Mosquitoes are the greatest killers of mankind, and diseases caused by mosquitoes continue to induce major public health and socio-economic burden in many parts of the world (notably in the tropical sub-regions). This dissertation contributes in providing deeper qualitative insights into the transmission dynamics and control of some mosquito-borne diseases of major public health significance, such as malaria and dengue. The widespread use of chemical insecticides, in the form of long-lasting insecticidal nets (LLINs) and indoor residual spraying, has led to a dramatic decline in malaria burden in endemic areas for the period 2000-2015. This prompted a concerted global effort aiming for malaria eradication by 2040. Unfortunately, the gains recorded are threatened (or not sustainable) due to it Anopheles resistance to all the chemicals embedded in the existing insecticides. This dissertation addresses the all-important question of whether or not malaria eradication can indeed be achieved using insecticides-based control. A novel mathematical model, which incorporates the detailed Anopheles lifecycle and local temperature fluctuations, was designed to address this question. Rigorous analysis of the model, together with numerical simulations using relevant data from endemic areas, show that malaria elimination in meso- and holo-endemic areas is feasible using moderate coverage of moderately-effective and high coverage of highly-effective LLINs, respectively. Biological controls, such as the use of sterile insect technology, have also been advocated as vital for the malaria eradication effort. A new model was developed to determine whether the release of sterile male mosquitoes into the population of wild adult female Anopheles mosquito could lead to a significant reduction (or elimination) of the wild adult female mosquito population. It is shown that the frequent release of a large number of sterile male mosquitoes, over a one year period, could lead to the effective control of the targeted mosquito population. Finally, a new model was designed and used to study the transmission dynamics of dengue serotypes in a population where the Dengvaxia vaccine is used. It is shown that using of the vaccine in dengue-naive populations may induce increased risk of severe disease in these populations.

The purpose of this thesis is to accurately simulate the surface brightness in various spectral emission lines of the HH 901 jets in the Mystic Mountain Formation of the Carina Nebula. To accomplish this goal, we gathered relevant spectral emission line data for [Fe II] 12660 Å, Hα 6563 Å, and [S II] 6720 Å to compare with Hubble Space Telescope observations of the HH 901 jets presented in Reiter et al. (2016). We derived the emissivities for these lines from the spectral synthesis code Cloudy by Ferland et al. (2017). In addition, we used WENO simulations of density, temperature, and radiative cooling to model the jet. We found that the computed surface brightness values agreed with most of the observational surface brightness values. Thus, the 3D cylindrically symmetric simulations of surface brightness using the WENO code and Cloudy spectral emission models are accurate for jets like HH 901. After detailing these agreements, we discuss the next steps for the project, like adding an external ambient wind and performing the simulations in full 3D.

This project attempts to create an accurate numerical simulation of the eastern limb of the HH 901 jet in the Mystic Mountain formation located in the Carina Nebula. Using a 3rd order accurate WENO numerical scheme in space, and a 3rd order accurate RK method in time, the temperature, density, radiative cooling, length, and average jet velocity of this astrophysical phenomenon were simulated based on observations made by Hubble Space Telescope and the work of Reiter and Smith (2013) and (2014). The results of this simulation are displayed in three figures, one each for temperature, radiative cooling, and density, which show a jet displaying morphology consistent with that of the HH 901 eastern limb without adjustment for stellar wind. Also discussed are the effects of different jet speeds, initial conditions, and pulse parameters on the shape and behavior of the simulated jets, as well as continuing work to be done on the simulation to enhance its accuracy and usefulness.

Rabies is an infectious viral disease. It is usually fatal if a victim reaches the rabid stage, which starts after the appearance of disease symptoms. The disease virus attacks the central nervous system, and then it migrates from peripheral nerves to the spinal cord and brain. At the time when the rabies virus reaches the brain, the incubation period is over and the symptoms of clinical disease appear on the victim. From the brain, the virus travels via nerves to the salivary glands and saliva.

