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Recent documents in Mathematics Faculty Researchen-usWed, 29 Jul 2020 02:44:35 PDT3600Local Lagged Adapted Generalized Method of Moments Dynamic Process
https://mds.marshall.edu/mathematics_faculty/45
https://mds.marshall.edu/mathematics_faculty/45Mon, 27 Jul 2020 13:50:36 PDT
Aspects of a local lagged adapted generalized method of moments (LLGMM) dynamic process are described herein. In one embodiment, the LLGMM process includes obtaining a discrete time data set as past state information of a continuous time dynamic process over a time interval, developing a stochastic model of the continuous time dynamic process, generating a discrete time interconnected dynamic model of local sample mean and variance statistic processes (DTIDMLSMVSP) based on the stochastic model, and calculating a plurality of admissible parameter estimates for the stochastic model using the DTIDMLSMVSP. Further, in some embodiments, the process further includes, for at least one of the plurality of admissible parameter estimates, calculating a state value of the stochastic model to gather a plurality of state values, and determining an optimal admissible parameter estimate among the plurality of admissible parameter estimates that results in a minimum error among the plurality of state values.
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Gangaram S. Ladde et al.A variational principle for discontinuous potentials
https://mds.marshall.edu/mathematics_faculty/44
https://mds.marshall.edu/mathematics_faculty/44Tue, 12 May 2020 13:16:32 PDT
Let $X$ be a compact space, $f\colon X \to X$ a continuous map, and $\Lambda \subset X$ be any $f$-invariant subset. Assume that there exists a nested family of subsets $\{\Lambda_l\}_{l \geq 1}$ that exhaust $\Lambda$, that is $\Lambda_l \subset\Lambda_{l+1}$ and $\Lambda =\bigcup_{l \geq 1} \Lambda_l$. Assume that the potential $\varphi \colon X \to \mathbb{R}$ is continuous on the closure of each $\Lambda_l$ but not necessarily continuous on $\Lambda$. We define the topological pressure of $\varphi$ on $\Lambda$. This definition is shown to have a corresponding variational principle. We apply the topological pressure and variational principle to systems with non-zero Lyapunov exponents, countable Markov shifts, and unimodal maps.
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Anna MummertThe thermodynamic formalism for almost-additive sequences
https://mds.marshall.edu/mathematics_faculty/43
https://mds.marshall.edu/mathematics_faculty/43Tue, 12 May 2020 13:16:21 PDT
We study the nonadditive thermodynamic formalism for the class of almost-additive sequences of potentials. We define the topological pressure PZ(Φ) of an almost-additive sequence Φ, on a set Z. We give conditions which allow us to establish a variational principle for the topological pressure. We state conditions for the existence and uniqueness of equilibrium measures, and for subshifts of finite type the existence and uniqueness of Gibbs measures. Finally, we compare the results for almost-additive sequences to the thermodynamic formalism for the classical (additive) case [10] [11] [3], the sequences studied by Barreira [1], Falconer [5], and that of Feng and Lau [7], [6].
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Anna MummertStudying the Recovery Procedure for the Time-dependent Transmission Rate(s) in Epidemic Models
https://mds.marshall.edu/mathematics_faculty/42
https://mds.marshall.edu/mathematics_faculty/42Fri, 08 May 2020 07:33:48 PDT
Determining the time-dependent transmission function that exactly reproduces disease incidence data can yield useful information about disease outbreaks, including a range potential values for the recovery rate of the disease and could offer a method to test the “school year” hypothesis (seasonality) for disease transmission. Recently two procedures have been developed to recover the time-dependent transmission function, β(t), for classical disease models given the disease incidence data. We first review the β(t) recovery procedures and give the resulting formulas, using both methods, for the susceptible-infected-recovered (SIR) and susceptible-exposed-infected-recovered (SEIR) models. We present a modification of one procedure, which is then shown to be identical to the other. Second, we explore several technical issues that appear when implementing the procedure for the SIR model; these are important when generating the time-dependent transmission function for real-world disease data. Third, we extend the recovery method to heterogeneous populations modeled with a certain SIR-type model with multiple time-dependent transmission functions. Finally, we apply the β(t) recovery procedure to data from the 2002–2003 influenza season and for the six seasons from 2002–2003 through 2007–2008, for both one population class and for two age classes. We discuss the consequences of the technical conditions of the procedure applied to the influenza data. We show that the method is robust in the heterogeneous cases, producing comparable results under two different hypotheses. We perform a frequency analysis, which shows a dominant 1-year period for the multi-year influenza transmission function(s).
