I will present the problem of adequate data subsampling for asymptotically consistent parametric estimation of unobservable stochastic differential equations (SDEs) when data are generated by multiscale dynamic systems approximated by these SDEs. The challenge is that the approximation accuracy is scale dependent, and degrades at very small scales. Data from multiscale dynamics systems, namely the Additive Triad model, will be used to illustrate this subsampling problem. I will also indicate the general framework for estimation under this indirect observability and present practical numerical techniques to identify the correct subsampling regime to construct bias-corrected estimators.
Then, I will show conditions for the exponential convergence of concentration to its uniform solution in the corresponding PDE model for chemical reaction-diffusion networks. Conditions obtained from the PDE model give an estimate for the maximal compartment size for space discretization in the stochastic model.
This is a joint work with Hans G. Othmer and Likun Zheng at the University of Minnesota.
The cytoskeleton of dividing cells is highly dynamic with microtubules stochastically transitioning between states of growth and shortening. In this dynamic environment "primitive" polymeric machines can generate force. In eukaryotic cells, chromosomes move to the cell equator by attaching to multiple dynamic microtubules. Attachment is mediated by complex multi-protein scaffolds called kinetochores. In this talk, we present a mathematical model for force generation at the microtubule/kinetochore interface in eukaryotic cells. Movement is modeled using a jump-diffusion process that incorporates both biased diffusion due to microtubule lattice binding by kinetochore elements as well as thermal ratchet forces due to microtubule polymerization against the kinetochore plate. A key result is that kinetochore motors obey nonlinear force-velocity relations. Finally, time permitting, we extend our modeling to explore how polymeric assemblies might facilitate the motility of the circular chromosome of Caulobacter Crescentus.
Ecological systems may exhibit complex dynamics, yet the spatial and temporal scales over which these play out make them difficult to explore experimentally. An alternative approach is to develop models based on detailed biological information about the systems and then fit them to observational data using nonlinear time-series techniques. I will give two examples of this approach, both involving systems with alternative states. The first is the dynamics of midges in Lake Myvatn, Iceland, which show fluctuations with amplitudes >105 yet with irregular period. A nonlinear time-series analysis demonstrates that these dynamics could be caused by the system having two states, a stable point and a stable cycle, with the irregular period caused by the population stochastically jumping from the domain of one state to the other. The second example is the dynamics of salvinia, an aquatic weed, and the salvinia weevil that was introduced into the billabongs of Kakadu to control the weed. Here the alternative states are two environmentally (seasonally) forced cycles, one in which salvinia is kept in check by the weevil and one in which it escapes. Understanding complex ecological dynamics may improve our management of vigorously fluctuating natural systems.
Waterborne diseases cause over 3.5 million deaths annually, with cholera alone responsible for 3-5 million cases/year and over 100,000 deaths/year. Many waterborne diseases exhibit multiple characteristic timescales or pathways of infection, which can be modeled as direct and indirect transmission. A major public health issue for waterborne diseases involves understanding the modes of transmission in order to improve control and prevention strategies. One question of interest is: given data for an outbreak, can we determine the role and relative importance of direct vs. environmental/waterborne routes of transmission? We examine these issues by exploring the identifiability and parameter estimation of a differential equation model of waterborne disease transmission dynamics. We use a novel differential algebra approach together with several numerical approaches to examine the theoretical and practical identifiability of a waterborne disease model and establish if it is possible to determine the transmission rates from outbreak case data (i.e. whether the transmission rates are identifiable).
Our results show that both direct and environmental transmission routes are identi?able, though they become practically unidenti?able with fast water dynamics. Adding measurements of pathogen shedding or water concentration can improve identi?ability and allow more accurate estimation of waterborne transmission parameters, as well as the basic reproduction number. Parameter estimation for a recent outbreak in Angola suggests that both transmission routes are needed to explain the observed cholera dynamics. I will also discuss some ongoing applications to the current cholera outbreak in Haiti.
Alternans, a long-short alternation of cardiac action potential durations, emerges as a period-doubling bifurcation under rapid pacing. Detecting alternans or bifurcation of the cardiac restitution has been a major task in prevent heart disease. We developed a new stochastic protocol and a regression method to approximate the full dynamics in a time interval. We also discuss the propagation of alternans in 1D cardiac fiber.
In a given population there are usually more than two configurations of sex chromosomes vs. phenotypical gender differentiation. A common configuration is XX chromosomes for females and XY chromosomes for males; variations of this theme (e.g. XY female) occur naturally at low frequencies. There is a vast family of such variations as a result of environmental intervention. In this talk I will present a general formulation for multi-sexual populations using hypermatrices. I will present a method to compare the asymptotic behavior (i.e. who goes to extinction first, if at all) of these competitive dynamical systems of different dimension under certain conditions of biological relevance.