A mathematical model is developed for the spread of rabies in a spatially distributed fox population to model the spread of the rabies epizootic through middle Europe that occurred in the second half of the 20th century. The model considers both territorial and wandering rabid foxes and includes a latent period for the infection. Since the model assumes these two kinds of rabid foxes, it is a system of both partial differential and integral equations (with integration

over space and, occasionally, also over time). To study the spreading speeds of the rabies epidemic, the model is reduced to a scalar Volterra-Hammerstein integral equation, and space-time Laplace transform of the integral equation is used to derive implicit formulas for the spreading speed. The spreading speeds are discussed and implicit formulas are given for latent periods of fixed length, exponentially distributed length, Gamma distributed length, and log-normally distributed length. A number of analytic and numerical results are shown pertaining to the spreading speeds.

Further, a numerical algorithm is described for the simulation

of the spread of rabies in a spatially distributed fox population on a bounded domain with Dirichlet boundary conditions. I propose the following methods for the numerical approximation of solutions. The partial differential and integral equations are discretized in the space variable by central differences of second order and by

the composite trapezoidal rule. Next, the ordinary or delay differential equations that are obtained this way are discretized in time by explicit

continuous Runge-Kutta methods of fourth order for ordinary and delay differential systems. My particular interest

is in how the partition of rabid foxes into

territorial and diffusing rabid foxes influences

the spreading speed, a question that can be answered by purely analytic means only for small basic reproduction numbers. I will restrict the numerical analysis

to latent periods of fixed length and to exponentially

distributed latent periods.

The results of the numerical calculations

are compared for latent periods

of fixed and exponentially distributed length

and for various proportions of territorial

and wandering rabid foxes.

The speeds of spread observed in the

simulations are compared

to spreading speeds obtained by numerically solving the analytic formulas

and to observed speeds of epizootic frontlines

in the European rabies outbreak 1940 to 1980.

The most advanced social insects, the eusocial insects, form often large societies in which there is reproductive division of labor, queens and workers, have overlapping generations, and cooperative brood care where daughter workers remain in the nest with their queen mother and care for their siblings. The eusocial insects are composed of representative species of bees and wasps, and all species of ants and termites. Much is known about their organizational structure, but remains to be discovered.

The success of social insects is dependent upon cooperative behavior and adaptive strategies shaped by natural selection that respond to internal or external conditions. The objective of my research was to investigate specific mechanisms that have helped shaped the structure of division of labor observed in social insect colonies, including age polyethism and nutrition, and phenomena known to increase colony survival such as egg cannibalism. I developed various Ordinary Differential Equation (ODE) models in which I applied dynamical, bifurcation, and sensitivity analysis to carefully study and visualize biological outcomes in social organisms to answer questions regarding the conditions under which a colony can survive. First, I investigated how the population and evolutionary dynamics of egg cannibalism and division of labor can promote colony survival. I then introduced a model of social conflict behavior to study the inclusion of different response functions that explore the benefits of cannibalistic behavior and how it contributes to age polyethism, the change in behavior of workers as they age, and its biological relevance. Finally, I introduced a model to investigate the importance of pollen nutritional status in a honeybee colony, how it affects population growth and influences division of labor within the worker caste. My results first reveal that both cannibalism and division of labor are adaptive strategies that increase the size of the worker population, and therefore, the persistence of the colony. I show the importance of food collection, consumption, and processing rates to promote good colony nutrition leading to the coexistence of brood and adult workers. Lastly, I show how taking into account seasonality for pollen collection improves the prediction of long term consequences.

Need-based transfers (NBTs) are a form of risk-pooling in which binary welfare exchanges

occur to preserve the viable participation of individuals in an economy, e.g. reciprocal gifting

of cattle among East African herders or food sharing among vampire bats. With the

broad goal of better understanding the mathematics of such binary welfare and risk pooling,

agent-based simulations are conducted to explore socially optimal transfer policies

and sharing network structures, kinetic exchange models that utilize tools from the kinetic

theory of gas dynamics are utilized to characterize the wealth distribution of an NBT economy,

and a variant of repeated prisoner’s dilemma is analyzed to determine whether and

why individuals would participate in such a system of reciprocal altruism.

