The Dynamical Behaviors of a Stochastic Mumps Infectious Disease Model.

ABSTRACT The study of infectious disease dynamics plays an important role in reflecting the transmission mechanisms of epidemics. Compared with the traditional statistical methods, understanding dynamics of an infectious disease can better make people understand some global characteristics of epidemic and help in designing appropriate strategies to control diseases. Early studies of mumps mainly focused on deterministic models. However, environmental noises are inevitable during the spread of infectious diseases. This paper extends a mumps transmission model from a deterministic to a stochastic framework and explores the dynamical behaviors of the model by constructing suitable Lyapunov functions. Our model is a six‐dimensional stochastic model. The construction of suitable Lyapunov functions is very challenging. Firstly, we show that this model has a unique global positive solution for any positive initial values. Secondly, we compute the basic reproduction numbers and present sufficient conditions for the existence of a unique ergodic stationary distribution and the extinction of the disease. Finally, we perform numerical simulations and sensitivity analysis for exploring the effect of some parameters and the white noises on the behavior of the model. The theoretical results can provide necessary guidelines to public health administrators for controlling and preventing diseases. [ABSTRACT FROM AUTHOR]