Stochastic multi-group epidemic SVIR models: Degenerate case

This work considers a multi-group epidemic SIR models with vaccination. The fluctuation of the environment is taken into account by introducing both color noise and white noise to a compartmental model. Unlike existing results on stochastic multigroup models, which were not successful in finding the reproduction numbers and fully classify the longtime behaviors of the models, we will provide a formula for the reproduction number R-0 of our model and will show that the disease is persistent if R-0 > 1 while the disease will be eradicated if R0(0) <= 1. We also provide the explicit formulae for R0 in some special cases. The formulae will be useful to determine the herb immunity threshold of the disease, which can be used to make right and timely policy to control a disease.