Showing 1 - 5 of 5 Items

Date: 2022-01-01

Creator: Sonia K. Shah

Access: Open access

The Nelder-Mead optimization method is a numerical method used to find the minimum of an objective function in a multidimensional space. In this paper, we use this method to study functions - specifically functions with three-dimensional graphs - and create images of the basin of attraction of the function. Three different methods are used to create these images named the systematic point method, randomized centroid method, and systemized centroid method. This paper applies these methods to different functions. The first function has two minima with an equivalent function value. The second function has one global minimum and one local minimum. The last function studied has several minima of different function values. The systematic point method is a reliable method in particular scenarios but is extremely sensitive to changes in the initial simplex. The randomized centroid method was not found to be useful as the basin of attraction images are difficult to understand. This made it particularly troublesome to know when the method was working effectively and when it was not. The systemized centroid method appears to be the most precise and effective method at creating the basin of attraction in most cases. This method rarely fails to find a minimum and is particularly adept at finding global minima more effectively compared to local minima. It is important to remember that these conclusions are simply based off the results of the methods and functions studied and that more effective methods may exist.

Date: 2020-01-01

Creator: Alanna Haslam

Access: Open access

In light of concerns about climate change, there is interest in how sustainable management can maintain the resilience of ecosystems. We use flow-kick dynamical systems to model ecosystems subject to a constant kick occurring every τ time units. We classify the stability of flow-kick equilibria to determine which management strategies result in desirable long-term characteristics. To classify the stability of a flow-kick equilibrium, we classify the linearization of the time-τ map given by the time-τ map of the variational equation about the equilibrium trajectory. Since the variational equation is a non-autonomous linear differential equation, we conjecture that the asymptotic stability classification of each instantaneous local linearization along the equilibrium trajectory indicates the stability of the variational time-τ map. In Chapter 3, we prove this conjecture holds when all of the asymptotic and transient behavior of the instantaneous local linearizations is the same. To explore whether the conjecture holds in general, we ask: To what degree can transient behavior differ from asymptotic behavior? Under what conditions can this transient behavior accumulate asymptotically? In Chapter 4, we develop the radial and tangential velocity framework to characterize transient behavior in autonomous linear systems. In Chapter 5, we use this framework to construct an example of a non-autonomous linear system whose time-τ map has asymptotic behavior that differs from the asymptotic behavior of each instantaneous linear system that composes it. Future work seeks to determine whether this constructed example can arise as a variational equation, and thus provide a counterexample for our conjecture.

Date: 2015-05-01

Creator: Stephen Ligtenberg

Access: Open access

Date: 2021-01-01

Creator: Juliana C. Taube

Access: Open access

SARS-CoV-2, the virus that causes COVID-19, has caused significant human morbidity and mortality since its emergence in late 2019. Not only have over three million people died, but humans have been forced to change their behavior in a variety of ways, including limiting their contacts, social distancing, and wearing masks. Early infectious disease models, like the classical SIR model by Kermack and McKendrick, do not account for differing contact structures and behavior. More recent work has demonstrated that contact structures and behavior can considerably impact disease dynamics. We construct a coupled disease-behavior dynamical model for SARS-CoV-2 by incorporating heterogeneous contact structures and decisions about masking. We use a contact network with household, work, and friend interactions to capture the variation in contact patterns. We allow decisions about masking to occur at a different time scale from disease spread which dramatically changes the masking dynamics. Drawing from the field of game theory, we construct an individual decision-making process that relies on perceived risk of infection, social influence, and individual resistance to masking. Through simulation, we find that social influence prevents masking, while perceived risk largely drives individuals to mask. Underlying contact structure also affects the number of people who mask. This model serves as a starting point for future work which could explore the relative importance of social influence and perceived risk in human decision-making.

Date: 2014-05-01

Creator: Lauren A Skerritt

Access: Open access

In the American lobster (Homarus americanus), neurogenic stimulation of the heart drives fluxes of calcium (Ca2+) into the cytoplasm of a muscle cell resulting in heart muscle contraction. The heartbeat is completed by the active transport of calcium out of the cytoplasm into extracellular and intracellular spaces. An increase in the frequency of calcium release is expected to increase amplitude and duration of muscle contraction. This makes sense because an increase in cytoplasmic calcium should increase the activation of the muscle contractile elements (actin and myosin). Since calcium cycling is a reaction-diffusion process, the extent to which calcium mediates contraction amplitude and frequency will depend on the specific diffusion relationships of calcium in this system. Despite the importance of understanding this relationship, it is difficult to obtain experimental information on the dynamics of cytoplasmic calcium. Thus, we developed a mathematical diffusion model of the myofibril (muscle cell) to simulate calcium cycling in the lobster cardiac muscle cell. The amplitude and duration of the force curves produced by the model empirically mirrored that of the experimental data over a range of calcium diffusion coefficients (1-16), nerve stimulation durations (1/6-1/3 of a contraction period), and frequencies (40-80 Hz). The characteristics that alter the response of the lobster cardiac muscle system are stimulation duration (i.e., burst duration), burst frequency, and the rate of calcium diffusion into the cell’s cytoplasm. For this reason, we developed protocols that allow parameters representing these characteristics in the calcium-force model to be determined from isolated whole muscle experiments on lobster hearts (Phillips et al., 2004). These parameters are used to predict variability in lobster heart muscle function consistent with data recorded in experiments. Within the physiological range of nerve stimulation parameters (burst duration and cycle period), calcium increased the cell’s force output for increased burst duration. For example, increased duration of stimulation increased the muscle contraction period and vice versa. In terms of diffusion, a slower rate of calcium diffusion out of the sarcoplasmic reticulum decreased both the calcium level and the contraction duration of the cell. Finally, changes in stimulation frequency did not produce changes in contraction amplitude and duration. When considered in conjunction with experimental stimulations using lobster heart muscle cells, these data illustrate the prominent role for calcium diffusion in governing contraction-relaxation cycles in lobster hearts.