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- 1. Universityof Wolverhampton October 2014 Mathematical Modelling of Infectious Disease: A Stochastic Approach Matthew Bickley 0903642
- 2. 1 Faculty of Science & Engineering School of Mathematics & Computer Science Dissertation Title: Mathematical Modelling of Infectious Disease: A Stochastic Approach Student Name: Matthew Bickley Student ID: 0903642 Supervisor: Nabeil Maflahi Award Title: MSc Mathematics Presented in partial fulfilment of the assessment requirements for the above award. This work or any part thereof has not previously been presented in any form to the University or to any other institutional body whether for assessment or for other purposes. Save for any acknowledgements, references and/or bibliographies cited in the work, I confirm that the intellectual content of the work is the result of my own efforts and of no other person. It is acknowledged that the author of any dissertation work shall own the copyright. However, by submitting such copyright work for assessment, the author grants to the University a perpetual royalty-free licence to do all or any of those things referred to in section 16(i) of the Copyright Designs and Patents Act 1988 (viz: to copy work; to issue copies to the public; to perform or show or play the work in public; to broadcast the work or to make an adaptation of the work). Signature: Date: 03/10/2014
- 3. 2 i. Dissertation Declaration This document must accompany all dissertation document submissions PLEASE READ THIS VERY CAREFULLY. The University considers seriously all acts of Academic Misconduct, which by definition are dishonest and in direct opposition to the values of a learning community. Misconduct may result in penalties ranging from the failure of the assessment to exclusion from the University. Further help and guidance can be obtained from your academic tutor or from our guide on How to avoid Academic Misconduct – available at http://www.wlv.ac.uk/skills By submitting this document for assessment you are confirming the following statements I declare that this submission is my own work and has not been copied from someone else or commissioned to another to complete. Any materials used in this work (whether from published sources, the internet or elsewhere) have been fully acknowledged and referenced and are without fabrication or falsification of data. I have adhered to relevant ethical guidelines and procedures in the completion of this assignment. I have not allowed another student to have access to or copy from this work. This work has not been submitted previously. By this declaration I confirm my understanding and acceptance that – 1. The University may use this work for submission to the national plagiarism detection facility. This searches the internet and an extensive database of reference material, including other students’ work and available essay sites, to identify any duplication with the work you have submitted. Once your work has been submitted to the detection service it will be stored electronically in a database and compared against work submitted from this and other Universities. The material will be stored in this manner indefinitely. 2. In the case of project module submissions, not subject to third party confidentiality agreements, exemplars may be published by the University Learning Centre. I have read the above, and declare that this is my work only, and it adheres to the standards above. Signature: Date: 03/10/2014 Print Name: MATTHEW BICKLEY Student ID: 0903642 a signed copy prior to marking
- 4. 3 ii. Acknowledgments I would like to thank my family for putting up with me during my academic career so far, in the last five years since I started my undergraduate degree at Keele University, but especially the past two years during my PGCE and postgraduate studies which has been difficult for not just me, but my family as a whole. I would also like to thank Nabeil Maflahi for supervising this dissertation and making sure my ideas stay on track and that I get to the point of what I am trying to show, alongside putting up with my random chats and lengthy meetings when I am worrying that all is wrong. I would also like to thank the other lecturers at the University of Wolverhampton who have taught me various high-level topics in the past year to add on to my always improving mathematical knowledge. I would like to thank my lecturers at Keele University, specifically Martin Parker, David Bedford, Neil Turner and Douglas Quinney, amongst others who all taught me the main areas of mathematics that I love. Finally, I would also like to thank my Bro, Ian. He has been there for me since I meet him at Keele throughout all of my trials and tribulations. He has been and still is in a similar position to me since we met, on a personal, academic and career basis. We have been through a lot together and he has always been there when things were awesome and when they were not so great. Without him, I may not have had the confidence and vision to make sure I get what I want from my career. He is my Bro and that will never change. Thanks, Bro. Maybe Ellen too…
- 5. 4 iii. Abstract Mathematical models are very useful for representing a real world problem. In particular, these models can be used to simulate the spread of an infectious disease through a population. However, standard approaches using deterministic rates for the flow of individuals from state to state can pose problems when the disease in question becomes more complex. Numerical methods can be used to help analyse a system, but this still does not take away from the fact that some factors are ignored to allow for solutions to be found. Stochastic processes have inherent properties that allow for random events to take place. Markov chains, specifically, allow modelling of an individual within a system where transitions are described by probabilities of changing state. The probabilities can be calculated from real world data and due to this, will incorporate a multitude of extra influences that the deterministic model had to assume insignificant. After analysing three diseases using standard ordinary differential equations, these three diseases were then also modelled using a Markov chain. Comparing the results of these analyses, we find that the stochastic approach did not fundamentally give better or worse results than that of the traditional method. Although the stochastic method does include random effects and other factors not explicitly accounted for before, it appears as if the original assumptions made were justified and made no real different to the results. One of the adapted models was hugely inaccurate with respect to real life cases, but this was due to the fact that the original model was also poor. Further work would be to stochastically model diseases with Markov chains which cannot be solved analytically using standard techniques and incorporating more and more complexity to each model to allow for the most accurate results possible.
- 6. 5 iv. Contents i. Dissertation Declaration......................................................................................................2 ii. Acknowledgments ...............................................................................................................3 iii. Abstract............................................................................................................................4 iv. Contents ...........................................................................................................................5 1. Introduction .........................................................................................................................8 2. Deterministic Modelling – an overview............................................................................10 3. Deterministic Modelling – analysis...................................................................................12 3.1. Model 1 – Chickenpox...............................................................................................12 3.1.1. Compartmental model ........................................................................................12 3.1.2. Assumptions .......................................................................................................13 3.1.3. Representing the model ......................................................................................14 3.1.4. Finding the steady states.....................................................................................15 3.1.5. Finding 𝑹𝟎..........................................................................................................17 3.1.6. Analysing the data ..............................................................................................17 3.2. Model 2 – Measles .....................................................................................................20 3.2.1. Compartmental model ........................................................................................20 3.2.2. Assumptions .......................................................................................................22 3.2.3. Representing the model ......................................................................................23 3.2.4. Finding the steady states.....................................................................................24 3.2.5. Finding 𝑹𝟎..........................................................................................................26 3.2.6. Analysing the data ..............................................................................................27 3.3. Model 3 – H1N1 ........................................................................................................30 3.3.1. Compartmental model ........................................................................................30 3.3.2. Assumptions .......................................................................................................32 3.3.3. Representing the model ......................................................................................33 3.3.4. Finding the steady states.....................................................................................35 3.3.5. Finding 𝑹𝟎..........................................................................................................37 3.3.6. Analysing the data ..............................................................................................38 4. Overview of deterministic models.....................................................................................41
- 7. 6 5. Stochastic Processes – an overview ..................................................................................44 6. Stochastic Modelling – adaptations and analysis ..............................................................48 6.1. Model 1 adaptation – Chickenpox.............................................................................48 6.1.1. Stochastic model.................................................................................................48 6.1.2. Transition matrix ................................................................................................49 6.1.3. 𝒏-step transition..................................................................................................53 6.1.4. Limiting distribution...........................................................................................54 6.2. Model 2 adaptation – Measles ...................................................................................56 6.2.1. Stochastic model.................................................................................................56 6.2.2. Transition matrix ................................................................................................57 6.2.3. 𝒏-step transition..................................................................................................62 6.2.4. Limiting distribution...........................................................................................64 6.3. Model 3 adaptation – H1N1.......................................................................................67 6.3.1. Stochastic model.................................................................................................67 6.3.2. Transition matrix ................................................................................................69 6.3.3. 𝒏-step transition..................................................................................................73 6.3.4. Limiting distribution...........................................................................................74 7. Results and model comparisons ........................................................................................77 7.1. Model 1 comparisons.................................................................................................79 7.2. Model 2 comparisons.................................................................................................80 7.3. Model 3 comparisons.................................................................................................80 8. Conclusions .......................................................................................................................82 8.1. Conclusion and critical evaluation.............................................................................82 8.2. Further work...............................................................................................................84 9. References .........................................................................................................................86 10. Bibliography ..................................................................................................................92 11. Appendices ....................................................................................................................93 11.1. Appendix A ............................................................................................................93 11.1.1. Appendix A1.......................................................................................................93 11.1.2. Appendix A2.......................................................................................................94 11.1.3. Appendix A3.......................................................................................................95 11.1.4. Appendix A4.......................................................................................................97 11.1.5. Appendix A5.......................................................................................................98
- 8. 7 11.1.6. Appendix A6.....................................................................................................100 11.1.7. Appendix A7.....................................................................................................101 11.1.8. Appendix A8.....................................................................................................102 11.1.9. Appendix A9.....................................................................................................103 11.1.10. Appendix A10...............................................................................................104 11.1.11. Appendix A11...............................................................................................105 11.2. Appendix B...........................................................................................................106 11.2.1. Appendix B1.....................................................................................................106 11.2.2. Appendix B2.....................................................................................................109 11.2.3. Appendix B3.....................................................................................................112 11.3. Appendix C...........................................................................................................114
- 9. 8 1. Introduction Mathematical models are a widely used system to represent problems in the real world. They can be implemented represent almost anything from elements of the natural sciences; such as physics, engineering and biology (PHAST 2011, Adeleye 2014), to computer science (Neves and Teodoro 2010) and social sciences including business and economics (Reiter 2014) and sociology (Inaba 2014). They can also represent how parts of language can be used (Pande 2012) and how education can adapt (Stohlmann et al. 2012). Mathematical models are primarily used to study the effect of specific components and parameters involved in the models, and use this and further analysis to predict behaviour and outcomes of some initial input. Models that represent biological situations can again be specified into particular areas; one of which has been and will continue to be vitally important to the real world – mathematically modelling infectious disease. Diseases have existed as long as life itself and will only continue to be a problem to humans and other organisms for a long time to come. It is therefore imperative that diseases can be outsmarted – that is to say we need to be able to know, or at least predict with some certainty, exactly how a disease behaves. This is where mathematical models come in. Using past data and specific discovered or known traits of a particular disease, we can apply modelling knowledge to calculate how the disease will spread, how infectious or deadly it may be, and how best to ensure the disease has the minimal effect on the population. However, these infections do not always behave the same from one outbreak to another, even when looking at the same specified strain of the disease (Schmidt-Chanasit 2014, in Hille 2014). Other factors which may be very difficult or impossible to model can affect the solutions in an unpredictable way. These may be but are not limited to environmental, temporal, climate or population effects. This is where stochastic processes can be applied. Stochastic processes are methods which inherently involve some amount of probability or randomness, which diseases can show signs of when a small number of individuals infected
- 10. 9 cause an outbreak (Spencer 2007). In this way, the solution to a stochastically-involved system gives probabilities that certain events will have occurred, and exactly what this means in the long-term. Although this can initially give the impression of misleading outcomes, it in fact shows the variety that the system can produce, giving us a better overview of the situation and then hopefully allowing us to pick the best real-world solution. This study will look at a variety of diseases, from those which are not deadly but can cause uncomfortable living, to those which have high mortality rates and or can reoccur in the same individual over a period of time. At first, a standard albeit complex deterministic approach will be used for each disease – a standard mathematical model. Each model will then be adapted to include a stochastic basis for the disease in question, hopefully showing a similar or if possible, better picture of the disease’s traits. Every model will be compared to real- world data to see the appropriateness of each model and then to each other, to see overall if a stochastic approach is better and if so, to what extent compared to the complexity increase from standard systems. This will bring together knowledge about mathematical modelling (of diseases) with that of stochastic models and processes. This application assumes that stochastic properties can be applied to diseases that are usually modelled using standard deterministic approaches which can help either represent the models better, in an alternative way giving the same or similar results or give entirely different information which could be used advantageously. For the analysis of disease in this research, three diseases will be picked that can be represented by a standard, deterministic mathematical model. The diseases were picked from a list that can be modelling using standard techniques (CMMID 2014). Chickenpox, which causes no (or few) deaths from the disease was picked to start with, followed by measles, a disease with deaths and maternal immunity, and then the final disease, the H1N1 virus was picked to allow for disease reoccurrence in the model and also to allow modelling and comparisons of a disease which is usually almost non-existent at most times, but then becomes very prevalent within populations when outbreaks occur.
