Global dynamics of a pine wilt disease transmission model with nonlinear incidence rates

In this study, we investigate a pine wilt transmission model with nonlinear incidence rates. The stability of the system is analyzed for disease-free and endemic equilibria. It is proved that the global dynamics are completely by the basic reproduction number _R0$R_0$. If _R0$R_0$ is less than one, the disease-free equilibrium is globally asymptotically stable, and in such a case, the endemic equilibrium does not exist. If _R0$R_0$ is greater than one, the disease persists and the unique endemic equilibrium is globally asymptotically stable.     

Analysis of a SEIV epidemic model with a nonlinear incidence rate

In this paper, a SEIV epidemic model with a nonlinear incidence rate is investigated. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. It is shown that if the basic reproduction number _R0<1${\mathfrak{R}}_0<1$, the disease-free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist. Moreover, we show that if the basic reproduction number _R0>1${\mathfrak{R}}_0>1$, the disease is uniformly persistent and the unique endemic equilibrium of the system with saturation incidence is globally asymptotically stable under certain conditions.     

Global dynamics of an SEI model with acute and chronic stages

A model with acute and chronic stages in a population with exponentially varying size is proposed. An equivalent system is obtained, which has two equilibriums: a disease-free equilibrium and an endemic equilibrium. The stability of these two equilibriums is controlled by the basic reproduction number _R0$R_0$. When _R0<1$R_0<1$, the disease-free equilibrium is globally stable. When _R0>1$R_0>1$, the disease-free equilibrium is unstable and the unique endemic equilibrium is locally stable. When _R0>1$R_0>1$ and γ=0,α=0$\gamma =0,\alpha =0$, the endemic equilibrium is globally stable in Γ⁰${\Gamma }^0$.     

Complete global analysis of an SIRS epidemic model with graded cure and incomplete recovery rates

In this paper, applying two types of Lyapunov functional techniques to an SIRS epidemic model with graded cure and incomplete recovery rates, we establish complete global dynamics of the model whose threshold parameter is the basic reproduction number _R0$R_0$ such that the disease-free equilibrium is globally asymptotically stable when _R0?1$R_0?1$, and the endemic equilibrium is globally asymptotically stable when _R0>1$R_0>1$.     

A novel time delayed HIV/AIDS model with vaccination & antiretroviral therapy and its stability analysis

In this paper, we propose a novel time delayed HIV/AIDS mathematical model and further analyze the effect of vaccination and ART (antiretroviral therapy) on this time delayed model, in which the time delay is due to the strong immune response to AIDS for the HIV-infected-aware because of the good physical conditions. We introduce the different stages of the period of AIDS infection having different abilities of transmitting disease, which reflects the developing progress of AIDS infection more realistically. By using suitable Lyapunov functionals and the LaSalle invariant principle, we obtain the basic reproduction number _R0${\mathfrak{R}}_0$ and derive that if _R0<1${\mathfrak{R}}_0<1$ and some parameters satisfy a given condition, the disease-free equilibrium is globally asymptotically stable, while the disease will be died out. Numerical simulations are carried out to verify the obtained stability criteria and demonstrate the effect of the vaccination rate and _R0${\mathfrak{R}}_0$ and the ART on the infective individuals.     

Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response

In this paper, we investigate the dynamical behavior of a virus infection model with delayed humoral immunity. By using suitable Lyapunov functional and the LaSalle?s invariance principle, we establish the global stabilities of the two boundary equilibria. If _R0<1$R_0<1$, the uninfected equilibrium _E0$E_0$ is globally asymptotically stable; if _R1<1<_R0$R_1<1<R_0$, the infected equilibrium without immunity _E1$E_1$ is globally asymptotically stable. When _R1>1$R_1>1$, we obtain the sufficient conditions to the local stability of the infected equilibrium with immunity _E2$E_2$. The time delay can change the stability of _E2$E_2$ and lead to the existence of Hopf bifurcations. The stabilities of bifurcating periodic solutions is also studied. We check our theorems with numerical simulations in the end.     

