MMCS Turkey Flu Pandemic Project

Transcription

1 MMCS Turkey Flu Pandemic Project This is a group project with 2 people per group. You can chose your own partner subject to the constraint that you must not work with the same person as in the banking project. Each person in the group will get the same mark for the project. The project is in two parts, A and B. The assessment will be based on two reports, one for each part. The report for part A is due at 14:00 on 15th March and is worth 40%. The report for part B is due at 14:00 on 24th March and is worth 60%. In part A you will be asked to create a simple model of the spread of a flu like epidemic and use this to understand the factors that effect the proportion of the population becoming infected. Two cases will be considered. The first case will develop a homogeneous model in which each member of the population has the same susceptibility to the disease and the same ability to spread the disease. The second case will split the population into two groups, adults and children, each of which has its own properties. In part B you will be asked to plan cost effective measures to limit the damage in the U.K. from and outbreak of a new Turkey Flu virus that is threatening to become a world wide pandemic among humans. The plan will have to take into account uncertainties of the properties of the new virus. Part B will use software developed in part A. 1

2 MMCS : Part A: Modelling Epidemics 2 Part A: Modelling Epidemics When a new strain of flu appears some of the population will be susceptible to it (i.e. able to catch it) and some will be immune. When a susceptible person catches the flu they go though the following 5 states in order: 1. Susceptible: Person has not caught the virus but is not immune to it. 2. Incubating: Person has caught the virus but has no symptoms and is not infectious. 3. Infectious but symptomless: Person shows no symptoms but can infect others. 4. Infectious with symptoms: Person shows symptoms including sneezing and raised temperature and can infect others. 5. Immune (or dead): Person has recovered (or is dead) and is not able to infect others Susceptible Incubate Symptoms Immune Catch Infectious In some diseases a person may become susceptible again after a period of time, or may become susceptible to new variant of the disease. In this study we assume that doesn t happen within the time scale of the epidemic. Also in some diseases the person may loose their symptoms but still be infectious, but this is not the case in this study. At the start of an epidemic some of the population may already be immune to the disease because of previous exposure to related diseases or due to inborn genetic reasons. A compartmental model is often used to analyse epidemics. The population is split into a series of compartments and it is assumed that all people in one compartment of the model are in the same state and indistinguishable. The simplest model is a Homogeneous Model. This has a single compartment for each of the disease state 1 to 5 above. Every individual in a compartment is assumed to have the same properties. People move between compartments as the infection progresses. time

3 MMCS : Part A: Modelling Epidemics 3 Susceptible Incubating 1 Without symptoms F t I t S t x C t symptoms With t 1 µ µ D t = F t + U t Infectious µ U t Immune An epidemic develops over continuous time and this time evolution can be modelled with differential equations. However the rate of infection varies with the time of day and it becomes very complex to model this accurately. If the proportions in the different compartments do not change significantly between days, then a sensible approximation is to model only daily changes in the proportions. This leads to difference equations with a 1 day time step, and this is what will be done in this project s d i Proportions The graph above shows the course of the epidemic for the case R 0 t = 2.0 (see later) and when all of the population is initially susceptible and the average length of an infection is 5 days. Days

4 MMCS : A1 - Homogeneous model 4 A1 - Homogeneous model Some useful notation: S t = number susceptible at the start of day t C t = number incubating at the start of day t F t = number infectious but without symptoms at the start of day t U t = number infectious and with symptoms at the start of day t D t = number infections at the start of day t I t = number immune at the start of day t V t = number who have or have had the flu up and including day t Rt 0 = value of R 0 (see below) on day t N = population size (which we assume to be constant) x t = proportion of susceptibles who catch flu in day t µ = probability that an infectious person will not be infectious one day later We will use lower case to represent the proportion of a quantity. For example s t = S t /N = proportion of the population that is susceptible at the start of day t. In the above model we are assuming that µ is independent of how long a person has had flu, and this is a good enough approximation for this study. In our model we shall assume that the incubation period is one day and the period without symptoms is one day. So including the incubation day the infection will on average last 1+ 1/µ days. (Note that if we did want to allow µ to depend on the time since the start of the infection, or to allow the infectivity of the flu to depend on this time, then the infectious compartments in the model would have to be subdivided into compartments each corresponding to a different time since the start of the infection.) The Basic Reproduction Number, denoted by R 0, is defined to be the average number of secondary cases a typical single infected case will cause in a population with no immunity to the disease and where there is no intervention to control the infection. For example if R 0 = 2.0 and if s 1 = 1.0 (i.e. all the population is susceptible), then on average a person with flu introduced into the population will during the whole period of their infection pass it on to 2.0 others. As a result the epidemic will grow. The average number of people that a person with flu infects is proportional to the proportion of the population who are susceptible. Hence if later in the epidemic only 40% of the population is susceptible (i.e. s = 0.4), then a person with the flu will pass it on to on average R 0 s = 0.8 others. As a result the epidemic will decline.

