  Most real networks are characterized by connectivity patterns that evolve in
time following complex, non-Markovian, dynamics. Here we investigate the impact
of this ubiquitous feature by studying the Susceptible-Infected-Recovered (SIR)
and Susceptible-Infected-Susceptible (SIS) epidemic models on activity driven
networks with and without memory (i.e., Markovian and non-Markovian). We show
that while memory inhibits the spreading process in SIR models, where the
epidemic threshold is moved to larger values, it plays the opposite effect in
the case of the SIS, where the threshold is lowered. The heterogeneity in tie
strengths, and the frequent repetition of connections that it entails, allows
in fact less virulent SIS-like diseases to survive in tightly connected local
clusters that serve as reservoir for the virus. We validate this picture by
evaluating the threshold of both processes in a real temporal network. Our
findings confirm the important role played by non-Markovian network dynamics on
dynamical processes
