This presentation is divided into three parts where we analyze three different epidemiological models.

In the first part we analyze a periodically-forced system SIR model. We prove that the condition R0 < 1 is not enough to guarantee the elimination of the disease. Using the Theory of rank-one attractors, we prove persistent strange attractors for an open subset in the space of parameters where R0 < 1.

In the second part of the seminar we study a SIR model with constant vaccination and the bifurcations it unfolds. We prove explicitly that the endemic equilibrium is a codimension two singularity in the parameter space (R0, p), where R0 is the basic reproduction number and p is the proportion of Susceptible individuals successfully vaccinated at birth. We outline explicitly the bifurcation curves it undergoes. The analytical expressions of the bifurcation curves as a function of R0 and p estimate the proportion of vaccinated individuals necessary to guarantee the elimination of the disease. 

Finally, in the third and final part we explore a SIR model with pulse vaccination subject to seasonal variations in the disease transmission rate. We identify the conditions for the existence and global stability of a disease-free periodic solution. We show that for Rp > 1, our model exhibits a globally stable periodic solution. We present the bifurcation diagrams (T, p) as well as the associated bifurcations. Under seasonality and low vaccine coverage, the system reveals unpredictable dynamics (due to chaotic behavior caused by the suspension of topological horseshoes).

This seminar will be in portuguese and is a joint work with Alexandre Rodrigues (ISEG, CEMAPRE).