  We propose a tractable epidemic model that includes containment measures. In
the absence of containment measures, the epidemics spread exponentially fast
whenever the infectivity rate is positive, $\lambda>0$. The containment
measures are modeled by considering a time-dependent modulation of the bare
infectivity $\lambda$ leading to effective infectivity that decays in time for
each infected individual, mimicking for instance the combined effect of the
asymptomatic onset of the disease, testing policies and quarantine. We consider
a wide range of temporal kernels for effective infectivity and we investigate
the effect of the considered containment measures. We find that not all kernels
are able to push the epidemic dynamics below the epidemic threshold, with some
containment measures only able to reduce the rate of the exponential growth of
newly infected individuals. We also propose a pandemic model caused by a
growing number of separated foci.
