The Applied Mathematics Seminar has talks on a wide range of topics, including but not limited to approximation theory and practice, numerical linear algebra, numerical optimization, numerical aspects of computer science, theoretical and applied partial differential equations and their numerical solutions, and mathematical biological models.
For more information contact Hayriye Gulbudak.
For the Spring 2022 semester we will meet on Zoom at 4:30 pm on Tuesdays.
For more information or connection details contact Hayriye Gulbudak.
For the Fall 2021 semester we will meet on Zoom at 4:30 pm on Thursdays.
For more information or connection details contact Hayriye Gulbudak.
16 September 2021
Data driven modeling of COVID-19 first wave outbreak in the United States
Abstract: Each state in the United States exhibited a unique response to the COVID-19 outbreak, along with variable levels of testing, leading to different actual case burdens in the country. Via per-capita testing dependent ascertainment rates, along with case and death data, we fit a minimal epidemic model for each state. We estimate infection-level responsive lockdown/self-quarantine entry and exit rates (representing government and behavioral reaction), along with the true number of cases as of May 31, 2020. Ultimately, we provide error-corrected estimates for commonly used metrics such as infection fatality ratio and overall case ascertainment for all 55 states and territories considered, along with the United States in aggregate, in order to correlate outbreak severity with first wave intervention attributes and suggest potential management strategies for future outbreaks. We observe a theoretically predicted inverse proportionality relation between outbreak size and lockdown rate, with the scale dependent on the underlying reproduction number, and simulations suggesting a critical population quarantine "half-life'' of 30 days independent of other model parameters. In addition to these public health implications, we will introduce and review the methods of parameter estimation, identifiability analysis, and uncertainty quantification used in this work. Particular attention will be given to the challenges posed by noisy data and unreliable reporting.
28 October 2021
Dynamics of a two-strain cholera model with environmental component
Abstract: Cholera can survive, proliferate, and compete in the aquatic environment, necessary factors to consider when modeling its long-term dynamics. We combine and extend previously studied SIRP (infectious disease models incorporating pathogen environmental concentration) and multi-strains models to consider the strain diversity of cholera and more accurately analyze its transmission dynamics and long-term behavior (e.g. coexistence versus competitive exclusion). We focus on distinct serotypes, strains which differ only in cross-immunity after host infection. This model extends the single-strain model to a two-strain SIRP cholera model, considering two distinct serotypes of the bacteria, Ogawa and Inaba. Via typical deterministic modelling techniques, we derive and analyze disease-free, boundary, and coexistence equilibria for the model. Population persistence methods allow us to determine long-term behavior of the strains when our basic reproduction number is greater than one.
18 November 2021
Unification of Discrete and Continuous Coagulation-Fragmentation Equations
Abstract: Since their introduction, the study of coagulation-fragmentation equations has been divided into two types of structure: where the size of particles is assumed to be discrete or where the size of particles is assumed to be continuous. The former school of thought results in a system of ODEs while the latter results in the use of a single PDE. In this talk, I will formulate a size-structured coagulation-fragmentation equation in the space of Radon measures endowed with the bounded Lipschitz norm. I will show under this framework the model is well-posed and unifies the study of both the discrete and continuous coagulation-fragmentation equations.
For the Spring 2020 semester we will meet at 3:30 pm on Wednesday in 207 Maxim Doucet Hall.
29 January 2020
A Minimal Time Solution to the Firing Squad Synchronization Problem with Von Neumann Neighborhood of Extent Two
Dr. Kathryn Boddie
Abstract: Cellular automata provide a simple environment in which to study global behaviors. One example of a problem that utilizes cellular automata is the Firing Squad Synchronization Problem, first proposed in 1957. We will provide an overview of the standard Firing Squad Synchronization Problem and a commonly used technique in solving it. A new extension of the Standard Firing Squad Synchronization Problem to a different neighborhood definition – a Von Neumann neighborhood of extent 2 will be presented. An 8 state 651 rule minimal time solution to the extended problem will described and presented, along with an outline of the proof of the correctness of the solution.
5 February 2020
Traveling Pulses in a Lateral Inhibition Neural Network
Dr. Aijun Zhang
Abstract: The talk is to study the spatial propagating dynamics in a neural network of excitatory and inhibitory populations. Our study demonstrates the existence and nonexistence of traveling pulse solutions with a nonsaturating piecewise linear gain function. We prove that traveling pulse solutions do not exist for such neural field models with even (symmetric) couplings. The neural field models only support traveling pulse solutions with asymmetric couplings. We also show that such neural field models with asymmetric couplings will lead to a system of delay differential equations. We further compute traveling 1–bump solutions using the system of delay differential equations. Finally, we develop Evans functions to assess the stability of traveling 1–bump solutions.
12 February 2020
Global analysis of viral infection models with cell-to-cell transmissions and immune chemokines
Dr. Xiang-Sheng Wang
Abstract: We propose a viral infection model incorporating both cell-to-cell infection and immune chemokines. Based on experimental results in the literature, we make a basic assumption that the cytotoxic T lymphocytes (CTL) will move toward the location with more infected cells, while the diffusion rate of CTL is a decreasing function of the density of infected cells. We first establish the global existence of solutions via a priori energy estimates and Gagliardo-Nirenberg inequality. Next, we define two basic reproduction numbers and determine the global dynamics of the model system by Lyapunov functional techniques and LaSalle invariance principle.
