Investigating the spread of disease within a network of cities. Adrian Merville-Tugg. Bsc (Hons) in Computer Science. University of Bath

Transcription

1 Investigating the spread of disease within a network of cities Adrian Merville-Tugg Bsc (Hons) in Computer Science University of Bath 2006 I

2 Investigating the spread of disease within a network of cities Submitted by Adrian Merville-Tugg COPYRIGHT Attention is drawn to the fact that copyright of this thesis rests with its author. The intellectual Property Rights of the products produced as part of the project belong to the University of Bath (see This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with its author and that no quotation from the thesis and no information derived from it may be published without the prior written consent of the author. DECLARATION This dissertation is submitted to the University of Bath in accordance with the requirements of the degree of Batchelor of Science in the Department of Computer Science. No portion of the work in this dissertation has been submitted in support of an application for any other degree or qualification of this or any other university or institution of learning. Except where specifically acknowledged, it is the work of the author. signed... This thesis may be made available for consultation within the University Library and may be photocopied or lent to other libraries for the purposes of consultation. signed... II

3 Acknowledgements My Mother for her understanding and support throughout my studies. Dr Dan Richardson for his help and supervision. III

4 Abstract This project investigates the factors that affect the spread of disease within a network of cities by modelling a collection of settlements, their populations and the transport networks between them. The aim is to produce an interactive simulation with the intention of allowing the user to change the factors which affect disease spread. IV

5 1 Introduction 1 2 Literature Survey Social Networks Eliciting a personal network Network construction techniques Epidemiology Transport Network Construction Trip Generation Trip Distribution Mode Split Traffic Assignment Previous Work The scope of this Project Census, Transport and Land use Data 16 3 Requirements Analysis and Specification Scope of this Project Purpose of this Project Source of Requirements Requirements Specification Definitions Functional Requirements Non Functional Requirements Summary 22 Chapter Design and Implementation NetLogo Software Modelling Mapping the cities Limitations of the choices made Agents Limitations of the agents in the model 32 V

6 4.2.3 Trip Purpose & Trip Generation Trip Distribution Implementation of the Gravity Model Limitations of the Gravity Model Mode Split Limitations Disease Limitations Summary 43 Chapter Tests and Experiments Setting up and Calibration Populations Ratio of Vehicles Likelihood of a Journey Examples of the Simulation 48 Example 1 Survival 49 Example 2 - Death 50 Example 3 Immunity 51 Example 4 Herd Immunity Investigating how infectivity affects the spread of disease 55 Running the Simulation Results T-Test Conclusion Investigating how chance of immunity affects the spread of disease 61 Running the Simulation Results T Test Conclusion 67 Chapter Conclusions 68 Bibliography 70 Appendix 1: Statistics 72 Appendix 2: Source Code 78 VI

7 Appendix 3: Calculation of Gravities 92 VII

8 Chapter 1 1 Introduction Understanding the spread of disease within a human population is important as it allows us to better plan strategies such as vaccination policies or quarantine to limit the damage done to a population by a disease. The dense social-contact networks characteristic of urban areas form a perfect fabric for fast, uncontrolled disease propagation [4. Human social interaction is very complex and is one of the main factors by which disease is spread. When people live in close proximity to each other, the risk of disease spread - both infectious and non-infectious - increases [1. The infectivity and type of transmission of a disease will determine how quickly a disease travels through a population and the type of people that it affects. Transportation networks also allows certain diseases to be spread far away from where they originated and thus increase the probability that the disease will gain hold of another subsection of the population in a different area. Transportation also allows other diseases to be introduced into a population from elsewhere and due to the close proximity experienced on most public transport, provides another place for infective diseases to be spread. Transport networks are very important in understanding the spread of disease as they constrain social networks. 1

9 Chapter 2 2 Literature Survey 2.1 Social Networks Social network theory explores the interaction between people in a population by modelling the relationships between them. In an example population such as one in a British city, relationships most commonly take the form of family, friends, coworkers, partner and other similar relations. Each relation (e.g. family, friend ) will have their own relationships with others in the population, some of which will overlap with the first person s relations. In this way a complex map forms very quickly connecting many people from across the whole population. An example of a social network, (James Moody) 2

10 The Small world phenomenon is a hypothesis that states that everyone in the world can be linked through a small number of relations. This number is generally stated between 5 and 8 and was investigated by Stanley Milgram who conducted an experiment and found that an average citizen living in the United States passing a parcel through their acquaintances (not using the postal system) to another citizen living in the United States usually succeeded with an average of 6 intermediaries. It is therefore not hard to imagine that with the extra people that the average person comes into contact with each day, who is not already a relation, the spread of an infectious disease can be fairly rapid and widespread across a population. Contact between two people from a population can have varying intensities, for example having a conversation or kissing. Different diseases will therefore be limited in their transmission from person to person depending on the type of contact made. Meredith Rolfe describes two key properties of personal networks, the average size and the density, both contained in each of three types of commonly described social networks. These are acquaintance networks, regular contact or support and core personal [10. Acquaintance networks are described as being very large containing ,000 people and include people who would be recognised and called by first name. Acquaintance networks are likely to have small densities (the chance of acquaintances knowing each other are relatively low so that there will be little cross-over). Regular contact or support networks are much smaller with an average of people and include co-workers and other people with whom regular social contact is made. Core personal networks are small networks which consist of the people who are considered to be the closest and usually contain 0-10 members. When constructing a social model for my simulation the questions which I will have to address are: How do I elicit the size of a person s networks? and, How are links in the network constructed? 3

11 2.1.1 Eliciting a personal network Estimating the size of a personal network is challenging and most attempts have involved surveys or questions of some kind. Questions which generate names are the most usual and follow along the lines of Who are your best friends?, Who do you go out with on a Saturday night?, Who would you ask to watch your house if you were away [10. Although I believe that this is a good place to start, most people are forgetful as soon as they are asked a question and the accuracy of data from this kind of survey would not be guaranteed. Other studies have tried approaches which involve asking people to estimate how many people that they know in a certain subpopulation (e.g. diabetics) [11. From this an estimate of the maximum network size can be calculated. Hill and Dunbar estimated the social network size in western society based on the exchange of Christmas cards [11. People were asked to fill in a questionnaire about the Christmas cards which they sent, where each card was given an attribute which included: distance, relationship, social status (individual, couple, family ), time of last contact, emotional closeness (classed from 0-10). They estimated the average network size to be (+/- 68.0). Total network sizes estimated from Christmas card lists are remarkably close to the value of 150 predicted for human social group size based on the relationship between group size and brain size across primates. [11 Other common methods of eliciting network size include forming call graphs (communication based network) which would be hard to obtain data for outside of telecommunication companies and networks which are similar to call graphs but are subject to privacy laws. Collaboration networks are another option and are graphs formed from nodes created whenever a member collaborates with another. Examples of collaboration could be actors in a film together or academics writing a paper together. [ Network construction techniques Constructing links between personal networks is also challenging and there are several approaches which are used. The makers of BioWar state that people are more likely to communicate with people that are similar to them and that one would more 4

12 likely to interact with family members, friends and acquaintances than to complete strangers.[1 A random approach to forming links between members of population would be a good start however this is a little too simple and can lead to some problems as it is not accurate. Random links would be expected to form an evenly distributed set of links across the domain of members, however in real life this is not the case. As stated above people are more like to interact with family members, friends and acquaintances. These set of relations would also have more chance of knowing each other, in other words the personal networks would have a low density which would not be the case in real life where there would be a greater overlap. As describe by Dliben-Nowell, the clustering coefficient of social networks is much higher than is predicted in a random graph where the clustering coefficient measures the probability that two people who have a common friend will themselves be friends. Typical social networks have clustering coefficients on the order of 10-1, which is orders of magnitude greater than the clustering coefficient that a random graph. [10, [12. A better method is to use a two-dimensional grid. This assumes overlap between the contacts of A and B if they are friends. (See Figure 1: from Social Networks and Simulations, Meredith Rolfe). Although this is a simple and common method of constructing networks, it has a couple of inconsistencies with actual social networks. [10, [12. Firstly it assumes a normal mixing of the population which is not realistic. In real life mixing is not as uniform and will depend on an individual s characteristics. Grid structures also do not allow for centrality within the population. This type of mixing also assumes that friends are shared uniformly which is also not realistic. [10 5

13 The small world method sets out to create networks that are likely to have the 6 degrees of separation property. In particular links are formed to random members of the population and not just friends. This helps to create larger links which can span large gaps in the population. It does not however solve the problem of overlapping friends, personal characteristics of a population s members or the uniform nature of the population s contacts. [10 Biased or structured random graphs allow members of the population to have differing numbers of friends. It also allows members to be more likely to have connections with other members with similar characteristics. Above are some of the issues which are present when modelling a social network. Although it must be noted that social networks rapidly evolve over time, with new friendships forming through random encounters, introductions by mutual friends, and meetings through common interests [12, a model for use in a simulation will serve as a snapshot of the network at a particular point and will be sufficient for use in my simulation. 6

