  We introduce a modified SIR model with memory for the dynamics of epidemic
spreading in a constant population of individuals. Each individual is in one of
the states susceptible (${\bf S}$), infected (${\bf I}$) or recovered (${\bf
R}$). In the state ${\bf R}$ an individual is assumed to stay immune within a
finite time interval. In the first part, we introduce a random life time or
duration of immunity which is drawn from a certain probability density
function. Once the time of immunity is elapsed an individual makes an
instantaneous transition to the susceptible state. By introducing a random
duration of immunity a memory effect is introduced into the process which
crucially determines the epidemic dynamics. In the second part, we investigate
the influence of the memory effect on the space-time dynamics of the epidemic
spreading by implementing this approach into computer simulations and employ a
multiple random walker's model. If a susceptible walker meets an infectious one
on the same site, then the susceptible one gets infected with a certain
probability. The computer experiments allow us to identify relevant parameters
for spread or extinction of an epidemic. In both parts, the finite duration of
immunity causes persistent oscillations in the number of infected individuals
with ongoing epidemic activity preventing the system from relaxation to a
steady state solution. Such oscillatory behavior is supported by real-life
observations and cannot be captured by standard SIR models.
