  Memory has a great impact on the evolution of every process related to human
societies. Among them, the evolution of an epidemic is directly related to the
individuals' experiences. Indeed, any real epidemic process is clearly
sustained by a non-Markovian dynamics: memory effects play an essential role in
the spreading of diseases. Including memory effects in the
susceptible-infected-recovered (SIR) epidemic model seems very appropriate for
such an investigation. Thus, the memory prone SIR model dynamics is
investigated using fractional derivatives. The decay of long-range memory,
taken as a power-law function, is directly controlled by the order of the
fractional derivatives in the corresponding nonlinear fractional differential
evolution equations. Here we assume "fully mixed" approximation and show that
the epidemic threshold is shifted to higher values than those for the
memoryless system, depending on this memory "length" decay exponent. We also
consider the SIR model on structured networks and study the effect of topology
on threshold points in a non- Markovian dynamics. Furthermore, the lack of
access to the precise information about the initial conditions or the past
events plays a very relevant role in the correct estimation or prediction of
the epidemic evolution. Such a "constraint" is analyzed and discussed.
