Asymptotic properties of a delayed SIR epidemic model with density dependent birth rate
Wanbiao Ma Yasuhiro Takeuchi
In this paper, we consider a delayed $SIR$ epidemic model with density dependent birth process. For the model with larger birth rate, we discuss the asymptotic property of its solutions. Furthermore, we also study the existence of Hopf bifurcation from the endemic equilibrium of the model and local asymptotic stability of the endemic equilibrium.
keywords: time delay Epidemic model asymptotic stability.
A dynamic model describing heterotrophic culture of chorella and its stability analysis
Yan Zhang Wanbiao Ma Hai Yan Yasuhiro Takeuchi
Chlorella is an important species of microorganism, which includes about 10 species. Chlorella USTB01 is a strain of microalga which is isolated from Qinghe River in Beijing and has strong ability in the utilization of organic compounds and was identified as Chlorella sp. (H. Yan etal, Isolation and heterotrophic culture of Chlorella sp., J. Univ. Sci. Tech. Beijing, 2005, 27:408-412). In this paper, based on the standard Chemostat models and the experimental data on the heterotrophic culture of Chlorella USTB01, a dynamic model governed by differential equations with three variables (Chlorella, carbon source and nitrogen source) is proposed. For the model, there always exists a boundary equilibrium, i.e. Chlorella-free equilibrium. Furthermore, under additional conditions, the model also has the positive equilibria, i.e., the equilibira for which Chlorella, carbon source and nitrogen source are coexistent. Then, local and global asymptotic stability of the equilibria of the model have been discussed. Finally, the parameters in the model are determined according to the experimental data, and numerical simulations are given. The numerical simulations show that the trajectories of the model fit the trends of the experimental data well.
keywords: carbon source differential equation Chlorella nitrogen source 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
Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response
Haiyin Li Yasuhiro Takeuchi
We investigate the dynamics of a non-autonomous and density dependent predator-prey system with Beddington-DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered. First, we derive a sufficient condition of permanence by comparison theorem, at the same time we propose a weaker condition ensuring some positive bounded set to be positive invariant. Next, we obtain two existence conditions for positive periodic solution by Brouwer fixed-point theorem and by continuation theorem, where the second condition is weaker than the first and gives the existence range of periodic solution. Further we show the global attractivity of the bounded positive solution by constructing Lyapunov function. Similarly, we have sufficient condition of global attractivity of boundary periodic solution.
keywords: Density dependent predator periodic solution. Beddington-DeAngelis functional response permanence
Mathematical modeling on helper T cells in a tumor immune system
Yueping Dong Rinko Miyazaki Yasuhiro Takeuchi
Activation of CD$8^+$ cytotoxic T lymphocytes (CTLs) is naturally regarded as a major antitumor mechanism of the immune system. In contrast, CD$4^+$ T cells are commonly classified as helper T cells (HTCs) on the basis of their roles in providing help to the generation and maintenance of effective CD$8^+$ cytotoxic and memory T cells. In order to get a better insight on the role of HTCs in a tumor immune system, we incorporate the third population of HTCs into a previous two dimensional ordinary differential equations (ODEs) model. Further we introduce the adoptive cellular immunotherapy (ACI) as the treatment to boost the immune system to fight against tumors. Compared tumor cells (TCs) and effector cells (ECs), the recruitment of HTCs changes the dynamics of the system substantially, by the effects through particular parameters, i.e., the activation rate of ECs by HTCs, $p$ (scaled as $\rho$), and the HTCs stimulation rate by the presence of identified tumor antigens, $k_2$ (scaled as $\omega_2$). We describe the stability regions of the interior equilibria $E^*$ (no treatment case) and $E^+$ (treatment case) in the scaled $(\rho,\omega_2)$ parameter space respectively. Both $\rho$ and $\omega_2$ can destabilize $E^*$ and $E^+$ and cause Hopf bifurcations. Our results show that HTCs might play a crucial role in the long term periodic oscillation behaviors of tumor immune system interactions. They also show that TCs may be eradicated from the patient's body under the ACI treatment.
