Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds, and demonstrate $L^1$ convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.Comment: 34 page

Epidemiological impact of waning immunization on a vaccinated population

This is an epidemiological SIRV model based study that is de- signed to analyze the impact of vaccination in containing infection spread, in a 4-tiered population compartment comprised of susceptible, infected, recov- ered and vaccinated agents. While many models assume a lifelong protection through vaccination, we focus on the impact of waning immunization due to conversion of vaccinated and recovered agents back to susceptible ones. Two asymptotic states exist, the \disease-free equilibrium" and the \endemic equi- librium" and we express the transitions between these states as function of the vaccination and conversion rates and using the basic reproduction number. We nd that the vaccination of newborns and adults have dierent consequences on controlling an epidemic. Also, a decaying disease protection within the re- covered sub-population is not sucient to trigger an epidemic on the linear level. We perform simulations for a parameter set modelling a disease with waning immunization like pertussis. For a diusively coupled population, a transition to the endemic state can proceed via the propagation of a traveling infection wave, described successfully within a Fisher-Kolmogorov framework

Endothelial cells regulate p53-dependent apoptosis of neural progenitors after irradiation

Endothelial cells represent an important component of the neurogenic niche and may regulate self-renewal and differentiation of neural progenitor cells (NPCs). Whether they have a role in determining the apoptotic fate of NPCs after stress or injury is unclear. NPCs are known to undergo p53-dependent apoptosis after ionizing radiation, whereas endothelial cell apoptosis after irradiation is dependent on membrane acid sphingomyelinase (ASMase) and is abrogated in sphingomyelin phosphodiesterase 1 (smpd1)- (gene that encodes ASMase) deficient mice. Here we found that p53-dependent apoptosis of NPCs in vivo after irradiation was inhibited in smpd1-deficient mice. NPCs cultured from mice, wild type (+/+) or knockout (−/−), of the smpd1 gene, however, demonstrated no difference in apoptosis radiosensitivity. NPCs transplanted into the hippocampus of smpd1−/− mice were protected against apoptosis after irradiation compared with those transplanted into smpd1+/+ mice. Intravenous administration of basic fibroblast growth factor, which does not cross the blood–brain barrier, known to protect endothelial cells against apoptosis after irradiation also attenuated the apoptotic response of NPCs. These findings provide evidence that endothelial cells may regulate p53-dependent apoptosis of NPCs after genotoxic stress and add support to an important role of endothelial cells in regulating apoptosis of NPCs after injury or in disease

Endothelial Membrane Remodeling Is Obligate for Anti-Angiogenic Radiosensitization during Tumor Radiosurgery

While there is significant interest in combining anti-angiogenesis therapy with conventional anti-cancer treatment, clinical trials have as of yet yielded limited therapeutic gain, mainly because mechanisms of anti-angiogenic therapy remain to a large extent unknown. Currently, anti-angiogenic tumor therapy is conceptualized to either "normalize" dysfunctional tumor vasculature, or to prevent recruitment of circulating endothelial precursors into the tumor. An alternative biology, restricted to delivery of anti-angiogenics immediately prior to single dose radiotherapy (radiosurgery), is provided in the present study.Genetic data indicate an acute wave of ceramide-mediated endothelial apoptosis, initiated by acid sphingomyelinase (ASMase), regulates tumor stem cell response to single dose radiotherapy, obligatory for tumor cure. Here we show VEGF prevented radiation-induced ASMase activation in cultured endothelium, occurring within minutes after radiation exposure, consequently repressing apoptosis, an event reversible with exogenous C(16)-ceramide. Anti-VEGFR2 acts conversely, enhancing ceramide generation and apoptosis. In vivo, MCA/129 fibrosarcoma tumors were implanted in asmase(+/+) mice or asmase(-/-) littermates and irradiated in the presence or absence of anti-VEGFR2 DC101 or anti-VEGF G6-31 antibodies. These anti-angiogenic agents, only if delivered immediately prior to single dose radiotherapy, de-repressed radiation-induced ASMase activation, synergistically increasing the endothelial apoptotic component of tumor response and tumor cure. Anti-angiogenic radiosensitization was abrogated in tumors implanted in asmase(-/-) mice that provide apoptosis-resistant vasculature, or in wild-type littermates pre-treated with anti-ceramide antibody, indicating that ceramide is necessary for this effect.These studies show that angiogenic factors fail to suppress apoptosis if ceramide remains elevated while anti-angiogenic therapies fail without ceramide elevation, defining a ceramide rheostat that determines outcome of single dose radiotherapy. Understanding the temporal sequencing of anti-angiogenic drugs and radiation enables optimized radiosensitization and design of innovative radiosurgery clinical trials

Spectral degeneracy and escape dynamics for intermittent maps with a hole, Nonlinearity

Abstract We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds and demonstrate L 1 convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations

Turing patterns from dynamics of early HIV infection

We have developed a mathematical model for in-host virus dynamics that includes spatial chemotaxis and diffusion across a two-dimensional surface representing the vaginal or rectal epithelium at primary HIV infection. A linear stability analysis of the steady state solutions identified conditions for Turing instability pattern formation. We have solved the model equations numerically using parameter values obtained from previous experimental results for HIV infections. Simulations of the model for this surface show hot spots of infection. Understanding this localization is an important step in the ability to correctly model early HIV infection. These spatial variations also have implications for the development and effectiveness of microbicides against HIV

Escape and metastability in deterministic and random dynamical systems

Dynamical systems that are close to non-ergodic are characterised by the existence of subdomains or regions whose trajectories remain confined for long periods of time. A well-known technique for detecting such metastable subdomains is by considering eigenfunctions corresponding to large real eigenvalues of the Perron-Frobenius transfer operator. The focus of this thesis is to investigate the asymptotic behaviour of trajectories exiting regions obtained using such techniques. We regard the complement of the metastable region to be a ‘hole’, and show in Chapter 2 that an upper bound on the escape rate into the hole is determined by the corresponding eigenvalue of the Perron-Frobenius operator. The results are illustrated via examples by showing applications to uniformly expanding maps of the unit interval. In Chapter 3 we investigate a non-uniformly expanding map of the interval to show the existence of a conditionally invariant measure, and determine asymptotic behaviour of the corresponding escape rate. Furthermore, perturbing the map slightly in the slowly expanding region creates a spectral gap. This is often observed numerically when approximating the operator with schemes such as Ulam’s method. We investigate the asymptotic scaling of the spectral gap as the perturbation vanishes. In Chapter 4 we consider escape rate from random sets under the action of random dynamics and prove a result analogous to that of Chapter 2. We also show, under fairly weak assumptions, that in Oseledets subspaces Lyapunov exponents with respect to different norms are equal. The results are applied to Rychlik random dynamical systems. Finally, Chapter 5 deals with the main themes of the earlier chapters in the settings of deterministic and random shifts of finite type. There, we demonstrate methods to decompose shifts into complementary subshifts of large entropy. Much of the material in this thesis has either appeared in a scientific journal or has been submitted to one for publication