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 dynamics of a viral infection model with a latent period and Beddington-DeAngelis response

In this paper, we study the global dynamics of a viral infection model with a latent period. The model has a nonlinear function which denotes the incidence rate of the virus infection in vivo. The basic reproduction number of the virus is identified and it is shown that the uninfected equilibrium is globally asymptotically stable if the basic reproduction number is equal to or less than unity. Moreover, the virus and infected cells eventually persist and there exists a unique infected equilibrium which is globally asymptotically stable if the basic reproduction number is greater than unity. The basic reproduction number determines the equilibrium that is globally asymptotically stable, even if there is a time delay in the infection.     

Global dynamics of SIS models with transport-related infection

To understand the effect of transport-related infection on disease spread, an epidemic model for several regions which are connected by transportation of individuals has been proposed by Cui, Takeuchi and Saito [J. Cui, Y. Takeuchi, Y. Saito, Spreading disease with transport-related infection, J. Theoret. Biol. 239 (2006) 376-390]. Transportation among regions is one of the main factors which affects the outbreak of diseases. The purpose of this paper is the further study of the local asymptotic stability of the endemic equilibrium and the global dynamics of the system. Sufficient conditions are established for global asymptotic stability of the endemic equilibrium. Permanence is also discussed. It is shown that the disease is endemic in the sense of permanence if and only if the endemic equilibrium exists. This implies that transport-related infection on disease can make the disease endemic even if all the isolated regions are disease free.     

Global dynamics of a two-patch SIS model with infection during transport

A two-patch SIS model is formulated and studied. The standard incidence rate and mass-action incidence rate are used within each patch and during transport, respectively. The basic reproduction number is calculated and the global dynamics is investigated. The simulation results show the influence of travel rates, the different dynamics by using standard incidence rate and mass-action incidence rate. The importance of border screening is also explored by numerical simulation.     

Global dynamics of a class of SEIRS epidemic models in a periodic environment

In this paper, we study a class of periodic SEIRS epidemic models and it is shown that the global dynamics is determined by the basic reproduction number R₀ which is defined through the spectral radius of a linear integral operator. If R₀<1, then the disease free periodic solution is globally asymptotically stable and if R₀>1, then the disease persists. Our results really improve the results in [T. Zhang, Z. Teng, On a nonautonomous SEIRS model in epidemiology Bull. Math. Biol. 69 (8) (2007) 2537-2559] for the periodic case. Moreover, from our results, we see that the eradication policy on the basis of the basic reproduction number of the time-averaged system may overestimate the infectious risk of the periodic disease. Numerical simulations which support our theoretical analysis are also given.     

Qualitative and bifurcation analysis using a computer virus model with a saturated recovery function

In this paper, we introduce a saturated treatment function into the computer virus propagation model, where the treatment function is limited for increasing number of infected computers. By carrying out global qualitative and bifurcation analysis, it is shown that the system exhibits some new and complicated behaviors: if the basic reproduction number is larger than unity, the number of infected computers will show persistent behavior, either converging to some positive constant or oscillating; and if the basic reproduction number is below unity, the model may exhibit complicated behaviors including: (i) backward bifurcation; (ii) almost sure virus eradication where the number of infective computers tends to zero for all initial positions except the interior equilibria; (iii) ‘‘oscillating’’ backward bifurcation where either the number of infective computers oscillates persistently, if the initial position lies in a region covering the stable virus equilibrium, or virus eradication, if the initial position lies outside this region; (iv) virus eradication for all initial positions if the basic reproduction number is less than a turning point value.     

Global analysis of a vector-host epidemic model with nonlinear incidences

In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number R₀. It is shown that if R₀ 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 is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively.     

一类具有空间异质和扩散的SVIR模型的全局稳定性

考虑了一类具有空间异质和反应扩散的SVIR传染病模型.当基本再生数等于1时,假定扩散系数为常数,证明了系统的无病平衡态是全局渐近稳定的.     

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.     

Threshold dynamics for a cholera epidemic model with periodic transmission rate

A cholera epidemic model with periodic transmission rate is presented. The basic reproduction number is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the cholera eventually disappears if the basic reproduction number is less than one. And if the basic reproduction number is greater than one, there exists a positive periodic solution which is globally asymptotically stable. Numerical simulations are provided to illustrate analytical results.     

Global dynamics of a cell mediated immunity in viral infection models with distributed delays

In this paper, we investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in vivo. The model has two distributed time delays describing time needed for infection of cell and virus replication. Our model admits three possible equilibria, an uninfected equilibrium and infected equilibrium with or without immune response depending on the basic reproduction number for viral infection R₀ and for CTL response R₁ such that R₁<R₀. It is shown that there always exists one equilibrium which is globally asymptotically stable by employing the method of Lyapunov functional. More specifically, the uninfected equilibrium is globally asymptotically stable if R₀?1, an infected equilibrium without immune response is globally asymptotically stable if R₁?1<R₀ and an infected equilibrium with immune response is globally asymptotically stable if R₁>1. The immune activation has a positive role in the reduction of the infection cells and the increasing of the uninfected cells if R₁>1.     

A viral infection model with immune circadian rhythms

A viral infection model with immune circadian rhythms is investigated in this paper. By employing the persistence theory, we establish a threshold between the extinction and the uniform persistence of the disease. These results can be used to explain the oscillation behaviors of virus population, which were observed in chronic HBV or HCV carriers. Further, numerical simulations indicate that the dynamics of the lytic component of cytotoxicity T cells (CTLs) is crucial to the outcome of a viral infection.