Over time, tumor treatment resistance inadvertently develops when androgen de-privation therapy (ADT) is applied to metastasized prostate cancer (PCa). To combat tumor resistance, while reducing the harsh side effects of hormone therapy, the clinician may opt to cyclically alternates the patient’s treatment on and off. This method,known as intermittent ADT, is an alternative to continuous ADT that improves the patient’s quality of life while testosterone levels recover between cycles. In this paper,we explore the response of intermittent ADT to metastasized prostate cancer by employing a previously clinical data validated mathematical model to new clinical data from patients undergoing Abiraterone therapy. This cell quota model, a system of ordinary differential equations constructed using Droop’s nutrient limiting theory, assumes the tumor comprises of castration-sensitive (CS) and castration-resistant (CR)cancer sub-populations. The two sub-populations rely on varying levels of intracellular androgen for growth, death and transformation. Due to the complexity of the model,we carry out sensitivity analyses to study the effect of certain parameters on their outputs, and to increase the identifiability of each patient’s unique parameter set. The model’s forecasting results show consistent accuracy for patients with sufficient data,which means the model could give useful information in practice, especially to decide whether an additional round of treatment would be effective.

Efforts to treat prostate cancer have seen an uptick, as the world’s most commoncancer in men continues to have increasing global incidence. Clinically, metastatic
prostate cancer is most commonly treated with hormonal therapy. The idea behind
hormonal therapy is to reduce androgen production, which prostate cancer cells
require for growth. Recently, the exploration of the synergistic effects of the drugs
used in hormonal therapy has begun. The aim was to build off of these recent
advancements and further refine the synergistic drug model. The advancements I
implement come by addressing biological shortcomings and improving the model’s
internal mechanistic structure. The drug families being modeled, anti-androgens,
and gonadotropin-releasing hormone analogs, interact with androgen production in a
way that is not completely understood in the scientific community. Thus the models
representing the drugs show progress through their ability to capture their effect
on serum androgen. Prostate-specific antigen is the primary biomarker for prostate
cancer and is generally how population models on the subject are validated. Fitting
the model to clinical data and comparing it to other clinical models through the
ability to fit and forecast prostate-specific antigen and serum androgen is how this
improved model achieves validation. The improved model results further suggest that
the drugs’ dynamics should be considered in adaptive therapy for prostate cancer.

Dengue is a mosquito-borne arboviral disease that causes significant public health burden in many trophical and sub-tropical parts of the world (where dengue is endemic). This dissertation is based on using mathematical modeling approaches, coupled with rigorous analysis and computation, to study the transmission dynamics and control of dengue disease. In Chapter 2, a new deterministic model was designed and used to assess the impact of local fluctuation of temperature and mosquito vertical (transvasorial) transmission on the population abundance of dengue mosquitoes and disease in a population. The model, which takes the form of a deterministic system of nonlinear differential equations, was parametrized using data from the Chiang Mai province of Thailand. The disease-free equilibrium of the model was shown to be globally-asymptotically stable when a certain epidemiological quantity is less than unity. Vertical transmission was shown to only have marginal impact on the disease dynamics, and its effect is temperature-dependent. Dengue burden in the province is maximized when the mean monthly temperature lie in the range [26-28] C. A new deterministic model was designed in Chapter 3 to assess the impact of the release of Wolbachia-infected mosquitoes on curtailing the mosquito population and dengue disease in a population. The model, which stratifies the mosquito population in terms of sex and Wolbachia-infection status, was rigorously analysed to characterize the bifurcation property of the model as well as the asymptotic stability of the various disease-free equilibria. Simulations, using Wolbachia-based mosquito control from Queensland, Australia, showed that the frequent release of mosquitoes infected with the bacterium can lead to the effective control of the local wild mosquito population, and that such effective control increases with increasing number of Wolbachia-infected mosquitoes released (up to 90% reduction in the wild mosquito population, from their baseline values, can be achieved). It was also shown that the well-known feature of cytoplasmic incompatibility has very little effect on the effectiveness of the Wolbachia-based mosquito control.

