Thesis (Ph. D.)--University of Washington, 2006. / Vita. Includes bibliographical references (p. 107-114).

Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (leaves [297]-301).

The thesis is based on the use of mathematical modeling and analysis to gain insightinto the transmission dynamics of malaria in a community. A new deterministic model for assessing the role of age-structure on the disease dynamics is designed. The model undergoes backward bifurcation, a dynamic phenomenon characterized by the co-existence of a stable disease-free and an endemic equilibrium of the model when the associated reproduction number is less than unity. It is shown that adding age-structure to the basic model for malaria transmission does not alter its essential qualitative dynamics. The study is extended to incorporate the use of anti-malaria drugs. Numerical simulations of the extended model suggest that for the case when treatment does not cause drug resistance (and the reproduction number of each of the two strains exceed unity), the model undergoes competitive exclusion. The impact of various effectiveness levels of the treatment strategy is assessed.

The thesis is based on the use of mathematical modeling and analysis to gain insightinto the transmission dynamics of malaria in a community. A new deterministic model for assessing the role of age-structure on the disease dynamics is designed. The model undergoes backward bifurcation, a dynamic phenomenon characterized by the co-existence of a stable disease-free and an endemic equilibrium of the model when the associated reproduction number is less than unity. It is shown that adding age-structure to the basic model for malaria transmission does not alter its essential qualitative dynamics. The study is extended to incorporate the use of anti-malaria drugs. Numerical simulations of the extended model suggest that for the case when treatment does not cause drug resistance (and the reproduction number of each of the two strains exceed unity), the model undergoes competitive exclusion. The impact of various effectiveness levels of the treatment strategy is assessed.

<div>Public health policies increasingly rely on complex models that need to approximate epidemics realistically and be consistent with the available data. Choosing appropriate simplifying assumptions is one of the critical challenges in disease modeling. In this thesis, we focus on some of these assumptions to show how they impact model outcomes. </div><div>In this thesis, an ODE model with a gamma-distributed infectious period is studied and compared with an exponentially distributed infectious period. We show that, for childhood diseases, isolating infected children is a possible mechanism causing oscillatory behavior in incidence. This is shown analytically by identifying a Hopf bifurcation with the isolation period as the bifurcation parameter. The threshold value for isolation to generate sustained oscillations from the model with gamma-distributed isolation period is much more realistic than the exponentially distributed model.</div><div><br></div><div>The consequences of not modeling the spectrum of clinical symptoms of the 2014 Ebola outbreak in Liberia include overestimating the basic reproduction number and effectiveness of control measures. The outcome of this model is compared with those of models with typical symptoms, excluding moderate ones. Our model captures the dynamics of the recent outbreak of Ebola in Liberia better, and the basic reproduction number is more consistent with the WHO response team's estimate. Additionally, the model with only typical symptoms overestimates the basic reproduction number and effectiveness of control measures and exaggerates changes in peak size attributable to interventions' timing.</div><div><br></div>

The aim of this thesis is to define and study harvesting models of fish populations. These models are applied to particular fish species e.g., haddock and cod. The thesis is divided into five chapters: The first chapter is considered as an introductory one. In it, basics of fish biology and the recruitment process are defined. Two simple recruitment models known by the names Ricker and Beverton-Holt are used. In the second chapter the generalized Leslie model or Usher model is introduced. In section 2.2, some matrix theory is presented. For this matrix model, the net reproductive number is defined and studied in section 2.3. It turns out to be more useful than the spectral radius. In section 2.4, this study is extended to nonlinear matrix models. The nonlinearity, however, is defined only by the recruitment process. This allows to determine the equilibrium components. Finally section 2.5, the local stability of nonlinear matrix models is analyzed. Harvesting of such general matrix model is defined in chapter 3. We distinguish three different harvesting models (selective, net and semicontinuous harvesting models). In chapter 4, these harvesting models are then applied to concrete fish populations and analyzed with respect to its various parameters. In chapter 5, the stability is studied again along the lines of the paper of Levin, Goodyear [18]. The key results in this study are: 1) The maximum sustainable yields for selective harvesting and net harvesting are rather close. 2) Semicontinuous harvesting is more realistic harvesting models. 3) From a quantitative point of view, the choice of the recruitment function is important. 4) Harvesting process increases mortality and stability when we used Ricker recruitment model. 5) Stability of populations always holds if we use Beverton-Holt recruitment model.

