Applied mathematics

This thesis is focused on studying the population dynamics of a predator-prey system in a patchy environment, taking anti-predation responses into consideration. Firstly, we conduct mathematical analysis on the equilibrium solutions of the system. Using techniques from calculus we show that particular steady state solutions exist when the parameters of the system meet certain criteria. We then show that a further set of conditions leads to the local stability of these solutions. The second step is to extend the existing mathematical analysis by way of numerical simulations. We use octave to confirm the previous results, as well as to show that more complicated dynamics can exist, such as stable oscillations. We consider more complex and meaningful functions for nonlinear dispersal between patches and nonlinear predation, and show that the proposed model exhibits behaviours we expect to see in a population model.

Author Keywords: Anti-predation response, Asymptotic stability, Dispersal, Patch model, Population dynamics, Predator-prey

Ionizing organic chemicals are recognized as constituting a large fraction of the organic chemicals of commerce. Many governments internationally are engaged in the time-consuming and expensive task of chemical risk assessment for the protection of human and environmental health. There are standard models that are consistently used to supplement experimental and monitoring data in such assessments of non-ionizing organics by both government regulators and industry stakeholders. No such standard models exist for ionizing organics. Equilibrium distribution models, the foundational equations within multimedia environmental fate models for non-ionizing organics, were developed for the standard series of biphasic systems: air-water, particle-water, air-particle and organic-aqueous phases within living tissue. Multiple chemical species due to the ionization reaction were considered for each system. It was confirmed that, under select conditions, the properties of the neutral parent are sufficient to predict the overall distribution of the organic chemical. Complications due to biotransformation and paucity of identifiable equilibrium distribution data for ionizing organics limited the development of the model for living tissues. However, the equilibrium distributions of ionizing organics within this biotic system were shown to correlate with the abiotic sediment-water system. This suggests that the model developed for particle-water systems should be adaptable to the biotic system as model input and test data become available. Observational data for soil- and sediment- water systems, i.e., particle-water systems, allowed the development of a primarily non-empirical distribution equation for mono-protic acids; this model was almost entirely theoretically derived. The theoretical approach to model development allowed a quantitative assessment of the role of the neutral ion pair, resulting from the complexation of the organic anion with metal cations. To demonstrate the model's potential usefulness in governmental screening risk assessments, it was applied to a broad range of mono-protic organics including drugs and pesticides using standard property estimation software and generic inputs. The order-of-magnitude agreement between prediction and observation typical of the existing models of non-ionizing organics was generally achieved for the chemicals tested. The model was sensitive to the octanol-water partition coefficient of the most populous species. No calibration set was used in the development of any of the models presented.

Author Keywords: bioconcentration, chemical equilibrium, environmental modelling, ionizing organic, sorption

Comprehending the dynamics of infectious diseases is very important in formulating public health policies to tackling their prevalence. Mathematical epidemiology (ME) has played a very vital role in achieving the above. Nevertheless, classical mathematical epidemiological models do not explicitly model the behavioural responses of individuals in the presence of prevalence of these diseases. Economic epidemiology (EE) as a field has stepped in to fill this gap by integrating economic and mathematical concepts within one framework. This thesis investigated two issues in this area. The methods employed are the standard linear analysis of stability of dynamical systems and numerical simulation. Below are the investigations and the findings of this thesis:

Firstly, an investigation into the stability properties of the equilibria of EE

models is carried out. We investigated the stability properties of modified EE systems studied by Aadland et al. [6] by introducing a parametric quadratic utility function into the model, thus making it possible to model the maximum number of contacts made by rational individuals to be determined by a parameter. This parameter in particular influences the level of utility of rational individuals. We have shown that if rational individuals have a range of possible contacts to choose from, with the maximum of the number of contacts allowable for these individuals being dependent on a parameter, the variation in this parameter tends to affect the stability properties of the system. We also showed that under the assumption of permanent recovery for

disease coupled with individuals observing or not observing their immunity, death

and birth rates can affect the stability of the system. These parameters also have

effect on the dynamics of the EE SIS system.

Secondly, an EE model of syphilis infectivity among &ldquo men who have sex with men &rdquo (MSM) in detention centres is developed in an attempt at looking at the effect of behavioural responses on the disease dynamics among MSM. This was done by explicitly incorporating the interplay of the biology of the disease and the behaviour of the inmates. We investigated the stability properties of the system under rational expectations where we showed that: (1) Behavioural responses to the prevalence of

the disease affect the stability of the system. Therefore, public health policies have the tendency of putting the system on indeterminate paths if rational MSM have complete knowledge of the laws governing the motion of the disease states as well as a complete understanding on how others behave in the system when faced with risk-benefit trade-offs. (2) The prevalence of the disease in the long run is influenced by incentives that drive the utility of the MSM inmates. (3) The interplay between the dynamics of the biology of the disease and the behavioural responses of rational MSM tends to put the system at equilibrium quickly as compared to its counterpart (that is when the system is solely dependent on the biology of the disease) when subjected to small perturbation.

