Browsing by Author "Docrat, Raeesa"

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    Generating functions in branching processes and birth and death processes
    (2018) Docrat, Raeesa
    "If I have seen further, it is by standing upon the shoulders of giants" -Sir Isaac Newton Birth and death processes and branching processes are areas of stochastic processes that can be applied to a multitude of elds and disciplines, such as biology, economics and engineering. In general, the analysis of distributions and moments of these processes are di cult to obtain in explicit form. The use of generating functions makes computation much easier. The dissertation looks at the theory, application and innovation of using generating functions in the analysis of branching and birth and death processes.
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    Stochastic modelling of the spread of infectious diseases
    (2022) Docrat, Raeesa
    Background The spread of infectious diseases is a world-wide problem that has a greater impact on low-income countries. Mathematical modelling is a useful tool to better understand these diseases and to plan prevention and interventions. These models can either be deterministic or stochastic. Generally, deterministic models are used as they are easier to understand and predict average behaviour. Stochastic models are probabilistic models which incorporate random fluctuations over time. Stochastic models allow the calculation of probabilities and likelihoods of possible progressions of an infectious disease. These models can be built to include many aspects and features of infectious diseases. One such inclusion is the spatial nature inherent in the spread of infectious diseases. The inclusion of a spatial component for the spread of infectious diseases is needed, especially in South Africa, where it is spatially heterogeneous, to accurately analyse and efficiently allocate resources to where it is most needed. Methods This thesis begins with a review of both deterministic and stochastic compartmental models to illustrate the uses of these models in the context of disease modelling. The spatial autocorrelation of the spread of TB in South Africa is then studied. The results, although intuitive, confirm the existence of spatial autocorrelation and hence the need for models which accounts for the spatial v heterogeneity. A spatio-stochastic model in discrete time and space is then developed using binomial chain models as a basis. The spatial and temporal differences are modelled in the number of contacts a person would encounter. The spatio-stochastic models are then further extended to account for different modelling scenarios: closed spatial populations, spatial populations with interactions, but no migrations and finally a spatial model with migrations and interactions from other spatial units. This developed model is then compared to existing models in a simulation study. Lastly, the developed model is applied to a measles data set. Results The simulation study illustrates the differences in the outcome of different cases of the spatio-stochastic model. The disease develops and progresses very differently, depending on the modelling case chosen. This highlights the importance of choosing the correct model. The simulation study demonstrated the differences between other binomial chain based models, namely the ReedFrost and the Tuckwell Williams model. The inclusion of spatial heterogeneity produced outcomes that varied from the two non-spatial models. Lastly, the application of the model to a measles data set confirms that the inclusion of the spatial component allows a better fit of the models to the data. Conclusion The spatio-stochastic models are flexible and can be tailored to the dynamics of a particular disease’s spatial spread. This aspect makes these models very useful in understanding and analysing the spatial spread of the infectious disease. The developed spatio-stochastic model can be applied to a wide variety of cases. The fact that it is stochastic allows not only the ability to predict average behaviour, but also to calculate probabilities related to vital questions in planning and prevention. Questions related to the probability of outbreak in a particular area or which areas are likely to be hot spot areas can now be addressed. The modelling of the spread of infectious diseases using a spatial model, like the model developed in this thesis, is important to make informed decisions on how to stop and prevent further infection. The spatial models can provide information on sub-populations which can be used to allocate resources accordingly. It is also useful in identifying the effects of different interventions like vaccines. Due to globalisation and the level of connection in society, the spatio-stochastic modelling of infectious diseases is vital.