Process algebra for epidemiology: evaluating and enhancing the ability of PEPA to describe biological systems. Soufiene Benkirane University of Stirling Abstract Modelling is a powerful method for understanding complex systems, which works by simplifying them to their most essential components. The choice of the components is driven by the aspects studied. The tool chosen to perform this task will determine what can be modelled, the maximum number of components which can be represented, as well as the analyses which can be performed on the system. Performance Evaluation Process Algebra (PEPA) was initially developed to tackle computer systems issues. Nevertheless, it possesses some interesting properties which could be exploited for the study of epidemiological systems. PEPA's main advantage resides in its capacity to change scale: the assumptions and parameter values describe the behaviour of a single individual, while the resulting model provides information on the population behaviour. Additionally, stochasticity and continuous time have already proven to be useful features in epidemiology. While each of these features is already available in other tools, to find all three combined in a single tool is novel, and PEPA is proposed as a useful addition to the epidemiologist's toolbox. Moreover, an algorithm has been developed which allows converting a PEPA model into a system of Ordinary Differential Equations (ODEs). This provides access to countless additional software and theoretical analysis methods which enable the epidemiologist to gain further insight into the model. Finally, most existing tools require a deep understanding of the logic they are based on and the resulting model can be difficult to read and modify. PEPA's grammar, on the other hand, is easy to understand since it is based on few, yet powerful concepts. This makes it a very accessible formalism for any epidemiologist. The objective of this thesis is to determine precisely PEPA's ability to describe epidemiological systems, as well as extend the formalism when required. This involved modelling two systems: the bubonic plague in prairie dogs, and measles in England and Wales. These models were chosen as they exhibit a good range of typical features, allowing to thoroughly test PEPA. All features required in each of these models have been analysed in detail, and a solution has been provided for representing each of these features. While some of them could be expressed in a straightforward manner, PEPA did not provide the tools to express others. In those cases, we determined methods to approach the desired behaviour, and the limitations of said methods were carefully analysed. In the case of models with a structured population, PEPA was extended to simplify their expression and facilitate the writing process of the PEPA model. The work also required the development of an algorithm to derive ODEs adapted to the type of models encountered. Finally, the PEPAdum software was developed to assist the modeller in the generation and analysis of PEPA models, by simplifying the process of writing a PEPA model with compartments, performing the aver- age of stochastic simulations and deriving and explicitly providing the ODEs using the Stirling Amendment. 4 Acknowledgements I would like to thank all those who have contributed to the success of this PhD project. First of all, I would like to thank Dr Carron Shankland, for her unrivalled dedication as principal supervisor of this project. Her enthusiasm, knowledge as well as her guidance have been greatly appreciated, and the quality of this thesis was greatly improved thanks to her advice. Next, a big thank you goes to Dr Rachel Norman, who, as second supervisor, subjected each bit of my thesis to further critique and thereby enabled me to further improve my work. I would also like to thank my fellow PhD students for their helpful advice and support through- out the past three years. In particular, thank you Ros for rekindling the dying flame of our "PG- Tips" advice sessions for postgraduate students; thank you Claire for your organisational talent; and thanks Andy and Jesse for your critical responses to my presentations. Further thanks go to the members of the administrative team, Grace, Linda and Heather. Some people, while not being physically present, helped me as best they could. My parents and my sister first of all, who have helped me get to where I am today. Joris, who patiently encouraged 5 and supported me through good times and bad times; V´ero,Alex and Erwann who came to see me despite the distance, cold and Scottish rain. Last but not least, Gwen has given me her support each and every day of this thesis, no matter how hard it got, with unsurpassed patience. I doubt that I would have achieved this without her. 6 Contents 1 Introduction 13 1.1 PEPA . 