Stochastic Epidemic Models with Vaccination and Hybrid Perturbations: A Survey

The objective of this work is to analyze some of the sophisticated stochastic epidemiology models that have emerged recently which go beyond the conventional deterministic modeling by including stochasticity, non-linear transmissions, and vaccination effects. This article discusses how the elements of randomness in terms of environment change, behavioral modification, imperfect vaccines, and reduced immunity are included in modern day models. Moreover, this paper also discusses some of the sophisticated hybrid models including diffusion dynamics, levy jumps, and Markovian switching representing the occurrence of sudden shock in the form of epidemiological events as well as the environmental changes. Some comparison among diffusion dynamics, vaccination model, and hybrid stochastic models was made in relation to extinction and persistence conditions, stochastic threshold theories, stability analysis, and the existence of fixed measures influencing them. However, it was found that even with advancements, there is a lack of consistency in the research being done today.