Infectious diseases continue to pose a major threat to human health on a global scale. The current COVID-19 pandemic and other recent outbreaks such as SARS (2002) and Ebola (2013) highlight the immense burden they can impose on society. Statistical models have been established as an essential tool for understanding the outbreak of epidemics and their transmission dynamics. Here, a discrete time model with branching process characteristics was developed to model the spreading of an epidemic through a susceptible population with the aim of producing a model as realistic to true events as possible. A key characteristic of the model is the time varying infectivity allowed for each infected. This was incorporated within a framework of Bayesian inference and an adaptive MCMC scheme was used to infer the model parameters, specifically the reproduction number R. The transmission dynamics of an epidemic are rarely homogeneous and both a super-spreading events model and super-spreader model were also developed to capture the uneven transmission patterns of real-life epidemics. Simulations of all models were carried out to aid understanding of the model dynamics involved and the inferred parameters were plotted to ensure correctness. The long-term, overall aim of this work is to get a better understanding of the underlying dynamics of epidemic outbreaks. Such an understanding could be used to inform public policy and control strategies with the goal of mitigating the worst effects of infectious diseases and epidemics.