19 February 2020
Modeling contact tracing in emerging outbreaks with application to Ebola and novel Coronavirus
Dr. Cameron Browne
Abstract: Mathematical models can be important tools during emerging infectious disease outbreaks. Current/recent outbreaks of Ebola (EVD) and the novel Coronavirus (COVID-19) have reinforced the need for quantifying disease spread and control efforts. Contact tracing and isolation are critical first line control and surveillance measures employed by public health authorities. In this talk, I will discuss some mathematical models for the 2014-2015 Ebola outbreaks with analysis of contact tracing, and possible applications to current EVD and COVID-19 outbreaks. In particular, we will derive formulae for reproduction numbers and outbreak final size relations, along with considering methods for utilizing data to parameterize models.
26 February 2020
4 March 2020
Selection-Mutation Models on the Space of Measures
Dr. Azmy Ackleh
Abstract: We first consider the long time behavior of a selection-mutation model formulated on the space of finite signed measures. The selection- mutation kernel is described by a family of measures which allows to handle continuous and discrete kernels under the same setting. We establish permanence results for the full model and we study the limiting behavior of the solution even when the fittest trait is not unique. We show that for the pure selection case the solution converges to a Dirac measure centered at the fittest trait and that in the case of small mutation and discrete trait space there exists an equilibrium that attracts any initial condition that is positive at the fittest trait. We then consider a selection-mutation model with an advection term that models fast evolution. We rescale the kernel to obtain a diffusively rescaled selection-mutation model. We prove that if the rescaled selection-mutation kernel converges to a pure selection kernel then the solution of the diffusively rescaled model converges to a solution of an advection-diffusion equation.
11 March 2020
Graphical modelling of co-evolution within structured, cooperative biosystems: Just how social is HIV?
Dr. Brittany Rife Magalis
University of Florida, Emerging Pathogens Institute
Abstract: Cooperativity exists at virtually every biological level, ranging from the altruistic nature of human beings to the intramolecular biochemical interactions that work to maintain molecular structure and function. At a microorganismal level, biofilms are the most studied cooperative biosystem, with structural and functional heterogeneity that works in the form of mutualistic interactions to increase fitness of the population as a whole. This social perspective has recently been embraced in the study of viruses as well (e.g, hepatitis c virus), with growing evidence that virus-virus interactions are pervasive and that our understanding of these is imperative for improvement of treatment strategies. Approaches to measuring and predicting cooperativity have relied on phenotypic changes in response to in vitro selection; however, there exists a cooperativity footprint in genetic sequence evolution, whereby compensatory mutations can be mapped in space and time to identify genomic sites required for molecular interaction within structured populations. Human immunodeficiency virus (HIV) continues to evolve within the host over a long period of time (years), adapts to a wide variety of tissues and cell types, and is exceptional at avoiding the host immune response, begging the question – does the ability of this virus to persist partly depend on a cooperative mechanism among tissue-resident viral populations? Co-evolving networks of sites within the envelope gene were identified across viral subpopulations in the S[imian]IV-infected macaque model, suggesting additional statistical evaluation using probabilistic graphical modeling is worthwhile.
18 March 2020
1 April 2020
Dr. Alex Farrel
University of Arizona
25 March 2020
University of Houston
8 April 2020
Dr. Longfei Li
15 April 2020
22 April 2020
Leah Kaisler (student presentation)
29 April 2020
Caleb MacDonald (student presentation)
For the Fall 2019 semester we will meet at 3:30 pm on Wednesday in 207 Maxim Doucet Hall.
18 September 2019
Persistence and Extinction of Nonlocal Dispersal Evolution Equations in Moving Habitats
Dr. Aijun Zhang
Abstract: The talk is devoted to the study of persistence and extinction of a species modeled by nonlocal dispersal evolution equations in moving habitats with a moving speed. Such evolution equations can be used in modeling the effects of global climate change on populations. It is shown that the species becomes extinct if the moving speed is larger than the so called spreading speed (minimal wave speed), where it is determined by the maximum linearized growth rate function. If the moving speed is smaller than spreading speed, it is shown that the persistence of the species depends on the patch size of the habitat, namely, the species persists if the patch size is greater than some critical number and in this case, there is a traveling wave solution, and it becomes extinct if the patch size is smaller than this critical number.
25 September 2019
Modeling Plankton Population Dynamics in a River in two-dimensions via the NPZ model and Upwind Differencing
Caleb Macdonald (Student presentation)
Abstract: Nutrient-Phytoplankton-Zooplankton (NPZ) models are relatively simple models for the interaction and population dynamics of the eponymous organisms. One of the central questions in plankton ecology is what drives these dynamics. Because plankton ecological niches are based, in part, on water temperature and nutrient composition, and since plankton are quick to evolve, plankton make excellent indicators of the effects of climate change on aquatic ecosystems. Since plankton cannot, by definition,swim against current, and inhabit ecosystems with strong water flow, NPZ(D) (detritus) models are often one-way coupled to an advection-diffusion equation in two or three dimensions. Though much more complex models exist many of these are essentially a series of linked NPZ or NPZD models which account for multiple functional groups of plankton, each with their own birth, death, and diffusion rates and interaction terms. Here, we give theoretical and numerical results for a specific instance of the NPZ model, coupled to an advection-diffusion equation that approximates a river at a particular depth, and discuss ecological implications.
