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archives@tulane.edu / 1 / Harley Hanes

Compartmentalization is a general principle in biological systems which is observable on all size scales, ranging from organelles inside of cells, cells in histology, and up to the level of groups, herds, swarms, meta-populations, and populations. Compartmental models are often used to model such phenomena, but such models can be both highly nonlinear and difficult to work with. Fortunately, there are many significant biological systems that are amenable to linear compartmental models which are often more mathematically accessible. Moreover, the biology and mathematics is often so intertwined in such models that one can be used to better understand the other. Indeed, as we demonstrate in this paper, linear compartmental models of migratory dynamics can be used as an exciting and interactive means of introducing sophisticated mathematics, and conversely, the associated mathematics can be used to demonstrate important biological properties not only of seasonal migrations but also of compartmental models in general. We have found this approach to be of great value in introducing derivatives, integrals, and the fundamental theorem of calculus. Additionally, these models are appropriate as applications in a differential equations course, and they can also be used to illustrate important ideas in probability and statistics, such as the Poisson distribution.

Due to their convenience, computers have become a standard in society and therefore, need the utmost care. It is convenient and useful to model the behavior of digital virus outbreaks that occur, globally or locally. Compartmental models will be used to analyze the mannerisms and behaviors of computer malware. This paper will focus on a computer worm, a type of malware, spread within a business network. A mathematical model is proposed consisting of four compartments labeled as Susceptible, Infectious, Treatment, and Antidotal. We shall show that allocating resources into treating infectious computers leads to a reduced peak of infections across the infection period, while pouring resources into treating susceptible computers decreases the total amount of infections throughout the infection period. This is assuming both methods are receiving resources without loss. This result reveals an interesting notion of balance between protecting computers and removing computers from infections, ultimately depending on the business executives' goals and/or preferences.

Threshold dynamics used to control the spread of the disease in infectious disease phenomena has an overwhelming importance and interest in mathematical epidemiology. One of the famous threshold quantity is known to be the basic reproduction ratio. Its formulation as well as computation is the main concern of infectious diseases. The aim of this thesis is to analyze the basic reproduction ratio in both autonomous and periodic systems via defining R0 as the spectral radius of the next generation operator. This thesis presents the vector host model for the diseases Dengue fever and avian influenza. As emerging of the diseases shows periodicity, systems of periodic ordinary differential equations are considered for both types of diseases. Simple implementation of the time-averaged systems gives rise to the comparison of these with the periodic systems. Thus, we investigate the occurence of the existence of underestimation or overestimation of the basic reproduction ratio in timeaveraged systems.

