Effect of seasonality on the dynamics of an imitation--based vaccination model with public health intervention.

We extend here the game-theoretic investigation made by d'Onofrio et al (2012) on the interplay between private vaccination choices and actions of the public health system (PHS) to favor vaccine propensity in SIR-type diseases. We focus here on three important features. First, we consider a SEIR--type disease. Second, we focus on the role of seasonal fluctuations of the transmission rate. Third, by a simple population--biology approach we derive - with a didactic aim - the game theoretic equation ruling the dynamics of vaccine propensity, without employing 'economy--related' concepts such as the payoff. By means of analytical and analytical--approximate methods, we investigate the global stability of the of disease--free equilibria. We show that in the general case the stability critically depends on the `shape' of the periodically varying transmission rate. In other words, the knowledge of the average transmission rate (ATR) is not enough to make inferences on the stability of the elimination equilibria, due to the presence of the class of latent subjects. In particular, we obtain that the amplitude of the oscillations favors the possible elimination of the disease by the action of the PHS, through a threshold condition. Indeed, for a given average value of the transmission rate, in absence of oscillations as well as for moderate oscillations, there is no disease elimination. On the contrary, if the amplitude exceeds a threshold value, the elimination of the disease is induced. We heuristically explain this apparently paradoxical phenomenon as a beneficial effect of the phase when the transmission rate is under its average value: the reduction of transmission rate (for example during holidays) under its annual average over--compensates its increase during periods of intense contacts. We also investigate the conditions for the persistence of the disease. Numerical simulations support the theoretical predictions. Finally, we briefly investigate the qualitative behavior of the non--autonomous system for SIR--type disease, by showing that the stability of the elimination equilibria are, in such a case, determined by the ATR.