Compartmental Systems

CompartmentalSystems is a Python package to deal with compartmental models of the form

\[\frac{d}{dt}\,x(t) = B(x(t),t)\,x(t) + u(t).\]

Since most computations are based on the state transition operator \(\Phi\) that solves

\[\frac{d}{dt}\,\Phi(t,s) = B(t)\,\Phi(t,s),\quad \Phi(s,s) = \bf{I},\]

nonlinear models need to be linearized in the first step. Then the package provides numerical computation of

  • age

    • compartmental age densities

    • system age densities

    • compartmental age mean and higher order moments

    • system age mean and higher order moments

    • compartmental age quantiles

    • system age quantiles

  • transit time

    • forward and backward transit time densities

    • backward transit time mean and higher order moments

    • forward and backward transit time quantiles

Main modules

(comprehensive list under Module Index)

smooth_reservoir_model

Module for symbolical treatment of smooth reservoir models.

smooth_model_run

Module for numerical treatment of piece-wise continuous reservoir models.

start_distributions

Module for computing age distributions or moments thereof to be used as start distributions in subsequent simulations.

Important Note

\(B(t)=(b_{ij}(t))\) is supposed to be a compartmental matrix for all times \(t\):

  • \(b_{ii}(t)\leq0\) for all \(i\)

  • \(b_{ij}(t)\geq0\) for \(i\neq j\)

  • \(\sum\limits_{i=1}^d b_{ij}(t)\leq 0\) for all \(j\)

Indices and tables