In this two part talk I will summarize work from my Ph.D. thesis, then introduce some ongoing projects as an MBI Postdoctoral Fellow and part of OSU's Aquatic Ecology Laboratory (AEL). The first part of this talk will focus on an infectious disease in house finches (Carpodacus mexicanus) and other wild birds caused by the pathogen Mycoplasma gallisepticum. After introducing the biological system, I will present results from a mathematical model of the immune-pathogen interaction which address the immune system's role in mediating disease symptoms and controlling infection. For the second part of the talk, we will shift gears and consider population dynamics in the context of simple aquatic food webs. I will start off with a brief but general introduction of the biology. I will then present results from model that combines consumer-resource (predator-prey) and host-parasite interactions. These results describe the consequences of some unexpected connections between consumer-resource and host-parasite interactions, as motivated by recent empirical findings from the study of Daphnia (a kind of freshwater zooplankton) their parasites and Daphnia's algal food source. The last part of the talk will introduce two ongoing projects with Stuart Ludsin and others at the AEL. The first of these focuses on the role of hypoxia in shaping disease risk among fish. The second investigates the importance of an aquatic larval insect (phantom midges; family Chaoboridae) in freshwater lakes and reservoirs in Ohio by modelling how they affect the dynamics of those ecosystems.
In this study, we use multi-stage cell lineages model, which include stem cell and multiple progenitor cell stages, to study how feedback regulation from different growth controls homeostasis of tissue growth and generation of a robust spatial stratification. ODE and PDE models have been presented for the multi-stage cell lineages. Our analysis shows how negative feedbacks enhance the stability of steady states and inter-regulation among different growth factors are responsible for developing spatial stratification. We also showed that the feedback on cell cycle from the growth factor is important for forming temporary "stem cell niche" during the development of the tissue.
Most phytoplankton movement is passive and occurs through either sinking/ floating (depending on their density relative to water) or through turbulent diffusion. As they move vertically in the water column, phytoplankton experience gradients in critical environmental factors, such as light intensity and nutrient concentrations. The rate at which phytoplankton move across these gradients can be critical to their persistence and vertical distribution. Grazing can also play a critical role in dictating where in the water column phytoplankton are found. However, theoretical models of critical sinking and diffusion rates either do not explicitly consider grazing loss or treat it as vertically homogenous, thus making it independent of movement. In nature, however, grazing intensity is often vertically heterogeneous. Despite its common occurrence, how such grazing heterogeneity influences critical rates of phytoplankton movement is not well understood. Here we put forth some basic predictions regarding phytoplankton persistence and spatial heterogeneity of grazing, using a reaction-diffusion-advection model. We introduce some new ideas to investigate the combined effects of advection, diffusion, and heterogeneous grazing pressure on the persistence of phytoplankton and to determine the unique number of critical sinking/buoyant rates that are specified by the inclusion of depth dependent mortality that is a result of heterogeneous predation.
PhyloPTE/P (Phylogeny with Path to Event, in People) is a method to bridge the gap between large, gene sequencing based (and, in the near future, other *omic based) studies and phylogenetically-driven approaches developed in other fields, for example epistatic-effect detection using comparison of phylogenetic tree reconstructions for different genes. PhyloPTE/P should be of interest to a wide audience of investigators, including those in biomedical informatics or medical genomics, as well as those in systems biology or evolutionary biology, and serve as a software platform to foster collaboration between the two areas.
Cholera, a waterborne diarrheal disease, is a major public health threat in many parts of the world. It is spread via direct contact with infected individuals as well as indirectly through a contaminated water source. Cholera dynamics can be described by the SIWR model, a modified SIR model incorporating an equation to track the concentration of the pathogen in the water (W) and the additional water transmission pathway. Factors affecting both transmission rates are likely to vary among different populations. Here we consider a multi-patch SIWR model, specifically a system of non-mixing patches sharing a common water source, and explore the effect of heterogeneity in transmission on the spread of the disease, as well as the implications for control.
For a given biochemical network of interest it is often desirable to estimate its reaction constants. I shall discuss several different approaches to rate constants estimation from partial trajectory data. The presentation will discuss the LSE as well Bayesian and MLE approaches as well as possible conditions on the data process which guarantee identifiability and estimators consistency. We shall also consider ways of approximating the likelihood of a partially observed biochemical network with certain other likelihoods (e.g., Gaussian) for which inference problem is simplified.
Clustering data into groups of similarity is well recognized as an important step in many diverse applications, including biomedical imaging, data mining and bioinformatics. Well known clustering methods, dating to the 70's and 80's, include the K-means algorithm and its generalization, the Fuzzy C-means (FCM) scheme, and hierarchical tree decompositions of various sorts. More recently, spectral techniques have been employed to much success. However, with the inundation of many types of data sets into virtually every arena of science, it makes sense to introduce new clustering techniques which emphasize geometric aspects of the data, the lack of which has been somewhat of a drawback in most previous algorithms.