- 11. 10 2. Deterministic Modelling – an overview “A model is a simplified version of something that is real.” (Schichl 2004 p.28). Taking this and applying it to a mathematically specific interpretation gives that a mathematical model is a simplified version of a real-world event that helps us to solve an existing problem. Mathematical models of infectious diseases are those which help us see what a disease is doing and how best to combat the infection. William Hamer and Ronald Ross were the earliest known pioneers of infectious disease modelling, giving a basis to today’s current models in the early part of the twentieth century. Within the next two decades, further research, data and scientific knowledge had allowed them, along with other researchers to develop their own epidemic model which showed a clear and logical relationship between individuals in a population – they could be susceptible, infected or immune. This is the root of the susceptible-infected-recovered (SIR) model (Weiss 2013). This simplistic model is effectively just three states, where an individual fits into one and only one. Movement of individuals between the states is described by deterministic parameters that help calculate the rate of change of the sizes, or proportions, of these states. Analysing this gives an easily interpretable system which shows the dynamics of individuals moving through the system, which in turn, shows the basic properties that a disease exhibits to cause an outbreak. This model can be applied to a wide variety of diseases, but mainly those which transfer when humans, or any other animals, directly come into contact with each other via touch or close- proximity sneezing and coughing, for example (Weiss 2013). So modelling this way does not (easily) allow for diseases which are purely airborne or waterborne, for example, to be simulated. Adaptations would be needed, making new types of systems to represent these types of infections. Focusing just on those diseases which can transmit when the host in question is coming into contact with another allows for much simpler analysis yet efficacy. In the last century, these models have been developed to include many more compartments, extra parameters that depend on disease traits and far more complicated systems of equations due to this. Extra states have been added in certain systems such as ‘exposed’, where
- 12. 11 individuals belong who have the disease but are not infectious yet, ‘carrier’, where individuals belong who can pass on the disease but never themselves suffer from it (and likely keep the disease for life) and ‘quarantine’, where individuals who have the disease are placed so as to no longer pass on the disease but still need time to recover. Limitations on these states can also be introduced, such as a space limit in quarantines so that once the state is ‘full’, no more individuals can enter and must stay put in these rest of the system. Parameters involved in the system normally depend on the compartments considered, but some are added to existing models such as a vaccination rate or a separate birth rate that takes into account all individuals born with the disease. These representations can also be extended from a standard system of linear ordinary differential equations with deterministic, explicit parameters, to more complex systems such as partial differential equations which look at spatial variances and systems that have implicit or time-based parameters. The more complicated the original disease is, the more complex the system of equations must be to accurately represent the virus. However, analytical solutions are only feasible on the simpler of models. Numerical methods must be used on the most complex, which have their advantages and disadvantages. Some converge to solutions very quickly but the sizes of errors during the calculations are larger than others. Methods with the smallest of errors are the best possible approximation to the solutions of these systems, but convergence is slow and the method, albeit doable, is intricate.
- 13. 12 3. Deterministic Modelling – analysis 3.1. Model1 – Chickenpox To start with, modelling a disease than can be represented with a very simple model will give a strong basis for the later, adapted model and comparison. Chickenpox, otherwise known medically as varicella, which is a relatively basic and mild disease caused by the varicella- zoster virus (VZV) spreads quickly and easily between individuals (NHS 2014a) but is rather simplistic to model – it only very rarely (directly) causes deaths in the human population. Symptoms of chickenpox are similar to that of influenza and often include sickness, high temperatures, aching muscles, headaches, loss of appetite and most obviously, rashes of spots which causes itching and irritation (NHS 2014b). These spots are the main way of the infection spreading from person to person – and open spots are especially infective. Close contact with individuals who have chickenpox themselves is the main cause of mass spread of the disease, so reducing contact will lower the possibility that any individual will be able to pass it on. 3.1.1. Compartmental model Assuming the simplicity of this model, we can represent it with one of the simplest compartmental models; that being, one that only includes one extra state besides the usual three. This is the ‘susceptible-exposed-infectious-recovered’ model, or the SEIR model. This is an adaptation of the ‘susceptible-infected-recovered’ model, or the SIR model, as it allows for individuals who gain chickenpox to harbour the disease before being able to spread it (exposed). Also, the ‘infected’ state is renamed ‘infectious’, to lessen the confusion as to what individuals are capable of doing with the disease in each state. This fits with chickenpox, as individuals will go through this stage before infecting others. After looking at how the infection spreads, we can construct the first standard black-box model (Fig. 3.1. and Fig. 3.2.):
- 14. 13 Figure 3.1.: Compartmental model for chickenpox Or more mathematically: Figure 3.2.: Compartmental model for chickenpox with mathematical symbols 3.1.2. Assumptions The following assumptions can be made about this model with this disease. The aim of these assumptions is to make the model viable and to enable a solution to be found, when at the same time not taking away too much from the original traits of the disease and general complexity of the disease transmission. Development rate Transmission rate Susceptible Exposed Births Natural Deaths Natural Deaths Infectious Natural Deaths Recovery rate Recovered Natural Deaths 𝜎𝛽 𝑆 𝐸 𝜇 𝜇𝜇 𝐼 𝜇 𝛾 𝑅 𝜇
- 15. 14 No individual is born with the virus (all individuals initially enter the susceptible state). The population has a life expectancy of 𝜇, so that the natural mortality rate is 1 𝜇 . A closed population, so that the birth rate is also 1 𝜇 . The virus has a latency period of 1 𝜎 . A constant proportional population implies the individuals in the system add up to 1 (individuals either are susceptible, infected or have recovered, and nothing else): 𝑆 + 𝐸 + 𝐼 + 𝑅 = 1 (1) where 𝑆( 𝑡), 𝐸( 𝑡), 𝐼( 𝑡), 𝑅( 𝑡) ≥ 0, ∀𝑡 and we simplify 𝑆( 𝑡) = 𝑆, 𝐸( 𝑡) = 𝐸, 𝐼( 𝑡) = 𝐼 and 𝑅( 𝑡) = 𝑅. 3.1.3. Representing the model Using the above assumptions, with 𝛽, the transmission rate being the product of the contact rate and probability of a successful transmission, 𝛾, the recovery rate, and 𝜎, the development rate, we can construct the following ordinary differential equations (ODEs) to represent the rate of change of each compartment in the model: 𝑑𝑆 𝑑𝑡 = 𝜇 − 𝛽𝑆𝐼 − 𝜇𝑆 (2) 𝑑𝐸 𝑑𝑡 = 𝛽𝑆𝐼 − 𝜎𝐸 − 𝜇𝐸 (3) 𝑑𝐼 𝑑𝑡 = 𝜎𝐸 − 𝛾𝐼 − 𝜇𝐼 (4) 𝑑𝑅 𝑑𝑡 = 𝛾𝐼 − 𝜇𝑅 (5) where 𝛽, 𝜎, 𝛾, 𝜇 > 0.
- 16. 15 3.1.4. Finding the steady states At the steady states to this system of equations, there will be no change to the proportions represented by 𝑆, 𝐼 and 𝑅 as time continues to pass. Therefore, the rates of change are equal to zero. Hence: 𝑑𝑆 𝑑𝑡 = 0, 𝑑𝐸 𝑑𝑡 = 0, 𝑑𝐼 𝑑𝑡 = 0, 𝑑𝑅 𝑑𝑡 = 0 Therefore: 𝜇 − 𝛽𝑆𝐼 − 𝜇𝑆 = 0 (6) 𝛽𝑆𝐼 − 𝜎𝐸 − 𝜇𝐸 = 0 (7) 𝜎𝐸 − 𝛾𝐼 − 𝜇𝐼 = 0 (8) 𝛾𝐼 − 𝜇𝑅 = 0 (9) Rearranging all of (6) to (9) to make 𝐼 the subject gives: 𝐼 = 𝜇(1 − 𝑆) 𝛽𝑆 (10) 𝐼 = ( 𝜎 + 𝜇) 𝐸 𝛽𝑆 (11) 𝐼 = 𝜎𝐸 𝛾 + 𝜇 (12) 𝐼 = 𝜇𝑅 𝛾 (13) If 𝐼 = 0 (no virus present), (10) ⟹ 𝑆 = 1 (12) ⟹ 𝐸 = 0 (13) ⟹ 𝑅 = 0 Checking in (1), 𝑆 = 1, 𝐸 = 0, 𝐼 = 0 and 𝑅 = 0 clearly satisfy 𝑆 + 𝐸 + 𝐼 + 𝑅 = 1.