The incidence of Type 1 diabetes is growing yearly. Worryingly, the aetiology of the disease is inconclusive. What is known is that the total number of affected individuals, as well as the severity and number of associated complications are growing for this chronic disease. With increasing complications due to severity, length of exposure, and poor control, the disease is beginning to consume an increasingly major portion of healthcare costs to the extent that it poses major economic risks in several nations. Research has shown that intensive insulin therapy aimed at certain minimum glycosylated haemoglobin threshold levels reduces the incidence of complications by up to 76% compared to conventional insulin therapy. Moreover, the effects of such intensive therapy regimes over a 6.5y duration persists for at least 10y after, a so called metabolic memory. Thus, early intervention can slow the momentum of complications far more easily than later intervention. Early, safe, intensive therapy protocols offer potential solutions to the growing social and economic effects of diabetes. Since the 1970s, the artificial endocrine pancreas has been heralded as just this type of solution. However, no commercial product currently exists, and ongoing limitations in sensors and pumps have resulted in, at best, modest clinical advantages over conventional methods of insulin administration or multiple daily injection. With high upfront costs, high costs of consumables, significant complexity, and the extensive infrastructure and support required, these systems and devices are only used by 2-15% of individuals with Type 1 diabetes. Clearly, there is an urgent need to address the large majority of the Type 1 diabetes population using conventional glucose measurement and insulin administration. For these individuals, current conventional or intensive therapies are failing to deliver recommended levels of glycaemic control. This research develops an understanding of clinical glycaemic control using conventional insulin administration and glucose measurement techniques in Type 1 diabetes based on a clinically validated in silico virtual patient simulation. Based on this understanding, a control protocol for Type 1 diabetes that is relatively simple and clinically practical is developed. The protocol design incorporates physiological modelling and engineering techniques to adapt to individual patient clinical requirements. By doing so, it produces accurate, patient-specific recommendations for insulin interventions. Initially, a simple, physiological compartmental model for the pharmacokinetics of subcutaneously injected insulin is developed. While the absorption process itself is subject to significant potential variability, such models enable a real-time estimation of plasma insulin concentration. This information would otherwise be lacking in the clinical environment of outpatient Type 1 diabetes treatment due to the inconvenience, cost, and laboratory turnaround for plasma insulin measurements. Hence, this validated model offers significant opportunity to optimise therapy selection. An in silico virtual patient simulation tool is also developed. A virtual patient cohort is developed on patient data from a representative cohort of the broad diabetes population. The simulation tool is used to develop a robust, adaptive protocol for prandial insulin dosing against a conventional intensive insulin therapy, as well as a controls group representative of the general diabetes population. The effect on glycaemic control of suboptimal and optimal, prandial and basal insulin therapies is also investigated, with results matching clinical expectations. To gauge the robustness of the developed adaptive protocol, a Monte Carlo analysis is performed, incorporating realistic and physiological errors and variability. Due to the relatively infrequent glucose measurement in outpatient Type 1 diabetes, a method for identifying the diurnal cycle in effective insulin sensitivity and modelling it in retrospective patient data is also presented. The method consists of identifying deterministic and stochastic components in the patient effective insulin sensitivity profile. Circadian rhythmicity and sleep-wake phases have profound effects on effective insulin sensitivity. Identification and prediction of this rhythm is of utmost clinical relevance, with the potential for safer and more effective glycaemic control, with less frequent measurement. It is thus a means of further enhancing any robust protocol and making it more clinically practical to implement. Finally, this research presents an entire framework for the realistic, and rapid development and testing of clinical glycaemic control protocols for outpatient Type 1 diabetes. The models and methods developed within this framework allow rapid and physiological identification of time-variant, patient-specific, effective insulin sensitivity profiles. These profiles form the responses of the virtual patient and can be used to develop and robustly test clinical glycaemic control protocols in a broad range of patients. These effective insulin sensitivity profiles are also rich in dynamics, specifically those circadian in nature which can be identified, and used to provide more accurate glycaemic prediction with the potential for safer and more effective control.

Orientador: Andriana Susana Lopes de Oliveira Campanharo / Resumo: Os Modelos Baseados em Indivíduos (MBI’s) têm sido crescentemente empregados na modelagem de processos infecciosos. Um MBI consiste de uma estrutura na qual ocorrem interações entre um certo número de indivíduos, cujo comportamento é determinado por um conjunto de características que evoluem estocasticamente no tempo. Estudos recentes têm mostrado que as redes complexas constituem um suporte natural para o estudo da propagação de uma doença. Redes complexas são descritas por um conjunto de vértices (nós), arestas (conexões, ligações ou links) e algum tipo de interação entre os mesmos. Na formulação original do MBI e em modelos SIR (Suscetível, Infectado e Recuperado) e SEI (Suscetível, Exposto e Infectado), as relações entre os indivíduos são representadas por grafos completos, ou seja, todos os indivíduos estão conectados entre si. Como a topologia de uma rede real não pode ser descrita por uma rede puramente aleatória, nesse trabalho o MBI foi implementado de forma a incorporar modelos mais realísticos de redes de contato na propagação de uma doença infecciosa. De maneira geral, observou-se que redes complexas com diferentes topologias resultam em curvas de indivíduos suscetíveis, infectados e recuperados (ou suscetíveis, expostos e infectados) com diferentes comportamentos, e desta forma, que a evolução de uma dada doença, em particular a tuberculose, é altamente sensível à topologia de rede utilizada. Mais especificamente, observou-se que quanto maior o valor do comprimen... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: Individual-Based Models have been increasingly employed in the modeling of an infectious process. An IBM consists of a structure in which interactions occur between a certain number of individuals, whose behavior is determined by a set of characteristics that evolve stochastically in time. Recent studies have shown that complex networks are a natural framework for the study of a disease spread. Complex networks are described by a set of vertices (or nodes), edges (connections or links) and some type of interactions between them. In the original IBM approach and in SIR (Susceptible, Infected, Recovered) and SEI (Susceptible, Exposed and Infected) models, the relations between individuals are represented by complete graphs, that is, all individuals are connected to each other. Since the topology of a real network can not be described by a purely random network, in this work an IBM has been implemented in order to incorporate some realistic contact networks xvii models. In general, it was observed that complex networks with different topologies correspond to curves of susceptible, infected and recovered individuals (or susceptible, exposed and infected) with different behaviors, and thus, that the evolution of a given disease, in particular tuberculosis, is highly sensitive to a network topology. More specifically, it was observed that the higher the value of the mean jump length is, the faster the disease spreads and consequently, the higher is the number of infected individual... (Complete abstract click electronic access below) / Mestre