14 2.2 Epidemiology Epidemiology studies the causes, transmission, incidence and prevalence of health and disease in human populations [2. Studying epidemiology has importance in deciding strategies on how to deal with disease outbreaks before they become epidemics. For many infectious diseases, transmission occurs mainly between people who are collocated (simultaneously in the same location), and spread is due mainly to people's movement [4. When trying to understand the spread of a disease, factors such as: Infectivity which can be thought of as the probability that a disease will be passed on from an infected person to a non-infected person during contact between them. A higher infectivity the more likely it is that a person will be infected and so the quicker the disease will spread. The time taken for symptoms to occur can limit the effectiveness of a quarantine strategy or treatment of infected individuals. The longer the symptoms take to occur, the more contacts the infected person would probably have made and the higher the probability that the disease had been spread. For diseases with a short time between symptoms and occurring and death, the less opportunity there would be for treatment to occur. It is generally agreed that early detection is an important factor in stopping the spread of a disease. [4 Potential resistance of the disease. Resistance is describe by Jonathan Swinton in his dictionary of epidemiology as 'The reduction, due to genetic selection, of susceptibility of a parasite or its vector to chemotherapy' [7. As many parasites and bacteria have very small generation times, the are able to mutate rapidly and as such can form resistance to certain treatments. A notorious example of this is the influenza virus which mutates extremely quickly and new vaccines must be made and administered every year. If a disease is able to mutate quickly it could render the current vaccines obsolete 7

15 and the time taken to create a new vaccine could be costly in terms of the spread of the disease. Alternatives to mass vaccination involve isolating and/or vaccinating small subsets of individuals to ensure that the disease will spread only locally [4. For example, in 2003 the severe acute respiratory syndrome (SARS) spread rapidly causing at least 800 deaths although it was eventually contained successfully. Whether the methods used to control SARS are likely to be equally effective for future outbreaks of other emerging infectious diseases requires a more detailed understanding of the factors that make containment feasible even when effective vaccines or treatment are not available. [5 Potential immunity of the host. Immunity can occur in three ways: natural immunity (being born immune), acquired immunity after coming into contact with a pathogen and immunity after vaccine. Immunity in a population can limit a disease as it can no longer spread well through the population's network. Herd immunity is a term to describe the situation where only a percentage of the population need be vaccinated or otherwise immune to stop the spread of a virus and is what selective vaccination aims to achieve. Selective vaccination of contact persons of smallpox has contributed to its eradication on a global scale [8. An example of natural immunity can be found in areas badly affected by malaria. In these areas a mutation in peoples genes has caused a disease call sickle cell. Sickle cell affects the red blood cells, changing their shape and their ability to carry oxygen and so sickle cell usually kills its victims while they are still children. As sickle cell is a recessive gene it can occur in a less acute form called sickle cell trait. People suffering from sickle cell trait will have the symptoms of anemia but carrying this gene also gives them resistance to malaria. This has caused the population in these areas to have high occurrences of sickle cell trait as people without it usually die from malaria. [23 If the time taken to die is short it can limit the ability for a person to pass on the disease as they are less likely to come into contact with people as if the time taken for them to die was longer. If the time taken to die is short then it 8

16 can also limit the ability to be able to treat an infected person, however this is also in balance with the time taken for symptoms to occur, as it is likely that individuals will only receive treatment after symptoms have occurred, assuming that a vaccination program is not already in place. This information is important when trying to plan strategies to contain a disease. These factors are also variable from disease to disease and so each should be changed when modeling a specific disease. 2.3 Transport Network Construction Modelling transport networks involves finding factors which affect the rate and destinations of travel. These factors are affected by social behaviour, work trends, land use, distance between two locations, the cost to travel between two locations, catchment areas and services (e.g. shopping centres), physical constraints (e.g. roads, sea, mountains), time of day and many others. Once these factors have been found and simplified to a relevant degree, they can be combined into a model or simulation which will represent a transport network. This transport network will show the rates of travel between nodes or places in the model. The main issues in deriving such a model are being able to simplify and combine these factors in a way which will give an accurate representation of real life. The import stages of transport network modelling [15,13: 1. Trip Purpose & Trip Generation 2. Trip Distribution 3. Mode Split 4. Traffic Assignment 9

17 2.3.1 Trip Generation This stage of the modelling deals with calculating the number of trips that a person makes in a given amount of time. This trip generation number is closely related to the trip purpose and so land use data is sometimes used to determine the type of economic activity that occurs in an area. The economic data can be combined with demographic information of the area to estimate the average number of trips that people will make. Census data is often used during this stage to give accurate population figures as this will greatly affect the number of trips that will be made in a certain area. Trip generation is most commonly thought of as how many trips will originate at each place? but can also be thought of as a trip frequency choice problem how many shopping (or other purpose) trips will be made by a certain person in a given amount of time? [13 Trip purpose includes reasons such as shopping trips, trips to work, trips to school, recreational trips and so on. It is easy to imagine that the type of trip purpose can affect its frequency, for example shopping trips may only occur, say, two to three times a week, whereas trips to work and school may in many cases occur 5 times a week. In some cases trip purpose will affect the mode of transport and so the length of trip and destination. For example in areas with high levels of industry there may be a higher proportion of lorries accounting for the traffic arriving and leaving the area. However in a suburban area with a high level of commuters there may be a higher proportion of trips made by train or other public transport. The purpose of a trip may also affect the type and number of contacts which a person experiences during the trip; this is important in terms of disease spread. For example social trips may involve a larger number of contacts at a higher level of intimacy than a shopping trip, therefore one may be at greater risk of contracting a disease on a social trip. Personal factors can also affect trip productions [13. These include: income, car 10

18 ownership, household structure, family size, value of land, residential density and accessibility. Due to time constraints this project will use national transport statistics data which gives the average number of trips per person per mode of transport per year to estimate the trip generation for the model. It must be taken into account that if the model was to be made more accurate and reliable then the above factors would have to be taken into account Trip Distribution Once the trip generation has been estimated one must decide how these trips are to be distributed around available destinations. Trip distribution can be affected by many factors such as distance to destination, services provided by a destination, shopping centres, companies, transport links, physical factors, tourism etc. Trying to balance these factors is important during this process. There are a few ways of modelling this however the simplest and most common of these is the gravity model. The gravity model is criticised by some for not having much grounding in the economic theory. It must also be noted that most gravity models do not take into account social factors in much detail, which may be important in further work on this project if the simulation was to be made more accurate. However the gravity model seems to work well in simulating the actions of large groups where choices must be made. As this project is more focused on the general movement of people from city to city the gravity model is the most appropriate. The form of the gravity model is as follows [13: T ij = α P i P j d ij 2 Where: T ij trip distribution between i and j 11

19 α balancing factor Pi Population of i Pj Population of j d ij distance or cost to travel between i and j More complicated alternatives to the simpler gravity model include the doubly constrained model and the entropy-maximising approach. [ Mode Split For the purposes of this project, mode split is important as it is possibly related to the trip purpose and therefore the type of interaction that the person will have during that trip. Different modes of transport will also allow infection spread during transit if they are public. Factors which may affect the mode include car availability, possession of a driving licence, household structure, income, residential density [13. The National Transport Statistics department has published statistics which give some transport use by age and gender, this is useful as it gives an idea of the cross section of people using certain transport and so an idea of trip purpose. This project will attempt to differentiate between the infection rates which a person will be subject to when using a certain mode of transport Traffic Assignment This is the process of deciding which route a trip will take after the trip generation, distribution and mode split has been calculated. This is an important stage in more detailed models as it allows traffic flows to be modelled along certain routes and is useful for traffic planning departments when deciding where to add new traffic lights, roads etc. This is however unnecessary in the simulation aimed for in this project and due to the complexity this will not be added. 12

20 2.4 Previous Work Two major simulations using disease spread in social networks have been EpiSims and BioWar. EpiSims models the spread of a disease in an urban area modelling the whole social and transport network in the city on a second-by-second basis. EpiSims was based on TranSims, the transport simulator. EpiSims main advantage is the extremely high detail it models. This high detail lets procedures such as closing schools and tactics for quarantine be investigated and their effect on disease spread analysed. BioWar is a simulation tool used for modelling the effects of biological and chemical attacks. Both EpiSims and BioWar use heterogeneous population mixing, in other words they use social network characteristics and individual agent decisions to dictate the mixing in the population. Both simulations also use real data from census, and land-use data to model their cities. The makers of BioWar say that although EpiSims uses heterogeneous population mixing, it lets normal non-attack day transportation patterns drive social networks while the reverse is true in reality: agent behaviours drive social networks and social networks drive, even while being constrained by them, transportation patterns. They say that transportation networks constrain but do not define social networks, which EpiSims seems to have done (having been based on a transport simulator). They also criticise EpiSims for modelling disease spread by disease load (the amount of disease per unit population), which may lead to incorrect results [6. BioWar has many of the features of EpiSims; it also implements some of the other features that may have an effect on a biological or chemical attack such as geographic and weather factors. Transport routes were modelled in great detail by the Los Alamos National Laboratory in their TranSims project. TranSims produces estimates of social networks based on the assumption that the transportation infrastructure constrains people's 13