2 October 2019
Traveling wave solution of a diffusive viral infection model with time delay
Srijana Ghimire (Student presentation)
Abstract: In this talk, we study the traveling waves of a diffusive viral infection model with time delay. To establish the existence result, we first consider a perturbed system and construct suitable upper and lower solutions so that Schauder fixed point theorem can be applied. Next, we use a limiting argument together with Lyapunov functional techniques to find a traveling wave solution which connects the infection-free equilibrium and the endemic equilibrium.
9 October 2019
Viral diffusion and cell-to-cell transmission: mathematical analysis and simulation study
Dr. Xiangsheng Wang
Abstract: We propose a general model to investigate the joint impact of viral diffusion and cell-to-cell transmission on viral dynamics. The mathematical challenge lies in the fact that the model system is partially degenerate and the solution map is not compact. First, we identify the basic reproduction number as the spectral radius of the sum of two linear operators corresponding to direct and indirect transmission modes. It is well-known that viral mobility may induce infection in low-risk regions. However, as the diffusion coefficient increases, we prove that the basic reproduction number actually decreases, which indicates that faster viral movements may result in a lower level of viral infection. By an innovative construction of Lyapunov functionals, we further demonstrate that the basic reproduction number is the threshold parameter that determines the global picture of viral dynamics. Numerical simulation supports our theoretical results and suggests an interesting phenomenon: the boundary layer and the internal layer may occur when the diffusion parameter tends to zero. This talk is based on a joint work with Zongwei Ma (Visiting student at UL Lafayette), Hongying Shu (Shaanxi Normal University, China) and Lin Wang (University of New Brunswick, Canada).
16 October 2019
A stable and accurate algorithm for a generalized Kirchhoff-Love plate model
Thuy Anh Duong Nguyen (Student presentation)
Abstract: An efficient and accurate numerical algorithm is developed to solve a generalized Kirchhoff-Love plate model subject to three physical boundary conditions: (i)clamped; (ii) pinned; and (iii) free. The numerical approach consists of a predictor-corrector time-stepping method in conjunction with a standard finite difference spatial discretization. Von-Neumann Analysis is performed to determine a stable time step for the scheme. A sequence of benchmark problems with increasing complexity are considered to demonstrate the numerical properties of the algorithm; the results confirm the stability and 2nd-order accuracy of the algorithm. Comparison with existing numerical methods reveals the efficiency of the new method.
23 October 2019
Using Markov Mixture Models to Estimate Continuous-Time Rates from Discrete-Time Data with Application to Malaria Incidence and Recovery Rates
Dr. Celeste Vallejo
Ohio State University
Abstract: Sampling a continuous time two-state stochastic process at discrete times and calculating transition probability matrices for each pair of consecutive observation times yields a time series of two-wave panel data; i.e. interval censoring. Estimating transition rates for the underlying continuous time process requires that we identify a time series of continuous time models whose transition probabilities at the observation times match those in the observed transition matrices. Empirical and theoretical literature over the past 43 years assesses whether or not the observed transition matrices are embeddable in the class of continuous time Markov chains, and if so, transition rates are calculated within that class of models. We show that non-Markov embeddable matrices are embeddable in the class of two-component mixtures of continuous time chains, but that apriori constraints are required to estimate transition rates in the resulting under-identified system. Depending on the imposed constraints, the rates in the mixture model may either be identified, or partially identified with resulting restricted ranges of non-uniqueness of transition rates. We apply this methodology to estimate incidence and recovery rates from malaria infection in the Garki district of northern Nigeria in the 1970s. We are able to assess the impact of combinations of indoor residual spraying and distinct drug administration regimens where such evaluations had previously been regarded as not doable as a consequence of non-Markov embeddability of observed transition matrices.
29 October 2019 (date and location change)
This week the seminar will take place 29 October from 3:30-4:30 in Maxim Doucet Hall, room 208.
6 November 2019
Connecting population genetics and dynamics in rapidly evolving pathogen systems
Dr. Cameron Browne
Abstract: Genomic data offers a promising avenue for exploring evolutionary dynamics, with particular application to rapidly evolving pathogens. Demographic information may be inferred through a population genetic measure known as effective population size, Ne, which may also reveal combined effects of ecological and evolutionary interactions on the biological system. In this talk, I will present two collaborative projects where high-dimensional dynamical host-pathogen models are connected to Ne through simulating genetic diversity and compared to measurements of Ne in experimental data. First, I utilize network ODE systems to describe the interaction between several immune cell populations and viral "quasi-species" sampled from experiments of the simian immunodeficiency virus (SIV)-infected macaque model of HIV infection. The second work concerns a mixed transmission dynamic model of the Cholera outbreak in Haiti, where aquatic reservoirs actively contribute to the epidemic. The mathematical models can recapitulate and shed light on certain aspects of the data. However there are fundamental questions on how to connect population genetics and dynamics in a unified "eco-evolutionary" modeling framework. This important problem in mathematical biology will likely require a combination of deterministic, stochastic, and even measure-valued, dynamical system approaches, along with effective numerical and statistical methods.
