The first part of the thesis investigates the behavior of stochastic Hopf bifurcations. A topic which is well studied through invariant measures. These are typically based on the governing stochastic differential equations. When modeling real-life systems, the exact equations are not necessarily known, which renders the traditional methods from the literature not directly applicable. This thesis focuses on how to apply the theory to relatively short time series, which can arise in systems such as some oscillatory protein translation systems. This is done through a time embedding, and coordinate transformation, which results in an empirical probability distribution in which curvature can be used as a measure of whether the system is above the Hopf bifurcation. Lastly, the limited amount of data necessitates the application of hypothesis testing to quantify the results. The second part considers the effect of spatial heterogeneity on the dynamics of disease spread exemplified through a SIR/SEIR (Susceptible,(Exposed), Infected and recovered) agent-based Covid model. Modeling of diseases took center stage during the Covid-19 pandemic, specifically during the early stages of waves, where predicting the severity of the height of the wave was crucial in determining the appropriate measures to ”flatten the curve”. We show that when fitting simulations from the spatial heterogeneous model with a homogeneous model, the homogeneous model overestimates the peak and total number of infected by at least a factor of two. In the last part, multiple other aspects of epidemic covid modeling are investigated. Including another way of looking at the effect of spatial heterogeneity on the number of covid cases, through population density in different municipalities. A mathematical approach and a simulated approach to approximating the effect of assembly bans. An approximation of the effect of contact tracing on the reproductive number based on Monte Carlo simulations. Lastly, a look at how to quantify the effect of non-pharmaceutical interventions of both pandemic control and impact on personal freedom, which is used to compare different ways of implementing non-pharmaceutical interventions.