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Anna MummertCampus Outbreak! Modeling Seasonal Influenza
https://mds.marshall.edu/mathematics_faculty/41
https://mds.marshall.edu/mathematics_faculty/41Thu, 07 May 2020 11:41:51 PDT
This PowerPoint-driven case study follows the progress of three undergraduate students as they attempt to model the rapid spread of an influenza outbreak to determine whether their local newspaper's claim that "40% of the campus has the flu" is accurate. The case introduces epidemiological modeling using a base model for a seasonal influenza outbreak written in the NetLogo programmable modeling environment. In class, students develop tests for the various parameters of the model, run simulations, and evaluate the output. The students then explore the impact of influenza control strategies (vaccination, isolation, and antiviral medications), and finish with the question of whether the continuing outbreak on campus could be a pandemic. The case is written at a basic level for a lower-level undergraduate lecture-style class, but can be adapted to upper-level courses as well. The case was piloted in four different microbiology courses. The simulations stimulated active discussion and the content worked well, whether it was used in a pre-nursing microbiology or upper-level immunology class.
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Marcia Harrison-Pitaniello et al.Modeling super-spreading events for infectious diseases: Case study SARS
https://mds.marshall.edu/mathematics_faculty/40
https://mds.marshall.edu/mathematics_faculty/40Thu, 07 May 2020 11:41:40 PDT
Super-spreading events for infectious diseases occur when some infected individuals infect more than the average number of secondary cases. Several super-spreading individuals have been identiﬁed for the 2003 outbreak of severe acute respiratory syndrome (SARS). We develop a model for super spreading events of infectious diseases, which is based on the outbreak of SARS. Using this model we describe two methods for estimating the parameters of the model, which we demonstrate with the small-scale SARS outbreak at the Amoy Gardens, Hong Kong, and the large-scale outbreak in the entire Hong Kong Special Administrative Region. One method is based on parameters calculated for the classical susceptible - infected - removed (SIR) disease model. The second is based on parameter estimates found in the literature. Using the parameters calculated for the SIR model, our model predicts an outcome similar to that for the SIR model. On the other hand, using parameter estimates from SARS literature our model predicts a much more serious epidemic.
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Thembinkosi Mkhatshwa et al.Controlling viral outbreaks: Quantitative strategies
https://mds.marshall.edu/mathematics_faculty/39
https://mds.marshall.edu/mathematics_faculty/39Mon, 27 Apr 2020 11:19:20 PDT
Preparing for and responding to outbreaks of serious livestock infectious diseases are critical measures to safeguard animal health, public health, and food supply. Almost all of the current control strategies are empirical, and mass culling or “stamping out” is frequently the principal strategy for controlling epidemics. However, there are ethical, ecological, and economic reasons to consider less drastic control strategies. Here we use modeling to quantitatively study the efficacy of different control measures for viral outbreaks, where the infectiousness, transmissibility and death rate of animals commonly depends on their viral load. We develop a broad theoretical framework for exploring and understanding this heterogeneity. The model includes both direct transmission from infectious animals and indirect transmission from an environmental reservoir. We then incorporate a large variety of control measures, including vaccination, antivirals, isolation, environmental disinfection, and several forms of culling, which may result in fewer culled animals. We provide explicit formulae for the basic reproduction number, R_{0}, for each intervention and for combinations. We evaluate the control methods for a realistic simulated outbreak of low pathogenic avian influenza on a mid-sized turkey farm. In this simulated outbreak, culling results in more total dead birds and dramatically more when culling all of the infected birds.
