On the final size of epidemics in random environment

• SIR and SEIR classical ODEs systems are defined in what called random environment.
• The basic reproduction number is defined for such as model.
• A final epidemic size result is shown to be valid a.s. with respect to R0 and the initial infected proportions.

This paper extends the final size result of the classical SIR epidemic model in constant and periodic environments to random environment. Conditionally on the basic reproduction number R0 recently defined for random environment and the initial infected population fraction, we prove a final size result of an epidemic governed by the SIR model with time-depending parameters. The parameters are driven by an ergodic inhomogeneous time-periodic Markov process with finite state space. We also analyze the classical SEIR epidemic model in random environment.