13 November 2019
Global dynamics of discrete-time predator-prey models
Istiaq Hossain (student presentation)
Abstract: In the authors' previous work, conditions for the persistence and (local asymptotic) stability of the equilibria of two discrete-time predator-prey models was described. The first model described predator-prey dynamics, while the second model considered how this interaction may change when the prey population is exposed to a toxicant and is able to evolve in the response to the toxicant. This latter model incorporated an additional dimension that describes the dynamics of an evolving phenotypic trait representing the mean toxicant resistance possessed by the prey population. In this work, we explore the long-term population and evolutionary dynamics of both these models, including global stability of equilibria, the existence of cycles, and chaos. We establish conditions under which the interior equilibrium is global asymptotically stable. We study the global asymptotic stability of equilibria using perturbation analysis together with the construction of Lyapunov functions. In addition, we perform numerical studies to establish the existence and stability of cycles and generate bifurcation diagrams showing that the evolutionary model may exhibit chaotic behavior.
20 November 2019
Modeling With Measures
Rainey Lyons (student presentation)
Abstract: In this talk, I will introduce and motivate the idea of modeling with measure-valued functions. We will discuss some of the many benefits of Radon Measures for biological models and provide some examples. We will then discuss the analytical tools inherited from adjact feilds such as optimal transport. Finally, I will end the talk with a discussion of finite-difference schemes of a size/age structured population model in this space of Radon Measures. In particular, I will present a simple upwind scheme and a minmod flux limiter scheme and sketch the convergence proof for these two schemes. I will then present Numerical examples which demonstrate the order of the two schemes under different regularity of the initial conditions.
4 December 2019
Dr. Nicolas Saintier (University of Buenos-Aires, Argentina)
For the Spring 2019 semester we will meet at 3:30 pm on Tuesdays in 208 Maxim Doucet Hall.
22 January 2019
Special seminar in Oliver Hall
29 January 2019
Understanding species persistence under reoccurring and interacting disturbances
Abstract: An important focus for management and conservation is determining whether a species or a system of interacting species can sustain itself. This question becomes increasingly important as populations are exposed to various disturbances, both natural and anthropogenic, such as hurricanes, habitat fragmentation, toxicants, and invasive species. Here, we first develop a model to consider the effect of a reoccurring disturbance on population persistence. Disturbances are assumed to occur stochastically according to a two-state Markov chain with their frequency depending on the average length of effect of a disturbance and the average time between disturbances. Using this model, we derive an approximation for the stochastic growth rate that allows us to consider how changes in the parameters describing the disturbance may impact species persistence. This information can be used to help inform management decisions. We then consider case where populations are exposed to multiple types of disturbances. In particular, we examine how the interaction between disturbance types, that is whether the combined effect of the disturbances is greater or less than their additive effects, may alter persistence conditions.
5 February 2019
Competing Interactions, Patterns, and Traveling Waves in Discrete Systems
Abstract: We consider bistable lattice differential equations with competing first and second nearest neighbor interactions. We construct heteroclinic orbits connecting the stable zero equilibrium state with stable spatially periodic orbits of period p=2,3,4 using transform techniques and a bilinear bistable nonlinearity. We investigate the existence, global structure, and multiplicity of such traveling wave solutions. For smooth nonlinearities an abstract result on the persistence of traveling wave solutions is presented and applied to lattice differential equations with repelling first and/or second neighbor interactions and to some problems with infinite range interactions.
12 February 2019
19 February 2019
SIMM seminar in Oliver Hall
26 February 2019
Investigating the effect of migration on the outcomes competition between zebra and quagga mussels
Abstract: We develop and use a mathematical model of difference equations to investigate interactions and competitive outcomes between two invasive species of mollusks: zebra mussel (Dreissena polymorpha) and quagga mussel (Dreissena rostriformis bugensis). The model has both spatial structure (patches) and temporal structure (age: juvenile and adult individuals). We show that, when migration among patches does not take place, in each patch one species eliminates the other, as it settles at a constant population size. Further, we investigate both numerically and analytically how migration among patches might affect this outcome. Some mathematical tools used in the analysis of the model will also be introduced and discussed.
5 March 2019
Mardi Gras break - no meeting
12 March 2019
Endemic dynamics of foot-and-mouth disease viruses in their natural host, African buffalo
Department of Integrative Biology
Oregon State University
Abstract: Extremely contagious pathogens are among the most important global biosecurity threats because of their high burden of morbidity and mortality, and their violent outbreaks that are difficult to quell. Understanding the mechanisms that enable persistence of contagious pathogens in their host populations is thus a central problem in disease ecology. Foot-and-mouth-disease viruses (FMDV) are massively contagious and represent the most important livestock pathogen restricting international trade. In sub-Saharan Africa, wild buffalo (Syncerus caffer) act as a reservoir for FMDV.
For the first time, we estimated epidemiologic parameters from experimental data and constructed transmission models to investigate how FMDV achieves endemic persistence in its reservoir host populations. Our study reveals striking epidemiologic variation among southern African FMDV serotypes (SAT1, 2, 3). None of the serotypes likely persist due to transmission of acute infection among susceptible juveniles alone. Including transmission from carriers, which retain viable virus after initial infection, reliably rescues SAT1, the most infectious serotype, from fade-out, and may also enable persistence of SAT3 in large populations, but not of SAT2. Additional mechanisms, such as viral antigenic shift, loss of acquired immunity, or spillover from other ungulate species, may be required for the persistence of SAT2 and SAT3 in buffalo populations.
