# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

2010 , Volume 13 , Issue 1

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2010, 13(1): 1-25 doi: 10.3934/dcdsb.2010.13.1 +[Abstract](249) +[PDF](357.0KB)
Abstract:
This study presents a deterministic model for theoretically assessing the potential impact of an imperfect avian influenza vaccine (for domestic birds) in two avian populations on the transmission dynamics of avian influenza in the domestic and wild birds population. The model is analyzed to gain insights into the qualitative features of its associated equilibria. This allows the determination of important epidemiological thresholds such as the basic reproduction number and a measure for vaccine impact. A sub-model without vaccination is first considered, where it is shown that it has a globally-asymptotically stable disease-free equilibrium whenever a certain reproduction threshold is less than unity. Unlike the sub-model without vaccination, the model with vaccination undergoes backward bifurcation, a phenomenon associated with the co-existence of multiple stable equilibria. In other words, for the model with vaccination, the classical epidemiological requirement of having the associated reproduction number less than unity does not guarantee disease elimination in the model. It is shown that the possibility of backward bifurcation occurring decreases with increasing vaccination rate (for susceptible domestic birds). Further, the study shows that the vaccine impact (in reducing disease burden) is dependent on the sign of a certain threshold quantity (denoted by $\nabla_{\mathcal P}$). The vaccine will have positive or no impact if $\nabla_{\mathcal P}$ is less than or equal to unity. Numerical simulations suggest that the prospect of effectively controlling the disease in the avian population increases with increasing vaccine efficacy and coverage.
2010, 13(1): 27-57 doi: 10.3934/dcdsb.2010.13.27 +[Abstract](227) +[PDF](312.2KB)
Abstract:
In this study, we investigate the connection between impulse control and singular control. The dynamic systems are driven by Brownian motion with drift. For simplicity we consider only one-dimension problems, where we can perform explicit calculations. We will see that Quasi-Variational Inequalities (QVI) are the common tool to consider these problems. The two problems have interesting links. By some aspects, singular control problems appear as particular cases of impulse control problems; however an impulse control is a particular case of singular control. We can, in particular, approximate an optimal singular control by a minimizing sequence of impulse controls. We show that optimal singular controls are linked to reflected diffusions. Thanks to the one-dimensionality we completely solve the QVI by the two band approach.
2010, 13(1): 59-78 doi: 10.3934/dcdsb.2010.13.59 +[Abstract](228) +[PDF](625.3KB)
Abstract:
This paper concerns optimization of data traffic flows on a telecommunication network, modelled using a fluid - dynamic approach. Flows can be controlled adjusting traffic distribution and priority parameters. Two cost functionals are considered, which measure average velocity and average travelling time of packets.
First we address general optimal control problems, showing that existence of solutions is related to properties of packet loss probability functions.
A direct solution of the general optimal control problem corresponds to a centralized policy and is hard to achieve, thus we focus on a decentralized policy and provide solutions for a single node with two entering and two exiting lines and asymptotic costs. Such solutions permit to simulate the behaviour of decentralized algorithms for complex networks. Local optimization ensures very good results also for large networks. The latter is shown by case study of a test telecommunication network.
2010, 13(1): 79-100 doi: 10.3934/dcdsb.2010.13.79 +[Abstract](220) +[PDF](305.8KB)
Abstract:
In this paper we deal with diffusive relaxation limits of nonlinear systems of Euler type modeling chemotactic movement of cells toward Keller-Segel type systems. The approximating systems are either hyperbolic-parabolic or hyperbolic-elliptic. They all feature a nonlinear pressure term arising from a volume filling effect which takes into account the fact that cells do not interpenetrate. The main convergence result relies on energy methods and compensated compactness tools and is obtained for large initial data under suitable assumptions on the approximating solutions. In order to justify such assumptions, we also prove an existence result for initial data which are small perturbation of a constant state. Such result is proven via classical Friedrichs's symmetrization and linearization. In order to simplify the coverage, we restrict to the two-dimensional case with periodical boundary conditions.
2010, 13(1): 101-116 doi: 10.3934/dcdsb.2010.13.101 +[Abstract](245) +[PDF](2502.3KB)
Abstract:
We use the iterative grid redistribution method(IGR) to solve Prandtl equations and study the self-similar behavior of the singular solutions of them. The IGR method enables us to get more accurate solutions of Prandtl equations when they develop singularity. We also study the self-similar behavior of the singular solutions. Blow up rate and blow up profiles are derived and the results are verified by the numerical solutions.
2010, 13(1): 117-127 doi: 10.3934/dcdsb.2010.13.117 +[Abstract](191) +[PDF](159.7KB)
Abstract:
In this paper we consider $C^{0}$ random perturbations of a hyperbolic set of a flow. We show that the hyperbolic set is structurally semi-stable under such perturbations.
2010, 13(1): 129-156 doi: 10.3934/dcdsb.2010.13.129 +[Abstract](310) +[PDF](631.5KB)
Abstract:
This paper focuses on the characterization of delay effects on the asymptotic stability of some continuous-time delay systems encountered in modeling the post-transplantation dynamics of the immune response to chronic myelogenous leukemia. Such models include multiple delays in some large range, from one minute to several days. The main objective of the paper is to study the stability of the crossing boundaries of the corresponding linearized models in the delay-parameter space by taking into account the interactions between small and large delays. Weak, and strong cell interactions are discussed, and analytic characterizations are proposed. An illustrative example together with related discussions completes the presentation.
2010, 13(1): 157-174 doi: 10.3934/dcdsb.2010.13.157 +[Abstract](249) +[PDF](218.1KB)
Abstract:
In this paper we discuss the existence of traveling wave solutions for a nonlocal reaction-diffusion model of Influenza A proposed in Lin et. al. (2003). The proof for the existence of the traveling wave takes advantage of the different time scales between the evolution of the disease and the progress of the disease in the population. Under this framework we are able to use the techniques from geometric singular perturbation theory to prove the existence of the traveling wave.
Wei Wang and Yan Lv
2010, 13(1): 175-193 doi: 10.3934/dcdsb.2010.13.175 +[Abstract](283) +[PDF](268.3KB)
Abstract:
Dynamical behavior of the following nonlinear stochastic damped wave equations

