Uppsala universitet


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

Leif Gustafsson and Mikael Sternad

In collaboration with
Pär Sparén, Karolinska Institutet and
Juni Palmgren, Karolinska Instititet.

With applications to optimization of cancer screening programs, supported by Cancerfonden 2016-2018.

Overview Slides (pdf)

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.

Within this ongoing research progam, 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. We are also applying the methods and insights in cancer epidemology and in other areas.

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.

In the paper [8], stochastic continuous system simulation (CSS), that handels also discrete entities, is described. It is a generalized framework, with Poisson simulation at its core.

An implementation, in the form of a cloud-based simulation environment and a tool for statistical analysis of multiple simulations [R5] can be found at

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.

Deterministic versus stochastic modelling of population systems

In the paper [6], Poisson simulation is used as a tool to investigate the question to what extent the results from deterministic models of population models will correspond to the averages of the experiment outcomes from the corresponding stochastic simulation models. The investigation covers many causes of bias and it indicate that deterministic simulation will almost always generate some form of bias in the results. Modellers should be very restrictive in their use of deterministic modeling of populations. Disciplines dealing with population models (ecology, epidemology etc.) should not base theories and studies only on deterministic models.

A guide to population modelling for simulation

The paper [7] summarizes our research results so far. It outlines the fundamentals of a consistent theory of numerical modelling of a population system under study. The focus is on the systematic work to construct an executable simulation model. There are six fundamental choices of model category and model constituents to make. These choices have a profound impact on how the model is structured, what can be studied, possible introduction of bias, lucidity and comprehensibility, size, expandability, performance of the model, required information about the system studied and its range of validity.

The first choice concerns a discrete versus a continuous description of the population system under study – a choice that leads to different model categories. The second choice is the model representation (based on agents, entities, compartments or situations) used to describe the properties and behaviours of the objects in the studied population. Third, incomplete information about structure, transitions, signals, initial conditions or parameter values in the system under study must be addressed by alternative structures and statistical means. Fourth, the purpose of the study must be explicitly formulated in terms of the quantities used in the model. Fifth, irrespective of the choice of representation, there are three possible types of time handling: Event Scheduling, Time Slicing or Micro Time Slicing. Sixth, start and termination criteria for the simulation must be stated. The termination can be at a fixed end time or determined by a logical condition.

Population models can thereby be classified within a unified framework, and population models of one type can be translated into another type in a consistent way. Understanding the pros and cons for different choices of model category, representation, time handling etc. will help the modeller to select the most appropriate type of model for a given purpose and population system under study. By understanding the rules for consistent population modelling, an appropriate model can be created in a systematic way and a number of pitfalls can be avoided.

In [8] we focus on compartment-based models, and show how discrete and continuous entities can be handled within them.


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

[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.
In Pdf.

[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.
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.
Abstract and Paper (ScienceDirect) ; In Pdf.

[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)

[6] L. Gustafsson and M. Sternad,
When can a Deterministic Model of a Population System Reveal What Will Happen on Average?
Mathematical Biosciences, vol. 243, issue 1, May 2013, pp. 28-45.
Abstract and Paper (ScienceDirect) ; In Pdf.

[7] L. Gustafsson and M. Sternad,
A Guide to Population Modelling for Simulation.
Open Journal of Modelling and Simulation, no. 4, pp. 55-92, 2016.
DOI: 10.4236/ojmsi.2016.42007
In Pdf

[8] L. Gustafsson, M. Sternad and E. Gustafsson,
The Full Potential of Continuous System Simulation Modelling.
Open Journal of Modelling and Simulation, vol. 5, no. 4, pp.253-299, October 2017.
DOI: 10.4236/ojmsi.2017.54019
In Pdf


[R1] L. Gustafsson,
Poisson Simulation Outperforms Markov Simulation.
Technical Report R1002, Signals and Systems, Uppsala University,
vers. 2, May 2010.
In Pdf (341K).

[R4] L. Gustafsson and M. Sternad
The Poisson Simulation Approach to Combined Simulation.
Technical Report R091, Signals and Systems, Uppsala University,
Sept. 2009.
In Pdf.

[R5] E. Gustafsson
System Dynamics Statistics (SDS). A statistical tool for stochastic system dynamics modeling and simulation.
Masther Thesis, Department of Information Technnology, Uppsala University, February 2017, Report IT 17 013.
In Pdf.

An implementation, in the form of a cloud-based simulation environment, and a tool for statistical analysis of multiple simulations can be found at

Tutorials, Presentations:

[T1] L. Gustafsson and Mikael Sternad
A tutorial on the Poission simulation
approach to combined simulation.

September 2009.
paper in Pdf (230K).

[T2] L. Gustafsson
Stochastic Population Modelling and Simulation -
especially Poission Simulation.

Presentation slides, January 2009.
Slides in Pdf (940K).

[T3] L. Gustafsson
Stochastic Model Building and Simulation.
Laboratory Excercise at the Swedish University of Agricultural Sciences, Mar. 2006.
In Pdf (224K).

[T4] M. Sternad and L. Gustafsson
Unified Consistent Modelling of Populations of Discrete Entities for Simulation.
January 2019
Slides in Pdf (571K).