21 choices about where and when to perform activities [3. TranSims then estimates the positions of all the travellers on a second-by-second basis. This provides a seemingly accurate representation of the transport system in a city which can be used to help model a social network through which diseases can be propagated. TranSims like EpiSims is modelled in extremely high detail. The British Department for Transport has provided a Transport Analysis Guidance Website (WebTag). It is designed for any project which requires government approval, but it contains much useful information on modelling transport and will be valuable to this project. It provides especially useful information on trip distribution which this project will need to draw upon while constructing its gravity model. [16 The current threat of a disease outbreak seems to be very intense indeed. With the recent scare of SARS and the current threat that Avian Flu is spreading and might soon mutate and be able to infect humans it is no surprise that many governments and large organisations have invested large amounts of money and time in investigating the spread of diseases, not only in solitary cities as EpiSims and BioWar have done, but also across networks containing many global cities. With people being able to move across the world with such ease, an infectious disease outbreak in a certain city is surely a threat to all other cities in the world. This was shown in the SARS outbreak where although the first cases were suspected to be in China there were cases found in many other cities in the world including Torronto in Canada, a long distance away. Air travel is believed to play a large part in the spread of disease on a global scale and being able to better understand peoples movement through these transport networks was thought to be important in being able to deal with deadly diseases such as SARS. Luis Amaral and colleagues at Northwestern University in Illinois constructed a network of 27,051 links using transport data on flights. They found trends which could help health experts to better control the spread of such diseases, carried large distances by air travellers. Amaral and his colleagues found that the network they constructed was scale-free, in other words some nodes were more important than others. They also found that although some airports had similar levels of traffic, some were more importantly 14

22 linked in the network and that this reflected political and economic relationships between countries. It was these important hubs that one would most like to impose barriers to limit disease spread. [17. This concept is similar to the one this project aims to investigate, although it will be done on a smaller scale with road and rail being the vectors for disease spread instead of flights. The World Health Organisation (WHO) has done much work in analysing the effects of disease spread, especially after the SARS outbreak. One branch of WHO, The Global Outbreak Alert and Response Network (GOARN) were the first to identify the SARS causative agent and developed early response diagnostic tests. It consists of 11 laboratories which monitor the outbreak of global diseases and track their progress through cities. As disease spread is such a threat to the world s populations many countries have pooled resources including United Nations organisations such as UNICEF and the UN refugee agency, The Red Cross and many other organisations that have the power to contribute to international outbreak alert and response [ The scope of this Project The amount of money and other resources that have gone into investigating the global impact of disease spread; in solitary cities and even on a global scale is immense. It is hard to see how a single project, limited in time can even scratch the surface of the many governments and organisations seemingly trying to do the same thing but to a much greater depth. The subject is however very interesting and there are no concrete answers to the questions posed. There are many factors that may be changed and many more routes to explore. Most past simulations have concentrated on modelling the transport and social networks either within a specific city or in a network of global cities and have done this in great detail. This project will aim to model an area of land which will include several large cities and also include towns and villages of varying size. The model will aim to incorporate real transport, census and land use data into the simulation to construct the network. The aim will be to see, after this network is constructed, how 15

23 disease is able to spread throughout the network between the different sized settlements using the transport networks as the main method of disease transmission. 2.6 Census, Transport and Land use Data Both EpiSims and BioWar have used census and land use data. This is important as without accurate real life data to model with the results obtained from a simulation would be arbitrary. Luis Amaral and his team also used real data for air traffic and gained some interesting results. The British Department for Transport publishes many statistics for its road use which will be helpful in this project. Road statistics include average daily flow of vehicles, average trips per year per person including by main mode and purpose, average distance travelled by region of residence and many more listings. There is also data available for rail travel. From this data I hope to construct a realistic transport network model of a part of the United Kingdom. By the type of transport used I can ascertain as to the type of business certain people have in a certain area they are visiting and the types of contact that they will be likely to make (family, business...). The average distance travelled by region of residence will be useful in determining the spread of a person's social network depending on where they live. The government s national statistics department provides much census and land use data. There is also data on social aspects depending on area which will help me to form a model of the type of person living in a certain area and the type of contacts made from this I will hopefully be able to estimate the personal network size and density of the agents in my simulation. The national statistics department also provides information on population density which I will use to estimate the potential number of contacts made by an infected individual (also coupled with social data). [9. 16

24 Chapter 3 3 Requirements Analysis and Specification 3.1 Scope of this Project This project aims to combine aspects from social science, epidemiology and transport network modelling to create a simulation which shows and is able to predict the spread of a disease through a network of cities. Very detailed simulations of transport networks and disease spread have been done in the past, but they have mainly focused on the movements of people within a city and have not concentrated on the trends which occur in a larger system. Although this project will not go in to nearly as much detail as some of the previous work in the field, it will investigate the trends that occur when a collection of settlements of varying sizes are considered. The literature review explored social networks in great detail. Social networks are important to understand and it is an important concept to keep in mind when constructing an epidemic model. This project however is limited in time and resources and it was decided to greatly limit the degree to which social networks are included. 3.2 Purpose of this Project Once a suitable model of the cities and transport networks has been developed, it can be used to explore the different factors which affect the spread of a disease. 17

25 The factors which may affect the spread of a disease will be able to be manipulated by the user so that the results of running the simulation can be analysed easily. The purpose of this project is to provide a suitable base model for a group of cities, such that the spread of disease within this model can be observed and analysed. 3.3 Source of Requirements This project is investigative and thus a large proportion of the initial stage is modelling the situation well enough to have a useful simulation. The requirements in this section were derived from the literature review and more were added as the project progressed and the model became more concrete. The most important requirement for this project is to create a model of the situation sufficiently enough to be able to investigate the effect of changing disease spread parameters. 3.4 Requirements Specification The model will simulate people travelling between towns and cities for various purposes and spreading disease to people that come in close contact with individuals who are infected. There is no previous source code for me to build upon and so the simulation will have to be coded from scratch. The simulation consists of very separate subsections and so will be built incrementally starting with first the mapping of cities then adding inhabitants and movement and lastly adding disease. This section will describe the specific requirements that the model will meet. 18

26 These requirements will help insure that the model does what it is suppose to do and hopefully will achieve its aim of being able to investigate the spread of disease in the city network Definitions This sub-section contains a list of definitions which are used in the requirements document. Agent An agent is the representation of a human in the simulation. An agent can be infected, infect other agents, gain immunity, travel and die. Disease A disease is an attribute that an agent can have. An agent with a disease can infect other agents so that they have the disease, recover from the disease, become immune to the disease or die. Immunity An agent who gains immunity from a disease cannot be killed by the disease. It is left up to the user whether or not an immune agent may spread the disease. Infectivity Infectivity is that chance that an infected agent will transmit a disease to an uninfected agent. The higher the infectivity is the greater the chance of transmission. Map The map is the output of the graphical user interface which shows the layout of the settlements and the movement of the agents. Mode of transport The mode of transport is an attribute of an agent who is travelling. Different modes of transport will have different infectivity. 19

27 Mutation A disease which mutates can bypass any previous resistance that an agent had to it. This can be seen as a new disease. Recover An agent who recovers from a disease will no longer have the disease and may not infect others. It is left up to the user whether or not an agent who has recovered gains immunity. Settlement A settlement is the representation of a City or town in the simulation. A settlement may have a population of agents. User The user is the person who will run the simulation Functional Requirements These are the requirements that the simulation will have to fulfil to meet its objectives. User Interface The user must be able to view a map of the settlements The user must be able to save and open simulation files The user must be able to view agents moving around the map The user must be able to start and pause the simulation at any time It must be possible to view graphs of important data The user should be able to adjust attributes of the simulation from the interface without editing code 20

28 Cities It must be possible to add settlements to the map There must be a data structure which holds information on a settlements including distance from other settlements. Agents It must be possible to add agents to settlements Agents will have a citizenship to a specific settlements It must be possible for agents to visit other settlements which are not their home Agents should return home after their visits Agents will move around their home settlements when not travelling It must be possible for the user to change the probability that an agent will make a journey Agents who have a disease may infect agents who are not already infected Transport It must be possible for agents to travel to other settlements by different modes of transport The chance for the spread of disease (infectivity) will be affected by the mode of transport taken Disease Agents will be able to contract diseases from infected agents It must be possible for agents to become immune to disease It must be possible for a disease to mutate 21