In this talk, we focus on a slate of "random-walk" distances arising in the context of several weighted graphs formed from the data set, in a comprehensive generalized FCM framework, which allow to assign "fuzzy" variables to data points which respect in many ways their geometry. The method we present groups together data which are in a sense "well-connected", as in spectral clustering, but also assigns to them membership values as in FCM. We demonstrate the effectiveness and robustness of our method on several standard synthetic benchmarks and other standard data sets such as the IRIS and the YALE face data sets. This is joint work with Sijia Liu and Sunder Sethuraman.
Currently, efforts are underway to develop vaccines for several viral infections, including Human Immunodeficiency Virus type 1 (HIV-1) and Herpes Simplex Virus type 2 (HSV-2). In this talk, I will present the results of mathematical models that address vaccination strategies for these viral infections. I will demonstrate the use of these results to predict the impact of prevention efforts as well as to assess the mechanisms of virus-host interactions. I will also show how such studies can guide the development of future vaccines and other therapeutic interventions.
My primary goal in this talk will be to provide a summary of some current research interests with the hope of stimulating potential collaborations with other faculty and postdocs during my time at MBI. The general theme will concern the effective statistical description of a complex microbiological system consisting of a number of individual dynamical components with some structural interactions as well as with stochastic noise sources. I will briefly touch on the examples of molecular motors and swimming microorganisms, then describe in some more detail a recent study of synchrony in stochastically driven neuronal networks.
A brief introduction is presented to stochastic differential equations (SDEs) in mathematical biology. In particular, a procedure is described for deriving accurate SDE models for randomly varying biological dynamical systems. Next, several research projects involving SDE models in biology are briefly described. Specifically summarized are: an investigation of a schistosomiasis infection with biological control, a derivation of stochastic partial differential equations for size-and age-structured populations, and the development of SDE models for biological diversity. Finally, current/future work is pointed out.
A mathematical model which incorporates the spatial dispersal and interaction dynamics of mistletoes and birds is derived and studied to gain insights of the spatial heterogeneity in abundance of mistletoes. Fickian diffusion and chemotaxis are used to model the random movement of birds and the aggregation of birds due to the attraction of mistletoes respectively. The spread of mistletoes by birds is expressed by a convolution integral with a dispersal kernel. Two different types of kernel functions are used to study the model, one is Dirac delta function which reflects one extreme case that the spread behavior is local, and the other one is a general non-negative symmetric function which describes the nonlocal spread of mistletoes. When the kernel function is taken as the Dirac delta function, the threshold condition for the existence of mistletoes is given and explored in term of parameters. For the general non-negative symmetric kernel case, we prove the existence and stability of non-constant equilibrium solutions. Numerical simulations are conducted by taking specific forms of kernel functions. Our study shows that the spatial heterogeneous patterns of the mistletoes are related to the specific dispersal pattern of the birds which carry mistletoe seeds.
Remarkable progress in advanced microscopy has yielded unprecedented access to a path-wise observation of the diffusive behavior of bacteria, viruses, organelles and various invasive particulates in biological fluids. Upon inspection of the data one immediately notes, "That's not Brownian motion!" Perhaps not surprisingly, media such as human mucus are highly heterogeneous and exhibit significant viscoelastic properties. In this talk, I will provide a survey of recent experimental observations along with mathematical models that are currently in use. Wherever possible I will point out open problems in this burgeoning area of research.
Classical mathematical formulation of the dynamics of chemical reaction systems involves setting up and analyzing a system of ODEs, or PDEs if spatial effects are considered. However, a system may be sensitive to the stochasticity inherent in the mechanism of chemical reactions, for example due to having small numbers of molecules, or reaction rates which vary over several orders of magnitude. We consider such a reaction system in a cellular environment, and also impose a 'global' cell division mechanism, which adds noise to the concentrations of chemical species along a given lineage, and find parameter regimes for which this produces a qualitative change in the dynamics. We model these reaction and division processes as Jump Markov Processes, and discuss some toy models in which the stochasticity can allow the system to exhibit behavior that is not possible with a deterministic formulation. One such behavior is bistability, for which we find two processes that have similar macroscopic signatures but whose underlying causes are fundamentally different; one such case leads to the Large-Deviation theory of Freidlin and Wentzell. Such bistability is characteristic of many gene expression systems that effectively incorporate an ON/OFF switch, but the framework is very general and is applicable in other areas, such as population genetics, where bistability may represent alternating dominance of allelic types in a population. This is joint work with Lea Popovic of Concordia University (Montreal).