الفهرس 
Chapter 1 Mathematical IntroductionOnly 14 pages are availabe for public view
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Abstract
In this thesis, we generated and developed different statistical probabilistic and dynamical stochastic mathematical models to cover two engineering interdisciplinary which are Biomedical engineering and Electrical power engineering. In the area of biomedical engineering, we focused on generating and modifying different population-based dynamical models of infectious diseases, through using Caputo fractional order derivative and stochastic white noise as well as using curve-fitting models to predict daily confirmed cases in different countries.
The generated fractional-order deterministic and stochastic dynamical models are applied on COVID-19 virus data to predict virus transmission scenarios to help governments to understand and control this epidemic disease behavior. We started with modelling COVID-19 dynamics, deterministically and stochastically, through developing the default SEIR epidemiological model of infectious diseases to suit the new virus behavior. Then Establishing two different stochastic models with different assumptions to model COVID-19 dynamics under vaccinations. In the end, we generated forecasting models to predict daily virus infections using Fourier and the sum of sine-waves fitting models to be applied on low, medium, and high virus transmission rate countries.
In the area of Electrical power engineering, we started with modelling commercial wind turbines’ power curves using single and multiple probability density functions to predict accurately near actual wind turbines output powers at different input wind speed. Then, we generated a new algorithm for optimal allocation of wind turbines at sites based on probabilistic modelling of wind turbines capacity factors.
This thesis is organized as follows:
- Chapter 1 represents the required mathematical concepts, basic definitions of probability, fractional calculus, and stochastic differential equations as well as the required numerical algorithms to solve or optimize different proposed models in our study.
- Chapter 2 represents two main subsections. The first one, illustrates epidemiological diseases history, modelling principles and mathematical backgrounds. While the second one explains the background of wind speed data modelling and wind turbines electrical power modelling techniques which lie under the area of renewable energy resources. By the end of each subsection, recent literature review about both applications is also discussed.
- Chapter 3 represents the stochastic and deterministic epidemiological models that were developed to describe the COVID-19 transmission and then applied to the Egyptian case study.
- Chapter 4 represents two different stochastic and deterministic epidemiological models, with different assumptions, developed to describe the COVID-19 transmission under vaccinations and then applied to Saudi Arabia and Egypt case studies.
- Chapter 5 represents the predictive models established to forecast COVID-19 daily confirmed cases in different countries using Fourier and the sum of sinewaves fitting models.
- Chapter 6 represents modelling of different commercial wind turbines power curves with different manufacturers and ratings using single and composite cumulative density functions.
- Chapter 7 represents a new methodology for wind turbines optimal site matching using four generated probabilistic models of wind turbines capacity factors.
- Chapter 8 provides the conclusions and future works.