18 1.1.1 Background . 18 1.1.2 Syntax . 21 1.1.3 Apparent Rate . 24 1.1.4 Additional definitions . 26 1.2 BioPEPA . 27 1.3 Presentation of the running examples . 29 1.3.1 The Bubonic Plague in a community of Black-Tailed Prairie Dogs . 30 1.3.2 Measles in England and Wales between 1944 and 1964 . 31 1.4 Structure of the thesis . 33 7 2 First epidemiological models 35 2.1 Indirect Transmission . 37 2.2 Direct Transmission . 41 2.3 Summary . 45 3 Deriving ODEs from a PEPA model 46 3.1 Existing methods . 47 3.1.1 Jane Hillston Method . 47 3.1.2 The Stirling Amendment . 53 3.1.3 Other existing methods . 60 3.2 The Hillston Method and the Stirling Amendment compared . 62 3.2.1 Condition (6): an illustrative example . 69 3.3 The Dining Philosophers Problem . 71 3.3.1 Presentation of the problem . 72 3.3.2 Optimisation of the service . 74 3.4 PEPAdum: automatically deriving the ODEs from a PEPA model . 77 3.4.1 PEPAdum description . 77 3.4.2 PEPAdum architecture . 81 3.4.3 Limitations . 82 3.5 Summary . 82 8 4 Modelling disease transmission 84 4.1 The mirror subgroup . 85 4.2 Frequency dependent transmission . 91 4.3 Density dependent transmission . 96 4.4 Summary . 99 5 Modelling births and deaths 101 5.1 Introducing births and deaths to a PEPA model . 102 5.1.1 Including births and deaths in a constant population . 102 5.1.2 Introducing Ghosts . 103 5.1.3 Vertical transmission . 104 5.2 Modelling births and deaths directly . 104 5.3 Modelling births and deaths using limited resources . 107 5.4 Conclusion on births and deaths techniques . 113 5.5 Modelling a prairie dog town during a bubonic plague outbreak . 113 5.5.1 Description of the model . 114 5.5.2 Results . 117 5.5.3 Conclusion . 122 9 6 Introducing Timed Events to PEPA 124 6.1 Intervention . 125 6.2 Cyclic behaviour . 127 6.3 Control policies . 129 6.3.1 Vector culling . 129 6.3.2 Vaccination . 132 6.3.3 Quarantine . 134 6.4 Measles in Leeds between 1944 to 1964 . 137 6.4.1 Presentation of the model . 137 6.4.2 The parameters . 140 6.4.3 Results . 143 6.4.4 Conclusion . 146 6.5 Summary . 147 10 7 Modelling structured populations 149 7.1 PEPA and structured populations . 150 7.2 Grammar . 151 7.2.1 General overview . 152 7.2.2 Definition of the compartments . 154 7.2.3 Definition of the component types . 155 7.2.4 Definition of the model component . 159 7.2.5 Automatically deriving a PEPA model from a PEPA model with compartments163 7.3 Modelling the interaction between prairie dog towns affected by the bubonic plague 163 7.3.1 The model . 164 7.3.2 Results . 170 7.4 Adding space to measles . 176 7.4.1 The model . 177 7.4.2 Parameters . 184 7.4.3 Results . 186 7.5 Summary . 188 11 8 Conclusion 190 8.1 Thesis summary . 190 8.2 Strengths and weaknesses of PEPA when modelling epidemiological systems . 191 8.3 Current developments in PEPA . 194 8.4 Future work . 195 8.5 General conclusion . 199 12 Chapter 1 Introduction According to the World Health Organization [95], in 2002, about 19:1% of worldwide deaths, and 52:7% of deaths in Africa were caused by infectious and parasitic diseases. In order to avoid as many of them as possible, it is important to carry out research on the topic to understand, prevent, cure and reduce the impact of a given disease. To achieve these goals, different areas of expertise must be involved, ranging from medicine to geography, sociology, biology and mathematics. Mathematical modelling is an important part of this process of understanding the disease and its behaviour. For some, modelling is not a scientifically acceptable technique of describing biological systems. To a certain extent, these critics are right. A model will never be able to describe accurately a system involving living organisms, whether they are animals, plants, bacteria or cells. Yet, as the statistician George Box said [12, p.424]: \essentially, all models are wrong, but some are useful". A model, by definition, is an incomplete representation of a system. The simplification required to construct.