Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay
Jinliang Wang Lijuan Guan
A recent paper [H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete and Continuous Dynamical Systems - Series B, 12(2009), 511--524] presented a mathematical model for HIV-1 infection with intracellular delay and cell-mediated immune response. By combining the analysis of the characteristic equation and the Lyapunov-LaSalle method, they obtain a necessary and sufficient condition for the global stability of the infection-free equilibrium and give sufficient conditions for the local stability of the two infection equilibria: one without CTLs being activated and the other with. In the present paper, we show that the global dynamics are fully determined for $\Re_1<1<\Re_0$ and $\Re_1>1$ (Theorem 4.2 and Theorem 4.3) without other additional conditions. The approach used here, is to use a direct Lyapunov functional and Lyapunov-LaSalle invariance principle.
keywords: Lyapunov functional. Global stability Intracellular delay Immune response
Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility
Jinliang Wang Xianning Liu Toshikazu Kuniya Jingmei Pang

In this paper, we investigate the global asymptotic stability of multi-group SIR and SEIR age-structured models. These models allow the infectiousness and the death rate of susceptible individuals to vary and depend on the susceptibility, with which we can consider the heterogeneity of population. We establish global dynamics and demonstrate that the heterogeneity does not alter the dynamical structure of the basic SIR and SEIR with age-dependent susceptibility. Our results also demonstrate that, for age structured multi-group models considered, the graph-theoretic approach can be successfully applied by choosing an appropriate weighted matrix as well.

keywords: Age-structured model susceptibility asymptotic smoothness global attractor global stability
Sveir epidemiological model with varying infectivity and distributed delays
Jinliang Wang Gang Huang Yasuhiro Takeuchi Shengqiang Liu
In this paper, based on an SEIR epidemiological model with distributed delays to account for varying infectivity, we introduce a vaccination compartment, leading to an SVEIR model. By employing direct Lyapunov method and LaSalle's invariance principle, we construct appropriate functionals that integrate over past states to establish global asymptotic stability conditions, which are completely determined by the basic reproduction number $\mathcal{R}_0^V$. More precisely, it is shown that, if $\mathcal{R}_0^V\leq 1$, then the disease free equilibrium is globally asymptotically stable; if $\mathcal{R}_0^V > 1$, then there exists a unique endemic equilibrium which is globally asymptotically stable. Mathematical results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number. However, there is a necessary condition for successful elimination of disease. If the time for the vaccinees to obtain immunity or the possibility for them to be infected before acquiring immunity can be neglected, this condition would be satisfied and the disease can always be eradicated by some suitable vaccination strategies. This may lead to over-evaluating the effect of vaccination.
keywords: global stability. vaccination strategy distributed delays varying infectivity Epidemic model
Analysis of an HIV infection model incorporating latency age and infection age
Jinliang Wang Xiu Dong

There is a growing interest to understand impacts of latent infection age and infection age on viral infection dynamics by using ordinary and partial differential equations. On one hand, activation of latently infected cells needs specificity antigen, and latently infected CD4+ T cells are often heterogeneous, which depends on how frequently they encountered antigens, how much time they need to be preferentially activated and quickly removed from the reservoir. On the other hand, infection age plays an important role in modeling the death rate and virus production rate of infected cells. By rigorous analysis for the model, this paper is devoted to the global dynamics of an HIV infection model subject to latency age and infection age from theoretical point of view, where the model formulation, basic reproduction number computation, and rigorous mathematical analysis, such as relative compactness and persistence of the solution semiflow, and existence of a global attractor are involved. By constructing Lyapunov functions, the global dynamics of a threshold type is established. The method developed here is applicable to broader contexts of investigating viral infection subject to age structure.

