Applied & Industrial Mathematics

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Open Access
A Branch-and-Price Algorithm for Bin Packing Problem
(2015-12-16) Ataei, Masoud; Chen, Michael
Bin Packing Problem examines the minimum number of identical bins needed to pack a set of items of various sizes. Employing branch-and-bound and column generation usually requires designation of the problem-specific branching rules compatible with the nature of the pricing sub-problem of column generation, or alternatively it requires determination of the k-best solutions of knapsack problem at level kth of the tree. Instead, we present a new approach to deal with the pricing sub-problem of column generation which handles two-dimensional knapsack problems. Furthermore, a set of new upper bounds for Bin Packing Problem is introduced in this work which employs solutions of the continuous relaxation of the set-covering formulation of Bin Packing Problem. These high quality upper bounds are computed inexpensively and dominate the ones generated by state-of-the-art methods.
Open Access
A High Order Method for Analyzing the Electromagnetic Response of an Array of Thin Wires
(2019-11-22) Xue, Qunxing; Haslam, Michael C
The problem of evaluating the electromagnetic response of wire-antenna systems by means of integral equations is one of great importance and significant mathematical complexity. The literature on this subject is extensive, including contributions from the 19th century work of Pocklington, the classic works of King, to some of the most recent work by Davies et.al., Bruno and Haslam and others. In this thesis, we develop a new high order numerical method to treat the problem corresponding to a parallel array of thin straight wires. A number of significant difficulties arise in this problem as a result of certain singularities and near-singularities that are inherent in its integral equation formulation. In particular, no satisfactory quadrature methods exist for the high-order evaluation of the integrals which arise from the thin wire equations when two wires in an array are separated by a small distance but finite distance. Such a configuration requires the evaluation of integrals whose integrands are logarithmically singular not only at a point inside the domain of integration, but also at a point just outside of the domain integration. The quadrature formulas we derive as a main contribution of this thesis explicitly treat both of these cases. A full numerical implementation of our algorithm was developed and results corresponding to high, moderate and low excitation frequencies are presented.
Open Access
Dynamics of Naive and Memory CD4 T-cells in Chronically Infected HIV Patients Post-Injection of Down-Modulated CCR5 Memory CD4-Cells
(2016-11-25) Raad, Angie; Heffernan, Jane M.
HIV/AIDS, a sexually transmitted diseases continues to affect the lives of millions of individuals worldwide. This retrovirus targets CD4 T-cell populations, the main driver of the immune system by using the chemokine co-receptor 5 (CCR5). Despite the success of the highly active antiretroviral therapy in reconstituting the immune system, HIV infected individuals still suffer from low CD4 T-cell counts. Recently, researchers were able to highlight the success of immunotherapy in restoring the CD4 T-cell count. To further, investigate such importance, our collaborators at case Western University injected CCR5-down-modulated memory CD4 T-cells into 9 chronically infected HIV patients. Using a linear transitions from the naive to the effector memory state, a non linear ordinary differential equation model was used to model the experiment. Various data fitting techniques in Matlab Stan and Monolix software were used to estimate the model parameters (proliferation, death, transition and birth rates) before and after the initiation of the treatment to study the change of the cell dynamics. Our fittings have indicated an increase in the memory stem and na\"{\i}ve cell lifespan post-clinical trial. Using sensitivity analysis, we showed that the na\"{\i}ve cell birth rate from the thymus lambda, the memory stem cell proliferation rate p_ST and the central memory cell death rate d_C played an important role in restoring the CD4 T-cell count. A stochastic model for the CD4 T-cells population was developed to examine if fluctuations from the stochastic simulation were able to capture the experimental data measurements. The findings of this study indicates the importance of looking further into how modified CD4 T-cells are able to restore the T-cell counts which thereby decrease the HIV virus pool and help HIV patients to maintain a low level of the virus and most importantly a high level of T-cell count.
Open Access
Even-Odd Cycled High-Order S-FDTD Method for Maxwell's Equations and Application to Coplanar Waveguides
(2017-07-27) Sarai, Maninder Kaur; Liang, Dong
In this thesis, a new even-odd cycled high-order splitting finite difference time domain scheme for Maxwell's equations in two dimensions is developed. The scheme uses fourth order spatial difference operators and even-odd time step technique to make it more accurate in both space and time. The scheme is energy-conserved, unconditionally stable and very efficient in computation. We analyze in detail the stability, dispersion and phase error for the scheme. We also prove its energy conservation, convergence of high order accuracy and convergence of divergence free approximation. Numerical experiments confirm the theoretical analysis results. Further, the developed scheme is applied to computations of the grounded coplanar waveguides, the elevated CPW and the complex transitions between CPW and rectangular waveguides.