Stability analysis of an HIV/AIDS epidemic model with treatment

An HIV/AIDS epidemic model with treatment is investigated. The model allows for some infected individuals to move from the symptomatic phase to the asymptomatic phase by all sorts of treatment methods. We first establish the ODE treatment model with two infective stages. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are completely determined by the basic reproduction number _ℜ0${\Re }_0$. If _ℜ0≤1${\Re }_0\le 1$, the disease-free equilibrium is globally stable, whereas the unique infected equilibrium is globally asymptotically stable if _ℜ0>1${\Re }_0>1$. Then, we introduce a discrete time delay to the model to describe the time from the start of treatment in the symptomatic stage until treatment effects become visible. The effect of the time delay on the stability of the endemically infected equilibrium is investigated. Moreover, the delay model exhibits Hopf bifurcations by using the delay as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the results.     

Analysis of the permanence of an SIR epidemic model with logistic process and distributed time delay

In this paper, we study the dynamics of an SIR epidemic model with a logistic process and a distributed time delay. We first show that the attractivity of the disease-free equilibrium is completely determined by a threshold $R_0$. If $R_0?1$, then the disease-free equilibrium is globally attractive and the disease always dies out. Otherwise, if $R_0>1$, then the disease-free equilibrium is unstable, and meanwhile there exists uniquely an endemic equilibrium. We then prove that for any time delay $h>0$, the delayed SIR epidemic model is permanent if and only if there exists an endemic equilibrium. In other words, $R_0>1$ is a necessary and sufficient condition for the permanence of the epidemic model. Numerical examples are given to illustrate the theoretical results. We also make a distinction between the dynamics of the distributed time delay system and the discrete time delay system.     

Stability of the virus dynamics model with Beddington–DeAngelis functional response and delays

A virus dynamics model with Beddington–DeAngelis functional response and delays is introduced. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and the LaSalle invariance principle, we show that the infection-free equilibrium is globally asymptotically stable if _R0?1$R_0?1$ and the chronic-infection equilibrium is globally asymptotically stable if _R0>1$R_0>1$. Numerical simulations are also given to explain our results.     

Global dynamics of a mathematical model for HTLV-I infection of CD4 T-cells

In this paper, a mathematical model for HILV-I infection of CD4⁺ T-cells is investigated. The force of infection is assumed be of a function in general form, and the resulting incidence term contains, as special cases, the bilinear and the saturation incidences. The model can be seen as an extension of the model [Wang et al. Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression, Math. Biosci. 179 (2002) 207-217; Song, Li, Global stability and periodic solution of a model for HTLV-I infection and ATL progression, Appl. Math. Comput. 180(1) (2006) 401-410]. Mathematical analysis establishes that the global dynamics of T-cells infection is completely determined by a basic reproduction number _R0${\mathfrak{R}}_0$. If _R0?1${\mathfrak{R}}_0?1$, the infection-free equilibrium is globally stable; if _R0>1${\mathfrak{R}}_0>1$, the unique infected equilibrium is globally stable in the interior of the feasible region.     

Global properties of a general dynamic model for animal diseases: A case study of brucellosis and tuberculosis transmission

Animal diseases such as brucellosis and tuberculosis can be transmitted through an environmentally mediated mechanism, but the topics of most modeling work are based on infectious contact and direct transmission, which leads to the limited understanding of the transmission dynamics of these diseases. In this paper, we propose a new deterministic model which incorporates general incidences, various stages of infection and a general shedding rate of the pathogen to analyze the dynamics of these diseases. Under the biologically motivated assumptions, we derive the basic reproduction number _R0$R_0$, show the uniqueness of the endemic equilibrium, and prove the global asymptotically stability of the equilibria. Some specific examples are used to illustrate the utilization of our results. In addition, we elaborate the epidemiological significance of these results, which are very important for the prevention and control of animal diseases.     

Global stability of a multiple infected compartments model for waterborne diseases

In this paper, mathematical analysis is carried out for a multiple infected compartments model for waterborne diseases, such as cholera, giardia, and rotavirus. The model accounts for both person-to-person and water-to-person transmission routes. Global stability of the equilibria is studied. In terms of the basic reproduction number $R_0$, we prove that, if $R_0⩽1$, then the disease-free equilibrium is globally asymptotically stable and the infection always disappears; whereas if $R_0>1$, there exists a unique endemic equilibrium which is globally asymptotically stable for the corresponding fast–slow system. Numerical simulations verify our theoretical results and present that the decay rate of waterborne pathogens has a significant impact on the epidemic growth rate. Also, we observe numerically that the unique endemic equilibrium is globally asymptotically stable for the whole system. This statement indicates that the present method need to be improved by other techniques.     

