Analysis and Simulation of a Discrete Epidemic Model with Vaccination Hysteresis

This paper presents and studies a discrete normalized epidemic SIRS (Susceptible-Infectious-Recovered- Susceptible) model with susceptible recruitment and eventual joint vaccination efforts. The normalization to unity of the three above subpopulations is performed, prior to the discretization process, related to the total population consisting of the sum of the above three subpopulations. The disease transmission rate is also normalized, taking into account the total population, so that the epidemic model is of "true-mass" type in the sense that the transmission rate decreases as the total population increases. The vaccination control is implemented on the normalized model under a hysteresis rule such that the vaccination effort is removed under weak disease incidence, and reinforced under its large incidence according to a linear feedback rule proportional to the infectious subpopulation. A constant vaccination value along a certain time interval, which equalizes its value acquired at its last switch of the hysteresis function, is applied on a time interval in the cases of disease incidence in the population which could be considered of medium intensity. Some numerical simulations are performed to check the control method efficiency.