Uppsala universitet

RESEARCH ON SIMULATION OF POPULATION MODELS


Signals and Systems Group, Dept. of Engineering Sciences,
Uppsala University

Researchers:
Leif Gustafsson, UU and Karolinska Institutet and
Mikael Sternad Signals and Systems, UU.


The problem of modelling and simulating population models

Population models are models of dynamic systems with an integer number of discrete entities (individuals), such as plants, animals, patients, vehicles, molecules, atoms, data packages or entities of any kind. Such models are frequently used in systems biology, ecology, epidemiology, demography and queuing systems, and are also important in physics, chemistry, traffic planning, production and many other fields.

In this project, we are constructing a systematic methodology for designing simulation models for population systems, a methodology that enables the user to pick an appropriate technique for the problem and purpose at hand.

The crucial task in population modelling is to preserve the characteristics of interest of the system under study, in particular four fundamental properties:

  • The integer non-negative quality of the entities in the population.
  • The continuous nature of time, which should at least be sufficiently well approximated in the model.
  • The structural and temporal relations creating the dynamics of the system.
  • The irregularly occurring events of the system, which have to be preserved by an appropriate probabilistic representation in the model, because they cannot be described in detail.
We often wish to convert a population model into an (executable) simulation model that can be used for model experiments. There are mainly three ways of obtaining models of populaton systems, that are usable for simulation:
  1. In micro models (discrete event models, individual-based models, agent-based models), every entity, with its attributes and conditional behaviour, is represented. This is the formulation most close to the original problem. It may however be computationally infeasible when a very large number of entities need to be simulated.
  2. In a macro model, the entities are aggregated into different compartments, so that each compartment (state variable) holds a non-negative number of entities with prescribed properties. One question that arises is then how to design models that capture the probabilistic aspects in appropriate ways. A second question is how to design a macro simulation models so that their output closely resembles that of a micro model. Another question is when deterministic models might be used.
  3. Finally, in state-based models, such as markov models, every combinatorial possibility of the system is represented as a state. Such a formulation may provide important theoretical insights, but it would mostly lead to very impractical simulation models.


Poisson simulation: An efficient approach for stochastic macro simulation

Poisson simulation is a technique for constructing macro models that can be used efficiently to simulate population models. It is a stochastic extension of (deterministic) Continuous System Simulation.

Poisson simulation is based on a form of differential/difference equation that models change in the number of entities in a compartment by Poisson processes that are fuctions of the state variables and time. Such a mechanism is appropriate in a very wide variety of situations. The method was first proposed by L. Gustafsson in 2000 [1], [2] and is discussed further in the paper [3]. It is there shown to be closely related to micro modelling, but it enables a much more efficient execution by aggregating entities. It therefore represents an additional useful tool in the modellers toolbox. The method is sometimes called the tau-leap method.

In the report [R1], Poisson simulation models are shown to be far superior to state-based Markov models in many respects, such as computational complexity and conceptual complexity.

Many application examples of using Poisson modelling and simulation can be found in the book chapter [5] below. A Powersim implementation for simulation and parameter estimation is described in [R3] and a Matlab implementation is described in [R4].


Consistence between micro simulation, macro simulation and state-based (Markov) approaches

As outlined above, a population system can be modelled using a micro model focusing on the individual entities, a macro model where the entities are aggregated into compartments, or a state-based model where each possible discrete state in which the system can exist is represented.

However, the concepts, building blocks, procedural mechanisms and the time handling for theses approaches are very different. For the results and conclusions from studies based on micro, macro and state-based models to be consistent (contradiction-free), a number of modelling issues must be understood and appropriate modelling procedures applied.

The paper [4] presents a uniform approach to micro, macro and state-based population modelling so that these different types of models produce consistent results and conclusions. In particular, we demonstrate the procedures (distributional, attributal and combinational expansions) necessary to keep these three types of models consistent.

We also show that the different time handling methods usually used in micro, macro and state-based models can be regarded as different integration methods that can be applied to any of these modelling categories. The result is free choice in selecting the modelling approach and the time handling method most appropriate for the study without distorting the results and conclusions.


Publications:

[1] L. Gustafsson,
Poisson Simulation - A Method for Generating Stochastic Variations in Continuous System Simulation.
Simulation, vol. 74, no. 5 (2000), pp. 264-274.

[2] L. Gustafsson,
Poisson Simulation as an Extension of Continuous System Simulation for the Modeling of Queuing Systems.
Simulation, vol. 79, no. 9 (2003), pp. 528-541.

[3] L. Gustafsson and M. Sternad,
Bringing Consistency to Simulation of Population Models - Poisson Simulation as a Bridge between Micro and Macro Simulation.
Mathematical Biosciences, vol. 209 (2007), pp. 361-385.
doi: 10.1016/j.mbs.2007.02.004.
Paper (ScienceDirect); In Pdf (322K).

[4] L. Gustafsson and M. Sternad,
Consistent Micro, Macro and State-Based Population Modelling.
Mathematical Biosciences, vol. 225, no. 2, June 2010, pp 94-107.
doi: 10.1016/j.mbs.2010.02.003.
Abstract and Paper (ScienceDirect) ; In Pdf (481K).

[5] L. Gustafsson,
Studying Dynamic and Stochastic Systems using Poisson Simulation.
In: H. Liljenström and U. Svedin, (Eds.), Micro - Meso - Macro: Addressing Complex Systems Couplings. World Scientific Publishing Company, Singapore, 2005, pp. 131-170.
Book chapter online (Google Books)

Reports:

[R1] L. Gustafsson,
Poisson Simulation Outperforms Markov Simulation.
Technical Report R0902, Signals and Systems, Uppsala University,
vers. 2, May 2010.
In Pdf (341K).

[R2] L. Gustafsson,
Tools for Statistical Handling of Poisson Simulation: Documentation of STOCRES and PARMEST.
Technical Report Biometri 2004:01 Department of Biometry and Engineering, The Swedish University of Agricultural Sciences (SLU), 2004.
In Pdf (367K).

[R3] T. Hedqvist,
Wanda for MATLAB
Methods and Tool for Statistical Handling of Poisson Simulation.
Master of Science Thesis in Engineering, Signals and Systems, Uppsala University, September 2004.
In Pdf (568K).

Tutorials, Presentations:

[T1] L. Gustafsson
Poisson Simulation -
Realization of continuous time dynamic and stochastic processes.
Presentation slides, May 2000.
Slides in Pdf (274K).

[T2] L. Gustafsson and Mikael Sternad
A tutorial on the Poission simulation
approach to combined simulation.
September 2009.
paper in Pdf (230K).

[T3] L. Gustafsson
Stochastic Population Modelling and Simulation -
especially Poission Simulation.
Presentation slides, January 2009.
Slides in Pdf (940K).

[T4] L. Gustafsson
Stochastic Model Building and Simulation.
Laboratory Excercise at the Swedish University of Agricultural Sciences, Mar. 2006.
In Pdf (224K).