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Anna Mummert et al.Get the News Out Loudly and Quickly: Modeling the Influence of the Media on Limiting Infectious Disease
https://mds.marshall.edu/mathematics_faculty/38
https://mds.marshall.edu/mathematics_faculty/38Mon, 27 Apr 2020 11:19:10 PDT
During outbreaks of infectious diseases with high morbidity and mortality, individuals closely follow media reports of the outbreak. Many will attempt to minimize contacts with other individuals in order to protect themselves from infection and possibly death. This process is called social distancing. Social distancing strategies include restricting socializing and travel, and using barrier protections. We use modeling to show that for short-term outbreaks, social distancing can have a large influence on reducing outbreak morbidity and mortality. In particular, public health agencies working together with the media can significantly reduce the severity of an outbreak by providing timely accounts of new infections and deaths. Our models show that the most effective strategy to reduce infections is to provide this information as early as possible, though providing it well into the course of the outbreak can still have a significant effect. However, our models for long-term outbreaks indicate that reporting historic infection data can result in more infections than with no reporting at all. We examine three types of media influence and we illustrate the media influence with a simulated outbreak of a generic emerging infectious disease in a small city. Social distancing can never be complete; however, for a spectrum of outbreaks, we show that leaving isolation (stopping applying social distancing measures) for up to 4 hours each day has modest effect on the overall morbidity and mortality.
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Anna Mummert et al.A Perspective on Multiple Waves of Influenza Pandemics
https://mds.marshall.edu/mathematics_faculty/37
https://mds.marshall.edu/mathematics_faculty/37Mon, 27 Apr 2020 11:18:58 PDTBackground: A striking characteristic of the past four influenza pandemic outbreaks in the United States has been the multiple waves of infections. However, the mechanisms responsible for the multiple waves of influenza or other acute infectious diseases are uncertain. Understanding these mechanisms could provide knowledge for health authorities to develop and implement prevention and control strategies. Materials and Methods: We exhibit five distinct mechanisms, each of which can generate two waves of infections for an acute infectious disease. The first two mechanisms capture changes in virus transmissibility and behavioral changes. The third mechanism involves population heterogeneity (e.g., demography, geography), where each wave spreads through one sub-population. The fourth mechanism is virus mutation which causes delayed susceptibility of individuals. The fifth mechanism is waning immunity. Each mechanism is incorporated into separate mathematical models, and outbreaks are then simulated. We use the models to examine the effects of the initial number of infected individuals (e.g., border control at the beginning of the outbreak) and the timing of and amount of available vaccinations. Results: Four models, individually or in any combination, reproduce the two waves of the 2009 H1N1 pandemic in the United States, both qualitatively and quantitatively. One model reproduces the two waves only qualitatively. All models indicate that significantly reducing or delaying the initial numbers of infected individuals would have little impact on the attack rate. Instead, this reduction or delay results in a single wave as opposed to two waves. Furthermore, four of these models also indicate that a vaccination program started earlier than October 2009 (when the H1N1 vaccine was initially distributed) could have eliminated the second wave of infection, while more vaccine available starting in October would not have eliminated the second wave.
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Anna Mummert et al.The modal logic of Reverse Mathematics
https://mds.marshall.edu/mathematics_faculty/36
https://mds.marshall.edu/mathematics_faculty/36Wed, 15 Apr 2020 10:11:36 PDT
The implication relationship between subsystems in Reverse Mathematics has an underlying logic, which can be used to deduce certain new Reverse Mathematics results from existing ones in a routine way. We use techniques of modal logic to formalize the logic of Reverse Mathematics into a system that we name s-logic. We argue that s-logic captures precisely the "logical" content of the implication and nonimplication relations between subsystems in Reverse Mathematics. We present a sound, complete, decidable, and compact tableau-style deductive system for s-logic, and explore in detail two fragments that are particularly relevant to Reverse Mathematics practice and automated theorem proving of Reverse Mathematics results.