19 March 2019
Blow-up and Quenching in Caputo Fractional Reaction Diffusion Equation
Abstract: Mathematical models using fractional derivative has been demonstrated as a better and economic models than its counterpart with integer order derivatives. However the analysis of fractional differential equation particularly blow-up and quenching behavior is more complicated than the analysis of blow-up/quenching behavior of the solution of equations of integer order. The reason being that the tools commonly used for integer order differential equations are not available for fractional differential equations in general. One approach is to compare the solution of the fractional differential equations to that of the corresponding ordinary differential equations, which are either solvable explicitly or the behavior of the solution is known. This is the main approach we have taken in the study of blow-up/quenching problems for fractional differential equations. The quenching problems for ordinary caputo fractional differential equation and time dependent caputo-fractional reaction diffusion equation in finite domain with a nonlinear source of Kawarada's type is studied and the solution is compared with the solution of the standard reaction diffusion equation. The results are extended for the equation with a general reaction term. Methods for the study of quenching of solution of fractional differential equation using the solution of the differential equation and lower solutions are developed. Sufficient conditions for quenching of the solution and the bounds for quenching time are obtained. The results are veried with some known examples. The blow-up problems for ordinary Caputo fractional differential equation and the time Caputo fractional reaction diffusion equation in one dimensional space is studied. Methods for the study of blow-up of solution of fractional differential equation using the solution of the differential equation and lower solutions are developed. Sufficient conditions for blow-up of the solution and the bounds for blow-up time are obtained. The results are verified with some known examples.
26 March 2019
SIMM seminar in Oliver Hall
2 April 2019
Peter Hinow, Prof. at UW Milwaukee.
9 April 2019
Hamiltonian Methods For Hyperbolic Equations With Fractional Diffusion
Jorge Macias Diaz
Universidad Autonoma de Aguascalientes
Abstract: In this talk, we consider an initial-boundary-value problem governed by a (1+1)-dimensional hyperbolic partial differential equation with constant damping that generalizes many nonlinear wave equations from mathematical physics. The model considers the presence of a spatial Laplacian of fractional order which is defined in terms of Riesz fractional derivatives, as well as the inclusion of a generic continuously differentiable potential. It is known that the undamped regime has an associated positive energy functional, and we show here that it is preserved throughout time under suitable boundary conditions. To approximate the solutions of this model, we propose a finite-difference discretization based on fractional centered differences. Some discrete quantities are proposed in this work to estimate the energy functional, and we show that the numerical method is capable of conserving the discrete energy under the same boundary conditions for which the continuous model is conservative. Moreover, we establish suitable computational constraints under which the discrete energy of the system is positive. The method is consistent of second order, and is both stable and convergent. The numerical simulations shown here illustrate the most important features of our numerical methodology.
16 April 2019
spring break - no meeting
23 April 2019
SIMM seminar in Oliver Hall
For the Fall 2018 semester we will meet at 3:30 pm on Tuesdays in 208 Maxim Doucet Hall.
25 September 2018
Global dynamics of a disease model with both direct and indirect transmissions
Abstract: We study the global dynamics of a disease model with both direct and indirect transmissions. Though the model system is nonlinear and it couples two transmission mechanisms, the basic reproduction number exhibits a nice linear property: it is simply the sum of two basic reproduction numbers for direct and indirect disease transmissions respectively. We further demonstrate that the basic reproduction number is a threshold parameter which characterizes the local and global dynamics of the model system. The key ideas (as well as main difficulties) in the proof will be explained in detail, and the presentation should be accessible to graduate and undergraduate students who have some basic knowledge of differential equations.
2 October 2018
Population Dynamics of Fisher-KPP Equations with Nonlocal Dispersal in Spatially Periodic Environment
Abstract: This talk deals with front propagation dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. We show that a general spatially periodic monostable equation with nonlocal dispersal has a unique spatially periodic positive stationary solution and has a spreading speed in every direction. In this talk, we also show that a spatially periodic nonlocal monostable equation with certain spatial homogeneity or small nonlocal dispersal distance has a unique stable periodic traveling wave solution connecting its unique spatially periodic positive stationary solution and the trivial solution in every direction for all speeds greater than the spreading speed in that direction. In the end, we will discuss some open problems in population dynamics.
9 October 2018
Picard’s Iterative Method for Caputo Fractional Differential Equations
Abstract: With fractional differential equations (FDEs) rising in popularity and methods for solving them still being developed, approximations to solutions of fractional initial value problems (IVPs) have great applications in related fields. This paper proves an extension of Picard’s Iterative Existence and Uniqueness Theorem to Caputo fractional ordinary differential equations, when the nonhomogeneous term satisfies the usual Lipschitz condition. As an application of our method, we have provided several numerical examples.
16 October 2018
October SIMM in Oliver Hall
23 October 2018
Continuous Data Assimilation from Scattered Spatial Observations in Time-Dependent PDEs
Abstract: Data assimilation is the task of combining mathematical models with observational data. In this work, we developed a new nonlinear data assimilation algorithm that extends the earlier approaches introduced by Azouani, Olson, and Titi (AOT) based on feedback control penalty for the solution of a system of partial differential equations. We improved the existing AOT algorithm by introducing a dynamic control process. Instead of using a constant parameter for the feedback control, we choose the parameter based on the current local state, the physical dynamic from the partial differential equations and the error estimate of the algorithm, which provides better feedback locally and leads a faster convergence rate comparing with the fixed feedback control. Also, we construct the interpolation from the discrete spatial data using a robust weighted least square interpolation method. The weight for each observation is determined by the distance to the observation location, which can be easily implemented over non-structured data in higher dimensions. As a proof-of-concept for these models, we have tested our algorithm on a number of test problems, including 1D KPP-Burgers' equation, 1D Kuramoto-Sivashinsky equation, and 2D shallow water equations.
