The CDC launched the National Plan to Eliminate Syphilis from the USA in October 1999 [4]. In order to reach this goal, a good understanding of the transmission dynamics of the disease is necessary. Based on a SIRS model Breban et al.  [3] provided some evidence that supports the feasibility of the plan proving that no recurring outbreaks should occur for syphilis. We study in this work a syphilis model that includes partial...

We present a nonasymptotic theorem for interacting particle approximations of unnormalized Feynman–Kac models. We provide an original stochastic analysis-based on Feynman–Kac semigroup techniques combined with recently developed coalescent tree-based functional representations of particle block distributions. We present some regularity conditions under which the -relative error of these weighted particle measures grows linearly with respect to the time horizon yielding what seems to be the first...

We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $$\left\{\begin{array}{c}-{\mathrm{div}\left(\right|x|}^{-\alpha p}{\left|\nabla u\right|}^{p-2}{\nabla u)=|x|}^{-(\alpha +1)p+\beta }\left(a{u}^{p-1}-f\left(u\right)-{\displaystyle \frac{c}{u^{\gamma }}}\right),x\in \Omega ,\hfill \\ u=0,x\in \partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a bounded smooth domain of ${ℝ}^N$ with $0\in \Omega$, $1<p<N$, $0\le \alpha <(N-p)/p$, $\gamma \in (0,1)$, and $a$, $\beta$, $c$ and $\lambda$ are positive parameters. Here $f:[0,\infty )\to ℝ$ is a continuous function. This model arises in the studies of population biology of one species with $u$ representing the concentration of the species. We discuss the existence of a positive solution when $f$ satisfies certain additional conditions. We use the method of sub-supersolutions...

The existence of a positive solution for the generalized predator-prey model for two species $$\begin{array}{c}\Delta u+u(a+g(u,v\left)\right)=0\text{in}\Omega ,\\ \Delta v+v(d+h(u,v\left)\right)=0\text{in}\Omega ,\\ u=v=0\text{on}\partial \Omega ,\end{array}$$ are investigated. The techniques used in the paper are the elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.

We investigate a Lotka-Volterra predator-prey model with state dependent impulsive effects, in which the control strategies by releasing natural enemies and spraying pesticide at different thresholds are considered. We present some sufficient conditions to guarantee the existence and asymptotical stability of semi-trivial periodic solutions and positive periodic solutions.

The paper is concerned with an extension of the classical relation between the flame speed and the curvature-flow stretch, valid only for high Lewis numbers (diffusively stable flames). At low Lewis numbers the corresponding flame-flow system suffers short-wavelength instability, making the associated initial value problem ill-posed. In this study the difficulty is resolved by incorporation of higher-order effects. As a result one ends up with a reduced model based on a coupled system of second-order...