Stewart et. al. reduced the proving of this conjecture to the proving of the following two other conjectures:
1) phase rigid property: Suppose phase relations on X_0 are rigid, then for each pair of phase-related cells, the signals they receive are also phase-related with the same phase-relation.
2) fully oscillatory property: In a transitive network, a hyperbolic periodic solution of an admissible vector field of the network is generically fully oscillatory (all cells are oscillatory on the periodic solution).
We show that these two conjectures are actually correct. These results are joint work with Marty Golubitsky and David Romano.
The model consists of a coupled system of partial differential equations in the partially healed region, with the wound boundary as a free boundary. The ECM is assumed to be viscoelastic, and the free boundary moves with the velocity of the ECM at the boundary. The model variables include the concentration of oxygen, PDGF and VEGF, the densities of macrophages, fibroblasts, capillary tips and sprouts, and the density and velocity of the ECM. Simulations of the model demonstrate how oxygen deficiency may limit macrophage recruitment to the wound-site and impair wound closure. The results are in general agreement with experimental findings in an animal model.
I will first give a solid mathematical foundation of multiple testing that is difficult to gather from literature. Then I will focus on two typically overlooked issues in testing of biomarkers. One issue is interpretation of an unconditional expectation error rate such as FDR, however it is controlled. The other issue, which has only recently come to light, is the ever popular permutation testing requires a strong assumption on the (unknown) joint distribution of biomarkers to control its error rate. These and other issues will be illustrated in the Genome-wide Association Studies (GWAS) setting.
However, there are other mechanisms responsible for genetic variation between species, and the most influential of them is the coalescent process, which explains how sequence variation can be retained in a population, and how each gene tree does not necessarily reflect the history of the species.
With next-generation sequencing becoming less expensive, there will be a massive influx of sequence data in the near future, and with multi-gene datasets, the effect of the coalescent process will be more important to take into consideration when estimating the species tree. A set of sequenced transcriptomes will have genes sampled randomly, with a high frequency of missing data for each gene when considered across all sampled species.
Here I present a simulation study on the effects of missing data on estimating the species tree from a set of gene trees when taking the coalescent process into consideration. We have examined the effects on species tree estimation from sampling several lineages per species, different degrees and patterns of missing data and recent and older speciations.
Infectious diseases have a long history of study within many subdisciplines of biology and mathematics. This talk will be an overview of two projects where dynamical systems theory is used to study two infectious disease systems. In each case, the multiple time scales naturally present in these system help simplify the mathematical analyses necessary for answering our motivating biological questions.
Noise in cardiac pacing cycles, for instance, the heart rate variability, has been observed and researched for decades. Contemporarily, various cardiac models have been constructed to investigate the electric activity of the cardiac cells. Yet there has not been a study on extracting information of the underlying dynamics if some noisy data are given. In this talk we will show a method to determine the cardiac restitution approximately in the range of the pacing cycles provided a series of noisy data. We assume the data are generated through some unknown mapping model with memory, and the memory is supposed to be hidden and not able to be detected.
Heterogeneity is a fundamental issue in mathematical epidemiology. We expect many factors influencing disease transmission to vary across populations and different spatial scales. Many results exist for the effect of heterogeneity on the spread of disease for SIR type models, where transmission occurs as a result of direct contact with infected individuals. Waterborne disease, such as cholera, may be spread through contact with a contaminated water source as well as through direct person-person transmission. We investigate the effect of heterogeneity in both transmission pathways on the value of the basic reproductive number R0 in multi-patch SIWR models, specifically a system of N patches sharing a common water source.
Several methods have been proposed for correlating genomic sequence patterns directly with phenotypes of similar organisms. However, the evolutionary relationships between organisms lead to non-independence among the sequences. A phylogenetic tree reconstruction uncovers sibling lineages where the phenotypes first start to differentiate, and, conditional on this tree, PhyloPTE adopts an additive hazard model to identify likely mutational paths along the tree as the phenotypes fully develop.
Overfishing, pollution and other environmental factors have greatly reduced commercially valuable stocks of fish. In a 2006 Science article, a group of ecologists and economists warned that the world may run out of seafood from natural stocks if overfishing continues at current rates. In this talk, we will explore the interaction between constant and periodic proportion harvest policies and recruitment dynamics. In case studies, we analyze these policies and illustrate how they might be applied to Gulf of Alaska Pacific halibut fishery and the Georges Bank Atlantic cod fishery based on harvest rates from 1975 to 2007.
In this talk, we use a generalized age-structured model to discuss the role that both compensatory (non-oscillatory) and overcompensatory (oscillatory) dynamics play in the long term dynamics of exploited fisheries. When each species is governed by compensatory dynamics via the Beverton-Holt model and the predator's response to species interaction is modeled using a linear function, we show that the predator-prey model exhibits a globally stable positive fixed point. In stark contrast, we show that when each species is governed by compensatory dynamics via the Beverton-Holt model and the predator response function is exponential, then the predator-prey model exhibits population oscillations.