  We consider strong two-body losses in bosonic gases trapped in
one-dimensional optical lattices. We exploit the separation of time scales
typical of a system in the many-body quantum Zeno regime to establish a
connection with the theory of the time-dependent generalized Gibbs ensemble.
Our main result is a simple set of rate equations that capture the simultaneous
action of coherent evolution and two-body losses. This treatment gives an
accurate description of the dynamics of a gas prepared in a Mott insulating
state and shows that its long-time behaviour deviates significantly from
mean-field analyses. The possibility of observing our predictions in an
experiment with $^{174}$Yb in a metastable state is also discussed.
