Abstract The system matrix eigenvalue with the largest real part (leading eigenvalue) of any input-output (IO) connectable compartmental model is real and visible (appears explicitly) in the impulse response, and thus it governs the asymptotic response of the model. Its visible multiplicity is calculated here by decomposing the model into strongly connected components and applying the Perron-Frobenius theorem. Cascade models and fully visible eigenvalues are defined, and it is shown that for any cascade model the leading eigenvalue is fully visible in the impulse response. A necessary and sufficient condition is given for full visibility of the leading eigenvalue of any IO-connectable model. As a corollary, if an IO-connectable compartmental model has one or more traps, the leading eigenvalue λ 1 = 0 always has visible multiplicity one.