In this thesis, we study the population dynamics of predator-prey interactions described by mathematical models with age/stage structures. We first consider fixed development times for predators and prey and develop a stage-structured predator-prey model with Holling type II functional response. The analysis shows that the threshold dynamics holds. That is, the predator-extinction equilibrium is globally stable if the net reproductive number of the predator $\mathcal{R}_0$ is less than $1$, while the predator population persists if $\mathcal{R}_0$ is greater than $1$. Numerical simulations are carried out to demonstrate and extend our theoretical results. A general maturation function for predators is then assumed, and an age-structured predator-prey model with no age structure for prey is formulated. Conditions for the existence and local stabilities of equilibria are obtained. The global stability of the predator-extinction equilibrium is proved by constructing a Lyapunov functional. Finally, we consider a special case of the maturation function discussed before. More specifically, we assume that the development times of predators follow a shifted Gamma distribution and then transfer the previous model into a system of differential-integral equations. We consider the existence and local stabilities of equilibria. Conditions for existence of Hopf bifurcation are given when the shape parameters of Gamma distributions are $1$ and $2$.

Mathematical models, that depict the dynamics of a cancer cell population growing out of the human body (in vitro) in unconstrained microenvironment conditions, are considered in this thesis. Cancer cells in vitro grow and divide much faster than cancer cells in the human body, therefore, the effects of various cancer treatments applied to them can be identified much faster. These cell populations, when not exposed to any cancer treatment, exhibit exponential growth that we refer to as the balanced exponential growth (BEG) state. This observation has led to several effective methods of estimating parameters that thereafter are not required to be determined experimentally. We present derivation of the age-structured model and its theoretical analysis of the existence of the solution. Furthermore, we have obtained the condition for BEG existence using the Perron-Frobenius theorem. A mathematical description of the cell-cycle control is shown for one-compartment and two-compartment populations, where a compartment refers to a cell population consisting of cells that exhibit similar kinetic properties. We have incorporated into our mathematical model the required growing/aging times in each phase of the cell cycle for the biological viability. Moreover, we have derived analytical formulae for vital parameters in cancer research, such as population doubling time, the average cell-cycle age, and the average removal age from all phases, which we argue is the average cell-cycle time of the population. An estimate of the average cell-cycle time is of a particular interest for biologists and clinicians, and for patient survival prognoses as it is considered that short cell-cycle times correlate with poor survival prognoses for patients. Applications of our mathematical model to experimental data have been shown. First, we have derived algebraic expressions to determine the population doubling time from single experimental observation as an alternative to empirically constructed growth curve. This result is applicable to various types of cancer cell lines. One option to extend this model would be to derive the cell cycle time from a single experimental measurement. Second, we have applied our mathematical model to interpret and derive dynamic-depicting parameters of five melanoma cell lines exposed to radiotherapy. The mathematical result suggests there are shortcomings in the experimental methods and provides an insight into the cancer cell population dynamics during post radiotherapy. Finally, a mathematical model depicting a theoretical cancer cell population that comprises two sub-populations with different kinetic properties is presented to describe the transition of a primary culture to a cell line cell population.

In this study, we consider the mechanism design problem for the optimal allocation of fishing quotas at different total allowable catch (TAC) levels. An age-structured fish population model is employed. Fishing technologies are embedded in the economic model as a key determinant. As a result, we showed that the quota allocation mechanism is important to minimize the impact of fishing on total fish biomass or achieve maximum sustainable yield (MSY). Moreover, we indicated technology-based optimality conditions for allocation of quotas at different TAC levels, which minimize the impact of fishing on total fish biomass or enable us to achieve MSY. Under the consideration that the fishermen fulfill their remaining quotas through capturing untargeted (less revenue-generating) fish after the targeted fish population is fully caught, the fix ratio of the catch of targeted fish to untargeted fish is not valid anymore. Concordantly, we indicated technology-based optimal quota levels, including the interior solutions. In the EU, TACs are distributed among states according to the principle of &lsquo / relative stability&rsquo / which prescribes that the fishing quotas should be allocated based on historical catches of the EU states. In this context, rather than allocating the quotas based on historical catches, our main suggestion is that the structure of the fishing industry should be considered for allocation of quotas to provide the sustainability of EU fisheries and achieve responsible and effective management of the fishing industry in the EU.

abstract: 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. / Dissertation/Thesis / Ph.D. Mathematics 2013