Matching Items (30)
Description
Rabies disease remains enzootic among raccoons, skunks, foxes and bats in the United States. It is of primary concern for public-health agencies to control spatial spread of rabies in wildlife and its potential spillover infection of domestic animals and humans. Rabies is invariably fatal in wildlife if untreated, with a non-negligible incubation period. Understanding how this latency affects spatial spread of rabies in wildlife is the concern of chapter 2 and 3. Chapter 1 deals with the background of mathematical models for rabies and lists main objectives. In chapter 2, a reaction-diffusion susceptible-exposed-infected (SEI) model and a delayed diffusive susceptible-infected (SI) model are constructed to describe the same epidemic process -- rabies spread in foxes. For the delayed diffusive model a non-local infection term with delay is resulted from modeling the dispersal during incubation stage. Comparison is made regarding minimum traveling wave speeds of the two models, which are verified using numerical experiments. In chapter 3, starting with two Kermack and McKendrick's models where infectivity, death rate and diffusion rate of infected individuals can depend on the age of infection, the asymptotic speed of spread $c^\ast$ for the cumulated force of infection can be analyzed. For the special case of fixed incubation period, the asymptotic speed of spread is governed by the same integral equation for both models. Although explicit solutions for $c^\ast$ are difficult to obtain, assuming that diffusion coefficient of incubating animals is small, $c^\ast$ can be estimated in terms of model parameter values. Chapter 4 considers the implementation of realistic landscape in simulation of rabies spread in skunks and bats in northeast Texas. The Finite Element Method (FEM) is adopted because the irregular shapes of realistic landscape naturally lead to unstructured grids in the spatial domain. This implementation leads to a more accurate description of skunk rabies cases distributions.
ContributorsLiu, Hao (Author) / Kuang, Yang (Thesis advisor) / Jackiewicz, Zdzislaw (Committee member) / Lanchier, Nicolas (Committee member) / Smith, Hal (Committee member) / Thieme, Horst (Committee member) / Arizona State University (Publisher)
Created2013
Description
In complex consumer-resource type systems, where diverse individuals are interconnected and interdependent, one can often anticipate what has become known as the tragedy of the commons, i.e., a situation, when overly efficient consumers exhaust the common resource, causing collapse of the entire population. In this dissertation I use mathematical modeling to explore different variations on the consumer-resource type systems, identifying some possible transitional regimes that can precede the tragedy of the commons. I then reformulate it as a game of a multi-player prisoner's dilemma and study two possible approaches for preventing it, namely direct modification of players' payoffs through punishment/reward and modification of the environment in which the interactions occur. I also investigate the questions of whether the strategy of resource allocation for reproduction or competition would yield higher fitness in an evolving consumer-resource type system and demonstrate that the direction in which the system will evolve will depend not only on the state of the environment but largely on the initial composition of the population. I then apply the developed framework to modeling cancer as an evolving ecological system and draw conclusions about some alternative approaches to cancer treatment.
ContributorsKareva, Irina (Author) / Castillo-Chavez, Carlos (Thesis advisor) / Collins, James (Committee member) / Nagy, John (Committee member) / Smith, Hal (Committee member) / Arizona State University (Publisher)
Created2012
Description
Bacteriophage (phage) are viruses that infect bacteria. Typical laboratory experiments show that in a chemostat containing phage and susceptible bacteria species, a mutant bacteria species will evolve. This mutant species is usually resistant to the phage infection and less competitive compared to the susceptible bacteria species. In some experiments, both susceptible and resistant bacteria species, as well as phage, can coexist at an equilibrium for hundreds of hours. The current research is inspired by these observations, and the goal is to establish a mathematical model and explore sufficient and necessary conditions for the coexistence. In this dissertation a model with infinite distributed delay terms based on some existing work is established. A rigorous analysis of the well-posedness of this model is provided, and it is proved that the susceptible bacteria persist. To study the persistence of phage species, a "Phage Reproduction Number" (PRN) is defined. The mathematical analysis shows phage persist if PRN > 1 and vanish if PRN < 1. A sufficient condition and a necessary condition for persistence of resistant bacteria are given. The persistence of the phage is essential for the persistence of resistant bacteria. Also, the resistant bacteria persist if its fitness is the same as the susceptible bacteria and if PRN > 1. A special case of the general model leads to a system of ordinary differential equations, for which numerical simulation results are presented.