30 October 2018
Title to be announced
Diana Paola Lizarralde-Bejarano
EAFIT University, Medellin, Colombia
6 November 2018
Prey evolution of toxicant resistance and its effect on the predator-prey dynamics
Abstract: Continuous exposure to a toxicant may result in the evolution of toxicant resistance in relatively short-lived species. In this study, we investigate the effect of such an evolution of toxicant resistance in the prey population on the overall dynamics of a predator-prey system. We first derive and analyze a discrete time predator-prey model. We establish conditions for the existence and stability (local or global) of various equilibria, as well as conditions for the persistence of both the predator and prey populations. We then extend this model to an evolutionary model by applying the Darwinian evolutionary game theory methodology. This methodology couples the population dynamics with the dynamics of an evolving phenotypic trait, which we assume provides a measure for the level of toxicant resistance developed by the prey. The predator is impacted by prey evolution indirectly, through changes in prey density, and directly, through an assumed trade-off between toxicant resistance and the ability of the prey to escape predation. We study the dynamics of this model and establish conditions for when the prey is able to evolve toxicant resistance. In particular, we show that the evolution of toxicant resistance may allow both the predator and prey to persist when, without the evolution, both may go extinct.
13 November 2018
November SIMM in Oliver Hall
20 November 2018
Interval analysis for the treatment of uncertainty in epidemiological models based on systems of ordinary differential equations
Diana Paola Lizarralde-Bejarano
EAFIT University, Medellin, Colombia
Abstract: In epidemiological models based on ordinary differential equations systems (henceforth ODEs), knowledge and available information about the model parameters and the initial conditions are limited. This is especially true for models that simulated the transmission of infectious diseases. Also, there is inherent uncertainty in any measurement process. We propose to consider such uncertainty by defining parameters and initial conditions as closed real intervals. After that, we will use the VSPODE (Verifying Solver for Parametric ODEs) solver for parametric ODEs, which produce guaranteed bounds on the solutions of nonlinear dynamic systems with interval-valued initial states and parameters. On the other hand, to understand the meaning of model fit to interval data, we present the concept of strong compatibility between interval data and the parameters and initial conditions of the nonlinear system. Finally, given a numerical solution of our system and the initial interval data, we formulate a strategy and an optimization problem to find the set of parameters and initial conditions which produce the best model fit to the interval data.
For the Spring 2018 semester we will meet at 3:30 pm on Tuesdays in 201 Maxim Doucet Hall.
23 January 2018
Changes in population outcomes resulting from phenotypic evolution and environmental disturbances
Abstract: We develop an evolutionary game theoretic version of a general nonlinear matrix model that includes the dynamics of a vector of mean phenotypic traits subject to natural selection. For this evolutionary model, we use bifurcation analysis to establish the existence and stability of a branch of positive equilibria that bifurcates from the extinction equilibrium when the inherent growth rate passes through one. We then present an application to a daphnia model to demonstrate how the evolution of resistance to a toxicant may change persistence scenarios. We show that if the effects of a disturbance are not too large, then it is possible for a daphnia population to evolve toxicant resistance whereby it is able to persist at higher levels of the toxicant than it would otherwise. These results highlight the complexities involved in using surrogate species to examine toxicity. Time permitting, we will also consider a nonautonomous matrix model to examine the possible long-term effects of environmental disturbances, such as oils spills, floods, and fires, on population recovery. We focus on population recovery following a single disturbance, where recovery is defined to be the return to the pre-disturbance population size. Using methods from matrix calculus, we derive explicit formulas for the sensitivity of the recovery time with respect to properties of the population or the disturbance.
6 February 2018
Modeling Distinct Virus Infection Strategies in Virus-Microbe Systems
Abstract: Viruses of microbes, including bacterial viruses (phage), archaeal viruses, and eukaryotic viruses, can influence the fate of individual microbes and entire populations. Here, we model distinct modes of virus-host interactions and study their impact on the abundance and diversity of both viruses and their microbial hosts. We consider two distinct viral populations infecting the same microbial population via two different strategies: lytic and chronic. A lytic strategy corresponds to viruses that exclusively infect and lyse their hosts to release new virions. A chronic strategy corresponds to viruses that infect hosts and then continually release new viruses via a budding process without cell lysis. The chronic virus can also be passed on to daughter cells during cell division. The long-term association of virus and microbe in the chronic mode drives differences in selective pressures with respect to the lytic mode. We utilize invasion analysis of the corresponding nonlinear differential equation model to study the ecology and evolution of heterogenous viral strategies. We first investigate stability of equilibria, and characterize oscillatory and bistable dynamics in some parameter regions. Then, we derive fitness quantities for both virus types and investigate conditions for competitive exclusion and coexistence. In so doing we find unexpected results, including a regime in which the chronic virus requires the lytic virus for survival and invasion.
20 February 2018
Overcoming the added-mass instability for coupling incompressible flows and elastic beams
Abstract: A new partitioned algorithm is described for solving fluid-structure interaction (FSI) problems coupling incompressible flows with elastic structures undergoing finite deformations. The new algorithm, referred to as the Added-Mass Partitioned (AMP) scheme, overcomes the added-mass instability that has for decades plagued partitioned FSI simulations of incompressible flows coupled to light structures. Within a Finite-Difference framework, the AMP scheme achieves fully second-order accuracy and remains stable, without sub-time-step iterations, even for very light structures when added-mass effects are strong. The stability and accuracy of the AMP scheme is validated through mode analysis and numerical experiments. Aiming to extend the AMP scheme to an Finite-Element framework, we also develop an accurate and efficient Finite-Element Method for solving the Incompressible Navier-Stokes Equations with high-order accuracy up-to the boundary.