keywords: helper T cells periodic solutions Hopf bifurcation. Tumor immune system
Mathematical analysis of a HIV model with frequency dependence and viral diversity
Shingo Iwami Shinji Nakaoka Yasuhiro Takeuchi
We consider the effect of viral diversity on the human immune sys- tem with the frequency dependent proliferation rate of CTLs and the elimina- tion rate of infected cells by CTLs. The model has very complex mathematical structures such as limit cycle, quasi-periodic attractors, chaotic attractors, and so on. To understand the complexity we investigate the global behavior of the model and demonstrate the existence and stability conditions of the equilibria. Further we give some theoretical considerations obtained by our mathematical model to HIV infection.
keywords: frequency dependence viral diversity stability analysis. HIV model
Construction of Lyapunov functions for some models of infectious diseases in vivo: From simple models to complex models
Tsuyoshi Kajiwara Toru Sasaki Yasuhiro Takeuchi
We present a constructive method for Lyapunov functions for ordinary differential equation models of infectious diseases in vivo. We consider models derived from the Nowak-Bangham models. We construct Lyapunov functions for complex models using those of simpler models. Especially, we construct Lyapunov functions for models with an immune variable from those for models without an immune variable, a Lyapunov functions of a model with absorption effect from that for a model without absorption effect. We make the construction clear for Lyapunov functions proposed previously, and present new results with our method.
keywords: stability Lyapunov functions immunity. ordinary differential equations infectious diseases
Stability conditions for a class of delay differential equations in single species population dynamics
Gang Huang Yasuhiro Takeuchi Rinko Miyazaki
We consider a class of nonlinear delay differential equations,which describes single species population growth with stage structure. By constructing appropriate Lyapunov functionals, the global asymptotic stability criteria, which are independent of delay, are established. Much sharper stability conditions than known results are provided. Applications of the results to some population models show the effectiveness of the methods described in the paper.
keywords: single species Lyapunov functionals global stability. Delay differential equation
Population dynamics of sea bass and young sea bass
Masahiro Yamaguchi Yasuhiro Takeuchi Wanbiao Ma
This paper considers population dynamics of sea bass and young sea bass which are modeled by stage-structured delay-differential equations. It is shown that time delay can stabilize the dynamics. That is, as time delay increases, system becomes periodic and stable even if system without time delay is chaotic.
keywords: Delay equations local stability Lotka-Volterra equation.
Global stability for epidemic model with constant latency and infectious periods
Gang Huang Edoardo Beretta Yasuhiro Takeuchi
In recent years many delay epidemiological models have been proposed to study at which stage of the epidemics the delays can destabilize the disease free equilibrium, or the endemic equilibrium, giving rise to stability switches. One of these models is the SEIR model with constant latency time and infectious periods [2], for which the authors have proved that the two delays are harmless in inducing stability switches. However, it is left open the problem of the global asymptotic stability of the endemic equilibrium whenever it exists. Even the Lyapunov functions approach, recently proposed by Huang and Takeuchi to study many delay epidemiological models, fails to work on this model. In this paper, an age-infection model is presented for the delay SEIR epidemic model, such that the properties of global asymptotic stability of the equilibria of the age-infection model imply the same properties for the original delay-differential epidemic model. By introducing suitable Lyapunov functions to study the global stability of the disease free equilibrium (when $\mathcal{R}_0\leq 1$) and of the endemic equilibria (whenever $ \mathcal{R}_0>1$) of the age-infection model, we can infer the corresponding global properties for the equilibria of the delay SEIR model in [2], thus proving that the endemic equilibrium in [2] is globally asymptotically stable whenever it exists.
    Furthermore, we also present a review of the SIR, SEIR epidemic models, with and without delays, appeared in literature, that can be seen as particular cases of the approach presented in the paper.
keywords: epidemic model infectious period Global stability age-structure.

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