- 17. 16 This gives the trivial steady state: ( 𝑆, 𝐸, 𝐼, 𝑅) = (1,0, 0, 0) (14) If 𝐼 ≠ 0 (virus present), then Maple confirms the following solution (along with the trivial steady state): > Checking in (1), 𝑆 = ( 𝜎+𝜇)( 𝛾+𝜇) 𝛽𝜎 , 𝐸 = 𝜇( 𝛽𝜎−(𝜎+𝜇)(𝛾+𝜇)) 𝛽𝜎(𝜎+𝜇) , 𝐼 = 𝜇( 𝛽𝜎−(𝜎+𝜇)(𝛾+𝜇)) 𝛽(𝜎+𝜇)(𝛾+𝜇) and 𝑅 = 𝛾( 𝛽𝜎−(𝜎+𝜇)(𝛾+𝜇)) 𝛽(𝜎+𝜇)(𝛾+𝜇) satisfy 𝑆 + 𝐸 + 𝐼 + 𝑅 = 1 (manually checked). This solution along with some manual working leads to the non-trivial endemic steady state: ( 𝑆∗ , 𝐸∗ , 𝐼∗ , 𝑅∗) = ( ( 𝜎 + 𝜇)( 𝛾 + 𝜇) 𝛽𝜎 , 𝜇( 𝛽𝜎 − (𝜎 + 𝜇)(𝛾 + 𝜇)) 𝛽𝜎(𝜎 + 𝜇) , 𝜇( 𝛽𝜎 − (𝜎 + 𝜇)(𝛾 + 𝜇)) 𝛽(𝜎 + 𝜇)(𝛾 + 𝜇) , 𝛾( 𝛽𝜎 − (𝜎 + 𝜇)(𝛾 + 𝜇)) 𝛽(𝜎 + 𝜇)(𝛾 + 𝜇) ) (15)
- 18. 17 3.1.5. Finding 𝑹 𝟎 For a closed population (which we have here), the critical value at which the virus becomes an epidemic is when it exceeds 1 𝑆 . So, for this model: 𝑅0 = 1 𝑆 = 𝛽𝜎 ( 𝜎 + 𝜇)( 𝛾 + 𝜇) If 𝑅0 is above 1, the theory suggests that an epidemic starts: 𝑅0 = 𝛽𝜎 ( 𝜎 + 𝜇)( 𝛾 + 𝜇) > 1 ⟹ 𝛽𝜎 > ( 𝜎 + 𝜇)( 𝛾 + 𝜇) (16) Interpreting this inequality, by reducing the transmission rate or increasing the mortality rate or recovery rate, we can cause a disease to die out in this instance. Increasing the mortality rate is unethical within most populations, especially for those diseases affecting humans (culling of other species can be introduced on a case by case basis), and the recovery rate is generally unchangeable due to traits of the virus, along with the latency period due to the properties of the disease. Hence reducing 𝛽, so in turn, reducing the contact rate is the best way to prevent an epidemic in this model. Keeping those affected by the chickenpox away from others is by far the most effective way of reducing the spread. 3.1.6. Analysing the data Assuming a standard scenario for chickenpox (an average rate of spread), we take an estimate for the basic reproduction number, 𝑅0, as equal to 3.83 for England and Wales (Nardone et al. 2007). It is stated that this is just an estimate, with a 95% confidence interval (CI) given as (3.32– 4.49). By taking a mid-range value, we can model for the average spread in the United Kingdom (UK). The latency period, 1 𝜎 , is given as ten to fourteen days (Knott 2013), so we take the average of twelve days. The recovery period, 1 𝛾 , is also reported as ten to fourteen days (Lamprecht 2012), so again, we take the average of twelve days. Average life
- 19. 18 expectancy at birth in the UK is approximately 81 years (WHO 2012), so this is taken to be the value of 1 𝜇 . All parameters are rates, so all must be converted to the same units. Numerical analysis will be taken at daily time periods with step size ℎ = 1 5 over the period of 365 days, so all parameters are multiplied or divided as needed to give each value in terms of days. We can substitute these values into 𝑅0 = 𝛽𝜎 ( 𝜎+𝜇)( 𝛾+𝜇) and rearrange, giving us a 𝛽 value (transmission rate) of 0.319426 (six significant figures). This system has been analysed using the 4th-order Runge-Kutta method for a more robust approach, with an initial susceptible individual proportion, 𝑆(0) of 0.999 and infected proportion, 𝐼(0) of 0.001 as a starting point (see Appendix B1 for data). Figure 3.3.: Chickenpox 4th-order Runge-Kutta numerical analysis (with steady state values) As Fig. 3.3. shows, the chickenpox outbreak eventually subsides but does produce a worrying epidemic initially. After around forty days, the exposed and infectious proportion starts to rise rapidly and then both peaking just before day 100 at around 20%, giving approximately 40% of the population having chickenpox at this point. This would most certainly be classified as an epidemic if the outbreak spread across a national population, such as the entire UK. The 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 50 100 150 200 250 300 350 400 Proportionofpopulation Time (days) S E I R S* E* I* R*
- 20. 19 data again shows that a majority of people have the virus as early as day 100, as half the population have entered the recovered state at this point. From the year-long graph, it show the levels of all four compartments fluctuate somewhat, with 𝑆 ever decreasing and 𝑅 ever increasing, and 𝐸 and 𝐼 showing the peak where the outbreak is at its worst before lowering back down again. The 𝑆 and 𝑅 states eventually level out in the long-term, moving toward the respective dotted lines shown, giving a steady proportion of the population who are present in each state. By substituting in the values for 𝛽, 𝜎, 𝛾 and 𝜇 in (𝑆∗ , 𝐸∗ , 𝐼∗ , 𝑅∗ ) and using Maple to work them out (see Appendix A1), we can find out the behaviour of the model analytically as 𝑡 tends to infinity (also shown in Appendix B1 as 𝑡 → ∞): (𝑆∗ , 𝐸∗ , 𝐼∗ , 𝑅∗ ) = (0.261097,0.000299789,0.000299667,0.738304) So eventually, there is a constant 26.1% of the population that are susceptible, 73.9% that have had chickenpox at some point and recovered, and a negligible amount suffering from chickenpox at any time (< 0.06%).
- 21. 20 3.2. Model2 – Measles Now, a disease with an extra state is introduced to try to see if the adaptation later will work with a different basis model. Measles, otherwise known as morbilli, English measles or rubeola, and not to be confused with rubella (German measles), is a disease caused by a paramyxovirus. This disease is a little more complex to model as individuals are more often than not born with maternal immunity to measles which lasts sometime into the second year of life (Nicoara et al. 1999) and can cause deaths directly from the disease in the human population. Measles has a relatively high 𝑅0 value, meaning it is more likely to cause epidemics but spread can be reduced significantly by use of vaccinations (CDC 2014). Symptoms of measles are similar to that of a cold, with red eyes, high temperatures and spots appearing in the mouth and throat (NHS 2013). These spots are the main cause of the spread of measles; tiny droplets are ejected from the nose or mouth when sneezing or coughing respectively (NHS 2013), and then infect others close by or land on surfaces or objects. In turn, these are then touched the hands of others and through improper and irregular hand- washing, this causes the infection to successfully transmit. Being in close proximity with individuals who have measles is the main cause of mass infection, so removing contact and introducing vaccinations will lower the possibility that any one individual will be infected. 3.2.1. Compartmental model We can represent measles with a slightly more complex compartmental model; that being, one that only includes one extra state besides the previously used four. This is the ‘maternal immunity-susceptible-exposed-infectious-recovered’ model, or the MSEIR model. This is an adaptation of the SIR model, as it allows for individuals who gain measles to harbour the disease before spreading it (exposed). Again, the ‘infected’ state is renamed ‘infectious’, to lessen the confusion. Also, a maternal immunity state is added. Individuals born enter this category initially before be able to contract measles (being susceptible). This fits with measles, as individuals will have immunity for a small time before losing it and entering a standard SEIR part of the model. Vaccinations are also allowed in this model; individuals who are vaccinated are only those in the susceptible state and will move into a recovered
- 22. 21 position after the immunisation, preventing them catching the disease (all vaccinations are assumed to be 100% effective). After looking at how the infection spreads, we can construct the second standard black-box model (Fig. 3.4. and Fig. 3.5.): Figure 3.4.: Compartmental model for measles Or more mathematically: Figure 3.5.: Compartmental model for measles with mathematical symbols Recovery rate Transmission rate Exposed Infectious Recovered Maternal Immunity Susceptibl e Births (with passive immunity) Natural Deaths Natural Deaths Natural Deaths Virus Deaths Natural Deaths Virus Deaths Natural Deaths Development rate Loss of passive immunity Gain of active immunity (vaccination) 𝛾𝛽𝐼 𝐸 𝐼 𝑅𝑀 𝑆 𝜇 𝜇 𝜇 𝜇 𝜒 𝜇 𝜒 𝜇 𝜎𝛿 𝜅 𝑋
- 23. 22 When mathematically representing this model, we need to create a new state. A new death state, 𝑋, is introduced to account for those who die due to the disease only. 3.2.2. Assumptions The following assumptions can be made about this model with this disease. Again, the aim of these assumptions is to make the model viable and to enable a solution to be found, when at the same time not taking away too much from the original traits of the disease and general complexity of the disease transmission. No individual is born with the virus, but all are born with maternal immunity for some length of time to it (all individuals initially enter the maternal immunity state). The population has a life expectancy of 𝜇, so that the natural mortality rate is 1 𝜇 . A closed population, so that the birth rate is also 1 𝜇 . This only includes individuals born that replace those who die naturally, so the birth rate relates to every state except 𝑋 (so in the first ODE, 𝜇 multiplies every state but 𝑋 – see below). The virus’ mortality rate is 1 𝜒 , so individuals who die from the virus directly enter the 𝑋 state at this rate. The virus has a latency period of 1 𝜎 . Vaccinations are only administered to susceptible individuals, not those who may have the virus but are not showing symptoms or capable or passing the virus on (exposed) nor infants with passive immunity from their mother (maternal immunity). Vaccinations are assumed to be 100% effective from the moment they are administered (individuals who receive it cannot catch measles at any point in the future). Deaths caused by the virus have the same prevalence in both the exposed and infectious individuals (virus mortality is the same whether you are exposed or infectious).
- 24. 23 To allow for the constant proportionality of the model, those who die of the disease must be modelled to be allowed to ‘die again’ naturally. Although, this would not happen in real life, it allows the model to maintain the same amount of individuals in the system and those who do ‘die again’ from the virus death state will ‘re-enter’ as an individual with maternal immunity. The low rate of virus and natural deaths within the length of a long-term analysis makes this error relatively small, albeit still present. A constant proportional population implies the individuals in the system add up to 1 (individuals either have maternal immunity, are susceptible, exposed, infectious, have recovered or have died from measles, and nothing else): 𝑀 + 𝑆 + 𝐸 + 𝐼 + 𝑅 + 𝑋 = 1 (17) where 𝑀( 𝑡), 𝑆( 𝑡), 𝐸( 𝑡), 𝐼( 𝑡), 𝑅( 𝑡), 𝑋( 𝑡) ≥ 0, ∀𝑡 and we simplify 𝑀( 𝑡) = 𝑀, 𝑆( 𝑡) = 𝑆, 𝐸( 𝑡) = 𝐸, 𝐼( 𝑡) = 𝐼, 𝑅( 𝑡) = 𝑅 and 𝑋( 𝑡) = 𝑋. 3.2.3. Representing the model Using the above assumptions, again with 𝛽, the transmission rate being the product of the contact rate and probability of a successful transmission, 𝛾, the recovery rate, and 𝜎, the development rate, but additionally 𝛿, the rate at which individuals lose their (passive maternal) immunity, and 𝜅, the constant number of individuals immunised, we can construct the following ODEs to represent the rate of change of each compartment in the model: 𝑑𝑀 𝑑𝑡 = 𝜇 − 𝛿𝑀 − 𝜇𝑀 (18) 𝑑𝑆 𝑑𝑡 = −𝛽𝑆𝐼 + 𝛿𝑀 − 𝜅𝑆 − 𝜇𝑆 (19) 𝑑𝐸 𝑑𝑡 = 𝛽𝑆𝐼 − 𝜎𝐸 − 𝜇𝐸 − 𝜒𝐸 (20) 𝑑𝐼 𝑑𝑡 = 𝜎𝐸 − 𝛾𝐼 − 𝜇𝐼 − 𝜒𝐼 (21) 𝑑𝑅 𝑑𝑡 = 𝛾𝐼 + 𝜅𝑆 − 𝜇𝑅 (22) 𝑑𝑋 𝑑𝑡 = 𝜒𝐸 + 𝜒𝐼 − 𝜇𝑋 (23)
- 25. 24 where 𝛽, 𝜎, 𝛾, 𝛿, 𝜇, 𝜒, 𝜅 > 0. 3.2.4. Finding the steady states Again, at the steady states to this system of equations, there will be no change to the proportions represented by 𝑀, 𝑆, 𝐸, 𝐼, 𝑅 and 𝑋 as time continues to pass. Therefore, the rates of change are equal to zero. Hence: 𝑑𝑀 𝑑𝑡 = 0, 𝑑𝑆 𝑑𝑡 = 0, 𝑑𝐸 𝑑𝑡 = 0, 𝑑𝐼 𝑑𝑡 = 0, 𝑑𝑅 𝑑𝑡 = 0, 𝑑𝑋 𝑑𝑡 = 0 Therefore: 𝜇 − 𝛿𝑀 − 𝜇𝑀 = 0 (24) −𝛽𝑆𝐼 + 𝛿𝑀 − 𝜅𝑆 − 𝜇𝑆 = 0 (25) 𝛽𝑆𝐼 − 𝜎𝐸 − 𝜇𝐸 − 𝜒𝐸 = 0 (26) 𝜎𝐸 − 𝛾𝐼 − 𝜇𝐼 − 𝜒𝐼 = 0 (27) 𝛾𝐼 + 𝜅𝑆 − 𝜇𝑅 = 0 (28) 𝜒𝐸 + 𝜒𝐼 − 𝜇𝑋 = 0 (29) Rearranging (24) to make 𝑀 the subject gives: 𝑀 = 𝜇 𝛿 + 𝜇 This shows that 𝑀 is independent of any other states, and is only dependent on the parameters given from the virus itself. More specifically, 𝑀 is independent of 𝐼, whether directly or indirectly through another state, so the 𝑀 state will be the same for both steady states, regardless whether an infection is present (𝐼 ≠ 0) or not (𝐼 = 0).