Submitted by EDUARDO RIBEIRO PINTO (eduribeiro2@bol.com.br) on 2018-05-03T15:47:28Z No. of bitstreams: 1 dissertacao_Eduardo.pdf: 6068904 bytes, checksum: 4ff00adcd4667c6d7ed4bcfb5db2321a (MD5) / Approved for entry into archive by Sulamita Selma C Colnago null (sulamita@btu.unesp.br) on 2018-05-03T19:01:49Z (GMT) No. of bitstreams: 1 pinto_er_me_bot_int.pdf: 6068904 bytes, checksum: 4ff00adcd4667c6d7ed4bcfb5db2321a (MD5) / Made available in DSpace on 2018-05-03T19:01:49Z (GMT). No. of bitstreams: 1 pinto_er_me_bot_int.pdf: 6068904 bytes, checksum: 4ff00adcd4667c6d7ed4bcfb5db2321a (MD5) Previous issue date: 2018-02-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Os Modelos Baseados em Indivíduos (MBI’s) têm sido crescentemente empregados na modelagem de processos infecciosos. Um MBI consiste de uma estrutura na qual ocorrem interações entre um certo número de indivíduos, cujo comportamento é determinado por um conjunto de características que evoluem estocasticamente no tempo. Estudos recentes têm mostrado que as redes complexas constituem um suporte natural para o estudo da propagação de uma doença. Redes complexas são descritas por um conjunto de vértices (nós), arestas (conexões, ligações ou links) e algum tipo de interação entre os mesmos. Na formulação original do MBI e em modelos SIR (Suscetível, Infectado e Recuperado) e SEI (Suscetível, Exposto e Infectado), as relações entre os indivíduos são representadas por grafos completos, ou seja, todos os indivíduos estão conectados entre si. Como a topologia de uma rede real não pode ser descrita por uma rede puramente aleatória, nesse trabalho o MBI foi implementado de forma a incorporar modelos mais realísticos de redes de contato na propagação de uma doença infecciosa. De maneira geral, observou-se que redes complexas com diferentes topologias resultam em curvas de indivíduos suscetíveis, infectados e recuperados (ou suscetíveis, expostos e infectados) com diferentes comportamentos, e desta forma, que a evolução de uma dada doença, em particular a tuberculose, é altamente sensível à topologia de rede utilizada. Mais especificamente, observou-se que quanto maior o valor do comprimento do salto médio, mais rápida será a propagação da doença e, consequentemente, maior será o número de indivíduos infectados. / Individual-Based Models have been increasingly employed in the modeling of an infectious process. An IBM consists of a structure in which interactions occur between a certain number of individuals, whose behavior is determined by a set of characteristics that evolve stochastically in time. Recent studies have shown that complex networks are a natural framework for the study of a disease spread. Complex networks are described by a set of vertices (or nodes), edges (connections or links) and some type of interactions between them. In the original IBM approach and in SIR (Susceptible, Infected, Recovered) and SEI (Susceptible, Exposed and Infected) models, the relations between individuals are represented by complete graphs, that is, all individuals are connected to each other. Since the topology of a real network can not be described by a purely random network, in this work an IBM has been implemented in order to incorporate some realistic contact networks xvii models. In general, it was observed that complex networks with different topologies correspond to curves of susceptible, infected and recovered individuals (or susceptible, exposed and infected) with different behaviors, and thus, that the evolution of a given disease, in particular tuberculosis, is highly sensitive to a network topology. More specifically, it was observed that the higher the value of the mean jump length is, the faster the disease spreads and consequently, the higher is the number of infected individuals.