Further investigations into the stability and bifurcation of a discrete predator–prey model

In this paper we revisit a discrete predator–prey model with a non-monotonic functional response, originally presented in Hu, Teng, and Zhang (2011) [6]. First, by citing several examples to illustrate the limitations and errors of the local stability of the equilibrium points _E3$E_3$ and _E4$E_4$ obtained in this article, we formulate an easily verified and complete discrimination criterion for the local stability of the two equilibria. Here, we present a very useful lemma, which is a corrected version of a known result, and a key tool in studying the local stability and bifurcation of an equilibrium point in a given system. We then study the stability and bifurcation for the equilibrium point _E1$E_1$ of this system, which has not been considered in any known literature. Unlike known results that present a large number of mathematical formulae that are not easily verified, we formulate easily verified sufficient conditions for flip bifurcation and fold bifurcation, which are explicitly expressed by the coefficient of the system. The center manifold theory and Project Method are the main tools in the analysis of bifurcations. The theoretical results obtained are further illustrated by numerical simulations.     

Permanence and extinction for a nonautonomous SIRS epidemic model with time delay

In this paper, a nonautonomous SIRS epidemic model with time delay is studied. We introduce some new threshold values _R∗$R_{\ast }$ and R^∗$R^{\ast }$ and further obtain the disease will be permanent when _R∗>1$R_{\ast }>1$ and the disease extinct when R^∗<1$R^{\ast }<1$. Using the method of Liapunov functional, some sufficient conditions are derived for the global attractivity of the system. The known results are extended.     

Asymptotics of a Brownian ratchet for protein translocation

Protein translocation in cells has been modelled by Brownian ratchets  . In such models, the protein diffuses through a nanopore. On one side of the pore, ratcheting molecules bind to the protein and hinder it to diffuse out of the pore. We study a Brownian ratchet by means of a reflected Brownian motion _(Xt)t≥0${\left(X_t\right)}_{t\ge 0}$ with a changing reflection point _(Rt)t≥0${\left(R_t\right)}_{t\ge 0}$. The rate of change of _Rt$R_t$ is γ(_Xt−_Rt)$\gamma (X_t-R_t)$ and the new reflection boundary is distributed uniformly between _Rt−$R_{t-}$ and _Xt$X_t$. The asymptotic speed of the ratchet scales with γ^1/3${\gamma }^{1/3}$ and the asymptotic variance is independent of γ$\gamma$.     

Characterizations of hypersurfaces of a Danielewski type

Let k$k$ be a field of characteristic zero and R$R$ a factorial affine k$k$-domain. Let B$B$ be an affineR$R$-domain. In terms of locally nilpotent derivations, we give criteria for B$B$ to be R$R$-isomorphic to the residue ring of a polynomial ring R[_X1,_X2,Y]$R[X_1,X_2,Y]$ over R$R$ by the ideal (_X1X2−φ(Y))$(X_1{X}_2-\phi \left(Y\right))$ for φ(Y)∈R[Y]?R$\phi \left(Y\right)\in R\left[Y\right]?R$.     

Lieb–Thirring type inequalities for non-self-adjoint perturbations of magnetic Schrödinger operators

Let H:=_H0+V$H:=H_0+V$ and _H⊥:=_H0,⊥+V$H_{\perp }:=H_{0,\perp }+V$ be respectively perturbations of the unperturbed Schrödinger operators _H0$H_0$ on L²(R³)$L^2({\mathbb{R}}^3)$ and _H0,⊥$H_{0,\perp }$ on L²(R²)$L^2({\mathbb{R}}^2)$ with constant magnetic field of strength b>0$b>0$, and V a complex relatively compact perturbation. We prove Lieb–Thirring type inequalities on the discrete spectrum of H   and _H⊥$H_{\perp }$. In particular, these estimates give a priori information on the distribution of eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.