ASU Electronic Theses and Dissertations
This collection includes most of the ASU Theses and Dissertations from 2011 to present. ASU Theses and Dissertations are available in downloadable PDF format; however, a small percentage of items are under embargo. Information about the dissertations/theses includes degree information, committee members, an abstract, supporting data or media.
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Description
A leading crisis in the United States is the opioid use disorder (OUD) epidemic. Opioid overdose deaths have been increasing, with over 100,000 deaths due to overdose from April 2020 to April 2021. This dissertation presents two mathematical models to address illicit OUD (IOUD), treatment, and recovery within an epidemiological framework. In the first model, individuals remain in the recovery class unless they relapse. Due to the limited availability of specialty treatment facilities for individuals with OUD, a saturation treat- ment function was incorporated. The second model is an extension of the first, where a casual user class and its corresponding specialty treatment class were added. Using U.S. population data, the data was scaled to a population of 200,000 to find parameter estimates. While the first model used the heroin-only dataset, the second model used both the heroin and all-illicit opioids datasets. Backward bifurcation was found in the first IOUD model for realistic parameter values. Additionally, bistability was observed in the second IOUD model with the heroin-only dataset. This result implies that it would be beneficial to increase the availability of treatment. An alarming effect was discovered about the high overdose death rate: by 2038, the disease-free equilibrium would be the only stable equilibrium. This consequence is concerning because although the goal is for the epidemic to end, it would be preferable to end it through treatment rather than overdose. The IOUD model with a casual user class, its sensitivity results, and the comparison of parameters for both datasets, showed the importance of not overlooking the influence that casual users have in driving the all-illicit opioid epidemic. Casual users stay in the casual user class longer and are not going to treatment as quickly as the users of the heroin epidemic. Another result was that the users of the all-illicit opioids were going to the recovered class by means other than specialty treatment. However, the relapse rates for those individuals were much more significant than in the heroin-only epidemic. The results above from analyzing these models may inform health and policy officials, leading to more effective treatment options and prevention efforts.
ContributorsCole, Sandra (Author) / Wirkus, Stephen (Thesis advisor) / Gardner, Carl (Committee member) / Lanchier, Nicolas (Committee member) / Camacho, Erika (Committee member) / Fricks, John (Committee member) / Arizona State University (Publisher)
Created2022
Description
This dissertation consists of three papers about opinion dynamics. The first paper is in collaboration with Prof. Lanchier while the other two papers are individual works.
Two models are introduced and studied analytically: the Deffuant model and the Hegselmann-Krause~(HK) model.
The main difference between the two models is that the Deffuant dynamics consists of pairwise interactions whereas the HK dynamics consists of group interactions.
Translated into graph, each vertex stands for an agent in both models.
In the Deffuant model, two graphs are combined: the social graph and the opinion graph.
The social graph is assumed to be a general finite connected graph where each edge is interpreted as a social link, such as a friendship relationship, between two agents.
At each time step, two social neighbors are randomly selected and interact if and only if their opinion distance does not exceed some confidence threshold, which results in the neighbors' opinions getting closer to each other.
The main result about the Deffuant model is the derivation of a positive lower bound for the probability of consensus that is independent of the size and topology of the social graph but depends on the confidence threshold, the choice of the opinion space and the initial distribution. For the HK model, agent~$i$ updates its opinion~$x_i$ by taking the average opinion of its neighbors, defined as the set of agents with opinion at most~$\epsilon$ apart from~$x_i$. Here,~$\epsilon > 0$ is a confidence threshold.
There are two types of HK models: the synchronous and the asynchronous HK models.
In the former, all the agents update their opinion simultaneously at each time step, whereas in the latter, only one agent is selected uniformly at random to update its opinion at each time step.
The mixed model is a variant of the HK model in which each agent can choose its degree of stubbornness and mix its opinion with the average opinion of its neighbors.
The main results of this dissertation about HK models show conditions under which the asymptotic stability holds or a consensus can be achieved, and give a positive lower bound for the probability of consensus and, in the one-dimensional case, an upper bound for the probability of consensus.
I demonstrate the bounds for the probability of consensus on a unit cube and a unit interval.
ContributorsLi, Hsin-Lun (Author) / Lanchier, Nicolas (Thesis advisor) / Camacho, Erika (Committee member) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Motsch, Sebastien (Committee member) / Arizona State University (Publisher)
Created2021