From agent-based simulation and kinetic exchange models, it is found that regressive

NBT wealth redistribution acts as a cutting stock optimization heuristic that most efficiently

matches deficits to surpluses to improve short-term survival; however, progressive

redistribution leads to a wealth distribution that is more stable in volatile environments and

therefore is optimal for long-term survival. Homogeneous sharing networks with low variance

in degree are found to be ideal for maintaining community viability as the burden and

benefit of NBTs is equally shared. Also, phrasing NBTs as a survivor’s dilemma reveals

parameter regions where the repeated game becomes equivalent to a stag hunt or harmony

game, and thus where cooperation is evolutionarily stable.

Inverse problems model real world phenomena from data, where the data are often noisy and models contain errors. This leads to instabilities, multiple solution vectors and thus ill-posedness. To solve ill-posed inverse problems, regularization is typically used as a penalty function to induce stability and allow for the incorporation of a priori information about the desired solution. In this thesis, high order regularization techniques are developed for image and function reconstruction from noisy or misleading data. Specifically the incorporation of the Polynomial Annihilation operator allows for the accurate exploitation of the sparse representation of each function in the edge domain.

This dissertation tackles three main problems through the development of novel reconstruction techniques: (i) reconstructing one and two dimensional functions from multiple measurement vectors using variance based joint sparsity when a subset of the measurements contain false and/or misleading information, (ii) approximating discontinuous solutions to hyperbolic partial differential equations by enhancing typical solvers with l1 regularization, and (iii) reducing model assumptions in synthetic aperture radar image formation, specifically for the purpose of speckle reduction and phase error correction. While the common thread tying these problems together is the use of high order regularization, the defining characteristics of each of these problems create unique challenges.

Fast and robust numerical algorithms are also developed so that these problems can be solved efficiently without requiring fine tuning of parameters. Indeed, the numerical experiments presented in this dissertation strongly suggest that the new methodology provides more accurate and robust solutions to a variety of ill-posed inverse problems.

Predicting resistant prostate cancer is critical for lowering medical costs and improving the quality of life of advanced prostate cancer patients. I formulate, compare, and analyze two mathematical models that aim to forecast future levels of prostate-specific antigen (PSA). I accomplish these tasks by employing clinical data of locally advanced prostate cancer patients undergoing androgen deprivation therapy (ADT). I demonstrate that the inverse problem of parameter estimation might be too complicated and simply relying on data fitting can give incorrect conclusions, since there is a large error in parameter values estimated and parameters might be unidentifiable. I provide confidence intervals to give estimate forecasts using data assimilation via an ensemble Kalman Filter. Using the ensemble Kalman Filter, I perform dual estimation of parameters and state variables to test the prediction accuracy of the models. Finally, I present a novel model with time delay and a delay-dependent parameter. I provide a geometric stability result to study the behavior of this model and show that the inclusion of time delay may improve the accuracy of predictions. Also, I demonstrate with clinical data that the inclusion of the delay-dependent parameter facilitates the identification and estimation of parameters.

The goal of this thesis is to extend the astrophysical jet model created by Dr.
Gardner and Dr. Jones to model the surface brightness of astrophysical jets. We attempt to accomplish this goal by modeling the astrophysical jet HH30 in the spectral emission lines [SII] 6716Å, [OI] 6300Å, and [NII] 6583Å. In order to do so, we used the jet model to simulate the temperature and density of the jet to match observational data by Hartigan and Morse (2007). From these results, we derived the emissivities in these emission lines using Cloudy by Ferland et al. (2013). Then we used the emissivities to determine the surface brightness of the jet in these lines. We found that the simulated surface brightness agreed with the observational surface brightness and we conclude that the model could successfully be extended to model the surface brightness of a jet.