29 It must be possible for an agent to recover from a disease It must be possible for an agent to die from a disease It should be possible for the user to decide whether or not immune agents can spread disease It should be possible for the user to decide whether or not recovering from a disease grants further immunity from it It should be possible for the user to change the chance for mutation to occur It should be possible for the user to change the infectivity of a disease It should be possible for the user to change the chance for an infected agent to recover from the disease Analysis of Results It must be possible for the user to download results from running a simulation into a spreadsheet for further analysis It must be possible for the user to save the displayed map as an image file for future reference Non Functional Requirements These are optional requirements which would make the simulation more usable. It must be possible to transfer the simulation onto a more powerful machine so that it is possible to run simulations with more agents 3.5 Summary This section has described the requirements that the system will have to fulfil. It has also included information on the user interface and saving data for results analysis. The simulation described by the requirements above should be sufficient to model a range of diseases and its affects on settlement populations. As the user is allowed the 22

30 flexibility to change certain aspects of disease traits it should be possible to model different types of real life disease. 23

31 Chapter 4 4 Design and Implementation This section describes the design choices for the simulation and details of how they were implemented. It gives a detailed description of why these choices were made and the implications of them. 4.1 NetLogo Software Before implementation of the simulation began, careful consideration was needed in choosing the language to use. Many simulation toolkits already existed and so writing one from scratch, for the purposes of this project, did not seem plausible given the time constraints. The choice for this project was to use one variant of the logo family called NetLogo. NetLogo is specifically designed for multi-agent modelling and so was very suited for this simulation. Instructions can be given to thousands of agents concurrently and so if needed, and if a suitable computer was available, a very large simulation could be run. As this simulation deals with very large populations of cities this is a great advantage. NetLogo also has very extensive documentation and tutorials compared with other similar simulation toolkits which made it much easier to learn. NetLogo has been used in the past to investigate many social and network phenomena which made it easy to see how the simulation in this project could be developed. NetLogo is supported on Windows, Mac OS X and Linux and so it was easy to port the code from different machines without having to change it. 24

32 4.1.1 Hardware Requirements (NetLogo User Manual version 3.0.2) 25MB hard disk space Windows Windows NT, 98, ME, 2000, or XP 64 MB RAM (or probably more for NT/2000/XP) Mac OS X OS X version or later (10.3 or later is recommended) 128 MB RAM (256 MB RAM strongly recommended) Other platforms NetLogo should work on any platform on which a Java Virtual Machine, version or later, is available and installed. Version or later is preferred. 4.2 Modelling The modelling process was done iteratively. In this way each stage could be completed and tested before the next stage was started, this insured that no errors were introduced into the simulation. The model was created in the following order: 1. Map of settlements 2. Populations added to cities 3. Agent movement added 4. Gravity added to cities 5. Diseases added to agents 25

33 What is presented here is a detailed description of the choices made during the modelling process, how they were implemented and the limitations of the use of these choices. Note about probability: In this simulation probability controlled by sliders are calculated by 1/value of slider. In other words, the higher the value of the slider the lower the probability the action of the slider will be taken Mapping the cities This project aimed to include real life transport and population data to make the simulation more accurate and hopefully to give a better representation of what would happen if a disease was introduced into the populations. Therefore a collection of real settlements were chosen and modelled. The area chosen was Bath and its surrounding area, although it could have been any area with a collection of mixed sized settlements. The first decision was which settlements to include. Most of Britain s area is covered with small towns, villages and hamlets in between the larger cities. It would not be practical to choose every settlement. This would grow to be a very large problem during the gravity modelling stage as each settlement has to be related to every other settlement. We may consider the definitions of certain types of settlements. Town Settlement with a full range of shopping, educational, health, leisure, community and social services. It acts as an important focus for employment for itself and the surrounding smaller settlements. Village Settlement with a limited range of shopping, education community and/or social facilities. 26

34 Hamlet Collection of dwellings, in close proximity to each other, but lacking any local services or central focal point for a community. [21 As this project aims to investigate the spread of disease and thus relies heavily on the assumption of social interaction for disease spread we may ignore all hamlets as they do not have a major focal point for the community. We will assume the same for smaller villages. Larger villages and towns are assumed to have some sort of social facilities and therefore a larger degree of population mixing and interaction important for disease spread. Cities are assumed to have all of the traits of towns but to a higher degree. A suitable level of abstraction had to be taken, and this was decided to be any settlement with a population below 8000 people was ignored. Any settlement below this had a population that was considered negligible on the results of the simulation at this level of abstraction. Of course this cannot be proved in this project and so it is a limitation which would have to be considered when analysing results. It is most probable that the exclusion of the smaller towns, villages and hamlets would affect the movement of populations, but whether or not this would affect the overall trend of disease spread cannot be known. This model therefore considered the following settlements: Settlement Populations Bristol 345,800 Calne 13,606 Swindon 151,700 Westbury 11,500 Bath 66,200 Devizes 11,296 Chippenham 33,200 Wootton Bassett 11,043 Trowbridge 28,200 Corsham 10,549 Frome 24,500 Midsomer Norton 10,500 Warminster 17,400 Wells 10,400 27

35 Keynsham 15,500 Shepton Mallet 8,600 Melksham 14,204 Note: population estimates taken from the 2001 National Census and [21,22 The settlements were implemented using patches in NetLogo. In NetLogo the display is divided into sections. Each section can have certain attributes such as colour, heights, labels and many sections or patches can be grouped together. In this simulation large cities are a collection of patches while smaller settlements are single patches. The patches can then be referred in the code by name, such as telling agents to move to Bristol. An example of the implementation: set bristol patch-at ask bristol [ set pcolor ifelse-value (city_color = true) [red [black set city 0 ask neighbors4 [ set city 0 set pcolor pcolor-of bristol This would name the patch at coordinates (-12,6) to be called Bristol and then set the four neighbouring patches to be in Bristol s group. This area of 5 patches would then represent the city of Bristol in the simulation. After repeating this for all the cities, the display will show the patches on screen: 28

36 Limitations of the choices made Not all settlements in the area were modelled and so the effect of this exclusion on the simulation cannot be known. The model assumes a closed system of settlements. In other words it does not have any influences from outside the system modelled and does not take into account other major transport links such as airports. Any other settlement outside the model is assumed not to exist. Although this is a minor point, the implementation in NetLogo does not take into account the exact size of a city or its exact distance from other cities. At this level of abstraction is should not make much difference in the results, and the distance between all of the cities is modelled accurately in the gravity model described later in this chapter Agents As described above NetLogo is designed to run multi-agent simulations and this has a large library of in built functions that allows different behaviour and movement around the map. This project uses agents in NetLogo s to represent the movement of people in the simulation. As this project is limited in the amount of processing power that is available to it, the simulations that will be run will not include a one-to-one representation of the settlement populations to agents. In other words one agent in the NetLogo simulation may correspond to many people in real life. The simulation models its agents in the following manner: 29

37 turtles-own [ hometown isvisiting isathome destination arrived lengthofstayleft vehicle ;; boolean ;; boolean ;; type 'city' ;; boolean ;;integer ;; mode of transport agent will use isimmune infected timetodeath ;; boolean: true if agent is immune to disease ;; boolean: true if agent is infected with disease ;; integer: turns left to die This shows the global variables of the agents. Note that in NetLogo the agents are referred to as turtles. The agents hold information on their hometown as well as if they are travelling, and if so where to, how long for and by which mode of transport. They also hold information on whether or not they are infected or immune to a disease and how long they have until they die. The simulation differentiates between agents with different hometowns. Populations in the code are named by <name of city>-folk. So creating a population for Bristol is done by: create-bristol-folk bristolpop ;; population determined by slider ask bristol-folk [ set hometown 0 set color yellow The population is then scattered randomly in the relevant patches for its hometown by: 30

38 ask turtles [ set isathome true set infected false set isimmune false set timetodeath timetodie set size 0.2 let homecity hometown let p random-one-of patches with [city = homecity setxy pxcor-of p pycor-of p rt random-float 50 - random-float 50 jump 0.3 Due to the constraints of the project there had to be a certain level of abstraction in the modelling of the agents. Although the literature survey described a great deal of social interaction theory that has been used to good effect in other simulations this project did not have the time available to do enough analysis of social behaviour trends in the Bath area. This would have been a huge undertaking and may have taken a department a couple of years to get a simple model of (as in EpiSims). It was thus decided to make the simulation homogenous. In other words each of the agents is the same and will react in the same way in each situation. In this simulation there is no distinction between age, gender, ethnicity or any other variations that occur in real life. For the purposes of this project and at the level of abstraction that it hopes to consider, this acceptable. The agents modelled in this simulation do not have any attributes relating to social interaction. This means that the agents interactions are random and there is no variation within a city of disease spread. The behaviour of the agents is described as follows. Code is not included but may be found in the move function in the NetLogo source code. [Appendix If an agent is in its home town and not making a trip, then there is a probability that it will make a trip, based on the user selected variable chancetotravel. If an agent does decide to make a trip, then a destination city and vehicle will be selected by the gravity model and the user weighted variables defining mode split (please see section mode split later in this chapter). 31