6 March 2018
20 March 2018
27 March 2018
Estimation of Distributed Delays
Abstract: Delay differential equations have been studied for several decades as they arise in many applications. A common approach is to transform a distributed delay system to a related ordinary differential equation system via the ‘linear chain trick’. This is due to the fact that the term involv- ing the distributed delay can be replaced by a state variable that is coupled to other state variables in a linear system of ODEs. This transformation relies on the assumption that the delay follows a gamma distribution. We try to determine when one correctly or incorrectly assumes a gamma distribution and the implications of such assumption by estimating the parameters associated with the distribution followed by a time delay. The results when performing the estimations in the ODE system (this is equivalent to the delay system if a gamma distribution is assumed) are compared to the results in the delay system.
10 April 2018
17 April 2018
Models of Dynamic Virus and Immune Response Networks
Abstract: The dynamics of virus and immune response within a host can be viewed as a complex and evolving ecological system. For example, during HIV infection, an array of CTL immune response populations recognize specific epitopes (viral proteins) presented on the surface of infected cells to effectively mediate their killing. However HIV can rapidly evolve resistance to CTL attack at different epitopes, inducing a dynamic network of viral and immune response variants. We consider models for the network of virus and immune response populations. Our analysis provides insights on viral immune escape from multiple epitopes. In the “binary allele” setting, we prove that if the viral fitness costs for gaining resistance to each of n epitopes are equal and multiplicative, then the system of 2^n virus strains converges to a “perfectly nested network” with less than n+1 persistent virus strains. Overall, our results suggest that immunodominance is the most important factor determining viral escape pathway of HIV against multiple CTL populations. To conclude, I briefly discuss ongoing collaborative work to connect the models with intra-host SIV/immune response data and to extend analysis to coevolving virus/antibody populations.
24 April 2018
To be announced
For the Fall 2017 semester we will meet at 3:30 pm on Tuesdays in 201 Maxim Doucet Hall.
26 September 2017
An Introduction to Bernstein polynomials
Abstract: Bernstein polynomials have been playing a crucial role in approximation theory since the early 20th century. They were used to prove the Weierstrass theorem by S. Bernstein in 1910s, and later became the basis of Bézier Curves, which are now widely used in computer graphics to model smooth curves. With Bernstein polynomials, we can approximate a function over a finite domain and refine the approximation to any desired precision. In this presentation, we will look into the definition of Bernstein polynomials, some important properties of the polynomials, the Weierstrass approximation theorem, and Bézier Curves.
10 October 2017
The dynamics of an ion channel model
Abstract: We study a time dependent Poisson-Nernst-Planck system which arises from the model of ion channels. The objective is to understand how the ion concentrations are distributed in the channel if there are ion fluxes on the channel boundaries. Assuming that the Debye length is small relative to the channel length, we derive an asymptotic formula for the dynamic solution by matching outer and Debye layer solutions. It is interesting to note that for the time-dependent problem, the outer solution has a boundary layer that does not occur in time-independent problems.
31 October 2017
Competitive Exclusion through Discrete Time Models
7 November 2017
Numerical Algorithms for PDE-Constrained Optimization Problems
Southern Illinois University Edwardsville
Abstract: PDE-Constrained optimization problems arise in many different scientiffic and engineering applications. In this talk, we will first present several efficient optimize-then-discretize algorithms for iteratively solving the first-order optimality KKT system from both parabolic and wave PDE-Constrained optimal control problems. Second, we will discuss some interesting numerical issues regarding the discretize-then-optimize algorithms, which are also widely used in practice. Numerical results will be shown to illustrate the effectiveness of our proposed numerical algorithms.
For the Spring 2017 semester we will meet at 3:30 pm on Tuesdays in 201 Maxim Doucet Hall.
31 January 2017
Poisson-Nernst-Planck system with multiple ions
Abstract: We study the Poisson-Nernst-Planck (PNP) system with an arbitrary number of ion species with arbitrary valences in the absence of fixed charges. Assuming point charges and that the Debye length is small relative to the domain size, we derive an asymptotic formula for the steady-state solution by matching outer and boundary layer solutions. The case of two ionic species has been extensively studied, the uniqueness of the solution has been proved, and an explicit expression for the solution has been obtained. However, the case of three or more ions has received significantly less attention. Previous work has indicated that the solution may be nonunique and that even obtaining numerical solutions is a difficult task since one must solve complicated systems of nonlinear equations. By adopting a methodology that preserves the symmetries of the PNP system, we show that determining the outer solution effectively reduces to solving a single scalar transcendental equation. Due to the simple form of the transcendental equation, it can be solved numerically in a straightforward manner. Despite the fact that for three ions, previous studies have indicated that multiple solutions may exist, we show that all except for one of these solutions are unphysical and thereby prove the existence and uniqueness for the three-ion case.
