Mixed partition of unity methods and stochastic Gillespie algorithms for Transport-Reaction equations

Abstract

We introduce a new type of model framework which is part stochastic and part deterministic. The starting point is a finite size particle system within a single reaction volume, with type exchanges modelled by a contact process. Inside the reaction volume, each particle can interact with every other particle with the same probability. This is the setting of a classical reaction system simulated with a Gillespie algorithm. Such systems can be used to describe other than chemistry type exchanges, like an infection process, and therefore are already very versatile. Their advantage is that they are able to be used where small size effects can play a role, like extinction events, which are impossible to model with differential equations, including stochastic differential equations. However finite size and single reaction volume settings for reaction systems are too restrictive in other ways. We might like to add internal or external states to the particles. These states are coordinates in a position space. An example of an internal position/state space is age (since entering the system), an example for an external position/state space is geographical location. The particles then can also change their positions in these state spaces, according to some probability distribution which evolution is modelled deterministically. The classical example for a transport process is a partial differential equation like the heat equation, or more general parabolic advection-diffusion equations. We assume that the distribution of the particles in position space is not influencing the evolution of the probability distribution driving in turn the evolution of the particles’ positions. The model framework with its finite-size particle population approach can very accurately model situations where finite-size effects take place, however provides in addition detailed descriptions of both internal and external particle state spaces where needed. The framework can therefore be used in addition to traditional established models, like transport PDEs or internally structured population models, when the computation of the statistics of finite-size effects is important.

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