39 2. If an agent is making a trip, and the current patch is not the destination city, then the agent will set its heading towards the city and move forward. 3. If an agent is making a trip, and the current patch is the destination city then it will move at random around the city. If it by chance happens to move outside of the city then it will reposition its self inside the city. 4. If an agent is making a trip and its exceeds its length of stay in a city, then it will return home. On each step disease spread will be calculated for each infected agents and agents within close proximity to it Limitations of the agents in the model All the agents are modelled identically, in other words they are homogenous and so react in the same way. This takes away a lot of the potential for social interaction. Social interaction is not modelled in very high detail. The mixing of the agents is random, which maybe for this level of abstraction is acceptable. Modelling social interaction in more detail would be a very important step in increasing the potential for modelling a wider variety of diseases Trip Purpose & Trip Generation In this simulation trip purpose was not modelled. This is due to the lack of modelling of social interaction, without which trip purpose would have little impact. It is however represented lightly by using different modes of transport. As described in the later subsection mode split, agents using different modes of transport have different infectivity and thus this makes up slightly for not modelling trip purpose. As a further 32

40 extension or improvement to this project, adding trip purpose would probably be the first obvious task, along with modelling social interaction to some higher degree. The data provided by the National Transports Statistics department provides cross sections of different mode use which describes the types of people who use them. This could be used to good effect if the necessary social interaction was implemented. Trip generation was in reality modelled after the trip distribution using the gravity model was implemented (described in the next subsection). Trip generation was thought of as, how many trips will be made per person in a given amount of time? As the NetLogo simulation runs on steps and in each step every agent makes one move, the system had to be calibrated. These steps were used to represent time and the user controlled the space of time that passed by manipulating global variables such as: This gives the user the power to change how likely it is that in a give step that an agent will make a trip to another city. By a method of calibration described in the testing section in a later chapter, the number of agents making trips could be adjusted to fit that of real life data provided by the National Transport Statistics, who have published figures for the average number of trips made per year per person. [Appendix Trip Distribution The simulation used a gravity model as a basis to distribute the trips to the different cities. The gravity model used weighted probabilities that a trip would be made from a certain settlement to another. This probability was based on the population of both settlements and the distance between them. The size of the population in this simulation was used to measure the degree of importance of a settlement. The greater the population the more services, business and social facilities it was assumed to have, and so the greater draw it would have on the surrounding area. This is perfectly 33

41 reasonable and intuitive as one would expect that a city with a higher population would naturally have a greater diversity of services. As described in the literature survey, the gravity model can be summarised as follows, T ij = α P i P j d ij 2 Where: Tij trip distribution between i and j α balancing factor Pi Population of i Pj Population of j dij distance or cost to travel between i and j The effects of this model are probably explained best in plain English: 1. If two destination settlements are the same distance from a start settlement, then there will be more trips made to the larger of the two destination settlements. 2. If two start settlements are the same distance from a destination settlement, then there will be more trips made from the larger of the two start settlements. 3. The number of trips from one settlement to another is inversely proportional to the distance. If one destination city is closer to a start city than another destination city, and if both destination cities have the same sized populations, more trips will be made to the closer of the two destination cities. 34

42 Implementation of the Gravity Model First the populations of the settlements and the distance between them were calculated. The distances were calculated using the service from using the shortest route as oppose to the quickest. The distances were then multiplied and then the gravities worked out by dividing by the distance squared. The balancing factor used was α = 1/ The reason for this was to simplify the numbers used in the NetLogo matrix as much as possible without loosing any differentiation between gravities when rounded. If the matrix of gravities was simplified any further then gravities which were in reality different would be computed as though they were the same in the NetLogo algorithm. Leaving the gravities as large numbers would only make the NetLogo code more bulky and may slow down the computation. The following table shows the gravities calculated for the cities and the simplified gravities used in the NetLogo code. For a more detailed look at the stages see [Appendix 3: Calculation of Gravities. Bristol Swindon Bath Chippenham Trowbridge Bristol Swindon Bath Chippenham Trowbridge Frome Warminster Keynsham Melksham

43 Bristol Swindon Bath Chippenham Trowbridge Bristol Swindon Bath Chippenham Trowbridge Frome Warminster Keynsham Melksham The gravity table for NetLogo was altered from the original so that each line of the matrix [X 1 X 2 X 3 X 4 is represented as [0+ X 1 (X 2 + (0+ X 1 )) (X 3 + (X 2 + (0+ X 1 )))., in other words each element is the sum of all the previous elements. This makes the algorithm for choosing which city to send an agent to very much simpler. The algorithm in NetLogo was implemented so that the city chosen for an agent who has decided to make a trip was weighted by the gravity table produced, as shown above. to-report choose-city [homecity let x 0 ;;the city let randomnumber (random (item (numcit - 1) (item homecity gravity))) foreach (item homecity gravity) [ ifelse randomnumber <=? [ [set x (x + 1) ;; works out which city has been choosen report x end 36

44 The code above chooses a random number from 0 to the largest number in the row for the home city of the agent, which will be the last number in the matrix row as it is the sum of all the previous elements. The code then works out which city this random number correspond to. For example, if we have the following gravities for an agent whose home town is Bristol: Bristol Swindon Bath Chippenham Trowbridge Bristol There is no chance that the agent will travel to its own home town so there is 0 gravity in the Bristol slot. If the random number chosen was 7500, then the city chosen for the agent to travel to would be Chippenham, this is as: 7004 (Bath) < 7500 < 7523 (Chippenham). As each city in the matrix row has different intervals, by probability, cities with a larger interval will be chosen more often and so more agents from the start city will make trips to it. In this way a destination city is chosen for all agents who are making a trip Limitations of the Gravity Model The gravity model has little grounding in economic theory. No land use data was used when constructing the gravities. For the purposes of this project this is not a major limitation. If one wished to model the cities in more detail adding land use data to the model would help to increase the accuracy of agent movement within a city. Although the gravity model is good for large populations, the choices made by individual agents are likely to be much different for a predicted value. 37

45 The gravity model does not take into account the trip purpose, this would be important if modelling the social interaction in more detail, which would allow for different types of disease spread. The gravity model requires calibration which will occur in the testing section Mode Split Mode split are the proportions of the different types of transport that would be represented in the simulation. As the modelling of social interaction was limited, the mode split became important in varying the type of interaction that a person would experience during their trip. Although the split has suggested defaults, they are left as variables for the user the change: The suggested defaults were elicited from the National Statistics data on different modes of transport [Appendix 1: Statistics. There was however still some difficulties with these base numbers. First there was no direct data linking lorries and other types of transport as they are seen as the be mainly transporting goods, while other modes of transport are mainly for people. The modes of transport also had to be limited as including many other modes of transport e.g. motorbike, bicycle, foot, bus, taxi would have complicated the model to a degree above which is designed to represent. The simulation only represented the vehicles: lorry, car, van and train. As the simulation was designed to focus on the trends in disease spread across a network of 38

46 cities and not specifically in a single city itself, it was decided to exclude busses, pedestrians, cyclists, taxis as the average distance that people travel by these modes of transport was significantly less than that of car, van and train, with passengers travelling on average 3,469 miles by car/van per year and 384 by train, but only 36 miles by foot and 49 miles by taxi.[appendix 1: Statistics. This would indicate that modes such as taxis, foot and bus were used less frequently to travel between cities, and more often for short journeys. The decision to included lorries was to increase the diversity of vehicles in the simulation. The different modes of transport had the same implementation in NetLogo. The mode of transport was added as a variable in the agent structure called vehicle. The mode of transport was chosen for an agent who had decided to make a trip. This was done by choosing a random vehicle, using the weighted probabilities that a certain vehicle would be chosen decided by the user. Please see function to-report typeofvehicle in the NetLogo source code [Appendix 2: Source Code for full details. The type of vehicle chosen was then used to calculate which agents an infected agent would infect on each NetLogo step. The infectivity of each mode was specified by a matrix in the code. The choice for this project will be explained in the next section sub section Disease Limitations The simulation does not take into account trips made by passengers. In other words each vehicle in the simulation contains only one agent. The effect of this is that there is no mixing on specific modes of transport such as a train, although this is counteracted by the fact that agents will take the shortest distance to travel to their destination and will by probability bump into other agents making similar routes. 39