22 February 2017 (WEDNESDAY 2:30 Maxim Doucet Hall 208)
Directional Statistics for High Volatility and Big Data Science
Indian Statistical Institute, Kolkata, West Bengal, INDIA and
Augusta University, Augusta, Georgia
Abstract: In this era of emerging complex problems, both small and big data – linear and non-linear, exhibit challenging characteristics which need to be carefully modelled. Thus, multidisciplinary research in mathematical sciences has become indispensable. Marked presence of asymmetry, multimodality, high volatility, long and fat tails, non-linear dependency, etc. are common features of contemporary data. Notwithstanding pitfalls, ideas from several disciplines do enrich the contribution of the research work. Directional statistics is one such scientific “key technology” as which on one hand is developed from the conglomeration of the inductive logic of statistics, objective rigor of mathematics and the skills of numerical analysis of computer science. On the other hand, it possesses the richness to handle the need for providing statistical inference to a wide and emerging arena of applied sciences. Directional data (DD) in general refer to multivariate observations on variables with possibly linear, axial, circular and spherical components. Circular random variables are usually those which pertain to observations on directions, angles, orientations, etc. Data on periodic occurrences can also be cast in the arena of circular data. In general, DD may be mapped to smooth manifolds, e.g. circle, hyper-sphere, hyper-toroid, hyper-cylinder, or to axial and hyper-disc also. Analysis of such data sets differs markedly from those for linear ones due to the disparate topologies between the line and the circle. First, some methods of construction of probability distributions and statistical models for DD on smooth manifolds are presented. The need for applications of these abound for data in a variety of applied sciences. To illustrate this fact, we take up two important problems, one for linear and the other for directional data. With linear data, we take up the problem of obtaining probability distributions for modelling high volatility. The work of Mandelbrot has shown the appropriateness of the stable families of distributions for high volatility. However, in general, these families do not possess any analytical closed form for their probability density functions. This leads to the complexity of inference involving the parameters of such distributions. We overcome this problem of modelling high volatility data by appealing to the area of probability distributions for directional data. A new family of possibly multimodal, asymmetric and heavy-tail distribution is presented. The usual fat-tail, Cauchy and t, distributions are encompassed by this family and it has even tails comparable to that of the stable family. The second problem deals with data, possibly Big Data, on smooth manifolds. It is first noted that such data invariably exhibit multimodality and hence the possibility of underlying multiple component distributions. Thus, it would be prudent to “Divide and Conquer”, prior to proceeding for drawing statistical inference on such data. Here we deal with this problem by developing Hierarchical Unsupervised Learning or statistical Clustering techniques for manifold data. We illustrate our approach by a real-life example based on agricultural insurance data.
4 April 2017 (208 Maxim Doucet Hall)
Estimation of Distributed Delays
University of Louisiana at Lafayette
Abstract: Delay differential equations have been studied for several decades as they arise in many applications. A common approach is to transform a distributed delay system to a related ordinary differential equation system via the ‘linear chain trick’. This is due to the fact that the term involving the distributed delay can be replaced by a state variable that is coupled to other state variables in a linear system of ODEs. This transformation relies on the assumption that the delay follows a gamma distribution. We try to determine when one correctly or incorrectly assumes a gamma distribution and the implications of such assumption by estimating the parameters associated with the distribution followed by a time delay. The results when performing the estimations in the ODE system (this is equivalent to the delay system if a gamma distribution is assumed) are compared to the results in the delay system.
For the Fall 2016 semester we will meet at 3:30 pm on Tuesdays in 201 Maxim Doucet Hall.
13 September 2016
Traveling wave solutions of a diffusive epidemic model
University of Louisiana at Lafayette
We study the traveling wave solutions of a diffusive epidemic model with standard incidence. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and a minimal wave speed. Our proof is based on Schauder fixed point theorem and Laplace transform.
27 September 2016
The Dynamics of Vector-Borne Relapsing Diseases
University of Louisiana at Lafayette
Motivated by Tick-Borne Relapsing Fever (TBRF) we will be investigating the dynamics of various models for the spread of a relapsing disease by a vector. In particular we quantify the effect that relapses have on the disease spread and the how the number of relapses influence control strategies for the disease.
4 October 2016
Modeling Multi-Epitope HIV/CTL Immune Response Dynamics and Evolution
University of Louisiana at Lafayette
The CTL (Cytotoxic T Lymphocyte) immune response plays a large role in controlling HIV infection. CTL immune effectors recognize epitopes (viral proteins) presented on the surface of infected cells to mediate their killing. The immune system has an extensive repertoire of CTLs, however HIV can evolve resistance to attack at different epitopes. The ensuing arms race creates an evolving network of viral strains and CTL populations with variable levels of epitope resistance. Motivated by this, we formulate a general ODE model of multi-epitope virus-immune response dynamics. Some special cases for the HIV/CTL interaction network are considered, the case of a nested network and the general two-epitope case. We characterize the persistent viral strains and immune response in terms of system parameters and prove global properties of solutions via Lyapunov functions. The results are interpreted in the context of within-host HIV/CTL evolution and numerical simulations are provided. To conclude, we discuss extensions of the model to a PDE system which incorporates cell-infection age structure.
11 October 2016
Synchrony and the dynamic dichotomy in a class of matrix population models
University of Louisiana at Lafayette
In this talk, I will discuss the dynamics of a class of discrete-time structured population models called synchrony models. Synchrony models are characterized by the simultaneous bifurcation of a branch of positive equilibria and a branch of synchronous 2-cycles from the extinction equilibrium. These models exhibit a dynamic dichotomy in which the two steady states have opposite stability properties that are determined by the relative levels of competition in the population. I will also present an application that is motivated by observations of a population of cannibalistic gulls.