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Carl Mummert et al.On the strength of the finite intersection principle
https://mds.marshall.edu/mathematics_faculty/35
https://mds.marshall.edu/mathematics_faculty/35Wed, 15 Apr 2020 10:11:24 PDT
We study the logical content of several maximality principles related to the finite intersection principle (FIP) in set theory. Classically, these are all equivalent to the axiom of choice, but in the context of reverse mathematics their strengths vary: some are equivalent to ACA0 over RCA0, while others are strictly weaker and incomparable with WKL0. We show that there is a computable instance of FIP every solution of which has hyperimmune degree, and that every computable instance has a solution in every nonzero c.e. degree. In particular, FIP implies the omitting partial types principle (OPT) over RCA0. We also show that, modulo Σ 02 induction, FIP lies strictly below the atomic model theorem (AMT).
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Damir D. Dzhafarov et al.On the existence of a connected component of a graph
https://mds.marshall.edu/mathematics_faculty/34
https://mds.marshall.edu/mathematics_faculty/34Wed, 15 Apr 2020 10:11:14 PDT
We study the logical content of several maximality principles related to the finite intersection principle (FIP) in set theory. Classically, these are all equivalent to the axiom of choice, but in the context of reverse mathematics their strengths vary: some are equivalent to ACA_{0} over RCA_{0}, while others are strictly weaker and incomparable with WKL_{0}. We show that there is a computable instance of FIP every solution of which has hyperimmune degree, and that every computable instance has a solution in every nonzero c.e. degree. In particular, FIP implies the omitting partial types principle (OPT) over RCA_{0}. We also show that, modulo Σ ^{0}_{2} induction, FIP lies strictly below the atomic model theorem (AMT).
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Kirill Gura et al.Quantitative analysis of a stochastic SEITR epidemic model with multiple stages of infection and treatment
https://mds.marshall.edu/mathematics_faculty/33
https://mds.marshall.edu/mathematics_faculty/33Wed, 08 Jan 2020 14:02:24 PST
We present a mathematical analysis of the transmission of certain diseases using a stochastic susceptible-exposed-infectious-treated-recovered (SEITR) model with multiple stages of infection and treatment and explore the effects of treatments and external ﬂuctuations in the transmission, treatment and recovery rates. We assume external ﬂuctuations are caused by variability in the number of contacts between infected and susceptible individuals. It is shown that the expected number of secondary infections produced (in the absence of noise) reduces as treatment is introduced into the population. By deﬁning R_{T}_{,n}and ℛ_{T}_{,n}as the basic deterministic and stochastic reproduction numbers, respectively, in stage n of infection and treatment, we show mathematically that as the intensity of the noise in the transmission, treatment and recovery rates increases, the number of secondary cases of infection increases. The global stability of the disease-free and endemic equilibrium for the deterministic and stochastic SEITR models is also presented. The work presented is demonstrated using parameter values relevant to the transmission dynamics of Inﬂuenza in the United States from October 1, 2018 through May 4, 2019 inﬂuenza seasons.
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Olusegun M. Otunuga et al.Completing partial latin squares with one nonempty row, column, and symbol
https://mds.marshall.edu/mathematics_faculty/32
https://mds.marshall.edu/mathematics_faculty/32Tue, 10 Dec 2019 08:59:51 PST
Let r,c,s ∈{1,2,…,n} and let PP be a partial latin square of order n in which each nonempty cell lies in row r, column c, or contains symbol s. We show that if n ∉ {3, 4, 5} and row r, column c, and symbol s can be completed in P, then a completion of P exists. As a consequence, this proves a conjecture made by Casselgren and Häggkvist. Furthermore, we show exactly when row r, column c, and symbol s can be completed.
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Jaromy Kuhl et al.ϕ-symmetric Hamilton cycle decompositions of graphs
https://mds.marshall.edu/mathematics_faculty/31
https://mds.marshall.edu/mathematics_faculty/31Tue, 10 Dec 2019 08:59:46 PST
The existence of symmetric Hamilton cycle decompositions for complete graphs and cocktail party graphs has been defined and explored in recent work by Akiyama et al., Brualdi and Schroeder, and others. In these works, the notion of symmetry in cocktail party graphs K_{2m}−F was integrally tied to the missing 1-factor. In this paper, we generalize the notion of symmetric decompositions in two ways. First, we require only that F is symmetric and show that if F is not the invariant 1-factor under the symmetry action, then K_{2m}−F has a symmetric Hamilton cycle decomposition for every m ≥ 2. Second, we consider other actions as symmetry, apply such definitions to appropriate complete graphs and complete multipartite graphs, and classify the existence of Hamilton cycle decompositions with such symmetry.