47 4.2.6 Disease The aim of this project is to build a simulation capable of investigating the different factors which affect the spread of disease through a network of cities, it was thus important to have a good base simulation and a flexible model for disease attributes. The simulation included sliders and switches which were able to control the different aspects of disease such that different disease could be modelled. These were as follows: Sliders timetodie This adjusted the number of steps that it would take for an infected agent would die in. The time to die was set to an agent s timetodeath variable which was decremented by one each step if it was infected. chanceforrecovery This controls the chance that an infected agent would recover. This means that it would not be able to infect other agents, but it would however be able to contract a disease again. If an agent recovered then its time to die would be set back to the value on the timetodie slider above. chanceforimmunity This controls the chance that an agent would become immune to a disease. If an agent became immune to a disease its time to die would be set back to the value on the timetodie slider above. It would not be able to die from the disease, its timetodeath variable would not be decremented on each step. 40

48 chanceformutation This controls the chance that an agent who had immunity would not longer have immunity. This represents the mutation of a disease which could gain immunity to certain drugs or just mutate enough so that antibodies produced by the body in the disease s first attack would no longer be effective against it. Increasing the chance for mutation could represent a disease such as the influenza virus which has a high rate of mutation. Infectivity This controls the chance that an agent who is infected with a disease will infect other agents who are in the vicinity. The variable infectivity was chosen to represent other factors which could not be represented in a model of this abstraction. These factors would include varying types of transmission such as airborne, physical contact which could not be modelled without increasing the accuracy of the social model. Infectivity thus represents a large collection of other variables. The greater the infectivity, the greater the chance that an agent will pass a disease on to another agent. The total infectivity for a particular agent is worked out on each step by the type of vehicle that an agent is travelling combined with this user specified variable. The vehicle infectivities are included in the code as the following matrix: set vehicleinfectivity [ ;; lorry / car / van / train The code represents the infectivities of the different vehicles. The greater the number the less the chance of transmission. The choice for this project was to have trains the most common place for transmission of a disease, car second, van third and lorry fourth. This represents the usual overcrowding of trains, the high proportion of car use for leisure [Appendix 1: Statistics and the use 41

49 of lorrys for commercial deliveries and hence a low probability of social contact and transmission. Varying the infectivity does not change the relationship between each vehicle s chance for transmission. Switches immunecanspread This controls whether or not an agent who has gained immunity and has the disease is able to spread the disease to uninfected agents; if an agent can be a carrier to a disease without being affected by it. survivor-gain-immunity This controls whether or not an agent who has recovered from a disease will become immune to it. The figure below shows the user interface for controlling the factors affecting disease traits: Introducing disease into the populations The arrival of a disease, how the infection reaches the population under consideration [14, was implemented by infecting an agent at the start of the simulation at random. This would represent the introduction of a disease in real life at a certain fixed point and may represent a disease which had mutated from an already existing disease, or a 42

50 new disease which had been introduced via an agent who had entered the network through some other means, for example a person who had entered the country by airport and had been infected with a foreign disease Limitations The factors affecting disease were limited to the accuracy of the agent interaction in the model. As differing degrees of social interaction were not modelled it was not possible to mode different types of transmission. An ideal simulation would include different types of contact between the agents, such as business, friendly and intimate and also include different types of transmission such as touch, airborne or transference of bodily secretions. This varying population mixing would make the simulation more accurate and maybe show different trends than those that appear in this simulation. 4.3 Summary This section has gone into the details of how the requirements were implemented and why certain choices were made during the implementation. The section has also gone into some of the limitations of the choices made and possible implications of them in the results that will be obtained when running the simulation. In the next section where the simulation will be tested and the factors affecting disease spread investigated, care will have to be taken to include the limitations described in this section when making any conclusions. 43

51 Chapter 5 5 Tests and Experiments This section demonstrates how to set up the simulation to make it ready to run experiments. It then uses the simulation to investigate a few of the factors affecting the spread of a disease and attempts to suggest some conclusions, keeping in mind the limitations of the simulation mentioned in the previous chapter. 5.1 Setting up and Calibration Before any experiments could be carried out, the model had to be calibrated so that all other factors apart from the ones being investigated were set as a standard. Changing any other factors during the experiments would lead to incorrect results. The process of calibration was also to insure that the number of trips made was realistic. Calibration also gave the simulation some notion of time which would be important in monitoring how quickly a disease would be able to spread through the populations Populations The populations used for the settlements were elicited from the following census data: 44

52 Settlement Populations Bristol 345,800 Calne 13,606 Swindon 151,700 Westbury 11,500 Bath 66,200 Devizes 11,296 Chippenham 33,200 Wootton Bassett 11,043 Trowbridge 28,200 Corsham 10,549 Frome 24,500 Midsomer Norton 10,500 Warminster 17,400 Wells 10,400 Keynsham 15,500 Shepton Mallet 8,600 Melksham 14,204 Note: population estimates taken from the 2001 National Census and [21, 22 As the simulation was being run on a PC it was not practical to have each member of the populations represented by a single agent. What was important however was that the ratios of the settlement populations remained constant, as this is what is used in the gravity model. To keep the running times of the simulation down the following population sizes were used during the NetLogo simulation. Settlement Populations Bristol 115 Swindon 51 Bath 22 Chippenham 11 Trowbridge 9 Frome 8 Warminster 6 Keynsham 5 Melksham 5 Calne 5 Westbury 4 45

53 Devizes 4 Wootton 4 Bassett Corsham 4 Midsomer 4 Norton Wells 3 Shepton 3 Mallet Although these populations are relatively small, they keep the trends in the population sizes the same while also keeping the time taken to run a simulation down Ratio of Vehicles The proportion of different vehicles had to remain the same throughout the experiment as the different vehicles had different infectivities. Varying the ratio of vehicles during the experiments would be like varying the infectivity, and as this is a factor of disease it would skew the results. The decision of which proportions to use were taken from data gathered from the National Transport Statistics department. [Appendix 1: Statistics. The data that was needed for splitting the ratios of vehicles was not available so the ratios had to be estimated. The problems that existed were that: No data existed that represented the difference between the number of cars and vans, but included them under the same category. No data existed that compared directly the number of lorrys and other vehicles. It was also not possible to use the distance travelled by the different vehicles to elicit how many trips were made, as lorrys are mainly used for transporting goods and so in general travel much further than any commute to work, which is the most common for cars. 46

54 What is important however is that different types of transport do exist which lead to differing degrees of social interaction. What we may expect in real life is that there are more cars than vans and more vans than lorries. We have the figures for cars, vans and rail travel [Appendix 1: Statistics so we can attempt to give the following ratios as: Cars 0.7 Vans 0.2 Rail 0.03 Lorry 0.03 This assumes that there are three times more cars than vans and the same number of lorry as rail trips, and as mentioned above these are estimates. What is important is the fact that there is some variation in the number of trips made by each type of transport and that the infectivity of different vehicle s trips is different Likelihood of a Journey This variable was to introduce an element of time into the simulation. Adjusting this variable would alter the chance that an agent who is in its home town would decide to make a trip to another destination settlement. This was important in calibrating the model as getting the time and number of trips realistic was important for the model. The figures used for calibration were the total number of trips per person per year. This figure is set at 988 trips per person in 2004 and includes all reasons for trips (eg. leisure, commuting, shopping and so on). The simulation was run with various likelihoods of a journey and it was decided to use 1. This was to keep the running time of the simulation down, however on a more powerful machine this could be changed. The time taken for an average of 988 trips per person was taken and this gave the number of steps in the simulation that represented 1 year in real life. This figure stood at steps. 47

55 5.2 Examples of the Simulation We will now give some examples of running the simulation, the different outputs which are represented on the interface and some of the different results which may occur by varying different factors. Notes for running the simulation: 1. First adjust the switches and sliders that you wish to change. These could included the populations, ratios of vehicles, and disease factors 2. Click the setup cities button to setup the simulation 3. Click the add turtles button to add agents to the simulation 4. Click go to run the simulation, the button will then be depressed 5. Click go again to stop the simulation If you wish the see the agents trails (as if the agents all have paint on their feet), go to the command center at the bottom of the simulation window, select turtles and type pd, short for pen down. You may export data from a graph by right clicking on it and selecting export. 48

56 In the number of infected graph, green represents the total number in the population, red represents the total infected in the population and blue represents the number of immune individuals. Example 1 Survival We will first begin by showing a case where most of the population survive a disease. This can be achieved by setting the infectivity of the disease to be very low. In other words the basic reproduction number R 0 of a disease is less than 1. The basic reproduction number is the number of secondary cases which one case could generate during the infectious period if introduced into a completely susceptible population. [14. If the basic reproduction rate of a disease is less than 1 this means that the chance that an infected individual will pass on a disease to another uninfected individual, is on average less than 1, and so it is likely that the disease will die out. The output of the simulation was as follows: 49