- 26. 25 Rearranging all of (25) to (29) to make 𝐼 the subject and substituting 𝑀 gives: 𝐼 = 𝛿𝜇 − ( 𝛿 + 𝜇)( 𝜅 + 𝜇) 𝑆 𝛽( 𝛿 + 𝜇) 𝑆 (30) 𝐼 = ( 𝜎 + 𝜇 + 𝜒) 𝐸 𝛽𝑆 (31) 𝐼 = 𝜎𝐸 𝛾 + 𝜇 + 𝜒 (32) 𝐼 = 𝜇𝑅 − 𝜅𝑆 𝛾 (33) 𝐼 = 𝜇𝑋 − 𝜒𝐸 𝜒 (34) If 𝐼 = 0 (no virus present), (30) ⟹ 𝑆 = 𝛿𝜇 ( 𝛿+𝜇)( 𝜅+𝜇) (32) ⟹ 𝐸 = 0 (33) ⟹ 𝑅 = 𝜅𝑆 𝜇 = 𝛿𝜅 ( 𝛿+𝜇)( 𝜅+𝜇) (34) ⟹ 𝑋 = 𝜒𝐸 𝜇 = 0 Checking in (17) (using Maple), 𝑀 = 𝜇 𝛿+𝜇 , 𝑆 = 𝛿𝜇 ( 𝛿+𝜇)( 𝜅+𝜇) , 𝐸 = 0, 𝐼 = 0, 𝑅 = 𝛿𝜅 ( 𝛿+𝜇)( 𝜅+𝜇) and 𝑋 = 0 satisfy 𝑀 + 𝑆 + 𝐸 + 𝐼 + 𝑅 + 𝑋 = 1: > > > > > > >
- 27. 26 This gives the trivial steady state: ( 𝑀, 𝑆, 𝐸, 𝐼, 𝑅, 𝑋) = ( 𝜇 𝛿 + 𝜇 , 𝛿𝜇 ( 𝛿 + 𝜇)( 𝜅 + 𝜇) , 0, 0, 𝛿𝜅 ( 𝛿 + 𝜇)( 𝜅 + 𝜇) , 0) (35) This steady state can be simplified even further – no infection in the population makes vaccinations unnecessary. So we can set 𝜅 = 0, causing everyone to stay in the 𝑀 or 𝑆 states, with no-one moving to the 𝑅 state via the infection or vaccination. This makes the steady state become: ( 𝑀, 𝑆, 𝐸, 𝐼, 𝑅, 𝑋) = ( 𝜇 𝛿 + 𝜇 , 𝛿 𝛿 + 𝜇 , 0, 0, 0, 0) If 𝐼 ≠ 0 (virus present), then Maple confirms the following solution (along with the trivial steady state) (solution omitted due to size, see Appendix A2): > Checking in (17), 𝑀 = 𝑀∗ , 𝑆 = 𝑆∗ , 𝐸 = 𝐸∗ , 𝐼 = 𝐼∗ , 𝑅 = 𝑅∗ and 𝑋 = 𝑋∗ (see Appendix A2) satisfy 𝑀 + 𝑆 + 𝐸 + 𝐼 + 𝑅 + 𝑋 = 1 (manually checked). This solution along with some manual working leads to the non-trivial endemic steady state (see Appendix A2): ( 𝑀, 𝑆, 𝐸, 𝐼, 𝑅, 𝑋) = ( 𝑀∗ , 𝑆∗ , 𝐸∗ , 𝐼∗ , 𝑅∗ , 𝑋∗) (36) 3.2.5. Finding 𝑹 𝟎 For a closed population (which again, we have here), the critical value at which the virus becomes an epidemic is when it exceeds 1 𝑆 . So, for this model:
- 28. 27 𝑅0 = 1 𝑆 = 𝛽𝜎 𝜎𝛾 + ( 𝜇 + 𝜒)( 𝜎 + 𝛾 + 𝜇 + 𝜒) If 𝑅0 is above 1, the theory suggests that an epidemic starts: 𝑅0 = 𝛽𝜎 𝜎𝛾 + ( 𝜇 + 𝜒)( 𝜎 + 𝛾 + 𝜇 + 𝜒) > 1 ⟹ 𝛽𝜎 > 𝜎𝛾 + ( 𝜇 + 𝜒)( 𝜎 + 𝛾 + 𝜇 + 𝜒) (37) Interpreting this inequality, again, the simplest way of reducing the epidemic is to reduce the transmission rate (as this only appears on the left hand side so reducing this quantity will not affect the right hand side) or increasing the mortality rate or recovery rate (as these only appear on the right hand, so similar to above). As before, increasing the mortality rate is unethical and the recovery rate is generally unchangeable, along with the latency period in this new model due to the properties of the disease. Hence reducing 𝛽, so in turn, reducing the contact rate is the best way to prevent an epidemic in this model. Notice that the vaccination number does not appear in the value of 𝑅0, so the rate at which we vaccinate people has no effect of preventing the actual outbreak of an epidemic. It appears as if this value only helps prevent the spread once an epidemic has begun. Keeping those affected by measles away from other individuals in the population is by far the most effective way of reducing the spread. 3.2.6. Analysing the data Assuming a standard scenario for measles (an average rate of spread), we take an estimate for the basic reproduction number, 𝑅0, as between 12 and 18 for the United States (CDC 2014). By taking the mid-range value of 15, we can model for the average spread in the United States (US). The latency period, 1 𝜎 , is given as seven to fourteen days (CDC 2009c), so we take the average of 10.5 days. The recovery period, 1 𝛾 , is reported as three to five days plus a few days for the virus to completely subside (CDC 2009c), so again, we take the average of approximately seven days. Average life expectancy at birth in the US is approximately 79 years (WHO 2012), so this is taken to be the value of 1 𝜇 . Individuals lose maternal immunity sometime between twelve and fifteen months (Nicoara 1999), so we take the average of 13.5
- 29. 28 months. Individuals are vaccinated at approximately 91.9% coverage per year (CDC 2013) with a CI of (90.2% − 92.0%), so we can then work out a rate per day for 𝜅. Approximately one or two individuals per every thousand die directly due to measles (CDC 2009b) so we take the average of 1.5 per thousand and then convert this for the parameter 𝜒. All parameters are rates, so all must be converted to the same units. Numerical analysis will be taken at daily time periods with step size ℎ = 1 5 over the period of 365 days, so all parameters are multiplied or divided as needed to give each value in terms of days. We can substitute these values into 𝑅0 = 𝛽𝜎 𝜎𝛾+( 𝜇+𝜒)( 𝜎+𝛾+𝜇+𝜒) and rearrange, giving us a 𝛽 value (transmission rate) of 2.14431 (six significant figures). Again, this system has been analysed using the 4th-order Runge-Kutta method, with an initial maternally immune proportion, 𝑀(0) of 0.1, an initially susceptible proportion, 𝑆(0) of 0.899 and infected proportion, 𝐼(0) of 0.001 as a starting point (see Appendix B2 for data). Figure 3.6.: Measles 4th-order Runge-Kutta numerical analysis (with steady state values) As Fig. 3.6. shows, like chickenpox, the measles outbreak eventually subsides but again produces a worrying epidemic initially, and this time, far worse than chickenpox. After around only ten days, the exposed and infectious proportion starts to rise rapidly and then 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 50 100 150 200 250 300 350 400 Proportionofpopulation Time (days) M S E I R X M* S* E* I* R* X*
- 30. 29 both peaking just before day 25 or so at around 45% and 20% respectively, giving approximately 65% of the population having measles at this point. This would most certainly be classified as another epidemic in the population. The data again shows that a majority of people have the virus or have been vaccinated as early as day 40, as half the population have entered the recovered state at this point. From the year-long graph, it shows the levels of all compartments fluctuate somewhat, with 𝑀 always decreasing, 𝑆 mostly decreasing, 𝑅 mostly increasing and 𝑋 always increasing but only slightly, and 𝐸 and 𝐼 showing the peak where the outbreak is at its worst before lowering back down again. Like before and as predicted by the steady state, the all states eventually level out in the long-term, moving toward the respective dotted lines shown, giving a steady proportion of the population who are present in each state. By substituting in the values for 𝛽, 𝜎, 𝛾, 𝛿, 𝜇, 𝜒 and 𝜅 in (𝑀∗ , 𝑆∗ , 𝐸∗ , 𝐼∗ , 𝑅∗ , 𝑋∗ ) and using Maple to work them out (see Appendix A3), we can find out the behaviour of the model analytically as 𝑡 tends to infinity (also shown in Appendix B2 as 𝑡 → ∞): (𝑀∗ , 𝑆∗ , 𝐸∗ , 𝐼∗ , 𝑅∗ , 𝑋∗ ) = (0.0111266,0.0666666,0.000159502,0.000106306,0.921910,0.0000314982) So eventually, there is a constant 6.7% of the population that are susceptible, 92.2% that have had chickenpox at some point and recovered or have been vaccinated, and a negligible amount suffering from chickenpox at any time (< 0.026%). Approximately 0.0031% of the population will have died from measles in a constant long-term population and about 1.1% will have maternal immunity at any time.
- 31. 30 3.3. Model3 – H1N1 Finally, a disease with a standard SEIR basis is proposed for analysis but with added loss of natural immunity; that is, individuals can ‘loop’ in the model and gain the disease more than once after contracting the virus or being vaccinated. H1N1, known fully as influenza A (Haemagglutinin type 1-Neuraminidase type 1) or popularly (and incorrectly) as swine flu (Flu.gov 2014), has exactly this property and exhibits a once-per-lifetime (on average) mass- outbreak rate, which allows for easier analysis. The last two recorded major outbreaks were in 2009 and 1918 (known as Spanish flu). This disease is still complex to model as individuals can gain H1N1 twice or more and like measles, it can directly cause deaths in the human population. Symptoms of measles are that of seasonal flu, but much more extreme (Flu.gov 2014). The main contagious ways of passing the disease on is by coughing or sneezing, which directly enters another person’s body or is transmitted via surfaces and improper hygiene routines (CDC 2009d). Being in close proximity with individuals who have flu or related symptoms is the main cause of spread, so removing contact and making sure individuals get regular flu vaccinations will lower the chance that anyone will become sick. 3.3.1. Compartmental model We can represent measles with the most complex compartmental model presented in this research; that being, one that includes one extra state besides the standard three but allows for ‘looping’. This is the SEIR model again, but includes a rate that allows recovered individuals can become susceptible again, making it a slightly adjusted model known as the SEIRS model. This fits with H1N1, as individuals will have immunity from vaccinations or earned immunity from catching the disease for a time; individuals who are vaccinated are only those in the susceptible state and will move into a recovered position after the immunisation (again, all vaccinations are assumed to be 100% effective at the time of immunisation). After looking at how the infection spreads, we construct our final black-box model (Fig. 3.7. and Fig. 3.8.):
- 32. 31 Figure 3.7.: Compartmental model for H1N1 Or more mathematically: Figure 3.8.: Compartmental model for H1N1 with mathematical symbols When mathematically representing this model, we need to create three new states. Firstly, just as before, a new death state, 𝑋, appears to account for those who die due to the disease only. Secondly, a ‘bin’ state, 𝐵, is created on the right of the model to account for individuals who lose their active immunity and get H1N1 more than once. We could continue to represent this with a loop (as in the previous diagram) but it would be impossible to analyse exactly how 𝛿 𝜅 𝛾𝜎𝛽𝐼 𝜒 𝑆 𝐸 𝐼 𝜇 𝑅 𝜇 𝜒 𝜇 𝜇 𝜇 𝛿 𝑋 𝐷 𝐵 Loss of active immunity Gain of active immunity (vaccination) Recovery rate Development rate Transmission rate Virus Deaths Susceptible Exposed Infectious Births Recovered Natural Deaths Natural Deaths Natural Deaths Virus Deaths Natural Deaths
- 33. 32 many cases had occurred, let alone how many individuals got the disease at some point (at least once). The individuals who enter the ‘bin’ state are replaced by another identical individual in the system from a ‘dummy’ state, 𝐷, entering the susceptible state where they are vulnerable to the disease again. The rate that individuals leave the 𝑅 state to enter 𝐵 is exactly the same at which they leave the 𝐷 state and enter 𝑆, and is modelled to only allow each individual to leave 𝐷 and enter 𝑆 when the corresponding individual has left 𝑅 to enter 𝐵. The loop at the top is therefore removed, and the corresponding rate, 𝛿, is placed on the arrows in question. The dotted line along the bottom of the diagram represents that 𝐵 affects 𝐷 but no individual actually travels between the two states. No deaths occur in these states (as the individuals who die when in the ‘bin’ state are accounted for elsewhere in the system by another state) and the two new states are not needed to be included in the sum for constant proportionality (see final assumption in section 3.3.2.). 3.3.2. Assumptions The following assumptions can be made about this model with this disease. Again, the aim of these assumptions is to make the model viable and to enable a solution to be found, when at the same time not taking away too much from the original traits of the disease and general complexity of the disease transmission. This is the most complex of the three models (due to the possibility of ‘cycling’ round the model via loss of active immunity) so the assumptions below are comparatively more simple. No individual is born with the virus (all individuals initially enter the susceptible state). The population has a life expectancy of 𝜇, so that the natural mortality rate is 1 𝜇 . A closed population, so that the birth rate is also 1 𝜇 . The virus’ mortality rate is 1 𝜒 . The virus has a latency period of 1 𝜎 .
