Global stability for a heroin model with two distributed delays
Bin Fang Xue-Zhi Li Maia Martcheva Li-Ming Cai
Discrete & Continuous Dynamical Systems - B 2014, 19(3): 715-733 doi: 10.3934/dcdsb.2014.19.715
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
Global dynamics of a vector-host epidemic model with age of infection
Yan-Xia Dang Zhi-Peng Qiu Xue-Zhi Li Maia Martcheva
Mathematical Biosciences & Engineering 2017, 14(5&6): 1159-1186 doi: 10.3934/mbe.2017060

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
A cost-based comparison of quarantine strategies for new emerging diseases
Anuj Mubayi Christopher Kribs Zaleta Maia Martcheva Carlos Castillo-Chávez
Mathematical Biosciences & Engineering 2010, 7(3): 687-717 doi: 10.3934/mbe.2010.7.687
A classical epidemiological framework is used to provide a preliminary cost analysis of the effects of quarantine and isolation on the dynamics of infectious diseases for which no treatment or immediate diagnosis tools are available. Within this framework we consider the cost incurred from the implementation of three types of dynamic control strategies. Taking the context of the 2003 SARS outbreak in Hong Kong as an example, we use a simple cost function to compare the total cost of each mixed (quarantine and isolation) control strategy from a public health resource allocation perspective. The goal is to extend existing epi-economics methodology by developing a theoretical framework of dynamic quarantine strategies aimed at emerging diseases, by drawing upon the large body of literature on the dynamics of infectious diseases. We find that the total cost decreases with increases in the quarantine rates past a critical value, regardless of the resource allocation strategy. In the case of a manageable outbreak resources must be used early to achieve the best results whereas in case of an unmanageable outbreak, a constant-effort strategy seems the best among our limited plausible sets.
keywords: reproductive number SARS isolation cost-effectiveness analysis. quarantine Contact tracing
An age-structured two-strain epidemic model with super-infection
Xue-Zhi Li Ji-Xuan Liu Maia Martcheva
Mathematical Biosciences & Engineering 2010, 7(1): 123-147 doi: 10.3934/mbe.2010.7.123
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
Epidemic models with age of infection, indirect transmission and incomplete treatment
Liming Cai Maia Martcheva Xue-Zhi Li
Discrete & Continuous Dynamical Systems - B 2013, 18(9): 2239-2265 doi: 10.3934/dcdsb.2013.18.2239
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
Subthreshold coexistence of strains: the impact of vaccination and mutation
Maia Martcheva Mimmo Iannelli Xue-Zhi Li
Mathematical Biosciences & Engineering 2007, 4(2): 287-317 doi: 10.3934/mbe.2007.4.287
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
Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion
Yixiang Wu Necibe Tuncer Maia Martcheva
Discrete & Continuous Dynamical Systems - B 2017, 22(3): 1167-1187 doi: 10.3934/dcdsb.2017057

This paper investigates a two strain SIS model with diffusion, spatially heterogeneous coefficients of the reaction part and distinct diffusion rates of the separate epidemiological classes. First, it is shown that the model has bounded classical solutions. Next, it is established that the model with spatially homogeneous coefficients leads to competitive exclusion and no coexistence is possible in this case. Furthermore, it is proved that if the invasion number of strain $j$ is larger than one, then the equilibrium of strain $i$ is unstable; if, on the other hand, the invasion number of strain $j$ is smaller than one, then the equilibrium of strain $i$ is neutrally stable. In the case when all diffusion rates are equal, global results on competitive exclusion and coexistence of the strains are established. Finally, evolution of dispersal scenario is considered and it is shown that the equilibrium of the strain with the larger diffusion rate is unstable. Simulations suggest that in this case the equilibrium of the strain with the smaller diffusion rate is stable.

keywords: Evolution of dispersal competitive exclusion reproduction numbers invasion numbers epidemic model with diffusion

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