Volterra-type integral model for epidemic of COVID-19

A refined version of the stochastic Volterra integral model for an epidemic of the new corona virus SARS-CoV-2, causing the COVID-19 disease, is analysed. The system is constructed from the corresponding ordinary differential model by introducing the class of immunized individuals with the finite duration of immunity gained by vaccination. In addition, independent delays, represented as random time distribution functions, were introduced, which describe the duration of incubation, infectiousness, and immunity acquired through recovery from the disease or vaccination. Beside these refinements, the classical Ornstein–Uhlenbeck process is introduced as a model for the transmission coefficient describing the exposure of individuals from susceptible class. For this particular system, the existence and uniqueness of the positive global solution are proven. Furthermore, the conditions for extinction and persistence in mean of the disease are derived. The theoretical results are illustrated via numerical simulations.