- 34. 33 Vaccinations are only administered to susceptible individuals, not those who may have the virus but are not showing symptoms or capable or passing the virus on (exposed). Vaccinations are assumed to be 100% effective at the moment they are administered (however individuals who receive it could lose immunity and catch H1N1 at another point in the future). Deaths caused by the virus have the same prevalence in both the exposed and infectious individuals (virus mortality is the same whether you are exposed or infectious). As before, to allow for the constant proportionality of the model, those who die of the disease must be modelled to be allowed to ‘die again’ naturally. Although, this would not happen in real life, it allows the model to maintain the same amount of individuals in the system and those who do ‘die again’ from the virus death state will ‘re-enter’ as an individual with maternal immunity. The low rate of virus and natural deaths compared to the overall length of a long-term analysis makes this error relatively small, albeit still present. A constant proportional population implies the individuals in the system add up to 1 (individuals either are susceptible, exposed, infectious, have recovered or have died from H1N1, and nothing else, with those in the bin or dummy state being accounted for somewhere else in one of the compartments): 𝑆 + 𝐸 + 𝐼 + 𝑅 + 𝑋 = 1 (38) where 𝑆( 𝑡), 𝐸( 𝑡), 𝐼( 𝑡), 𝑅( 𝑡), 𝑋( 𝑡), 𝐷( 𝑡), 𝐵( 𝑡) ≥ 0, ∀𝑡 and we simplify 𝑆( 𝑡) = 𝑆, 𝐸( 𝑡) = 𝐸, 𝐼( 𝑡) = 𝐼, 𝑅( 𝑡) = 𝑅, 𝑋( 𝑡) = 𝑋, 𝐷( 𝑡) = 𝐷 and 𝐵( 𝑡) = 𝐵. 3.3.3. Representing the model Using these assumptions, again with 𝛽, the transmission rate being the product of the contact rate and probability of a successful transmission, 𝜎, the development rate, 𝛾, the recovery rate, 𝛿, the rate at which individuals lose their (active) immunity, and 𝜅, the constant number of individuals immunised, we can construct the following ODEs to represent the rate of change of each compartment in the model:
- 35. 34 𝑑𝑆 𝑑𝑡 = 𝜇 − 𝛽𝑆𝐼 + 𝛿𝑅 − 𝜅𝑆 − 𝜇𝑆 (39) 𝑑𝐸 𝑑𝑡 = 𝛽𝑆𝐼 − 𝜎𝐸 − 𝜇𝐸 − 𝜒𝐸 (40) 𝑑𝐼 𝑑𝑡 = 𝜎𝐸 − 𝛾𝐼 − 𝜇𝐼 − 𝜒𝐼 (41) 𝑑𝑅 𝑑𝑡 = 𝛾𝐼 − 𝛿𝑅 + 𝜅𝑆 − 𝜇𝑅 (42) 𝑑𝑋 𝑑𝑡 = 𝜒𝐸 + 𝜒𝐼 − 𝜇𝑋 (43) 𝑑𝐷 𝑑𝑡 = −𝛿𝑅 (44) 𝑑𝐵 𝑑𝑡 = 𝛿𝑅 (45) where 𝛽, 𝜎, 𝛾, 𝛿, 𝜇, 𝜒, 𝜅 > 0. The final two equations here ((44) and (45)) are not included in the constant proportionality equation (38) nor are they included in the solutions to the system (using Maple or otherwise). This is because setting those equations to zero would imply the 𝑅 state must be zero at any trivial or endemic state found. Although this does make sense in long term as everyone would eventually leave the 𝑅 state one way or another, either by dying naturally and then being replaced by a new individual in the system to account for the equal birth and death rates, or by losing their active immunity and ‘looping’ back to enter the system as a susceptible individual. As the terms in (44) and (45) are repeated from other equations, the system still holds that the sum of equations (39) to (43) is one. The system would also be unsolvable for 𝐷 or 𝐵 and these states do not appear explicitly in the system of ODEs (an infinite number of solutions). These final two equations are only included when solving the system numerically using the 4th-order Runge-Kutta method (see Appendix B3). The values of 𝐷 and 𝐵 will be a fraction of 𝑅 (multiplied by negative and positive delta respectively) and 𝐵 specifically will show the proportion of individuals having the possibility of getting the H1N1 virus more than once.
- 36. 35 3.3.4. Finding the steady states Again, at the steady states to this system of equations, there will be no change to the proportions represented by 𝑆, 𝐸, 𝐼, 𝑅 and 𝑋 as time continues to pass. Therefore, the rates of change are equal to zero. Hence: 𝑑𝑆 𝑑𝑡 = 0, 𝑑𝐸 𝑑𝑡 = 0, 𝑑𝐼 𝑑𝑡 = 0, 𝑑𝑅 𝑑𝑡 = 0, 𝑑𝑋 𝑑𝑡 = 0 Therefore: 𝜇 − 𝛽𝑆𝐼 + 𝛿𝑅 − 𝜅𝑆 − 𝜇𝑆 = 0 (46) 𝛽𝑆𝐼 − 𝜎𝐸 − 𝜇𝐸 − 𝜒𝐸 = 0 (47) 𝜎𝐸 − 𝛾𝐼 − 𝜇𝐼 − 𝜒𝐼 = 0 (48) 𝛾𝐼 − 𝛿𝑅 + 𝜅𝑆 − 𝜇𝑅 = 0 (49) 𝜒𝐸 + 𝜒𝐼 − 𝜇𝑋 = 0 (50) Rearranging all of (46) to (50) to make 𝐼 the subject gives: 𝐼 = 𝜇 + 𝛿𝑅 − ( 𝜅 + 𝜇) 𝑆 𝛽𝑆 (51) 𝐼 = ( 𝜎 + 𝜇 + 𝜒) 𝐸 𝛽𝑆 (52) 𝐼 = 𝜎𝐸 𝛾 + 𝜇 + 𝜒 (53) 𝐼 = ( 𝛿 + 𝜇) 𝑅 − 𝜅𝑆 𝛾 (54) 𝐼 = 𝜇𝑋 − 𝜒𝐸 𝜒 (55)
- 37. 36 If 𝐼 = 0 (no virus present), (53) ⟹ 𝐸 = 0 (55) ⟹ 𝑋 = 𝜒𝐸 𝜇 = 0 Rearranging (54) to give 𝑅 in terms of 𝑆 and then substituting into (51) gives: 𝑅 = 𝜅𝑆 𝛿+𝜇 ⟹ 0 = 𝜇+𝛿( 𝜅𝑆 𝛿+𝜇 )−( 𝜅+𝜇) 𝑆 𝛽𝑆 ⟹ 𝑆 = 𝛿+𝜇 𝛿+𝜅+𝜇 ⟹ 𝑅 = 𝜅 𝛿 + 𝜅 + 𝜇 Checking in (38) (using Maple), 𝑆 = 𝛿+𝜇 𝛿+𝜅+𝜇 , 𝐸 = 0, 𝐼 = 0, 𝑅 = 𝜅 𝛿+𝜅+𝜇 and 𝑋 = 0 satisfy 𝑆 + 𝐸 + 𝐼 + 𝑅 + 𝑋 = 1: > > > > > > This gives the trivial steady state: ( 𝑆, 𝐸, 𝐼, 𝑅, 𝑋) = ( 𝛿 + 𝜇 𝛿 + 𝜅 + 𝜇 , 0 ,0, 𝜅 𝛿 + 𝜅 + 𝜇 , 0) (56) Again, this steady state can be simplified further – as before, no infection in the population makes vaccinations unnecessary. So we can set 𝜅 = 0, causing everyone to stay in the 𝑆 state, with no-one moving to the 𝑅 state via the infection or vaccination. This makes the steady state become: ( 𝑆, 𝐸, 𝐼, 𝑅, 𝑋) = (1, 0, 0,0, 0)
- 38. 37 If 𝐼 ≠ 0 (virus present), then Maple confirms the following solution (along with the trivial steady state, solution omitted due to size, see Appendix A4): > Checking in (38), 𝑆 = 𝑆∗ , 𝐸 = 𝐸∗ , 𝐼 = 𝐼∗ , 𝑅 = 𝑅∗ and 𝑋 = 𝑋∗ satisfy 𝑆 + 𝐸 + 𝐼 + 𝑅 + 𝑋 = 1 (manually checked). This solution along with some manual working leads to the non-trivial endemic steady state (see Appendix A4): ( 𝑆, 𝐸, 𝐼, 𝑅, 𝑋) = ( 𝑆∗ , 𝐸∗ , 𝐼∗ , 𝑅∗ , 𝑋∗) (57) 3.3.5. Finding 𝑹 𝟎 For a closed population (which again, we have here), the critical value at which the virus becomes an epidemic is when it exceeds 1 𝑆 . So, for this model: 𝑅0 = 1 𝑆 = 𝛽𝜎 𝜎𝛾 + ( 𝜇 + 𝜒)( 𝜎 + 𝛾 + 𝜇 + 𝜒) If 𝑅0 is above 1, the theory suggests that an epidemic starts: 𝑅0 = 𝛽𝜎 𝜎𝛾 + ( 𝜇 + 𝜒)( 𝜎 + 𝛾 + 𝜇 + 𝜒) > 1 ⟹ 𝛽𝜎 > 𝜎𝛾 + ( 𝜇 + 𝜒)( 𝜎 + 𝛾 + 𝜇 + 𝜒) (58)
- 39. 38 This gives the exact same 𝑅0 value as the previous measles model, due to similar traits of the diseases and models (vaccination and immunity loss). Interpreting this inequality, again, by reducing the transmission rate or increasing the mortality rate or recovery rate, we can cause a disease to die out in this instance. Just as before, increasing the mortality rate is unethical and the recovery rate is generally unchangeable, along with the latency period in this new model due to the properties of the disease. Hence reducing 𝛽, so in turn, reducing the contact rate is the best way to prevent an epidemic in this model. Notice here that again the vaccination number does not appear in the value of 𝑅0, so the rate at which we vaccinate people has no effect of preventing the actual outbreak of an epidemic. It appears as if this value only helps prevent the spread once an epidemic has begun. Keeping those individuals with the H1N1 virus away from others is clearly yet again the most effective way of reducing the spread of any endemic that may occur. 3.3.6. Analysing the data Assuming a standard scenario for H1N1 (an average rate of spread with one epidemic per lifetime), we take an estimate for the basic reproduction number, 𝑅0, as 2.6 for the US (Barry 2009). The latency period, 1 𝜎 , is given between one and four days (Balcan et al. 2009, CDC 2010b), so we take the average of 2.5 days. The recovery period, 1 𝛾 , is reported as three to five days (Asp 2009), so again, we take the average of approximately four days. Average life expectancy at birth is the same as before in the US; 79 years (WHO 2012), so this is taken to be the value of 1 𝜇 . Individuals lose active immunity at around 5% per year (Greenberg et al. 2009), so we take this for our value of 𝛿. Individuals are vaccinated for H1N1 and flu at approximately 91 million vaccinations per year (Drummond 2010), so accounting for a daily rate and the population size of the US, we can then work out a rate per day for 𝜅. Approximately 12270 individuals died due to the 2009 outbreak (CDC 2010a), so again, we can find the rate of death per day for the parameter 𝜒. All parameters are rates, so all must be converted to the same units. Again, numerical analysis will be taken at daily time periods with step size ℎ = 1 5 over the period of 365 days, so all parameters are multiplied or divided as needed to give each value in terms of days. We can substitute these values into 𝑅0 =
- 40. 39 𝛽𝜎 𝜎𝛾+( 𝜇+𝜒)( 𝜎+𝛾+𝜇+𝜒) and rearrange, giving us a 𝛽 value (transmission rate) of 0.650147(six significant figures). Again, this system has been analysed using the 4th-order Runge-Kutta method, with an initial maternally immune proportion, 𝑆(0) of 0.999 and infected proportion, 𝐼(0) of 0.001 as a starting point (see Appendix B3 for data). This analysis includes 𝐵 and 𝐵∗ to show the rate at which individuals loop round the system. Figure 3.9.: H1N1 4th-order Runge-Kutta numerical analysis (with steady state values) As Fig. 3.9. shows, like the two diseases before, the H1N1 influenza outbreak eventually subsides but again, a worrying epidemic ensues. After around twenty days, the exposed and infectious proportion starts to rise rapidly and then both peaking at day 40 at around 12% and 18% respectively, giving approximately 20% of the population having this flu at this point. This would certainly be classified as another epidemic in the population. The data shows that a majority of people have the virus or have been vaccinated as early as day 50, as half the population have entered the recovered state at this point, however, the level of recovered people quickly drops back below 50% by about day 65. This is due to the fact that individuals are constantly losing active immunity to H1N1. From the year-long graph, it shows the levels of all compartments fluctuate somewhat but all have hit their respective 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 50 100 150 200 250 300 350 400 Proportionofpopulation Time (days) S E I R X B S* E* I* R* X* B*
