Global dynamics of a mathematical model for HTLV-I infection of CD4 T cells with delayed CTL response

Human T-cell leukaemia virus type I (HTLV-I) preferentially infects the CD4⁺ T cells. The HTLV-I infection causes a strong HTLV-I specific immune response from CD8⁺ cytotoxic T cells (CTLs). The persistent cytotoxicity of the CTL is believed to contribute to the development of a progressive neurologic disease, HTLV-I associated myelopathy/tropical spastic paraparesis (HAM/TSP). We investigate the global dynamics of a mathematical model for the CTL response to HTLV-I infection in vivo. To account for a series of immunological events leading to the CTL response, we incorporate a time delay in the response term. Our mathematical analysis establishes that the global dynamics are determined by two threshold parameters R₀ and R₁, basic reproduction numbers for viral infection and for CTL response, respectively. If R₀≤1, the infection-free equilibrium P₀ is globally asymptotically stable, and the HTLV-I viruses are cleared. If R₁≤1<R₀, the asymptomatic-carrier equilibrium P₁ is globally asymptotically stable, and the HTLV-I infection becomes chronic but with no persistent CTL response. If R₁>1, a unique HAM/TSP equilibrium P₂ exists, at which the HTLV-I infection is chronic with a persistent CTL response. We show that the time delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations and stable periodic oscillations. Implications of our results to the pathogenesis of HTLV-I infection and HAM/TSP development are discussed.     

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.     

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.     

Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates

In this paper, we present a new delay multigroup SEIR model with group mixing and nonlinear incidence rates and investigate its global stability. We establish that the global dynamics of the models are completely determined by the basic reproduction number R₀. It is shown that, if R₀?1, then the disease free equilibrium is globally asymptotically stable and the disease dies out; if R₀>1, there exists a unique endemic equilibrium that is globally asymptotically stable and thus the disease persists in the population. Finally, a numerical example is also discussed to illustrate the effectiveness of the results.     

Global analysis of an environmental disease transmission model linking within-host and between-host dynamics

In this paper, a multi-scale mathematical model for environmentally transmitted diseases is proposed which couples the pathogen-immune interaction inside the human body with the disease transmission at the population level. The model is based on the nested approach that incorporates the infection-age-structured immunological dynamics into an epidemiological system structured by the chronological time, the infection age and the vaccination age. We conduct detailed analysis for both the within-host and between-host disease dynamics. Particularly, we derive the basic reproduction number R₀ for the between-host model and prove the uniform persistence of the system. Furthermore, using carefully constructed Lyapunov functions, we establish threshold-type results regarding the global dynamics of the between-host system: the disease-free equilibrium is globally asymptotically stable when R₀ < 1, and the endemic equilibrium is globally asymptotically stable when R₀ > 1. We explore the connection between the within-host and between-host dynamics through both mathematical analysis and numerical simulation. We show that the pathogen load and immune strength at the individual level contribute to the disease transmission and spread at the population level. We also find that, although the between-host transmission risk correlates positively with the within-host pathogen load, there is no simple monotonic relationship between the disease prevalence and the individual pathogen load.     

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.     

The global dynamics of a model about HIV-1 infection in vivo

A four dimension ODE model is built to study the infection of human immunodeficiency virus (HIV) in vivo. We include in this model four components: the healthy T cells, the latent-infected T cells, the active-infected T cells and the HIV virus. Two types of HIV transmissions in vivo are also included in the model: the virus-to-cell transmission, and the cell-to-cell HIV transmission. There are two possible equilibriums: the healthy equilibrium, and the infected steady state. The basic reproduction number R ₀ is introduced. When R ₀ < 1, the healthy equilibrium is globally stable and when R ₀ > 1, the infected equilibrium exists and is globally stable. Through simulations, we find that, the cell-to-cell HIV transmission is very important for the final outcome of the HIV attacking. Some important clinical observations about the HIV infection situation in lymph node are also verified.      

Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission

The dynamics of multi-group SEIR epidemic models with distributed and infinite delay and nonlinear transmission are investigated. We derive the basic reproduction number R₀ and establish that the global dynamics are completely determined by the values of R₀: if R₀≤1, then the disease-free equilibrium is globally asymptotically stable; if R₀>1, then there exists a unique endemic equilibrium which is globally asymptotically stable. Our results contain those for single-group SEIR models with distributed and infinite delays. In the proof of global stability of the endemic equilibrium, we exploit a graph-theoretical approach to the method of Lyapunov functionals. The biological significance of the results is also discussed.     

Global stability for an HIV/AIDS epidemic model with different latent stages and treatment

An HIV/AIDS epidemic model with different latent stages and treatment is constructed. The model allows for the latent individuals to have the slow and fast latent compartments. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are determined by the basic reproduction number under some conditions. If R₀ < 1, the disease free equilibrium is globally asymptotically stable, and if R₀ > 1, the endemic equilibrium is globally asymptotically stable for a special case. Some numerical simulations are also carried out to confirm the analytical results.     

Analysis of SE τ IR ω S epidemic disease models with vertical transmission in complex networks

When the role of network topology is taken into consideration, one of the objectives is to understand the possible implications of topological structure on epidemic models. As most real networks can be viewed as complex networks, we propose a new delayed SEτ IRωS epidemic disease model with vertical transmission in complex networks. By using a delayed ODE system, in a small-world (SW) network we prove that, under the condition R0 ≤ 1, the disease-free equilibrium (DFE) is globally stable. When R0 > 1, the endemic equilibrium is unique and the disease is uniformly persistent. We further obtain the condition of local stability of endemic equilibrium for R0 > 1. In a scale-free (SF) network we obtain the condition R1 > 1 under which the system will be of non-zero stationary prevalence.