Epidemic delay models of infection progression

This paper is devoted to the models of epidemic progression based on delay differential equations, with time delays corresponding to the latency period, infection, and immunity loss.We prove existence and positiveness of solutions, and determine the basic reproduction number.Reduction of delay differential equations to integral equations allows the analysis of stationary solutions and their stability.In the case of immunity waning, periodic epidemic outbreaks emerge as a result of the instability of a positive stationary solution if the basic reproduction number exceeds some critical value.Vaccination can change epidemic dynamics, resulting in more complex aperiodic oscillations confirmed by some data on Influenza A in Norway.Optimal control allows for the minimization of epidemic cost by vaccination.These models are used to describe COVID-19 and seasonal influenza epidemic progressions.