keywords: HIV infection latency infection age global stability Lyapunov function
A note on global stability for malaria infections model with latencies
Jinliang Wang Jingmei Pang Toshikazu Kuniya
A recent paper [Y. Xiao and X. Zou, On latencies in malaria infections and their impact on the disease dynamics, Math. Biosci. Eng., 10(2) 2013, 463-481.] presented a mathematical model to investigate the spread of malaria. The model is obtained by modifying the classic Ross-Macdonald model by incorporating latencies both for human beings and female mosquitoes. It is realistic to consider the new model with latencies differing from individuals to individuals. However, the analysis in that paper did not resolve the global malaria disease dynamics when $\Re_0>1$. The authors just showed global stability of endemic equilibrium for two specific probability functions: exponential functions and step functions. Here, we show that if there is no recovery, the endemic equilibrium is globally stable for $\Re_0>1$ without other additional conditions. The approach used here, is to use a direct Lyapunov functional and Lyapunov- LaSalle invariance principle.
keywords: Global stability malaria infection latency distribution Lyapunov functional.
Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission
Jinliang Wang Jiying Lang Yuming Chen

In this paper, we are concerned with an age-structured HIV infection model incorporating latency and cell-to-cell transmission. The model is a hybrid system consisting of coupled ordinary differential equations and partial differential equations. First, we address the relative compactness and persistence of the solution semi-flow, and the existence of a global attractor. Then, applying the approach of Lyapunov functionals, we establish the global stability of the infection-free equilibrium and the infection equilibrium, which is completely determined by the basic reproduction number.

keywords: HIV infection cell-to-cell transmission latency equilibrium global stability Lyapunov functional
Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells
Jinliang Wang Jiying Lang Xianning Liu
In this paper, we formulate a viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. The model can describe the in vivo infection dynamics of many viruses such as HIV-I, HCV, and HBV, where the infected cells of eclipse stage can revert to the uninfected class. Under certain parameters range, we establish that the global stability of equilibria is completely determined by the basic reproduction number $\mathfrak{R}_0$, which give us a complete picture on their global dynamics.
keywords: global stability Nonlinear infection rate eclipse stage Lyapunov function.
Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations
Lili Liu Xianning Liu Jinliang Wang
The aim of this paper is to investigate the threshold dynamics of a heroin epidemic in heterogeneous populations. The model is described by a delayed multi-group model, which allows us to model interactions both within-group and inter-group separately. Here we are able to prove the existence of heroin-spread equilibrium and the uniform persistence of the model. The proofs of main results come from suitable applications of graph-theoretic approach to the method of Lyapunov functionals and Krichhoff's matrix tree theorem. Numerical simulations are performed to support the results of the model for the case where $n=2$.
keywords: delay graph-theoretic approach global stability. Multi-group heroin epidemic model heterogeneous populations
A multi-group SIR epidemic model with age structure
Toshikazu Kuniya Jinliang Wang Hisashi Inaba
This paper provides the first detailed analysis of a multi-group SIR epidemic model with age structure, which is given by a nonlinear system of $3n$ partial differential equations. The basic reproduction number $\mathcal{R}_0$ is obtained as the spectral radius of the next generation operator, and it is shown that if $\mathcal{R}_0 < 1$, then the disease-free equilibrium is globally asymptotically stable, while if $\mathcal{R}_0 >1$, then an endemic equilibrium exists. The global asymptotic stability of the endemic equilibrium is also shown under additional assumptions such that the transmission coefficient is independent from the age of infective individuals and the mortality and removal rates are constant. To our knowledge, this is the first paper which applies the method of Lyapunov functional and graph theory to a multi-dimensional PDE system.
keywords: SIR epidemic model age structure multi-group model the basic reproduction number.
A note on dynamics of an age-of-infection cholera model
Jinliang Wang Ran Zhang Toshikazu Kuniya
A recent paper [F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10, 2013, 1335--1349.] presented a model for the dynamics of cholera transmission. The model is incorporated with both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is proved to be a sharp threshold determining whether or not cholera dies out. The global stability for disease-free equilibrium and endemic equilibrium is proved by constructing suitable Lyapunov functionals. However, for the proof of the global stability of endemic equilibrium, we have to show first the relative compactness of the orbit generated by model in order to make use of the invariance principle. Furthermore, uniform persistence of system must be shown since the Lyapunov functional is possible to be infinite if $i(a, t)/i^* (a) =0$ on some age interval. In this note, we give a supplement to above paper with necessary mathematical arguments.
keywords: global stability Cholera model Lyapunov functional uniform persistence. age-of-infection

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