ContributorsHan, Zhun (Author) / Smith, Hal (Thesis advisor) / Armbruster, Dieter (Committee member) / Kawski, Matthias (Committee member) / Kuang, Yang (Committee member) / Thieme, Horst (Committee member) / Arizona State University (Publisher)
Created2012
Description
Serge Galams voting systems and public debate models are used to model voting behaviors of two competing opinions in democratic societies. Galam assumes that individuals in the population are independently in favor of one opinion with a fixed probability p, making the initial number of that type of opinion a binomial random variable. This analysis revisits Galams models from the point of view of the hypergeometric random variable by assuming the initial number of individuals in favor of an opinion is a fixed deterministic number. This assumption is more realistic, especially when analyzing small populations. Evolution of the models is based on majority rules, with a bias introduced when there is a tie. For the hier- archical voting system model, in order to derive the probability that opinion +1 would win, the analysis was done by reversing time and assuming that an individual in favor of opinion +1 wins. Then, working backwards we counted the number of configurations at the next lowest level that could induce each possible configuration at the level above, and continued this process until reaching the bottom level, i.e., the initial population. Using this method, we were able to derive an explicit formula for the probability that an individual in favor of opinion +1 wins given any initial count of that opinion, for any group size greater than or equal to three. For the public debate model, we counted the total number of individuals in favor of opinion +1 at each time step and used this variable to define a random walk. Then, we used first-step analysis to derive an explicit formula for the probability that an individual in favor of opinion +1 wins given any initial count of that opinion for group sizes of three. The spatial public debate model evolves based on the proportional rule. For the spatial model, the most natural graphical representation to construct the process results in a model that is not mathematically tractable. Thus, we defined a different graphical representation that is mathematically equivalent to the first graphical representation, but in this model it is possible to define a dual process that is mathematically tractable. Using this graphical representation we prove clustering in 1D and 2D and coexistence in higher dimensions following the same approach as for the voter model interacting particle system.
ContributorsTaylor, Nicole Robyn (Co-author) / Lanchier, Nicolas (Co-author, Thesis director) / Smith, Hal (Committee member) / Hurlbert, Glenn (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2013-05
Description
Since its isolation from a rhesus monkey in the Zika forest of Uganda in 1947, Zika virus (ZIKV) has spread into many parts of the world, causing major epidemics, notably in the Americas and some parts of Europe and Asia. The flavivirus ZIKV is primarily transmitted to humans via the bite of infectious adult female Aedes mosquitoes. In the absence of effective treatment or a safe and effective vaccine against the disease, control efforts are focused on effective vector management to reduce the mosquito population and limit human exposure to mosquito bites. The work in this thesis is based on the use of a mathematical model for gaining insight into the transmission dynamics of ZIKV in a population. The model, which takes the form of a deterministic system of nonlinear differential equations, is rigorously analyzed to gain insight into its basic qualitative features. In particular, it is shown that the disease-free equilibrium of the model is locally-asymptotically stable whenever a certain epidemiological quantity (known as the reproduction number, denoted by R0) is less than unity. The epidemiological implication of this result is that a small influx of ZIKV-infected individuals or vectors into the community will not generate a large outbreak if the anti-ZIKV control strategy (or strategies) adopted by the community can reduce and maintain R0 to a value less than unity. Numerical simulations of the model, using data relevant to ZIKV transmission dynamics in Puerto Rico, shows that a control strategy that solely focuses on killing immature mosquitoes (using highly efficacious larvicides) can lead to the elimination of ZIKV if the larvicide coverage (i.e., proportion of breeding sites treated with larvicides) is high enough (over 90%). Such elimination is also feasible using a control strategy that solely focuses on the use of insect repellents (as a means of personal protection against mosquito bites) if the coverage level of the insect repellent usage in the community is high enough (at least 70%). However, it is also shown that although the use of adulticides (i.e., using insecticides to kill adult mosquitoes) can reduce the reproduction number (hence, disease burden), it fails to reduce it to a value less than unity, regardless of coverage level. Thus, unlike with the use of larvicide-only or repellent-only strategies, the population-wide implementation of an adulticide-only strategy is unable to lead to ZIKV elimination. Finally, it is shown that the combined (integrated pest management) strategy, based on using all three aforementioned strategies, is the most effective approach for combatting ZIKV in the population. In particular, it is shown that even a moderately-effective level of this strategy, which entails using only 50% coverage of both larvicides and adulticides, together with about 45% coverage for a repellent strategy, will lead to ZIKV elimination. This moderately-effective combined strategy seems attainable in Puerto Rico.
