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Appl Math Lett 2021 Jan 15;111:106617. Epub 2020 Jul 15.

Department of Mathematics, Emory University, 400 Dowman Drive, Atlanta, GA 30322, USA.

We present an early version of a Susceptible-Exposed-Infected-Recovered-Deceased (SEIRD) mathematical model based on partial differential equations coupled with a heterogeneous diffusion model. The model describes the spatio-temporal spread of the COVID-19 pandemic, and aims to capture dynamics also based on human habits and geographical features. To test the model, we compare the outputs generated by a finite-element solver with measured data over the Italian region of Lombardy, which has been heavily impacted by this crisis between February and April 2020. Read More

Appl Math Lett 2020 Sep 25;107:106442. Epub 2020 Apr 25.

School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, PR China.

In this paper, we present an SEIR epidemic model with infectivity in incubation period and homestead-isolation on the susceptible. We prove that the infection-free equilibrium point is locally and globally asymptotically stable with condition We also prove that the positive equilibrium point is locally and globally asymptotically stable with condition Numerical simulations are employed to illustrate our results. In the absence of vaccines or antiviral drugs for the virus, our results suggest that the governments should strictly implement the isolation system to make every effort to curb propagation of disease during the epidemic. Read More

Appl Math Lett 2017 Jan 2;63:109-117. Epub 2016 Aug 2.

Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212; Departments of Pathology and Medicine, Massachusetts General Hospital and Harvard Medical School, Boston, MA 02114.

Randomized longitudinal clinical trials are the gold standard to evaluate the effectiveness of interventions among different patient treatment groups. However, analysis of such clinical trials becomes difficult in the presence of missing data, especially in the case where the study endpoints become difficult to measure because of subject dropout rates or/and the time to discontinue the assigned interventions are different among the patient groups. Here we report on using a validated mathematical model combined with an inverse problem approach to predict the values for the missing endpoints. Read More

Appl Math Lett 2015 Feb 16;40:97-101. Epub 2014 Oct 16.

University of California, Merced, School of Natural Sciences, 5200 N Lake Rd, Merced, CA 95343.

The nucleated polymerization model is a mathematical framework that has been applied to aggregation and fragmentation processes in both the discrete and continuous setting. In particular, this model has been the canonical framework for analyzing the dynamics of protein aggregates arising in prion and amyloid diseases such as as Alzheimer's and Parkinson's disease. We present an explicit steady-state solution to the aggregate size distribution governed by the discrete nucleated polymerization equations. Read More

Appl Math Lett 2015 Jan;43:10-18

Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8212 USA.

In the context of inverse or parameter estimation problems we demonstrate the use of statistically based model comparison tests in several examples of practical interest. In these examples we are interested in questions related to information content of a particular given data set and whether the data will support a more complicated model to describe it. In the first example we compare fits for several different models to describe simple decay in a size histogram for aggregates in amyloid fibril formation. Read More

Appl Math Lett 2015 Feb;40:84-89

Department of Mathematics, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC.

Many experimental systems in biology, especially synthetic gene networks, are amenable to perturbations that are controlled by the experimenter. We developed an optimal design algorithm that calculates optimal observation times in conjunction with optimal experimental perturbations in order to maximize the amount of information gained from longitudinal data derived from such experiments. We applied the algorithm to a validated model of a synthetic Brome Mosaic Virus (BMV) gene network and found that optimizing experimental perturbations may substantially decrease uncertainty in estimating BMV model parameters. Read More

Appl Math Lett 2013 Jul;26(7):794-798

Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh, NC, United States.

We formulated a structured population model with distributed parameters to identify mechanisms that contribute to gene expression noise in time-dependent flow cytometry data. The model was validated using cell population-level gene expression data from two experiments with synthetically engineered eukaryotic cells. Our model captures the qualitative noise features of both experiments and accurately fit the data from the first experiment. Read More

Appl Math Lett 2013 May 3;26(5):571-577. Epub 2013 Jan 3.

Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, North Carolina State University, Raleigh, NC.

We developed a series of models for the label decay in cell proliferation assays when the intracellular dye carboxyfluorescein succinimidyl ester (CFSE) is used as a staining agent. Data collected from two healthy patients were used to validate the models and to compare the models with the Akiake Information Criteria. The distinguishing features of multiple decay rates in the data are readily characterized and explained via time dependent decay models such as the logistic and Gompertz models. Read More

Appl Math Lett 2013 Jan;26(1):10-14

Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, N.C. State University, Raleigh, NC.

We formulate an optimal design problem for the selection of best states to observe and optimal sampling times and locations for parameter estimation or inverse problems involving complex nonlinear nonlinear partial differential systems. An iterative algorithm for implementation of the resulting methodology is proposed. Read More

Appl Math Lett 2012 Jan 10;26(1):51-56. Epub 2012 Apr 10.

Virginia Commonwealth University, 1015 Floyd Avenue, P.O. Box 843083, Richmond, VA 23284.

We formalize an algorithm for solving the L(1)-norm best-fit hyperplane problem derived using first principles and geometric insights about L(1) projection and L(1) regression. The procedure follows from a new proof of global optimality and relies on the solution of a small number of linear programs. The procedure is implemented for validation and testing. Read More

Appl Math Lett 2010 Dec;23(12):1412-1415

Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, North Carolina State University, Raleigh, NC 27695-8213 and INRIA Rocquencourt, Projet BANG, Domaine de Voluceau, 78153 Rocquencourt, France.

We present a general class of cell population models that can be used to track the proliferation of cells which have been labeled with a fluorescent dye. The mathematical models employ fluorescence intensity as a structure variable to describe the evolution in time of the population density of proliferating cells. While cell division is a major component of changes in cellular fluorescence intensity, models developed here also address overall label degradation. Read More

Appl Math Lett 2009 Nov;22(11):1778-1780

Biomathematics Research Centre, University of Canterbury, New Zealand,

We improve the lower bound on the extremal version of the Maximum Agreement Subtree problem. Namely we prove that two binary trees on the same n leaves have subtrees with the same ≥ c log log n leaves which are homeomorphic, such that homeomorphism is identity on the leaves. Read More

Appl Math Lett 2007 Sep;20(9):959-963

Centre for Mathematical Biology, Mathematical Institute, 24-29 St Giles', Oxford, OX1 3LB, UK.

There are two simple solutions to reaction-diffusion systems with limit-cycle reaction kinetics, producing oscillatory behaviour. The reaction parameter mu gives rise to a 'space-invariant' solution, and mu versus the ratio of the diffusion coefficients gives rise to a 'time-invariant' solution. We consider the case where both solution types may be possible. Read More