Impact of periodic vaccination in SEIRS seasonal model

We study three different strategies of vaccination in a SEIRS (Susceptible--Exposed--Infected--Recovered--Susceptible) seasonal forced model, which are: ($i$) continuous vaccination; ($ii$) periodic short time localized vaccination and ($iii$) periodic pulsed width campaign. Considering the first strategy, we obtain an expression for the basic reproduction number and infer a minimum vaccination rate necessary to ensure the stability of the disease-free equilibrium (DFE) solution. In the second strategy, the short duration pulses are added to a constant baseline vaccination rate. The pulse is applied according to the seasonal forcing phases. The best outcome is obtained by locating the intensive immunization at inflection of the transmissivity curve. There, a vaccination rate of $44.4\%$ of susceptible individuals is enough to ensure DFE. For the third vaccination proposal, additionally to the amplitude, the pulses have a prolonged time width. We obtain a non-linear relationship between vaccination rates and the duration of the campaign. Our simulations show that the baseline rates, as well as the pulse duration, can substantially improve the vaccination campaign effectiveness. These findings are in agreement with our analytical expression. We show a relationship between the vaccination parameters and the accumulated number of infected individuals, over the years and show the relevance of the immunisation campaign annual reaching for controlling the infection spreading. Regarding the dynamical behaviour of the model, our simulations shows that chaotic and periodic solutions, as well as bi-stable regions, depend on the vaccination parameters range.