ContributorsUrcuyo, Javier (Author) / Gumel, Abba (Thesis director) / Hackney Price, Jennifer (Committee member) / School of Mathematical and Natural Sciences (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
Description
Although extracellular throughout their lifecycle, trypanosomes are able to persist despite strong host immune responses through a process known as antigenic variation involving a large, highly diverse family of surface glycopro- tein (VSG) genes, only one of which is expressed at a time. Previous studies have used mathematical models to investigate the relationship between VSG switching and the dynamics of trypanosome infections, but none have explored the role of multiple VSG expression sites or the contribution of mosaic gene conversion events involving VSG pseudogenes.
ContributorsKoury, Michael Andrew (Author) / Taylor, Jesse (Thesis director) / Gumel, Abba (Committee member) / Department of Physics (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
The dissipative shallow-water equations (SWE) possess both real-world application and extensive analysis in theoretical partial differential equations. This analysis is dominated by modeling the dissipation as diffusion, with its mathematical representation being the Laplacian. However, the usage of the biharmonic as a dissipative operator by oceanographers and atmospheric scientists and its underwhelming amount of analysis indicates a gap in SWE theory. In order to provide rigorous mathematical justification for the utilization of these equations in simulations with real-world implications, we extend an energy method utilized by Matsumura and Nishida for initial value problems relating to the equations of motion for compressible, vsicous, heat-conductive fluids ([6], [7]) and applied by Kloeden to the diffusive SWE ([4]) to prove global time existence of classical solutions to the biharmonic SWE. In particular, we develop appropriate a priori growth estimates that allow one to extend the solution's temporal existence infinitely under sufficient constraints on initial data and external forcing, resulting in convergence to steady-state.
ContributorsKofroth, Collin Michael (Author) / Jones, Don (Thesis director) / Smith, Hal (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2017-05
Description
The Kingdom of Saudi Arabia (KSA), which hosts some of the largest mass gatherings of humans globally every year, has seen the emergence of two coronavirus pandemics, namely the 2012 middle eastern respiratory syndrome (MERS-CoV) and the 2019 SARS-CoV-2 pandemics. This dissertation contributes in providing deeper insight into the transmission dynamics and control of the two diseases in the Kingdom. A model for SARS-CoV-2 transmission dynamics, which incorporates the key features of the disease, was designed first of all. Its disease-free equilibrium was shown, using Lyapunov function theory, to be globally-asymptotically stable when the associated reproduction number is less than one. The model, which has a unique and locally-asymptotically stable endemic equilibrium (for a special case) when the reproduction threshold exceeds one, was fitted using observed data for the KSA. Global sensitivity analysis was carried out to identify the key parameters of the model that have the most influence on the disease burden in the Kingdom. The model was used to assess the population-level impacts of control and mitigation interventions. It was shown that a face mask use strategy, based on using masks of moderate to high efficacy, can lead to the elimination of the pandemic if the coverage in its usage is high enough. A model for the spread of MERS-CoV in the human and camel host populations was also designed, rigorously analysed, and fitted with data. The model was later extended to include the use of intervention measures, notably vaccination of humans and camels and the use of face mask by humans in public or when having frequent closed contacts with camels. The population-level impacts of these interventions, implemented in isolation or in combinations, were assessed. The study showed that focusing intervention resources on containing the MERS-CoV spread in the camel population would be more effective than on containing the spread in humans.
