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### Open Access Journals

DCDS-B

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.

MBE

*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.

MBE

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

DCDS-B

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.

DCDS-B

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.

MBE

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.

MBE

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

DCDS-B

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.

DCDS-B

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.

MBE

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.

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.

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