- 41. 40 steady level by about day 130. This type of disease is a faster-hitting but quicker-settling one, due to the traits of the virus and how it interacts within a population. By substituting in the values for 𝛽, 𝜎, 𝛾, 𝛿, 𝜇, 𝜒 and 𝜅 in (𝑆∗ , 𝐸∗ , 𝐼∗ , 𝑅∗ , 𝑋∗ ) and using Maple to work them out (see Appendix A5), we can find out the behaviour of the model analytically as 𝑡 tends to infinity (also shown in Appendix B3 as 𝑡 → ∞): (𝑆∗ , 𝐸∗ , 𝐼∗ , 𝑅∗ , 𝑋∗ ) = (0.384615,0.0574299,0.0918750,0.465119,0.000961296) Therefore eventually, there is a constant 38.5% of the population that are susceptible (through not having the virus or having it at least once and losing their active immunity), 46.5% that have had H1N1 at some point and then either recovered or have been vaccinated, but this time a significant amount of people suffering from it at any time (1.5%). Approximately 0.096% of the population will have died from H1N1 in a constant long-term population, proving it to be the most deadly of the three diseases in this research.
- 42. 41 4. Overview of deterministic models All three models show stable, long-term, analytical solutions. In comparison to the actual data from real life outbreaks, two of the three models hold up particularly well (Table 7.1. and Table 7.2.) and the latter two models predict some amount of death which can be used to help discuss the extent at which vaccination or removal from the population needs to be implemented. However, can these models be improved upon to provide better estimates? Or are there other ways to represent the same information to give the same output? Are there any advantages to doing this? All dynamical systems that are represented in a similar way to those above are subject to the ‘Evolution Rule’. That is, if the state currently occupied in the system has pre-defined rate at which the next state is entered, we define the system as deterministic. Whereas, if the change in state is given as a probability, so that there is a chance that the subject in question can go to one state or another, or even more than that, and not split off into both or any subset of them, this is defined as a stochastic (or random) system (Meiss 2007). For the same initial starting point, analysed at the same point in discrete time (whether a finite time period or an infinite time steady state value), the standard deterministic model will always give the same output, every time. This is due to the defined rates of the system, the parameters calculated before analysis and the very same ones used in the analytical solution to the ODEs. The stochastic model can, however, account for random processes which can occur during analysis. This is done by calculating non-deterministic probabilities which are then applied as transition probabilities between each state. Some of these may be one – forcing a state change in that situation, whereas some could be zero – meaning there is no chance of moving to the state in question. Other probabilities must between these two values, which is where the random chance element comes into play. For example, if an individual is in a state which has two exits; one to a state with probability of a third and the other with transition probability of two thirds, it is unknown at the next step which state the individual will be in. Of an overall population, though, the differences between analyses can be cancelled out to give a rather (hopefully) accurate description of the overall behaviour of a system on a large scale.
- 43. 42 One of the main problems with deterministic models is as stated before; the lack of accountability for chance from the same starting point. During different time periods, diseases can have different rates of infection and this makes it difficult to decide on exactly what the rate involved will be (Yorke and London 1973). This also is a problem when analysing diseases which have quiet periods and then sudden epidemics or those which have not occurred in a long time. Current deterministic models of some populations exhibiting certain diseases have problems when the population dynamics are not standard; for example, feral pig population dynamics in Australia vary over time and in space due to the populations searching for optimal feeding ground (Dexter 2003). This highly affects the outcomes of a deterministic system, such as when this population is suffering from food and mouth disease (FMD) and a model needs to account for exactly what point the dynamics are at and how they may change. This can be represented by probabilistic elements, hence the application of a stochastic model. However, applying these elements into the standard setup of deterministic ordinary differential equations by Pech and Hone (1988) and Pech and McIlroy (1990) turned out to be too complex – a stable equilibrium point was never reached for the model (Dexter 2003). This was mainly due to the fact that the feral pig dynamics vary wildly depending on two other factors – vegetation density and kangaroo density, alongside their own population’s density. Simulation of these densities when random processes can affect the outcome left a constant outbreak of FMD present in the system. Evaluation of the system leads to a suggestion that the combination of both deterministic and probabilistic elements side-by-side causes complex behaviour that is difficult to model and get meaningful and clear results from. Dexter agrees that the untested assumptions of his model could substantially change the behaviour in the dynamics – an intrinsic problem of all deterministic models. This is why I wish to test a purely stochastic approach. This way, I will have two comparable models on each disease, where any overlapping features that could cause common errors or deviations between the models are eliminated as much as possible. This will allow a comparison of which method seems better or easier, without underlying factors that are inherent in any sort of hybrid model becoming significant. Höhle et al. (2005) state that they wanted to take an SEIR model and extend it stochastically to analyse previously undertaken disease transmission experiments in a more detailed way. They do this by taking existing work on diseases involving Markov chains and applying it to
- 44. 43 data from Belgian classical swine fever virus (CSFV). Like Dexter (2003), but this time with a more standard SEIR model comparable to those presented earlier in this research, spatial elements are considered to account for dynamics within the populations affected by CSFV. This model gets very complex very quickly, and two models are proposed – one that deals with data with missing time elements and one that only looks at data with complete entries for all fields. The first model has stochastic elements which allow for contact heterogeneity; that is that not all contacts between pigs are the same and an element of probability is involved in whether the disease transmits or not. The results are positive – the contact heterogeneity model has a better statistical test outcome than the standard one proving that the added complexity has provided better reliability. The second model also shows a promising improvement on the standard deterministic-only model, but lacks the robustness of the first model; some data had to be estimated or based off less applicable data than the first model. Although the results show a good and efficient outcome in comparison to data, I argue that the complexity of both models outweigh the effectiveness of the results. That is, is the extra data and analysis required worth the relatively small improvements in reliability of data? Obviously, there is a place for these models – anything that improves on something previously existing is worth using, but sometimes, simplicity can be the best approach. Following on from my previous work, I wish to use only probabilistic elements to model the flow of an individual around a system. This will hopefully provide a similarly simple model to the deterministic ones above, if not even simpler, yet still show the essence of the disease that they are modelling.
- 45. 44 5. Stochastic Processes – an overview Stochastic processes are similar to traditional mathematical models, in the way that they attempt to represent some real-world problem in a way that can be understood, adapted and solved to provide a solution to the initial question. However, they differ in a major way – the analysis of the system. Where traditional deterministic models allow the parameters in the system to depend on time and or space as required, stochastic and probabilistic systems do not rely on time intrinsically, instead conditional on just the probabilities of any one event randomly occurring to an individual at any one time. Here, the time must be split up into distinct, discrete chunks to allow for the analysis of the system at any given point in time. Any system which must be or is chosen to be analysed using probabilistic theory in some way is classified as stochastic (Nelson 1985), and many standard application revolve around quantum theory; the underlying rules of physics which are based entirely on probabilistic chance. This leads to a few definitions which will allow the implementation of this method to analyse disease transmission: A random walk (RW) is defined as (Urwin 2011b): “A random process in discrete time steps { 𝑋 𝑛: 𝑛 = 0,1,2,… } such that it is only possible to move forward, backward or remain in the same state, always with the same probability.” A Markov chain (MC) is further defined as (Urwin 2011b): “A stochastic process in discrete time steps { 𝑋 𝑛: 𝑛 = 0,1,2, … } with either a finite or infinite state space, which has both the Markov property (MP) and the stationarity property (SP).” The Markov property (MP) is that the currently occupied state is only dependant on the immediately occupied previous state, and the stationarity (or homogeneity) property (SP) says that the probability of transitioning from any one state to another (or to itself) stays the same no matter which step the process is at (Urwin 2011b).
