Stationary distribution of a stochastic SEIR model with infectivity in the incubation period and homestead-isolation on the susceptible under regime switching

This paper is concerned with a stochastic SEIR model with infectivity in the incubation period and homestead-isolation on the susceptible, which is perturbed by white and colour noises. The model has a unique stationary distribution, which reflects the persistence of epidemics over a long period. Using the Has-minskii theorem and constructing stochastic Lyapunov functions with regime switching, we derive an important condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>R</mml:mi><mml:mn>0</mml:mn><mml:mi>s</mml:mi></mml:msubsup><mml:mo>.</mml:mo></mml:math> Comparing the expression for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>R</mml:mi><mml:mn>0</mml:mn><mml:mi>s</mml:mi></mml:msubsup><mml:mo>,</mml:mo></mml:math> we can see that if there is no environmental noise, then <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>R</mml:mi><mml:mn>0</mml:mn><mml:mi>s</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>.</mml:mo></mml:math> It ensures the asymptotic stability of the positive equilibrium <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>E</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math> of the corresponding deterministic system.