A note on the direction of the transcritical bifurcation in epidemic models

In classical epidemic models, it is common to observe that a disease-free equilibrium looses its stability for R0 = 1 and a transcritical bifurcation takes place. We analyze this aspect from the point of view of the mathematical structure of models, in order to assess which parts of the structure might be responsible of the direction of the transcritical bifurcation. We formulate a general criterion, which gives sufficient (resp. necessary) conditions for the occurrence of forward (resp. backward) bifurcations. The criterion, obtained as consequence of a well known analysis of the centre manifold for general epidemic models, is applied to several epidemic models taken from the literature