DCDS-B
Global stability of an age-structured SIRS epidemic model with vaccination
Geni Gupur Xue-Zhi Li
This paper focuses on the study of an age-structured SIRS epidemic model with a vaccination program. We first give the explicit expression of the reproductive number $ \mathcal{R}(\psi) $ in the presence of vaccine, and show that the infection-free steady state is locally asymptotically stable if $ \mathcal{R}(\psi)<1 $ and unstable if $ \mathcal{R}(\psi)>1 $. Second, we prove that the infection-free state is globally stable if the basic reproductive number $ \mathcal{R}_0 <1 $, and that an endemic equilibrium exists when the reproductive number $ \mathcal{R}(\psi)>1 $.
keywords: infection-free steady state endemic state global stability. vaccination Age-structured SIRS epidemic model reproductive number
DCDS-B
Global stability for a heroin model with two distributed delays
Bin Fang Xue-Zhi Li Maia Martcheva Li-Ming Cai
In this paper, we consider global stability for a heroin model with two distributed delays. The basic reproduction number of the heroin spread is obtained, which completely determines the stability of the equilibria. Using the direct Lyapunov method with Volterra type Lyapunov function, we show that the drug use-free equilibrium is globally asymptotically stable if the basic reproduction number is less than one, and the unique drug spread equilibrium is globally asymptotically stable if the basic reproduction number is greater than one.
keywords: distributed delay Heroin model Lyapunov function global stability. basic reproduction number
MBE
Global dynamics of a vector-host epidemic model with age of infection
Yan-Xia Dang Zhi-Peng Qiu Xue-Zhi Li Maia Martcheva

In this paper, a partial differential equation (PDE) model is proposed to explore the transmission dynamics of vector-borne diseases. The model includes both incubation age of the exposed hosts and infection age of the infectious hosts which describe incubation-age dependent removal rates in the latent period and the variable infectiousness in the infectious period, respectively. The reproductive number $\mathcal R_0$ is derived. By using the method of Lyapunov function, the global dynamics of the PDE model is further established, and the results show that the basic reproduction number $\mathcal R_0$ determines the transmission dynamics of vector-borne diseases: the disease-free equilibrium is globally asymptotically stable if $\mathcal R_0≤ 1$, and the endemic equilibrium is globally asymptotically stable if $\mathcal{R}_0>1$. The results suggest that an effective strategy to contain vector-borne diseases is decreasing the basic reproduction number $\mathcal{R}_0$ below one.

keywords: Age structure reproduction number global stability vector-borne disease Lyapunov function
DCDS-B
Epidemic models with age of infection, indirect transmission and incomplete treatment
Liming Cai Maia Martcheva Xue-Zhi Li
An infection-age-structured epidemic model with environmental bacterial infection is investigated in this paper. It is assumed that the infective population is structured according to age of infection, and the infectivity of the treated individuals is reduced but varies with the infection-age. An explicit formula for the reproductive number $ \Re_0$ of the model is obtained. By constructing a suitable Lyapunov function, the global stability of the infection-free equilibrium in the system is obtained for $\Re_0<1$. It is also shown that if the reproduction number $\Re_0>1$, then the system has a unique endemic equilibrium which is locally asymptotically stable. Furthermore, if the reproduction number $\Re_0>1$, the system is permanent. When the treatment rate and the transmission rate are both independent of infection age, the system of partial differential equations (PDEs) reduces to a system of ordinary differential equations (ODEs). In this special case, it is shown that the global dynamics of the system can be determined by the basic reproductive number.
keywords: Lyapunov function infection-age-structured. global stability Epidemic model
MBE
Subthreshold coexistence of strains: the impact of vaccination and mutation
Maia Martcheva Mimmo Iannelli Xue-Zhi Li
We consider a model for a disease with two competing strains and vaccination. The vaccine provides complete protection against one of the strains (strain 2) but only partial protection against the other (strain 1). The partial protection leads to existence of subthreshold equilibria of strain 1. If the first strain mutates into the second, there are subthreshold coexistence equilibria when both vaccine-dependent reproduction numbers are below one. Thus, a vaccine that is specific toward the second strain and that, in absence of other strains, should be able to eliminate the second strain by reducing its reproduction number below one, cannot do so because it provides only partial protection to another strain that mutates into the second strain.
keywords: latent stage coexistence strongly subthreshold coexistence vaccine enhanced pathogen polymorphism. multiple coexistence equilibria multiple endemic equilibria mutation backward bifurcation latent-stage progression age structure alternating stability vaccination
MBE
Global stability of an age-structured cholera model
Jianxin Yang Zhipeng Qiu Xue-Zhi Li
In this paper, an age-structured epidemic model is formulated to describe the transmission dynamics of cholera. The PDE model incorporates direct and indirect transmission pathways, infection-age-dependent infectivity and variable periods of infectiousness. Under some suitable assumptions, the PDE model can be reduced to the multi-stage models investigated in the literature. By using the method of Lyapunov function, we established the dynamical properties of the PDE model, and the results show that the global dynamics of the model is completely determined by the basic reproduction number $\mathcal R_0$: if $\mathcal R_0 < 1$ the cholera dies out, and if $\mathcal R_0 >1$ the disease will persist at the endemic equilibrium. Then the global results obtained for multi-stage models are extended to the general continuous age model.
keywords: Age of infection global stability Lyapunov function. cholera
MBE
An age-structured two-strain epidemic model with super-infection
Xue-Zhi Li Ji-Xuan Liu Maia Martcheva
This article focuses on the study of an age-structured two-strain model with super-infection. The explicit expression of basic reproduction numbers and the invasion reproduction numbers corresponding to strain one and strain two are obtained. It is shown that the infection-free steady state is globally stable if the basic reproductive number $ R_0 $ is below one. Existence of strain one and strain two exclusive equilibria is established. Conditions for local stability or instability of the exclusive equilibria of the strain one and strain two are established. Existence of coexistence equilibrium is also obtained under the condition that both invasion reproduction numbers are larger than one.
keywords: age-structured coexistence equilibrium two-strain epidemic model super-infection basic reproduction number stability. invasion reproduction number exclusive equilibrium

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