Dynamics of a stochastic semi-Markovian hybrid switching epidemiological model with latency

State transitions and time delays significantly influence disease dynamics. This study introduces a semi-Markovian hybrid process into a delayed Susceptible-Exposed-Infectious-Recovered (SEIR) model to effectively capture these state changes. The hybrid process incorporates a semi-Markovian framework with mode-dependent fixed dwell times for each state, while the delay accounts for the transition period from the exposed to the infectious stage. We establish the positivity and uniqueness of the model’s solutions. A critical value for each state is derived to assess the disease’s growth rate, indicating that the disease will become extinct in a state if the value is less than zero. Our findings reveal that the delay exerts dual effects on the critical value. We provide sufficient conditions for the stochastic stability of the model’s trivial solution and for the finite time stability of the disease, which evaluates its progression over a finite period. Results show that both stabilities rely heavily on the stationary distribution of the hybrid process. Numerical simulations are conducted to validate our theoretical results.