  Exact analytical solutions to the dissipative time-dependent Schr\"odinger
equation are obtained for a decaying two-state system with decay rates
$\Gamma_{1}$ and $\Gamma_{2}$ for levels with extremal spin projections. The
system is coherently driven with a pulse whose detuning is made up of two
parts: a time-dependent part (chirp) of hyperbolic-tangent shape and a static
part with real and imaginary terms. This gives us a wide range of possibilities
to arbitrarily select the interaction terms. We considered two versions which
led to decaying Demkov-Kunike (DK) models; the version in which the Rabi
frequency (interaction) is a time-dependent hyperbolic-secant function (called
decaying DK1 model) and the case when it is constant in time and never turns
off (decaying DK2 model). Our analytical solutions account for all possible
initial moments instead of only $t_{0}=0$ or $t_{0}=-\infty$ as for
non-decaying models and may be useful for experiments on level crossings. Two
complementary limits of the pulse detuning are considered and explored: the
limit of fast (i) and slow rise (ii). In the case (i), the coupling between
level positions in the first DK model collapses while the second DK model
reduces to a Rabi model (constant Hamiltonian), in the case (ii), both DK
models reduce to the LZ model. In both cases (i) and (ii), analytical
approximated solutions which conveniently approach the exact solutions are
derived.