57 We first setup the cities, added turtles, typed pd in the turtles command and then started the simulation. The simulation ran for a while as there was a single infected agent moving around the map (this is shown by the single red trail in the top left hand corner of the map). As there was a low infectivity rate, and hence a low R 0, the agent did not infect any others and so when the infected agent died the simulation halted with all the rest of the agents unaffected. Example 2 - Death We will now show a case where all of the agents in the simulation die. To do this we will increase the infectivity of the disease, and hence R 0. The output of the simulation was as follows: In this example, R 0 > 1; the average chance that the disease would spread from an infected individual to an uninfected individual was greater than 1. This meant that the disease spread rapidly throughout the population, shown by the steep gradient on the top right hand graph s red line (infection). As the simulation progressed, all of the 50

58 agents had been infected causing a pandemic. All of the agents eventually died from their disease and the simulation halted. Example 3 Immunity We will now show an example of agents in a population being able to become immune to a disease. To do this we will increase the chance for immunity. As discussed before, this variable represents an agent s ability to become immune to a disease through natural means, and through vaccinations. The output of the simulation was as follows: At first the infection spread throughout the population (shown by the steep gradient of the red line). This continued until all of the population was infected. However the immunity, the blue line at the bottom of the graph, of the population started to increase. As the simulation progressed some of the infected agents who were not immune started to die off. This continued until all of the infected agents who were not 51

59 immune were dead leaving a stable population of infected agents who were immune to the disease (shown as the blue line levels off). These agents can be seen as carriers and as discussed in the literature survey, this phenomenon is similar (although involving two diseases) to that of sickle cell trait, where carriers are not affected by malaria. [23. Example 4 Herd Immunity In this example we will need to run the simulation twice. We will not vary the disease variables, but we will need to enter a command at the command line later. One of the strengths of NetLogo is that it allows us direct access to the agents and variables through the command line. The variables are set and the output of the first simulation is as follows: As we can see, the disease spread rather rapidly causing the whole population to become infected and eventually killing the population. (This is shown by the steep gradient of the red infection line and the steep decline in the green population line). 52

60 We then ran the simulation again, keeping the variables the same, but this time vaccinating a proportion of the agents. We did this by executing the command: repeat 110[ask random-one-of turtles [ set isimmune true then ask turtles with [isimmune = true [set color blue and finally pd. This set 110 of the 263 agents to be immune to the disease. These agents would be chosen evenly from the different cities by probability. We then set the immune agents to blue and asked them to pen down so that we could see their trails. Our simulation now contains a mix of agents who are susceptible to the disease, agents who are now vaccinated against the disease and one agent who is infected. The output of the second simulation is as follows: It is immediately obvious that the simulation does not end with the death of all of the agents as before, but the map shows a higher percentage of uninfected agents (yellow) and immune agents (blue) but only a small percentage of infected agents (red). The blue line on the top right hand graph shows the level of immunity in the population. The red line (infection) stays very low below this blue line fluctuating only very 53

61 slightly. There are only a couple of infected agents at any one time. Note that the lifespan of the agents was 500 steps and the simulation ended at around 1440 steps with all of the infected agents dying off. This shows that an infected agent was only able to infect on average only 1 other agent. This shows a very interesting phenomenon called herd immunity. Herd immunity is when only a percentage of a population is vaccinated, but this stops widespread infection and keeps the disease at a low level. The simulation was run exactly as before, except for the vaccinated agents. Instead of the disease spreading to all of the agents and killing the whole population, the vaccinated agents slowed down the spread of the disease so that the number of infected individuals remained relatively low. It is important to note that not all of the agents were vaccinated but only 110 out of the 263. This shows that not all of the population need be vaccinated to stop a disease from becoming a pandemic. This is well known to governments and being able to target sections of populations without having to vaccinate everyone is important in the response times to diseases, as well as the cost of the vaccinations. Many strategies in combating epidemics and pandemics may take herd immunity into account, although it is important to note that the theory of herd immunity is disputed by some. [24 The NHS believe that previous vaccination programmes have caused herd immunity in the population and have caused some diseases such as smallpox to die out and expect diseases such as polio and measels to follow suit. When a disease dies out the WHO (World Health Organisation) can certify the world free of that disease. [25 54

62 5.3 Experiments Now that the model has been set up and calibrated, we may run some experiments which the simulation can be used for. This section will describe the experiments that were chosen, how they were conducted and factors which may limit their accuracy. The experiments will end in the following cases, 1. All the agents die from the disease 2. There are no longer any agents who are infected by the disease In either case the experiment will be considered to have terminated, and then statistical tests may be done on the data Investigating how infectivity affects the spread of disease This section will conduct experiments to investigate how the infectivity of the disease affects the spread of the disease and whether a higher or lower infectivity will cause the population to die out and if so, how this affects the time taken. In this simulation infectivity is modelled as the probability that a disease will be spread to an agent located in the same position, in other words, a disease with a higher infectivity will be more likely to infect another agent. We may expect therefore that increasing the infectivity will kill more agents in the simulation in a shorter time frame. We may describe this more formally, Hypothesis A disease with a higher infectivity will kill a population in less time. 55

63 Null Hypothesis Changing the infectivity of the disease will have no affect on the time taken to kill a population. We have now stated the hypothesis and the null hypothesis. Although it is impossible to prove a null hypothesis correct, it is relatively easy to prove a null hypothesis incorrect [19. We will do this later on using a t-test. As we are investigating the infectivity of the disease, this is the only factor that we will vary. We have already discussed how the system was calibrated and most of the factors were set. What are left are the variables which affect the disease and for this experiment are set as follows: Chance for immunity 500 Chance for mutation 500 Chance for recovery 1500 Time to die 445 Survivor gains immunity on Immune can spread off The variables were chosen at the above values to allow them not to interfere with the simulation, in other words, at the values set above the variables do not have any significant effect on the simulation like allowing agents to recover too easily or allowing all of the population to become immune to the disease. Infectivity will not remain constant however, but will be varied at a low (15), medium (75) and high (135) setting. A t-test will then be run to see if there is any significant difference in the number of deaths before the simulation terminates. 56

64 Running the Simulation The following screenshots show the output taken at different stages of the simulation. At first there seems to be an even number of infected agents (leaving red trails) and uninfected agents (leaving yellow trails). In all cases however, the infection spreads to all of the agents leaving a completely red map and killing off all the agents. 57

65 The following graphs show the trends for infection, along with the population size. The graphs are for low, medium and high infectivities respectively. Low infectivity Medium infectivity High Infectivity These output graphs alone show a difference in the trend of disease spread with varying infectivity. The trends appears to be with lower infectivity, the number of infected agents (red) climbs more steadily with time, whereas with higher infectivities there is a steep climb in the number of infected agents near the start of the simulation Results Note that all the simulations ended with the death of every agent. Low Medium High

66 We can thus calculate the variance of the results, Low Medium High We can thus calculate the means, Low Medium High Low Medium High (A graph showing the mean time to death for varying infectivities) As we can see from the graph above, there does appear to be a strong relation between the time taken to kill the population and the infectivity of the disease. This appears to be that the higher the infectivity the lower the time taken to kill the population. We may now conduct a t-test which is a statistical test for determining whether two sets of data are different from each other. By using the t-test we can determine whether the average time taken for all of the agents to die was statistically different enough from low, medium and high infectivities to be able to suggest with some confidence that the infectivity does affect the rate at which a population is killed by a disease. 59

67 As there are 3 sets of data, we will have to conduct 3 different t-tests, one to compare low-medium, medium-high and low-high values for infectivity T-Test The formula for the t-test is as follow: ( We may now work out the value of t for each combination. We may then look up the value of t in a statistical table. If the value of t that we calculate is greater than the value of t in the table then this means that there is a significant difference in the results. Note: the value of t looked up was using a degree of freedom 8 and probability p = This means that the value is statistically 99% accurate in its result, in other words there is 99% chance that our calculation holds for the real life phenomenon. This value of p was chosen so that we could be very certain that we could be comfortably certain that our results did show the trend that we describe from them. Low infectivity/ Medium infectivity Medium infectivity/ High infectivity Low infectivity/ High infectivity t value of t in table

