Epidemic threshold and ergodicity of an SEIR model with vertical transmission under the telegraph noise

An SEIR epidemic model incorporating both environmental and genetic factors is developed to investigate the impact of Markovian switching on the transmission dynamics of infectious diseases. Probabilistically, we show first that the basic reproduction number R₀ is a sharp threshold for the disease transmission: when R₀<1, the disease dies out almost surely; when R₀>1, the disease is persistent. We then obtain that the Markov process derived from the model is positive Harris recurrence if R₀>1, followed by the global attractivity of the Ω-limit set and the ergodicity of the Markov process. The theoretical results are applied to study the dynamics of rubella in China. This work has significantly simplified the analysis process in the existing literature. This research concludes that multiple waves of infections may be driven by the randomly environmental switching.