Stable and unstable endemic solutions in the seasonally forced SIR epidemic model

In this paper we analyze the seasonally forced SIR model. Compared with autonomous model, time dependence adds many difficulties due to complexities of mathematical tools deal with. The absence monotonicity properties and structure equations are marked features that hinder use commonly applied in dynamical systems. Our contribution can be summarized two items. First, provide sufficient conditions for local stability endemic solutions. main tool problem is a sharp criterion non-autonomous linear systems by Ortega [15]. Second, describe new mechanism construct unstable Generally speaking, strategy introduce Dirac deltas coefficients results allow us derive subtle epidemic lessons. For example, presence seasonal birth, transmission recovery peaks at same normally leads patterns.