The SIR can be expressed either as a system of nonlinear ordinary differential equations or as a nonlinear Volterra integral equation. In general, neither of these can be solved in closed form. In this thesis, it is shown that if we assume S(t) is a finite multi-exponential, i.e. function of the form S(t) = a+ ∑nk=1 rke-σkt or a logistic function which is an infinite-multi-exponential, i.e. function of the form S(t) = c + a/b+ewt, then we can have closed form solution. Also we will formulate a method to determine R0 the basic reproductive rate of an infection.

At the turn of the century, the City of Orlando initiated the "Neighborhood Horizon Program." This program involved local citizens to help improve their community resources by engaging in a process of planning where the problems associated with the communities were identified. Many residents favored to bring back the brick roads that were overlaid with asphalt concrete to provided better transportation in the mid 1900s. With majority of the neighborhood streets already bricked, removing asphalt ensured safety, served as a technique for slowing traffic, and added to the historical integrity. Since there were no official documentations available that stated the definite existence of bricks beneath the asphalt surface course, it would have been rather impossible to core hundreds of locations to ensure the whereabouts of these anomalies. Thus, without time delays and excessive coring costs, a nondestructive instrumentation of Ground Penetrating Radar (GPR) was employed in the detection of bricks. This geophysical survey system distinguishes materials based on their different electrical properties that depend upon temperature, density, moisture content and impurities by providing a continuous profile of the subsurface conditions. The Ground Penetrating Radar operates on the principle of the electromagnetic wave (EMW) theory. The main objectives of this study was to investigate the existing pavement by using Ground Penetrating Radar (GPR) in detecting the brick base and to analyze the performance of pavement system for fatigue and rutting. The results of this study will assist the City of Orlando in removing asphalt layer, rebuilding of brick roads, and facilitate in better zoning and planning of the city. The construction of controlled test area provided with a good sense of brick detection, which helped in precise locations bricks for sections of Summerlin Avenue, Church Street and Cherokee Drive. The project demonstrated a good sense of detecting the subsurface anomalies, such as bricks. The validation of the profile readings was near to a 100%.

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<p>Mathematical modelling has become a powerful tool used to predict the spread of infectious diseases in populations. Successful analysis and modeling of historical infectious disease data can explain changes in the pattern of past epidemics and lead to a better understanding of epidemiological processes. The lessons learned can be used to predict future epidemics and help to improve public healthstrategies for control and eradication.</p> <p>This thesis is focused on the analysis and modelling of smallpox dynamics based on the weekly smallpox mortality records in London, England, 1664-1930. Statistical analysis of these records is presented. A timeline of significant historical events related to changes in variolation and vaccination uptake levels and demographics was established. These events were correlated with transitions observed in smallpox dynamics. Seasonality of the smallpox time series was investigated and the contact rate between susceptible and infectious individuals was found to be seasonally forced. Seasonal variations in smallpox transmission and changes in their seasonality over long time scale were estimated. The method of transition analysis, which is used to predict qualitative changes in epidemiological patterns, was used to explain the transitions observed in the smallpox time series. We found that the standard SIR model exhibits dynamics similar to the more realistic Gamma distributed SEIR model if the mean serial interval is chosen the same, so we used the standard SIR model for our analysis. We conclude that transitions observed in the temporal pattern of smallpox dynamics can be explained by the changes in birth, immigration and intervention uptake levels.</p> / Doctor of Philosophy (PhD)

<p>Infectious diseases spreading in a human population can occasionally exhibit sudden transitions in their qualitative dynamics. Previous work has been very successful in predicting such transitions in New York City's measles incidence rates using the standard SIR model (susceptible, infected, recovered). This work relied on a dataset spanning 45 years, which we have extended to 93 years (1891-1984). We continue previous research in transition analysis on this larger dataset, and compare resonant and transient periods predicted to exist in NYC's measles incidence rates with those observed through a continuous wavelet transform of the data. We find good agreement between SIR predictions and observation, and in particular note the likely existence of previously unobserved hysteresis early in our new time-series.</p> / Master of Science (MSc)

<p>Modelling and analysis of recurrent infectious disease epidemics often depends on the reconstruction of a time-varying transmission rate from historical reports of cases or deaths. Statistically rigorous estimation methods for time-varying transmission rates exist but are too computationally demanding to apply to a time series longer than a few decades. We present a computationally ecient estimation method that is suitable for very long data sets. Our method, which uses a discrete-time approximation to the SIR model for infectious diseases, is easy to implement and outperforms the classic Fine and Clarkson estimation method.</p> / Master of Science (MSc)

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