Immunotherapy has received great attention recently, as it has become a powerful tool in fighting certain types of cancer. Immunotherapeutic drugs strengthen the immune system's natural ability to identify and eradicate cancer cells. This work focuses on immune checkpoint inhibitor and oncolytic virus therapies. Immune checkpoint inhibitors act as blocking mechanisms against the binding partner proteins, enabling T-cell activation and stimulation of the immune response. Oncolytic virus therapy utilizes genetically engineered viruses that kill cancer cells upon lysing. To elucidate the interactions between a growing tumor and the employed drugs, mathematical modeling has proven instrumental. This dissertation introduces and analyzes three different ordinary differential equation models to investigate tumor immunotherapy dynamics.

The first model considers a monotherapy employing the immune checkpoint inhibitor anti-PD-1. The dynamics both with and without anti-PD-1 are studied, and mathematical analysis is performed in the case when no anti-PD-1 is administrated. Simulations are carried out to explore the effects of continuous treatment versus intermittent treatment. The outcome of the simulations does not demonstrate elimination of the tumor, suggesting the need for a combination type of treatment.

An extension of the aforementioned model is deployed to investigate the pairing of an immune checkpoint inhibitor anti-PD-L1 with an immunostimulant NHS-muIL12. Additionally, a generic drug-free model is developed to explore the dynamics of both exponential and logistic tumor growth functions. Experimental data are used for model fitting and parameter estimation in the monotherapy cases. The model is utilized to predict the outcome of combination therapy, and reveals a synergistic effect: Compared to the monotherapy case, only one-third of the dosage can successfully control the tumor in the combination case.

Finally, the treatment impact of oncolytic virus therapy in a previously developed and fit model is explored. To determine if one can trust the predictive abilities of the model, a practical identifiability analysis is performed. Particularly, the profile likelihood curves demonstrate practical unidentifiability, when all parameters are simultaneously fit. This observation poses concerns about the predictive abilities of the model. Further investigation showed that if half of the model parameters can be measured through biological experimentation, practical identifiability is achieved.

Cancer is a disease involving abnormal growth of cells. Its growth dynamics is perplexing. Mathematical modeling is a way to shed light on this progress and its medical treatments. This dissertation is to study cancer invasion in time and space using a mathematical approach. Chapter 1 presents a detailed review of literature on cancer modeling.

Chapter 2 focuses sorely on time where the escape of a generic cancer out of immune control is described by stochastic delayed differential equations (SDDEs). Without time delay and noise, this system demonstrates bistability. The effects of response time of the immune system and stochasticity in the tumor proliferation rate are studied by including delay and noise in the model. Stability, persistence and extinction of the tumor are analyzed. The result shows that both time delay and noise can induce the transition from low tumor burden equilibrium to high tumor equilibrium. The aforementioned work has been published (Han et al., 2019b).

In Chapter 3, Glioblastoma multiforme (GBM) is studied using a partial differential equation (PDE) model. GBM is an aggressive brain cancer with a grim prognosis. A mathematical model of GBM growth with explicit motility, birth, and death processes is proposed. A novel method is developed to approximate key characteristics of the wave profile, which can be compared with MRI data. Several test cases of MRI data of GBM patients are used to yield personalized parameterizations of the model. The aforementioned work has been published (Han et al., 2019a).

Chapter 4 presents an innovative way of forecasting spatial cancer invasion. Most mathematical models, including the ones described in previous chapters, are formulated based on strong assumptions, which are hard, if not impossible, to verify due to complexity of biological processes and lack of quality data. Instead, a nonparametric forecasting method using Gaussian processes is proposed. By exploiting the local nature of the spatio-temporal process, sparse (in terms of time) data is sufficient for forecasting. Desirable properties of Gaussian processes facilitate selection of the size of the local neighborhood and computationally efficient propagation of uncertainty. The method is tested on synthetic data and demonstrates promising results.

