  We study a diffusive SIS epidemic model with the mass-action transmission
mechanism and show, under appropriate assumptions on the parameters, the
existence of multiple endemic equilibria (EE). Our results answer some open
questions on previous studies related to disease extinction or persistence when
$\mathcal{R}_0<1$ and the multiplicity of EE solutions when $\mathcal{R}_0>1$.
Interestingly, even with such a simple nonlinearity induced by the mass-action,
we show that the diffusive epidemic model may have an S-shaped or backward
bifurcation curve of EE solutions.