ContributorsAlatawi, Adel (Author) / Gumel, Abba (Thesis advisor) / Arizona State University (Publisher)
Created2023
Description
Synthetic biology (SB) has become an important field of science focusing on designing and engineering new biological parts and systems, or re-designing existing biological systems for useful purposes. The dramatic growth of SB throughout the past two decades has not only provided us numerous achievements, but also brought us more timely and underexplored problems. In SB's entire history, mathematical modeling has always been an indispensable approach to predict the experimental outcomes, improve experimental design and obtain mechanism-understanding of the biological systems. \textit{Escherichia coli} (\textit{E. coli}) is one of the most important experimental platforms, its growth dynamics is the major research objective in this dissertation. Chapter 2 employs a reaction-diffusion model to predict the \textit{E. coli} colony growth on a semi-solid agar plate under multiple controls. In that chapter, a density-dependent diffusion model with non-monotonic growth to capture the colony's non-linear growth profile is introduced. Findings of the new model to experimental data are compared and contrasted with those from other proposed models. In addition, the cross-sectional profile of the colony are computed and compared with experimental data. \textit{E. coli} colony is also used to perform spatial patterns driven by designed gene circuits. In Chapter 3, a gene circuit (MINPAC) and its corresponding pattern formation results are presented. Specifically, a series of partial differential equation (PDE) models are developed to describe the pattern formation driven by the MINPAC circuit. Model simulations of the patterns based on different experimental conditions and numerical analysis of the models to obtain a deeper understanding of the mechanisms are performed and discussed. Mathematical analysis of the simplified models, including traveling wave analysis and local stability analysis, is also presented and used to explore the control strategies of the pattern formation. The interaction between the gene circuit and the host \textit{E. coli} may be crucial and even greatly affect the experimental outcomes. Chapter 4 focuses on the growth feedback between the circuit and the host cell under different nutrient conditions. Two ordinary differential equation (ODE) models are developed to describe such feedback with nutrient variation. Preliminary results on data fitting using both two models and the model dynamical analysis are included.
ContributorsHe, Changhan (Author) / Kuang, Yang (Thesis advisor) / Wang, Xiao (Committee member) / Kostelich, Eric (Committee member) / Tian, Xiaojun (Committee member) / Gumel, Abba (Committee member) / Arizona State University (Publisher)
Created2021
Description
Malaria is a deadly, infectious, parasitic disease which is caused by Plasmodium parasites and transmitted between humans via the bite of adult female Anopheles mosquitoes. The primary insecticide-based interventions used to control malaria are indoor residual spraying (IRS) and long-lasting insecticide nets (LLINs). Larvicides are another insecticide-based intervention which is less commonly used. In this study, a mathematical model for malaria transmission dynamics in an endemic region which incorporates the use of IRS, LLINS, and larvicides is presented. The model is rigorously analyzed to gain insight into the asymptotic stability of the disease-free equilibrium. Simulations of the model show that individual insecticide-based interventions will not realistically control malaria in regions with high endemicity, but an integrated vector management strategy involving the use of multiple interventions could lead to the effective control of the disease. This study suggests that the use of larvicides alongside IRS and LLINs in endemic regions may be more effective than using only IRS and LLINs.
ContributorsJameson, Leah (Author) / Gumel, Abba (Thesis director) / Huijben, Silvie (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / School of Civic & Economic Thought and Leadership (Contributor)
Created2022-05