The analysis focuses on a two-population, three-dimensional model that attempts to accurately model the growth and diffusion of glioblastoma multiforme (GBM), a highly invasive brain cancer, throughout the brain. Analysis into the sensitivity of the model to

changes in the diffusion, growth, and death parameters was performed, in order to find a set of parameter values that accurately model observed tumor growth for a given patient. Additional changes were made to the diffusion parameters to account for the arrangement of nerve tracts in the brain, resulting in varying rates of diffusion. In general, small changes in the growth rates had a large impact on the outcome of the simulations, and for each patient there exists a set of parameters that allow the model to simulate a tumor that matches observed tumor growth in the patient over a period of two or three months. Furthermore, these results are more accurate with anisotropic diffusion, rather than isotropic diffusion. However, these parameters lead to inaccurate results for patients with tumors that undergo no observable growth over the given time interval. While it is possible to simulate long-term tumor growth, the simulation requires multiple comparisons to available MRI scans in order to find a set of parameters that provide an accurate prognosis.

Cancer is a worldwide burden in every aspect: physically, emotionally, and financially. A need for innovation in cancer research has led to a vast interdisciplinary effort to search for the next breakthrough. Mathematical modeling allows for a unique look into the underlying cellular dynamics and allows for testing treatment strategies without the need for clinical trials. This dissertation explores several iterations of a dendritic cell (DC) therapy model and correspondingly investigates what each iteration teaches about response to treatment.

In Chapter 2, motivated by the work of de Pillis et al. (2013), a mathematical model employing six ordinary differential (ODEs) and delay differential equations (DDEs) is formulated to understand the effectiveness of DC vaccines, accounting for cell trafficking with a blood and tumor compartment. A preliminary analysis is performed, with numerical simulations used to show the existence of oscillatory behavior. The model is then reduced to a system of four ODEs. Both models are validated using experimental data from melanoma-induced mice. Conditions under which the model admits rich dynamics observed in a clinical setting, such as periodic solutions and bistability, are established. Mathematical analysis proves the existence of a backward bifurcation and establishes thresholds for R0 that ensure tumor elimination or existence. A sensitivity analysis determines which parameters most significantly impact the reproduction number R0. Identifiability analysis reveals parameters of interest for estimation. Results are framed in terms of treatment implications, including effective combination and monotherapy strategies.

In Chapter 3, a study of whether the observed complexity can be represented with a simplified model is conducted. The DC model of Chapter 2 is reduced to a non-dimensional system of two DDEs. Mathematical and numerical analysis explore the impact of immune response time on the stability and eradication of the tumor, including an analytical proof of conditions necessary for the existence of a Hopf bifurcation. In a limiting case, conditions for global stability of the tumor-free equilibrium are outlined.

Lastly, Chapter 4 discusses future directions to explore. There still remain open questions to investigate and much work to be done, particularly involving uncertainty analysis. An outline of these steps is provided for future undertakings.

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.

Ideas from coding theory are employed to theoretically demonstrate the engineering of mutation-tolerant genes, genes that can sustain up to some arbitrarily chosen number of mutations and still express the originally intended protein. Attention is restricted to tolerating substitution mutations. Future advances in genomic engineering will make possible the ability to synthesize entire genomes from scratch. This presents an opportunity to embed desirable capabilities like mutation-tolerance, which will be useful in preventing cell deaths in organisms intended for research or industrial applications in highly mutagenic environments. In the extreme case, mutation-tolerant genes (mutols) can make organisms resistant to retroviral infections.

An algebraic representation of the nucleotide bases is developed. This algebraic representation makes it possible to convert nucleotide sequences into algebraic sequences, apply mathematical ideas and convert results back into nucleotide terms. Using the algebra developed, a mapping is found from the naturally-occurring codons to an alternative set of codons which makes genes constructed from them mutation-tolerant, provided no more than one substitution mutation occurs per codon. The ideas discussed naturally extend to finding codons that can tolerate t arbitrarily chosen number of mutations per codon. Finally, random substitution events are simulated in both a wild-type green fluorescent protein (GFP) gene and its mutol variant and the amino acid sequence expressed from each post-mutation is compared with the amino acid sequence pre-mutation.

This work assumes the existence of synthetic protein-assembling entities that function like tRNAs but can read k nucleotides at a time, with k greater than or equal to 5. The realization of this assumption is presented as a challenge to the research community.

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.