$\nu$utt$+u_t=$Δ$u+f(u)+$ε$\dot{W}$

on an open bounded domain $D\subset\R^n$, $1\leq n\leq 3$,, is studied in the sense of distribution for small $\nu,$ε$>0$. Here $\nu$ is the parameter that describes the singular perturbation. First, by a decomposition of Markov semigroup defined by (1), a stationary solution is constructed which describes the asymptotic behavior of solution from initial value in state space $H_0^1(D)\times L^2(D)$. Then a global measure attractor is constructed for (1). Furthermore under the case that the stochastic force is proportional to the square root of singular perturbation, that is ε$=\sqrt{\nu}$, we study the limit of the behavior of all the stationary solutions of (1) as $\nu\rightarrow 0$. It is shown that, by studying a continuity property on $\nu$ for the measure attractors of (1), any one stationary solution of the limit equation

$u_t=$Δ$u+f(u).$

is a limit point of a stationary solution of (1), as $\nu\rightarrow 0$.

2010, 13(1): 195-211 doi: 10.3934/dcdsb.2010.13.195 +[Abstract](268) +[PDF](649.2KB)
Abstract:
A disease transmission model of SIRS type with latent period and nonlinear incidence rate is considered. Latent period is assumed to be a constant $\tau$, and the incidence rate is assumed to be of a specific nonlinear form, namely, $\frac{kI(t-\tau)S(t)}{1+\alpha I^{h}(t-\tau)}$, where $h\ge 1$. Stability of the disease-free equilibrium, and existence, uniqueness and stability of an endemic equilibrium for the model, are investigated. It is shown that, there exists the basic reproduction number $R_0$ which is independent of the form of the nonlinear incidence rate, if $R_0\le 1$, then the disease-free equilibrium is globally asymptotically stable, whereas if $R_0>1$, then the unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region for the model in which there is no latency, and periodic solutions can arise by Hopf bifurcation from the endemic equilibrium for the model at a critical latency. Some numerical simulations are provided to support our analytical conclusions.
2010, 13(1): 213-227 doi: 10.3934/dcdsb.2010.13.213 +[Abstract](253) +[PDF](189.2KB)
Abstract:
We discuss the possible existence of uncountable smooth traveling wavefronts of a degenerate and singular parabolic equation in non-divergence form

$\frac{\partial u}{\partial t} =u^m$div$(|\nabla u|^{p-2}\nabla u)+u^qf(u),$

where $f(s)$ is a positive source taking logistic type as an example. A very interesting phenomenon is the presence of critical values $m_c$ and $q_c$ of the exponent $m$ and $q$. Precisely speaking, only for the case $m$<$m_c$ with $q\ge q_c$ can the family of smooth traveling wavefronts have minimal wave speed. We also discuss the regularity of smooth traveling wavefronts.

2010, 13(1): 229-248 doi: 10.3934/dcdsb.2010.13.229 +[Abstract](285) +[PDF](1304.6KB)
Abstract:
We show that the function $S_1(x)=\sum_{k=1}^\infty e^{-2\pi kx} \log k$ can be expressed as the sum of a simple function and an infinite series, whose coefficients are related to the Riemann zeta function. Analytic continuation to the imaginary argument $S_1(ix)=K_0(x) - iK_1(x)$ is made. For $x=\frac{p}{q}$ where $p$ and $q$ are integers with $p$<$q$, closed finite sum expressions for $K_0 (\frac{p}{q} )$ and $K_1 ( \frac{p}{q})$ are derived. The latter results enable us to evaluate Ramanujan's function $\varphi (x)=\sum_{k=1}^\infty (\frac{\log k}{k}-\frac{\log(k+x)}{k+x})$ for $x=-\frac{2}{3}, -\frac{3}{4},$ and $-\frac{5}{6},$ confirming what Ramanujan claimed but did not explicitly reveal in his Notebooks. The interpretation of a pair of apparently inscrutable divergent series in the notebooks is discussed. They reveal hitherto unsuspected connections between Ramanujan's $\varphi(x), K_0(x), K_1(x),$ and the classical formulas of Gauss and Kummer for the digamma function.

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