68 Conclusion The results above show that the value of t calculated for each of the different combinations of infectivities was greater than the value in the statistical table. The conclusions that we can draw from this is that there is a high chance (at least 99%) that changing the infectivity of a disease in our simulation affects the time taken for the total death of a population to occur. From this we may reject our null hypothesis, that Changing the infectivity of the disease will have no affect on the time taken to kill a population and accept our hypothesis A disease with a higher infectivity will kill a population in less time Investigating how chance of immunity affects the spread of disease This section will conduct an experiment to investigate how the chance of gaining immunity to a disease affects its spread. We will do this by investigating the number of people that survive a disease with different chances for immunity to occur. This simulation works by adjusting the chance that on a certain step in the simulation an infected agent will gain immunity to a disease. This represents the reality of both human factors such as immunisation and biological factors such as resistance and the body s ability to produce antibodies for a certain disease. We may expect that a disease that is easier to produce vaccinations for, or for a person to produce antibodies against will not kill as many people as a disease that has no vaccination available against it, or that the body cannot defend itself against. We may describe this more formally, Hypothesis A disease with a high chance of immunity against it will not kill as many people as a disease with a low chance of immunity, by the end on the simulation. 61

69 Null Hypothesis Varying the chance that an agent will become immune to a disease with not have any effect on the number of people killed by the end of the simulation. We have now stated the hypothesis and the null hypothesis, and as in the previous experiment we will try and disprove the null hypothesis by using a t-test. As before we must set all other variables, apart from the chance for immunity, so that they do not interfere with the accuracy of the experiments. Infectivity 90 Chance for mutation 500 Chance for recovery 1500 Time to die 445 Survivor gains immunity on Immune can spread off We have again chosen values for the other variables such that there are no prominent variables which will skew the results in a particular direction. We will be varying the chance for immunity and thus need to choose values. Having experimented roughly with the simulation it was decided that the values chosen be for low, medium and high values which were set at 5, 25 and 45 respectively. 62

70 Running the Simulation The following screenshots show the output taken at different stages of the simulation. These were taken from high chance of immunity; however the trends were similar for the other chances of immunity. At the start of the simulation, most of the agents were uninfected by the disease (yellow trails), although its spread starts to increase across the population. 63

71 As more agents become infected, there is a higher proportion of agents who become immune to the disease (blue). As the simulation comes to an end, most of the infected agents die off, leaving a stable population of immune agents. 64

72 The following graph shows the trend of infected agents (red), immune agents (blue) and the total population (green). The graph shows a steadily increasing level of immunity in the population which remains constant after the infected agents have died off, leaving a stable population of immune agents Results Note: All of the simulations terminated with all of the infected agents who were still alive becoming immune to the disease, and thus each of the simulations had a survival rate with part of the population surviving the disease. Low Medium High We can thus calculate the variance of the data, Low Medium High

73 And the means, Low Medium High low medium high As we can see from the graph of the means above, there does seem to be a change in the number of deaths in the population with the different chances for immunity. The trend seems to be, the lower the chance for immunity the higher the number of people who have died from the disease by the time the simulation had terminated T Test As in the previous experiment we will now conduct a t-test to give us some confidence that the results obtained really do show us that there is some change in the results obtained from the simulation, and the data was not different by chance. We will conduct a t-test for each of the following sets of data: low and medium, medium and high, and low and high. This will hopefully show us that each set of data is significantly different from each other to reject our null hypothesis. 66

74 Note: the value of t looked up was using a degree of freedom 8 and probability p = This means that the value is statistically 99% accurate in its result, in other words there is 99% chance that our calculation holds for the real life phenomenon. This value of p was chosen so that we could be very certain that we could be comfortably certain that our results did show the trend that we describe from them. Low immunity/ Medium immunity Medium immunity/ High immunity Low immunity/ High immunity t value of t in table Conclusion The results above show that the value of t calculated for each of the different combinations of chances for immunity was greater than the value in the statistical table. The conclusions that we can draw from this is that there is a high chance (at least 99%) that varying the chance for immunity against a disease in our simulation affects the number of deaths that occurred by the time the simulation terminated. Moreover, the number of deaths in the population dropped as the chance for immunity against the disease increased. We may therefore reject our null hypothesis, that Varying the chance that an agent will become immune to a disease with not have any effect on the number of people killed by the end of the simulation and accept our hypothesis A disease with a high chance of immunity against it will not kill as many people as a disease with a low chance of immunity, by the end on the simulation. 67

75 Chapter 6 6 Conclusions This project has constructed an interactive simulation for investigating the factors which affect disease spread within a network of cities. It has been designed from scratch and included elements of social network theory, epidemiology, transport network construction and also used real life transport and census data to try and make it more accurate. The model provides a user interface implemented in NetLogo to allow the user to conduct experiments to investigate the different factors that diseases can possess. In this respect this project has fulfilled its requirements and achieved what it set out to do successfully. This is demonstrated in the experiments section where it was used to investigate the affects of infectivity and immunity on disease spread, with interesting results. Constructing a simulation like the one this project has attempted is a difficult process however. There are infinitely many ways to model real life phenomena all of which may result in slightly different affects in the model. To this end I believe that it is almost impossible to ever be satisfied that a model truly represents real life. The question which must be asked is thus; is the model sufficiently realistic for our purposes? At a very abstract level, a model like the one presented in this project may be useful in studying the trends in the spread of disease in a network of cities, but if the model was needed for modelling how strategies such as quarantine procedures could stop the spread of a disease and minimise the damage done, for example, then many more factors would have to be looked at in much more detail. Deciding when 68

76 the simulation modelled real life well enough to make important real life decisions from running simulations would be a hard task. Due to the constant threat of epidemics such as SARS and Avian Flu, the problems of understanding the spread of infectious diseases and factors which can help limit their damage are ongoing. As diseases evolve and new problems such as resistance to vaccinations and antibiotics occur, it is important that organisations keep their strategies one step ahead of diseases, and thus the models too, will have to evolve. There is much work that can extend this project, much has already been mentioned on the limitations of this project which would be a good place to start. The most obvious and arguably the most important would be to model the social interaction in more detail. Projects such as EpiSims have already done this in very high detail. Modelling social interaction in more detail would allow diseases to be modelled more realistically. It would also be a good extension to the project to model land-use in greater detail which would allow economic situations to be modelled which greatly affect the movement of a population. Having modelled land-use in greater detail, the transport network could also be improved as in a more accurate model this would rely and landuse. Once the model was made more accurate it would be interesting to run the simulation on a powerful machine over a longer period of time, allowing more agents to be included in experiments. In conclusion, it is hard to see how a project on this scale could contribute to the vast amount of work already been done on the subject. However I believe that in this particular situation modelled and with the time constraints on the project some interesting points were explored. The project has been interesting from a personal point of view and particular enjoyment was drawn from being able to apply some of the theory of computer science in a practical situation mixing aspects from many other disciplines. 69

77 Bibliography [1 Carley, K et al. (2003). BioWar: Scalable Multi-Agent Social and Epidemiological Simulation of Bioterrorism Events [2 Gerstman, B. B. (2003). Epidemiology Kept Simple (2/e). New York: Wiley-Liss [3 Barrett, C. L. et al. (2000). TranSims: Transport Analysis Simulation System [4 Eubank, S. Guclu, H. et al. (2004) Modeling disease outbreaks in realistic urban social networks, [5 Fraser, C. Riley, S. et al. (2003) Factors that make an infectious disease outbreak controllable [6 Carley, K et al. (2003). BioWar: Scalable Agent-based Model of Bio attacks [7 Swinton, J. ( ). A dictionary of (ecological) epidemiology, Available from: [accessed January 2006 [8 Dietz K., Eichner M. (2002) Notes from talk: The effect of heterogeneous interventions on the spread of infectious diseases [9 Government National Statistics department Online: [accessed November 2005 [10 Rolfe, M. (2004) Social Networks and Simulations [11 Hill, R.A., Dunbar R. I. M. (2002) Social Network Size in Humans [12 Liben-Nowell, D. (2005) An Algorithmic Approach to Social Networks [13 Ortuzar. J, Willumsen, L.G, (1994) Modelling Transport (2/e), X [14 Denis Mollison, Epidemic Models: Their Structure and relation to Data, Cambridge University Press, [15 Beimborn. E, Kennedy. R, Schaefer.W,( 1996) Inside the Blackbox, Making Transportation Models Work for Livable Communities unknown publisher 70

78 [16 Government Department for Transport, Transport Analysis Guidance Website (WEBTAG). Available from: [accessed January 2006 [17 New Scientist news service (23 May 2005) Air-travel maths could limit spread of disease. Available from: [accessed January 2006 [18 World Health Organisation, Available from: [accessed November 2005 [19 Diamond. W.J, Practical Experiment Designs, VNB, [20 Montgomery. D.C, (2005), Design and Analysis of Experiments (6/e) X [21 Somerset Government [accessed January [22Town Guides [accessed January [23 Buford. J, (2004), Sickle Cell Haemoglobin and Malaria: An Adaptive Study of Natural Selection on an Infectious Disease [24 Gerstman. B. B, Epidemiology Kept Simple (2/e), Wiley, [25 NHS immunisation information, [accessed February

79 Appendix 1: Statistics Source: National Transport Statistics 72

80 73

81 74

82 Although the following data was not used during this project, it is nonetheless interesting to consider for future work. 75