- 46. 45 Therefore, any model consisting of a finite number of states, with constant probabilities and state transitions only relating between the originating and target states in question can be represented using a Markov chain. Looking at the original three compartmental models above, if the natural and virus mortality arrows become targets for new states alongside if the deterministic parameters that link each compartment are replaced with an assumed constant probability (adding in self targetting probability arrows) for the transition between the two states, then this model becomes a Markov chain. No state is needed for births as this method looks at each individual separately, so all individuals can be assumed to be born and susceptible to the disease from the outset. Each compartmental model needs one compartment for each ‘alive’ state; 𝑀, 𝑆, 𝐸, 𝐼 and 𝑅 are examples used in this research, and models not considered here that involved other compartments would need extra (such as for the carrier state 𝐶 in models which account for individuals who can pass on an infection without ever suffering from it themselves). Compartments are also needed for each ‘death’ state. These death states could be merged into one as once an individual enters a death state, they cannot leave, but for easier reading of results later, the death states are kept separate (except for the model presented in 6.3.). This is so the results clearly show the percentage of individuals who died having had the disease at one point, alongside rather than merged with the percentage who died never having caught it. If there are 𝑚 states, let 𝑝𝑖𝑗 be the probability of moving from state 𝑖 to state 𝑗 in a specified time interval, where 𝑖, 𝑗 ∈ {1,2, … , 𝑚}. This implies that the total of the values given on all of the arrows leaving any state must add exactly to one (Urwin 2011a): ∑ 𝑝𝑖𝑗 = 1, ∀𝑖 𝑚 𝑗=1 Some states are known as absorbing states. These are states that once entered cannot be left. This implies two things for an absorbing state 𝑖: 𝑝𝑖𝑖 = 1 and: 𝑝𝑖𝑗 = 0, ∀𝑖 ≠ 𝑗
- 47. 46 In the following models, all death states are clearly absorbing – once and individual enters them, they will stay in them throughout the entirety of the rest of the analysis (Urwin 2011a). Each compartmental model can be represented by something called a transition matrix, 𝑃. This is a square matrix made up of 𝑚 rows and 𝑚 columns, representing each of the finite 𝑚 states. By definition, just as above, each row of 𝑃 must add to one. Each element of the matrix gives the probability of leaving a state and entering a new one (or staying in the same state) in one discrete time step. That is, the element in the 𝑖 𝑡ℎ row and 𝑗 𝑡ℎ column of 𝑃 is the probability of leaving state 𝑖 and entering state 𝑗 (Urwin 2011a). The transition matrix for each model will look like this: 𝑃 = ( 𝑝11 𝑝12 ⋯ 𝑝1𝑚 𝑝21 𝑝22 ⋯ 𝑝2𝑚 ⋮ ⋮ ⋱ ⋮ 𝑝 𝑚1 𝑝 𝑚2 ⋯ 𝑝 𝑚𝑚 ) To measure the probability that any individual in this system is in state 𝑗 after 𝑛 steps given that they started in state 𝑖 can be calculated using the transition matrix. Matrix multiplication allows us to find any probability we wish. If 𝑃 is raised to the 𝑛𝑡ℎ power, the following matrix is produced: 𝑃 𝑛 = ( 𝑝11 𝑝12 ⋯ 𝑝1𝑚 𝑝21 𝑝22 ⋯ 𝑝2𝑚 ⋮ ⋮ ⋱ ⋮ 𝑝 𝑚1 𝑝 𝑚2 ⋯ 𝑝 𝑚𝑚 ) 𝑛 = ( 𝑝11 ( 𝑛) 𝑝12 ( 𝑛) ⋯ 𝑝1𝑚 ( 𝑛) 𝑝21 ( 𝑛) 𝑝22 ( 𝑛) ⋯ 𝑝2𝑚 ( 𝑛) ⋮ ⋮ ⋱ ⋮ 𝑝 𝑚1 ( 𝑛) 𝑝 𝑚2 ( 𝑛) ⋯ 𝑝 𝑚𝑚 ( 𝑛) ) Now, 𝑝𝑖𝑗 ( 𝑛) is not the same as 𝑝𝑖𝑗 𝑛 , the probability 𝑝𝑖𝑗 raised to the 𝑛𝑡ℎ power. 𝑝𝑖𝑗 ( 𝑛) is denoted as the probability of being in state 𝑗 after 𝑛 steps given that the individual started in state 𝑖. If we wish to know what state someone will be in after 2 steps, for example, we calculate 𝑃2 and then look at the appropriate element. Each of the following models will work out the probability an individual has the disease in question after one year, or 365 days. Each of the probabilities in 𝑃 will be calculated based on a daily probability, so each transition matrix will
- 48. 47 be raised to the 365 𝑡ℎ power to work out 𝑃365 (Urwin 2011b). Further multiplication will then be carried out to find out the limiting distribution (Urwin 2011c). The limiting distribution is the transition matrix 𝑃 raised to the 𝑛𝑡ℎ power, as 𝑛 → ∞ (Urwin 2011c). This will produce a matrix where any further matrix multiplication by any power of 𝑃 will not change the answer you get from the previous step: 𝑃 𝑛 ∗ 𝑃 𝑎 = 𝑃 𝑛+𝑎 = 𝑃 𝑛 , ∀𝑎 ≥ 1 This limiting distribution shows us when the disease will be steady in the population; where people will catch the disease at exactly the same rate as others are recovering. Obviously, in this model in this limiting matrix, each individual will end up dead in the long-term (end up in an absorbing state). But depending on which dead state they end up in shows us exactly what the probability of someone having the disease at some point was, and therefore, what the long-term steady proportion of any given population having the disease in question was.
- 49. 48 6. Stochastic Modelling – adaptations and analysis 6.1. Model1 adaptation– Chickenpox 6.1.1. Stochastic model The start of this method of infectious disease modelling includes a compartmental model, much in a way similar to the original deterministic model given in 3.1.1. However, instead of parameters given for each arrow, a probability of entering the state in question is given. This gives an initial compartmental model for chickenpox (Fig. 6.1. and Fig. 6.2.): Figure 6.1.: Stochastic model for chickenpox Susceptible Exposed Infectious Susceptible Deaths Exposed Deaths Infectious Deaths Recovered Recovered Deaths Probability of staying susceptible Probability of staying exposed Probability of staying infectious Probability of staying recovered Probability of infection Probability of development Probability of recovery Probability of staying dead Probability of staying dead Probability of staying dead Probability of staying dead Probability of natural death Probability of natural death Probability of natural death Probability of natural death
- 50. 49 Or more mathematically: Figure 6.2.: Stochastic model for chickenpox with mathematical symbols Arrows that are missing (such as 𝑝𝑆𝐼 ) imply that the probability of leaving state 𝑖 and entering state 𝑗 is zero (so 𝑝𝑆𝐼 = 0, for example). If at any time, an individual in state 𝑖 has the chance of staying in state 𝑖 (𝑝𝑖𝑖 > 0), then this is represented by a circular arrow that has the state it left as its target. 6.1.2. Transition matrix This can be represented in a square matrix, where each element corresponds to the relative probability of leaving the state and entering a new one, where the element in the 𝑖 𝑡ℎ row and 𝑗 𝑡ℎ column is the probability of leaving state 𝑖 and entering state 𝑗. Here, we need an eight-by- eight (8 × 8) matrix as we have eight states, so this gives us the matrix 𝑃 as follows: 𝑆 𝐸 𝐼 𝑊 𝑋 𝑌 𝑅 𝑍 𝑝𝑆𝑆 𝑝 𝐸𝐸 𝑝𝐼𝐼 𝑝 𝑅𝑅 𝑝𝑆𝐸 𝑝 𝐸𝐼 𝑝𝐼𝑅 𝑝 𝑌𝑌𝑝 𝑋𝑋 𝑝 𝑊𝑊 𝑝 𝑍𝑍 𝑝𝑆𝑊 𝑝𝐼𝑌𝑝 𝐸𝑋 𝑝 𝑅𝑍
- 51. 50 𝑃 = ( 𝑝𝑆𝑆 𝑝𝑆𝐸 𝑝𝑆𝐼 𝑝𝑆𝑅 𝑝𝑆𝑊 𝑝𝑆𝑋 𝑝𝑆𝑌 𝑝𝑆𝑍 𝑝 𝐸𝑆 𝑝 𝐸𝐸 𝑝 𝐸𝐼 𝑝 𝐸𝑅 𝑝 𝐸𝑊 𝑝 𝐸𝑋 𝑝 𝐸𝑌 𝑝 𝐸𝑍 𝑝𝐼𝑆 𝑝𝐼𝐸 𝑝𝐼𝐼 𝑝𝐼𝑅 𝑝𝐼𝑊 𝑝𝐼𝑋 𝑝𝐼𝑌 𝑝𝐼𝑍 𝑝 𝑅𝑆 𝑝 𝑅𝐸 𝑝 𝑅𝐼 𝑝 𝑅𝑅 𝑝 𝑅𝑊 𝑝 𝑅𝑋 𝑝 𝑅𝑌 𝑝 𝑅𝑍 𝑝 𝑊𝑆 𝑝 𝑊𝐸 𝑝 𝑊𝐼 𝑝 𝑊𝑅 𝑝 𝑊𝑊 𝑝 𝑊𝑋 𝑝 𝑊𝑌 𝑝 𝑊𝑍 𝑝 𝑋𝑆 𝑝 𝑋𝐸 𝑝 𝑋𝐼 𝑝 𝑋𝑅 𝑝 𝑋𝑊 𝑝 𝑋𝑋 𝑝 𝑋𝑌 𝑝 𝑋𝑍 𝑝 𝑌𝑆 𝑝 𝑌𝐸 𝑝 𝑌𝐼 𝑝 𝑌𝑅 𝑝 𝑌𝑊 𝑝 𝑌𝑋 𝑝 𝑌𝑌 𝑝 𝑌𝑍 𝑝 𝑍𝑆 𝑝 𝑍𝐸 𝑝 𝑍𝐼 𝑝 𝑍𝑅 𝑝 𝑍𝑊 𝑝 𝑍𝑋 𝑝 𝑍𝑌 𝑝 𝑍𝑍 ) Clearly, the probability of staying dead when already in a dead state is one, so: 𝑝 𝑊𝑊 = 𝑝 𝑋𝑋 = 𝑝 𝑌𝑌 = 𝑝 𝑍𝑍 = 1 And the probability of moving out of a dead state to any other must be zero, so: 𝑝 𝑊𝑆 = 𝑝 𝑊𝐸 = 𝑝 𝑊𝐼 = 𝑝 𝑊𝑅 = 𝑝 𝑊𝑋 = 𝑝 𝑊𝑌 = 𝑝 𝑊𝑍 = 0 𝑝 𝑋𝑆 = 𝑝 𝑋𝐸 = 𝑝 𝑋𝐼 = 𝑝 𝑋𝑅 = 𝑝 𝑋𝑊 = 𝑝 𝑋𝑌 = 𝑝 𝑋𝑍 = 0 𝑝 𝑌𝑆 = 𝑝 𝑌𝐸 = 𝑝 𝑌𝐼 = 𝑝 𝑌𝑅 = 𝑝 𝑌𝑊 = 𝑝 𝑌𝑋 = 𝑝 𝑌𝑍 = 0 𝑝 𝑍𝑆 = 𝑝 𝑍𝐸 = 𝑝 𝑍𝐼 = 𝑝 𝑍𝑅 = 𝑝 𝑍𝑊 = 𝑝 𝑍𝑋 = 𝑝 𝑍𝑌 = 0 There is no chance in this model that an individual can jump straight from being susceptible to being infectious, being susceptible to having recovered or from being exposed to having recovered, so: 𝑝𝑆𝐼 = 𝑝𝑆𝑅 = 𝑝 𝐸𝑅 = 0 Individuals also cannot travel backwards in the model, so: 𝑝 𝐸𝑆 = 𝑝𝐼𝑆 = 𝑝𝐼𝐸 = 𝑝 𝑅𝑆 = 𝑝 𝑅𝐸 = 𝑝 𝑅𝐼 = 0 Each compartment that is not a dead state has its own respective dead state; individuals who naturally die whilst susceptible go into the 𝑊 state, those who naturally die whilst exposed go into the 𝑋 state, those who naturally die whilst infectious go into the 𝑌 state and those who naturally die after having recovered go into the 𝑍 state. Therefore, for example, the probability of a susceptible individual dying and going into states 𝑋, 𝑌 or 𝑍 is zero (similar for states 𝐸, 𝐼 and 𝑅), so: 𝑝𝑆𝑋 = 𝑝𝑆𝑌 = 𝑝𝑆𝑍 = 0