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The main goal of this project is to investigate the behaviors of parameters used when modeling an epidemic. A stochastic SIHDRe model is used to simulate how an epidemic evolves over time. The SIHDRe model has nine parameters, and this project focuses on the infection rate (β) and the fraction of infections leading to deaths (FID), with all other parameters being considered known. Both parameters are time dependent. To estimate the two chosen parameters, this project uses synthetic data so that comparisons between estimations with true parameters are possible. A dynamic optimization procedure inspired by Model Predictive Control is utilized for the predictions. Using synthesized data from hospitalizations and deaths, a cost function is minimized to obtain estimations of the parameters. Only a subset of the time span, called a window, is considered for every parameter optimization. The parameters within the window are optimized and the window then moves forward in time defined by a time step until the parameters are optimized over the whole time span. To obtain error estimations of the parameters, synthetic bootstrapping is used, using optimized parameters to simulate new epidemics of which the parameters are optimized. The square difference between the new estimations compared to the original estimations can be used to obtain the standard deviation of the estimated parameters. This project also discusses how regularization parameters within the cost functions are chosen so that the estimated parameters will be most similar to the real parameter values, and end-of-data effects, i.e. increased uncertainty towards the end of a window, is also discussed. / Projektet undersöker hur olika parametrar till en epidemisk modell kan skattas. En stokastisk SIHDRe modell (Susceptible, Infected, Hospitalizalized, Dead, Recovered) används för att simulera hur en epidemi utvecklas över tid. SIDHRe modellen delar in populationen i olika grupper baserat på hur epidemin har påverkat dem, till exempel om de har blivit smittade eller om de har hamnat på sjukhus på grund av sjukdomen. Personer kan flyttas mellan olika grupper beroende på en rad parametrar samt storleken på de olika grupperna. Detta projekt fokuserar på att skatta två parameterar: β, som påverkar hur personer med risk för infektion blir smittade, samt FID som påverkar hur många infekterade som dör av sjukdomen. Modellen har nio parametrar totalt och alla andra parametrar anses kända. Projektet använder syntetisk data, som gör det möjligt att jämföra skattningar av parametrarna med deras sanna värden. Båda okända parametrarna är tidsberoende. För att bestämma parametervärdena används en dynamisk optimiseringsmetod. Data från antal individer inlagda på sjukhus samt antal döda anses känt och kan användas för att minimera en kostfunktion som har de okända parametrarna som inmatningsvärden genom att ändra dessa. Tidsspannet begränsas till en mindre del, det sägs att man ser ett fönster av hela tidsspannet. Fönstret startar vid den första tidspunkten och kostfunktionen minimiseras för inmatningsvärden inom fönstret. När detta är gjort flyttas fönstret ett kort tidsteg fram i tiden och optimiseringsprocessen återupprepas tills fönstret når slutet av hela tidsserien och alla parametervärden har uppskattats. Dessa skattade parametervärden kan sen jämföras med de sanna värdena. För att kunna uppskatta felet när parametervärdena bestäms används en metod kallad ”Synthetic Bootstrap”. Grundidén är att parameterna uppskattas en gång ochdenna uppskattning används sen som inmatningsvärde till epidemimodellen. Nya epidemier simuleras och baserat på dessa simuleringar, kan nya parametervärden estimeras. Dessa kommer att skilja i värde på grund av att modellen är stokastisk. De nya uppskattningarna jämförs sedan med de första uppskattningarna och en uppfattning om skillnaden mellan dessa kan sedan beskrivas som en standardavvikelse mellan de nya skattningarna och den första skattningen. Projektet diskuterar också val av olika regulariseringsparametrar för kostfunktionen. Dessa kontrollerar hur mycket de uppskattade värdena kan ändras från tidpunkt till tidpunkt genom att ett stort värde minskar möjliga ändringar och ett litet värde ökar dem. Ett fenomen som kallas ”end-of-data effects” diskuteras också och handlar om att osäkerheten växer i ett fönster ju längre in i fönstret man är.