  This paper gives a new, simple proof of the known fact that for contact
processes on general lattices, in the subcritical regime the expected number of
infected sites decays exponentially fast as time tends to infinity. The proof
also yields an explicit bound on the survival probability below the critical
recovery rate, which shows that the critical exponent associated with this
function is bounded from below by its mean-field value. The main idea of the
proof is that if the expected number of infected sites decays slower than
exponentially, then this implies the existence of a harmonic function that can
be used to show that the process survives for any lower value of the recovery
rate.