Open Access
Integrating Epigenetic Priors For Improving Computational Identification of Transcription Factor Binding Sites
(2016-09-20) Shoukat, Affan; Grigull, Jorg
Transcription factors and histone modifications play critical roles in tissue-specific gene expression. Identifying binding sites is key in understanding the regulatory interactions of gene expression. Nave computational approaches uses solely DNA sequence data to construct models known as Position Weight Matrices. However, the various assumptions and the lack of background genomic information leads to a high false positive rate. In an attempt to improve the predictive performance of a PWM, we use a Hidden Markov Model to incorporate chromatin structure, in particular histone modifications. The HMM captures physical interactions between distinct HMs. Indeed, the integration of sequence based PWM models and chromatin modifications improve the predictive ability of the integrative model.
Open Access
Local Volatility Model With Stochastic Interest Rate
(2015-12-16) Hu, Bing; Zhu, Huaiping
Many different models exist to describe the behaviour of asset prices and are used to value options on such an underlying asset. This report investigates the local volatility model in a stochastic interest rates framework. First, we derive the local volatility function for this model, which allows the local volatility surface to be exacted from the prices of traded call options. Next, we present numerical approaches to construct a local volatility surface based on finite difference approximation, Monte Carlo simulation and Lipschitz interpolation. Then, Monte Carlo simulation is applied to value options using the local volatility surface. Finally, a numerical implementation of the model and its results are reported and compared with real market data.
Open Access
Markovian Dynamics in Chromatin Loop Extrusion Factors
(2020-08-11) Eng, Kathleen; Grigull, Jorg
Markov properties can be used to model different dynamic processes at various stages of the loop extrusion process. The current methods are proposed to gain insight on how Markov models may illustrate chromatin behaviour once the appropriate observed data becomes available. We find that single molecule FRET experiments are able to identify the conformational states chromatin using Gaussian mixture models. The unbinding and binding rates of loop extrusion factors (LEFs) were applied in an immigration-death model, and found to play a role in influencing the frequency of loop extrusion. By including the additional parameter of the presence of nucleosomes with LEF binding on a strand of DNA, we find that the theoretical timescale of DNA exposure decreased upon LEF binding. The binding behaviour of LEFs is also dependent on the location of nucleosomes on a strand of DNA. This is modeled with the Gillespie algorithm to simulate LEF binding activity with single cell dynamics.
Open Access
Modelling HPV Vaccination and Screening Strategies to Optimize Treatment
(2015-12-16) Milwid, Rachael Michal; Heffernan, Jane
HPV is a common sexually transmitted infection found worldwide which can lead to serious health effects. While HPV has a high regression rate, if it does progress, it can cause various cancers (i.e. cervical, penile, throat). It is possible to minimize the mal-effects of HPV with tools such as screening, vaccination and treatment. Three sets of compartmental models were developed to study various aspects of HPV infection and progression. The first set of models studies which parameters are relevant in screening and vaccination programs and compares four different programs: a no intervention program, a screening only intervention program, a vaccination only program, and a screening and vaccination program. The second set of models compares various screening programs, including a co-screening program. The purpose of this set of models is to complete a cost analysis on the models, as well as to compare them epidemiologically. The third set of models studies the phenomenon of infection and re-infection with HPV. This chapter includes both single HPV type models and multi-type HPV models. All three sets of models lead to the same conclusions that HPV screening is essential in the minimization of HPV and cervical cancer. Furthermore, both screening and vaccination are essential in lowering the basic reproduction number.
Open Access
Optimal Investment in Deferred Income Annuities
(2018-11-21) Mauskopf, Adam Emanuel; Huang, Huaxiong
In this thesis we develop and analyze a technique for finding the optimal investment strategy for a deferred income annuity (DIA). We first present some initial background needed to understand what a DIA is. We then lay out a mathematical framework which will allow us to formalize the optimization process. The method and implementation of the optimization is explained, and the results are then analyzed. We then add an extra layer of complication to our model by allowing our optimal portfolio to contain more types of assets. The results of this new model are analyzed, and finally we mention possible extensions to this work.