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Michael W. SchroederCyclic matching sequencibility of graphs
https://mds.marshall.edu/mathematics_faculty/30
https://mds.marshall.edu/mathematics_faculty/30Tue, 10 Dec 2019 08:59:41 PST
We define the cyclic matching sequencibility of a graph to be the largest integer d such that there exists a cyclic ordering of its edges so that every d consecutive edges in the cyclic ordering form a matching. We show that the cyclic matching sequencibility of K_{2m}and K_{2m+1 }equals m − 1.
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Richard A. Brualdi et al.On the t-term rank of a matrix
https://mds.marshall.edu/mathematics_faculty/29
https://mds.marshall.edu/mathematics_faculty/29Tue, 10 Dec 2019 08:59:36 PST
For t a positive integer, the t-term rank of a (0,1)-matrix A is defined to be the largest number of 1s in A with at most one 1 in each column and at most t 1s in each row. Thus the 1-term rank is the ordinary term rank. We generalize some basic results for the term rank to the t-term rank, including a formula for the maximum term rank over a nonempty class of (0,1)-matrices with the same row sum and column sum vectors. We also show the surprising result that in such a class there exists a matrix which realizes all of the maximum terms ranks between 1 and t.
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Richard A. Brualdi et al.Patterns of alternating sign matrices
https://mds.marshall.edu/mathematics_faculty/28
https://mds.marshall.edu/mathematics_faculty/28Tue, 10 Dec 2019 08:59:30 PST
We initiate a study of the zero–nonzero patterns of n × n alternating sign matrices. We characterize the row (column) sum vectors of these patterns and determine their minimum term rank. In the case of connected alternating sign matrices, we find the minimum number of nonzero entries and characterize the case of equality. We also study symmetric alternating sign matrices, in particular, those with only zeros on the main diagonal. These give rise to alternating signed graphs without loops, and we determine the maximum number of edges in such graphs. We also consider n × n alternating sign matrices whose patterns are maximal within the class of all n × n alternating sign matrices.
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Richard A. Brualdi et al.On Hamilton cycle decompositions of <em>r</em>-uniform <em>r</em>-partite hypergraphs
https://mds.marshall.edu/mathematics_faculty/27
https://mds.marshall.edu/mathematics_faculty/27Tue, 10 Dec 2019 08:59:25 PST
The definition of edge-adjacency can be generalized in multiple ways to hypergraphs, and extended from that, cycles and Hamilton cycles. One such generalization of a Hamilton cycle is attributed to Kierstead and Katona. In a recent paper by Kuhl and Schroeder, Hamilton cycle decompositions of complete r-partite r-uniform hypergraphs are discussed, a conjecture was made that the necessary numerical conditions are sufficient, and was shown true for some cases. In this paper, the conjecture is proved using constructions involving Hamming codes, comparisons between the two constructions are made, and a classification of when they are equivalent is shown.
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Michael W. SchroederA bijection on classes enumerated by the Schröder numbers
https://mds.marshall.edu/mathematics_faculty/26
https://mds.marshall.edu/mathematics_faculty/26Tue, 10 Dec 2019 08:59:19 PST
We consider a sorting machine consisting of two stacks in series where the first stack has the added restriction that entries in the stack must be in decreasing order from top to bottom. The class of permutations sortable by this machine is known to be enumerated by the Schroder numbers. In this paper, we give a bijection between these sortable permutations of length n and Schroder paths of order n − 1: the lattice paths from (0, 0) to (n − 1, n − 1) composed of East steps (1, 0), North steps (0, 1), and Diagonal steps (1, 1) that travel weakly below the line y = x.
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Michael W. Schroeder et al.