Conservative deterministic Markov models in mathematical biology: uniqueness of steady states, reversibility and computational methods

Abstract

Ordinary differential equations are commonly used throughout the sciences to build mechanistic models of time-dependent processes. Often, such models are Markov models describing the time-evolution of different interconnected "states". When these models have no "sources" or "sinks", they naturally conserve the total population of the system. When each state is reachable (directly or indirectly) from any other state, these models are called irreducible. For many applications, a deterministic system of ordinary differential equations (ODE) is the most suitable modelling approach. We summarise important mathematical results which show that this irreducibility property guarantees the existence and uniqueness of global stable equilibria, and discuss the computationally-efficient implementation of such ODE-based models. We also discuss the condition of microscopic reversibility and show how it guarantees non-oscillatory behaviour, which enables additional efficiencies in computations. These properties and methods are demonstrated through example models of biological phenomena, where we demonstrate their importance for efficient model fitting and simulation.