Author Keywords: economic and mathematical epidemiology models, explosive path, indeterminate-path stability, numerical solution, health gap, saddle-path stability, syphilis,

Health literature has long noted a positive correlation between health and levels of education. Two competing theories have been advanced to explain this phenomenon: (1) education "causes" health by allowing individuals to process complex information and act on it; and, (2) education and health are merely correlated through some third underlying characteristic.

Determining which of these two theories is correct is of importance to public policy. But that task is empirically difficult because, from the standard, static perspective, the theories are observationally equivalent.

We exploit a way in which the two theories have different implications regarding the sort of behaviour we should observe over time. We use smoking as a measure of health behaviour and find that smoking rates between "high" and "low" educated individuals expand when information is hard to process, and then contract as it becomes more easily processable. This approach is then repeated using physical activity as a measure of health-related behaviour to address limitations of the smoking model.

Our novel approach to estimating the differences in the behavioural responses to changes in the processability of health-related information, across education groups, provides strong evidence in support of the view that education and health are causally linked.

Author Keywords: applied statistics, education, health economics, public health, public policy, smoking

In this thesis, we explore the application of the Sinc-Collocation method to an oceanography model. The model of interest describes a wind-driven current with depth-dependent eddy viscosity and is formulated in two different systems; a complex-velocity system and a real-value coupled system. In general, the Sinc-based methods excel over other traditional numerical methods due to their exponentially decaying errors, rapid convergence and handling problems in the presence of singularities at end-points. In addition, the Sinc-Collocation approach that we utilize exploits first derivative interpolation, whose integration is less sensitive to numerical errors. We present several model problems to demonstrate the accuracy, and stability of the method. We compare the approximate solutions determined by the Sinc-Collocation technique with exact solutions and also with those obtained by the Sinc-Galerkin approach in earlier studies. Our findings indicate that the method we utilized outperforms those used in past studies.

Author Keywords: Boundary Value Problems, Eddy Viscosity, Oceanography, Sinc Numerical Methods, Wind-Driven Currents

In this thesis, a new sinc-collocation method based upon derivative interpolation is developed for solving linear and nonlinear boundary value problems involving differential as well as integro-differential equations. The sinc-collocation method is chosen for its ease of implementation, exponential convergence of error, and ability to handle to singularities in the BVP. We present a unique method of treating boundary conditions and introduce the concept of the stretch factor into the conformal mappings of domains. The result is a method that achieves great accuracy while reducing computational cost. In most cases, the results from the method greatly exceed the published results of comparable methods in both accuracy and efficiency. The method is tested on the Blasius problem, the Lane-Emden problem and generalised to cover Fredholm-Volterra integro-differential problems. The results show that the sinc-collocation method with derivative interpolation is a viable and preferable method for solving nonlinear BVPs.

Author Keywords: Blasius, Boundary Value Problem, Exponential convergence, Integro-differential, Nonlinear, Sinc

The time-dependent Gross-Pitaevskii Equation, describing the movement of parti-

cles in quantum mechanics, may not be solved analytically due to its inherent non-

linearity. Hence numerical methods are of importance to approximate the solution.

This study develops a discrete scheme in time and space to simulate the solution

defined in a finite domain by using the Crank-Nicolson difference method and Sinc

Collocation Methods (SCM), respectively. In theory and practice, the time discretiz-

ing system decays errors in the second-order of accuracy, and SCMs are decaying

errors exponentially. A new SCM with a unique boundary treatment is proposed

and compared with the original SCM and other similar numerical techniques in time

costs and numerical errors. As a result, the new SCM decays errors faster than the

original one. Also, to attain the same accuracy, the new SCM interpolates fewer

nodes than the original SCM, which saves computational costs. The new SCM is

capable of approximating partial differential equations under different boundary con-

ditions, which can be extensively applied in fitting theory.

Author Keywords: Crank-Nicolson difference method, Gross-Pitaevskii Equation, Sinc-Collocation methods

We propose a stochastic weather model based on temperature, precipitation, humidity and wind speed for Qatar, as a representative arid region, in order to obtain simulated values for a drought index. As a drought index, the Reconnaissance Drought Index (RDI) is commonly accepted in agriculture and is used to measure drought severity. It can be used to price weather derivatives to help farmers reduce nancial losses from drought. RDI, which is the ratio of precipitation to evapotranspiration, is calculated by considering crop growth stages. The use of dierent crop coecient value depending on the growth stage to calculate evapotranspiration can provide improved values for RDI. Additionally, six calculation methods for evapotranspiration using weather data are investigated to obtain accurate values for RDI.

Author Keywords: Evapotranspiration, Markov chains, Mean reversion processes, Reconnaissance Drought Index, Stochastic dierential equations, Stochastic weather models