- 52. 51 𝑝 𝐸𝑊 = 𝑝 𝐸𝑌 = 𝑝 𝐸𝑍 = 0 𝑝𝐼𝑊 = 𝑝𝐼𝑋 = 𝑝𝐼𝑍 = 0 𝑝 𝑅𝑊 = 𝑝 𝑅𝑋 = 𝑝 𝑅𝑌 = 0 The reasons for having separate dead states is to make the analysis slightly easier later – those who end up in states 𝑋, 𝑌 or 𝑍 must have had the infection at some point to end up in there, whereas those who end up in state 𝑊 have not. The remaining eleven probabilities have values which are neither zero (impossible) nor one (certain). These can be calculated (estimated) as follows (all probabilities are rounded only at the final answer and are rounded to six significant figures): The average life expectancy of an individual in the United Kingdom (UK) is currently 81 years (WHO 2012). As each step of this model accounts for one day, 81 years in days is 81 years ∗ 365 days per year = 29565 days. Hence, the reciprocal of this gives the probability of dying naturally each day during the analysis of this model. An assumption to be made here is that the chances of dying naturally are equal regardless of whether the individual in question is currently susceptible, exposed, infectious or has recovered from chickenpox, so: 𝑝𝑆𝑊 = 𝑝 𝐸𝑋 = 𝑝𝐼𝑌 = 𝑝 𝑅𝑍 = 1 29565 ≅ 0.0000338238 Annually in the UK, there are approximately 57 cases of chickenpox per 10,000 people presented to and recorded by doctors and practice nurses (Fleming et al. 2007 p.14). Although this report is not taken from all hospitals, over 52 million patients were analysed (Fleming et al. 2007 p.9) which is a high proportion of the UK population so it is representative. Extrapolating this up to the entire population of the UK will give us the estimated number of cases in one year. So there are approximately 57 ∗ 64100000 10000 = 365370 cases each year. Dividing this number by 365 will give the number of cases per day, so 365370 365 ≅ 1001.01 cases per day. This number further divided by the population size will give the probability of an individual contracting chickenpox on any given day, so:
- 53. 52 𝑝𝑆𝐸 = 1001.01 64100000 ≅ 0.0000156164 The latency period of chickenpox is given to be somewhere between ten and fourteen days (Knott 2013), so taking the average of twelve days estimates the value of the reciprocal of 𝑝 𝐸𝐼 , so: 𝑝 𝐸𝐼 = 1 12 = 0.083̅ ≅ 0.0833333 The recovery period of chickenpox once contagious is also given to be somewhere between ten and fourteen days (Lamprecht 2012), so again, taking the average of twelve days estimates the value of the reciprocal of 𝑝𝐼𝑅 , so: 𝑝𝐼𝑅 = 1 12 = 0.083̅ ≅ 0.0833333 Now, all rows of the matrix must add up to one, so the final stationary probabilities can all be worked out using the values we already know (given to full decimal places): 𝑝𝑆𝑆 = 1 − 0.0000156164 − 0.0000338238 = 0.9999505598 𝑝 𝐸𝐸 = 1 − 0.0833333 − 0.0000338238 = 0.9166328762 𝑝𝐼𝐼 = 1 − 0.0833333 − 0.0000338238 = 0.9166328762 𝑝 𝑅𝑅 = 1 − 0.0000338238 = 0.9999661762 Combining all of the above results give the following final transition matrix (given to appropriate significant figures): 𝑃 = ( 0.999951 1.56 × 10−5 0 0 3.38 × 10−5 0 0 0 0 0.916633 0.0833 0 0 3.38 × 10−5 0 0 0 0 0.916633 0.0833 0 0 3.38 × 10−5 0 0 0 0 0.999966 0 0 0 3.38 × 10−5 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 )
- 54. 53 6.1.3. 𝒏-step transition The square of this matrix will give the probability that an individual will be in any given state after one day. That is, the element in the 𝑖 𝑡ℎ row and 𝑗 𝑡ℎ column of 𝑃2 , denoted 𝑝𝑖𝑗 (2) , gives the probability that an individual will be state 𝑗 at the end of the second day, if they started in state 𝑖. Following on from this, working out 𝑃365 will give all the probabilities that an individual will end in any desired state, given any initial starting state, at the end of the 365 𝑡ℎ day, or one year. Hence, using Maple (omitted due to size, see Appendix A6), this gives the 365-step matrix (given to appropriate significant figures): 𝑃365 = ( 0.982116 0.000184080 0.000184114 0.00524607 0.0122353 2.22 × 10−6 2.14× 10−6 3.04 × 10−5 0 1.6 × 10−14 5.28 × 10−13 0.987730 0 0.000405721 0.000405556 0.0114587 0 0 1.6 × 10−14 0.987730 0 0 0.000405721 0.0118643 0 0 0 0.987730 0 0 0 0.0122700 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 ) Now, assuming we start with a susceptible individual, we can just concentrate on the first row of 𝑃365 . The chance of any susceptible individual having had chickenpox at any point in the year of analysis is the sum of the following probabilities: 𝑝𝑆𝐸 (365) + 𝑝𝑆𝐼 (365) + 𝑝𝑆𝑅 (365) + 𝑝𝑆𝑋 (365) + 𝑝𝑆𝑌 (365) + 𝑝𝑆𝑍 (365) The first three probabilities of these six represent those who were susceptible but have ended day 365 being exposed, infectious or having recovered, respectively. The last three of these probabilities represent those who were susceptible but have ended the year having passed away naturally (not due to chickenpox directly) after being exposed, infectious or having recovered, respectively. 𝑝𝑆𝑆 (365) and 𝑝𝑆𝑊 (365) are not included in this sum as these represent those who were susceptible and ended the year still being susceptible (not having had chickenpox) and those who started susceptible but have died naturally before contracting the disease at any point.
- 55. 54 Hence, the probability that any individual in the UK contracts chickenpox in this year is: 𝑝𝑆𝐸 (365) + 𝑝𝑆𝐼 (365) + 𝑝𝑆𝑅 (365) + 𝑝𝑆𝑋 (365) + 𝑝𝑆𝑌 (365) + 𝑝𝑆𝑍 (365) = 0.000184080+ 0.000184114+ 0.00524607+ 0.00000221723 + 0.00000214164+ 0.0000303784 ≅ 0.00564900 This shows that each individual has a 0.56% probability of contracting chickenpox during year one. 6.1.4. Limiting distribution If we take 𝑃 to higher and higher powers, we can get towards the limiting distribution of this matrix system. 𝑃365 gives the probability of an individual, hence the percentage of a population, that will get the disease in one year. Raising this new matrix to the power 𝑛 will give the percentage of a constant population that will contract the disease at some point during 𝑛 years. As 𝑛 → ∞, the matrix ( 𝑃365) 𝑛 = 𝑃365∗𝑛 → 𝑃 𝑛 . So raising the original matrix to exceptionally high powers will give the steady proportion of a population (based on UK data) that will contract chickenpox. After some manual checking (using integer powers of 10 for 𝑛), both 𝑃365∗100000 = 𝑃36500000 and 𝑃365∗1000000 = 𝑃365000000 give the same matrix (using Maple, omitted due to size, see Appendix A7). This implies that the limiting distribution, 𝑃 𝑛 , is (to six significant figures): 𝑃 𝑛 = ( 0 0 0 0 0.684136 0.000128153 0.000128101 0.315608 0 0 0 0 0 0.000405721 0.000405556 0.999189 0 0 0 0 0 0 0.000405721 0.999594 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 )
- 56. 55 Some rounding errors have occurred during the calculation process due to the sheer size of the power the matrix 𝑃 has been raised to. However, rounding to significant figures shows that all rows of 𝑃 𝑛 still add up to one. From this, the result here shows that given enough time, all individuals will pass away (all elements in the first four columns are zero so no-one ends up in the 𝑆, 𝐸, 𝐼 or 𝑅 states), which is obvious. It also shows a steady percentage of 𝑝𝑆𝑊 ( 𝑛) = 68.4% of the population will have died directly from the 𝑆 state, so 68.4% of the population will never contract chickenpox. The sum of the other three non-zero values in this row gives the percentage of people who will get chickenpox (this can also be worked out using 1 − 𝑝𝑆𝑊 ( 𝑛) as all rows must add to one): 𝑝𝑆𝑋 ( 𝑛) + 𝑝𝑆𝑌 ( 𝑛) + 𝑝𝑆𝑍 ( 𝑛) = 0.000128153 + 0.000128101 + 0.315608 = 0.315864254 So 31.6% of the population will contract chickenpox in a steady population that stays constant for many years.
- 57. 56 6.2. Model2 adaptation– Measles 6.2.1. Stochastic model Just like the previous adaptation, the start of this model includes a compartmental model, much in a way similar to the original deterministic model given in 3.2.1., with a probability of entering the state in question is given instead of deterministic parameters. This gives an initial compartmental model for measles (Fig. 6.3. and Fig. 6.4.): Figure 6.3.: Stochastic model for measles Probability of staying susceptible Probability of staying exposed Probability of staying infectious Probability of staying recovered Probability of staying immune Probability of staying dead Probability of staying dead Probability of staying dead Probability of staying dead Probability of staying dead Probability of staying dead Probability of staying dead Prob. of natural death Prob. of natural death Prob. of natural death Prob. of natural death Prob. of virus death Prob. of natural death Prob. of virus death Probability of recovery Probability of infection Exposed Infectious RecoveredMaternal Immunity Susceptible Probability of development Probability of loss of passive immunity Probability of gain of active immunity (vaccination) Infectious Virus Deaths Recovered Deaths Maternal Deaths Susceptible Deaths Infectious Natural Deaths Exposed Natural Deaths Exposed Virus Deaths
- 58. 57 Or more mathematically: Figure 6.4.: Stochastic model for measles with mathematical symbols Again, arrows that are missing (such as 𝑝 𝑀𝐸 ) imply that the probability of leaving state 𝑖 and entering state 𝑗 is zero (so 𝑝 𝑀𝐸 = 0, for example). If at any time, an individual in state 𝑖 has the chance of staying in state 𝑖 (𝑝𝑖𝑖 > 0), then as before, this has been added and is represented by a circular arrow that has the state it left as its target. 6.2.2. Transition matrix This can be represented in a square matrix, where each element corresponds to the relative probability of leaving the state and entering a new one, where the element in the 𝑖 𝑡ℎ row and 𝑗 𝑡ℎ column is the probability of leaving state 𝑖 and entering state 𝑗. Here, we need a twelve- by-twelve (12 × 12) matrix as we have twelve states (five ‘alive’ states, five ‘natural death’ states and two ‘virus death’ states), so this gives us the matrix 𝑃 as follows: 𝑝𝑆𝑆 𝑝 𝐸𝐸 𝑝𝐼𝐼 𝑝 𝑅𝑅𝑝 𝑀𝑀 𝑝 𝑉𝑉 𝑝 𝑍𝑍 𝑝 𝑊𝑊 𝑝 𝑋𝑋 𝑝 𝑇𝑇 𝑝 𝑌𝑌 𝑝 𝑈𝑈 𝑝 𝑀𝑉 𝑝𝑆𝑊 𝑝 𝑅𝑍𝑝 𝐸𝑋 𝑝𝐼𝑈𝑝𝐼𝑌𝑝 𝐸𝑇 𝑝𝐼𝑅𝑝𝑆𝐸 𝐸 𝐼 𝑅𝑀 𝑆 𝑝 𝐸𝐼𝑝 𝑀𝑆 𝑝𝑆𝑅 𝑈 𝑍𝑉 𝑊 𝑌𝑋 𝑇

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