  Many physical and biological systems rely on the progression of material
through multiple independent stages. In viral replication, for example, virions
enter a cell to undergo a complex process comprising several disparate stages
before the eventual accumulation and release of replicated virions. While such
systems may have some control over the internal dynamics that make up this
progression, a challenge for many is to regulate behaviour under what are often
highly variable external environments acting as system inputs. In this work, we
study a simple analogue of this problem through a linear multicompartment model
subject to a stochastic input in the form of a mean-reverting
Ornstein-Uhlenbeck process, a type of Gaussian process. By expressing the
system as a multidimensional Gaussian process, we derive several closed-form
analytical results relating to the covariances and autocorrelations of the
system, quantifying the smoothing effect discrete compartments afford
multicompartment systems. Semi-analytical results demonstrate that feedback and
feedforward loops can enhance system robustness, and simulation results probe
the intractable problem of the first passage time distribution, which has
specific relevance to eventual cell lysis in the viral replication cycle.
Finally, we demonstrate that the smoothing seen in the process is a consequence
of the discreteness of the system, and does not manifest in system with
continuous transport. While we make progress through analysis of a simple
linear problem, many of our insights are applicable more generally, and our
work enables future analysis into multicompartment processes subject to
stochastic inputs.
