Epidemiological impact of a genital herpes type 2 vaccine for young females

2012-2013 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation

We show that Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an n-dimensional dynamical system. This extends results of Fiedler et al. [PRL 98, 114101 (2007)], who demonstrated that such feedback control can stabilize the UPO associated with a two-dimensional subcritical Hopf normal form. Pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization, as well as knowledge of the period of the targeted UPO. We apply feedback in the directions tangent to the two-dimensional center manifold. We parameterize the feedback gain by a modulus and a phase angle, and give explicit formulae for choosing these two parameters given the period of the UPO in a neighborhood of the bifurcation point. We show, first heuristically, and then rigorously by a center manifold reduction for delay differential equations, that the stabilization mechanism involves a highly degenerate Hopf bifurcation problem that is induced by the time-delayed feedback. When the feedback gain modulus reaches a threshold for stabilization, both of the genericity assumptions associated with a two-dimensional Hopf bifurcation are violated: the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied, and the real part of the cubic coefficient of the normal form vanishes. Our analysis of this degenerate bifurcation problem reveals two qualitatively distinct cases when unfolded in a two-parameter plane. In each case, Pyragas-type feedback successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of the original bifurcation point, provided that the phase angle satisfies a certain restriction.Comment: 35 pages, 19 figure

Mathematical modelling of Toxoplasma gondii transmission: A systematic review

Background: Toxoplasma gondii is a ubiquitous protozoan parasite that can infect virtually all warm-blooded animals. It is the causative agent of toxoplasmosis, a significant public health issue worldwide. Mathematical models are useful to study the transmission dynamics of T. gondii infection in different settings, and may be used to compare the effectiveness of prevention measures. Methods: To obtain an overview of existing mathematical models for transmission of T. gondii, a systematic review was undertaken. The review was conducted according to an a priori protocol and the results were reported according to the PRISMA guidelines. Specific search terms were developed and used in the search of three databases (Scopus, PubMed, and Embase). Results: In total, 484 unique records were retrieved from the systematic search. Among them, 15 studies that used mathematical models to study the transmission of T. gondii. These studies were categorized into four groups based on the primary aims: dynamics of transmission (n = 8), intervention (n = 5), spatial distribution (n = 1), and outbreak investigation (n = 1). Conclusions: Considering the high disease burden caused by T. gondii, the number of studies using mathematical models to understand the transmission dynamics of this parasite and to evaluate the effectiveness of intervention measures was only 15. This systematic review provides an overview of existing mathematical models and identifies the data gaps for model building. The results from this study can be helpful for further development of mathematical models and improved understanding of the transmission dynamics of T. gondii infection

Double Hopf bifurcation in delay differential equations

The paper addresses the computation of elements of double Hopf bifurcation for retarded functional differential equations (FDEs) with parameters. We present an efficient method for computing, simultaneously, the coefficients of center manifolds and normal forms, in terms of the original FDEs, associated with the double Hopf singularity up to an arbitrary order. Finally, we apply our results to a nonlinear model with periodic delay. This shows the applicability of the methodology in the study of delay models arising in either natural or technological problems

Stability and Hopf bifurcation for a cell population model with state-dependent delay

International audienceWe propose a mathematical model describing the dynamics of a hematopoietic stem cell population. The method of characteristics reduces the age-structured model to a system of differential equations with a state-dependent delay. A detailed stability analysis is performed. A sufficient condition for the global asymptotic stability of the trivial steady state is obtained using a Lyapunov-Razumikhin function. A unique positive steady state is shown to appear through a transcritical bifurcation of the trivial steady state. The analysis of the positive steady state behavior, through the study of a first order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation and gives criteria for stability switches. A numerical analysis confirms the results and stresses the role of each parameter involved in the system on the stability of the positive steady state