Open Access
Rosenbluth Algorithm Studies of Self-Avoiding Walks
(2015-12-16) Tabrizi, Mandana; VanRensburg, Esaias J. Janse
In this thesis we used self-avoiding walks as a model of linear polymers to study some of the most fundamental questions about polymers- namely the quantification of polymer entropy. We introduced scaling formulas for the number of walks and other polymer properties such as radius of gyration and end-to-end distance. Then, we calculated these quantities using a Monte Carlo simulation and estimated the critical exponents in the scaling formulas. There is a pressure field in the vicinity of a polymer and a particle placed close to the polymer will accelerate away from it due to the pressure gradient. The scaling of the pressure as a function of distance from the polymer and length of the polymer is determined and tested numerically. Also, we modeled the relationship between velocity and the position of the particle in the 2-dimensional lattice and estimated the limiting speed.
Open Access
The Effects of Pre-Movement on Large Building Evacuations
(2015-12-16) Farnell, Lisa Carroll Ruth; Madras, Neal; Chen, Shengyuan Michael
Evacuation times for buildings with a range of heights and occupant loads were generated by a computer simulation algorithm, assuming simultaneous start. Additional evacuation times were generated for the same buildings with pre-movement times assigned to building occupants. Pre-movement times were assigned based on uniform and gamma distributions. Building evacuation times with pre-movement were compared to those without, to determine the quantitative effects of pre-movement. Using regression analysis, equations were generated to predict the effects of pre-movement for given building heights and occupant loads. Regression equations were shown to reasonably predict the effects of pre-movement for the building cases used for the regression analysis. Additional simulations were performed with and without pre-movement for buildings with alternative heights and occupant loads. The regression function was applied to these additional simulations, and found to predict the effects of pre-movement in these building cases with some accuracy.
Open Access
The Impact of Population Heterogeneities and Disease Interventions on Herd Immunity: A Case Study of the COVID-19 Pandemic in Ontario
(2023-03-28) Liwag, Maria Geneva Roselle Marino; Moghadas, Seyed
In epidemiology, herd immunity refers to the population level of immunity required to prevent or extinguish a large disease outbreak. In models with homogeneously mixing assumptions and without demographic structures, the herd immunity level may be different from that in heterogeneous models. With the COVID-19 pandemic in Ontario as a case study, a comprehensive deterministic mathematical model of disease spread with age and contact pattern variations was developed to examine the required herd immunity for different variants and compare with theoretical values obtained using homogeneous assumptions. The effects of non-pharmaceutical (testing/isolation of silent infections) interventions and vaccination on epidemic progression and herd immunity were investigated. With the inclusion of age and contact pattern structures, the resulting herd immunity level required to end an epidemic under the assumptions of long-term protection (without re-infection) is lower than theoretical values, even for more transmissible variants. While waning immunity and re-infection results in an oscillation in herd immunity levels in the population, subsequent epidemic peaks are less amplified, suggesting that even with increased variant transmissibility, infections of any variant allow for population immunity to rise, leading to an endemic state.
Open Access
Understanding the effect of interventions on transmission dynamics of emerging disease: A case study of COVID-19 pandemic
(2021-11-15) Tariq, Mehreen; Moghadas, Seyed
Many countries implemented strict social distancing measures to reduce infections, hospitalizations, and deaths during the coronavirus disease 2019 (COVID-19) pandemic. We developed an age-structured deterministic compartmental model and parameterized it with recent COVID-19 estimates to evaluate the effect of self-isolation and stay-at-home orders on infections, hospitalizations, and deaths. The findings show that a 5-month stay-at-home order targeting older individuals ($\geq 50$ years) had the greatest reduction in hospitalizations (over $47\%$) and deaths (over $55\%$). A 5-month stay-at-home order for individuals $\geq 65$ years had the most hospitalizations (over 0.0087) and deaths (0.0027) averted per-person practising the stay-at-home order. School closures reduced the outcomes of interest if implemented for a longer duration. Due to the increase in infections post-lockdown (shown in scenario 2), the strategies tested in this study can be used to strategically lift lockdown orders and minimize the burden on healthcare systems until herd immunity is achieved (through vaccination).

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