at least one positive solution; and for λ > λ_*, it has at least two positive solutions.     

On the period of the limit cycles appearing in one-parameter bifurcations

The generic isolated bifurcations for one-parameter families of smooth planar vector fields {X_μ} which give rise to periodic orbits are: the Andronov-Hopf bifurcation, the bifurcation from a semi-stable periodic orbit, the saddle-node loop bifurcation and the saddle loop bifurcation. In this paper we obtain the dominant term of the asymptotic behaviour of the period of the limit cycles appearing in each of these bifurcations in terms of μ when we are near the bifurcation. The method used to study the first two bifurcations is also used to solve the same problem in another two situations: a generalization of the Andronov-Hopf bifurcation to vector fields starting with a special monodromic jet; and the Hopf bifurcation at infinity for families of polynomial vector fields.     

Organizing center for the bifurcation analysis of a generalized Gause model with prey harvesting and Holling response function of type III

The present note is an addendum to the paper of Etoua-Rousseau (2010) [1] which presented a study of a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . Complete bifurcation diagrams were proposed, but some parts were conjectural. An organizing center for the bifurcation diagram was given by a nilpotent point of saddle type lying on an invariant line and of codimension greater than or equal to 3. This point was of codimension 3 when b≠0, and was conjectured to be of infinite codimension when b=0. This conjecture was in line with a second conjecture that the Hopf bifurcation of order 2 degenerates to a Hopf bifurcation of infinite codimension when b=0. In this note we prove these two conjectures.     

Dynamical analysis and control strategies on malware propagation model

A variable infection rate is more realistic to forecast dynamical behaviors of malware (malicious software) propagation. In this paper, we propose a time-delayed SIRS model by introducing temporal immunity and the variable infection rate. The basic reproductive number R₀ which determines whether malware dies out is obtained. Furthermore, using time delay as a bifurcation parameter, some necessary and sufficient conditions ensuring Hopf bifurcation to occur for this model are derived. Finally, numerical simulations verify the correctness of theoretical results. Most important of all, we investigate the effect of the variable infection rate on the scale of malware prevalence and compare our model with stationary analytical model by simulation. According to simulating results, some strategies that control malware rampant are given, which may be incorporated into cost-effective antivirus policies for organizations to work quite well in practice.     

Hopf bifurcation in numerical approximation of the sunflower equation

In this paper we consider the numerical solution of the sunflower equation. We prove that if the sunflower equation has a Hopf bifurcation point ata =a ₀, then the numerical solution with the Euler-method of the equation has a Hopf bifurcation point ata _h =a ₀ +O(h).     

Spatially nonhomogeneous equilibrium in a reaction-diffusion system with distributed delay

We consider a reaction-diffusion system with general time-delayed growth rate and kernel functions. The existence and stability of the positive spatially nonhomogeneous steady-state solution are obtained. Moreover, taking minimal time delay τ as the bifurcation parameter, Hopf bifurcation near the steady-state solution is proved to occur at a critical value τ=τ₀. Especially, the Hopf bifurcation is forward and the bifurcated periodic solutions are stable on the center manifold. The general results are applied to competitive and cooperative systems with weak or strong kernel function respectively.     

Analysis and control of the bifurcation of Hodgkin–Huxley model

The Hodgkin–Huxley (HH) equations are parameterized by a number of parameters and show a variety of qualitatively different behaviors depending on the various parameters. This paper finds the bifurcation would occur when the leakage conductance g_l is lower than a special value. The Hopf bifurcation of HH model is controlled by applying a simple and unified state-feedback method and the bifurcation point is moved to an unreachable physiological point at the same time, so in this way an absolute bifurcation control is achieved. The simulation results demonstrate the validity of such theoretic analysis and control method. This new method could be a great help to the design of new closed-loop electrical stimulation systems for patients suffering from different nerve system dysfunctions.     

Analysis of a delayed HIV/AIDS epidemic model with saturation incidence

In this paper, a delayed HIV/AIDS epidemic model with saturation incidence is proposed and analyzed. The equilibria and their stability are investigated. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. It is found that if the threshold R ₀<1, then the disease-free equilibrium is globally asymptotically stable, and if the threshold R ₀>1, the system is permanent and the endemic equilibrium is asymptotically stable under certain conditions.