5 MMCS : Task A1: Software development Homogeneous model 5 The basic reproduction number for flu is different in summer and winter. This is mainly due to the effects of temperature and humidity on the survival of the virus and to differences in average interpersonal distance in summer and winter. Task A1: Software development Homogeneous model In task A1 assume the population is homogeneous, i.e. all age groups in the population have the same characteristics. Formulate a model of an epidemic as a set of difference equations and implement it in Java or some other programming language or in a system such as Matlab or Excel. Your system should be able to work for arbitrary (valid) values of µ, R 0 t, C 1, F 1, U 1, i 1 and N, and should calculate C t, F t, U t, D t, x t, i t and V t for all t. Answer the following questions using your program where appropriate. For Task A1 assume N = (the U.K. population) and unless stated otherwise assume that µ = 0.25, C 1 = 48, F 1 = 40 and U 1 = 200. The model should be run for 365 days. 1. Are the equations of your model linear or non-linear? 2. Goal: How does the proportion who are susceptible on day 1 affect the epidemic. Take Rt 0 = 2.0 throughout the year. Calculated the proportion of the population that catches the disease within 365 days for a range of different initial proportions of susceptibles, s 1 between 0.0 and 1.0. Display the results as a plot of the proportion of the population who get the flu during the year against the proportion initially susceptible. 3. Goal: How much of the population should be vaccinated to stop the epidemic? Assume i 1 = 0.0 (i.e. none of the population is immune) and that a vaccine was available at the start of the epidemic, and that every person vaccinated becomes immune. What is the minimum proportion of the population that needs to be vaccinated on day 1 to prevent a large growth in infections? 4. Goal: How does the length of the infection affect the epidemic? Compare the course of the epidemic when i 1 = 0.0 and R 0 t = 2.0 for the cases µ= 0.2 and µ= Plot s t, d t and i t and find V 365 for each case.

6 MMCS : Task A1: Software development Homogeneous model 6 5. Goal: How does the size of the initial infected population affect the epidemic? Compare the course of the epidemic when i 1 = 0.0 and R 0 t = 2.0 for the cases where C 1 = 48, F 1 = 40 and U 1 = 200 and where C 1 = 96, F 1 = 80 and U 1 = 400. Plot d t and i t and find V 365 for each case. 6. Goal: Can you distinguish initial behaviour of epidemics? Assume that R 0 t = 8.0 throughout the year and that i 1 = 0.75 (s ). Calculate the course of the epidemic over a year and plot the cumulative proportion who have caught the disease on a log scale as a function of time. On the same graph plot equivalent results for the case R 0 t = 2.0 throughout the year and i 1 = 0.0 (s 1.0). Explain the reason for the relation between these graphs. 7. Goal: What is the best value for R 0 for in summer, in a summer - winter epidemic? Assume that i 1 = 0.0 and that the value of R 0 t = 1.25 for the first 182 days (i.e. summer) and R 0 t = 2.0 for the next 183 days (i.e. winter). Find the total number who get flu in the year and plot the cumulative number on a log scale versus time. Assume that it is possible to control the value of Rt 0 during the summer within the range 1.1 to 1.6 (with the same value throughout the summer). (Note that value > 1.25 correspond to increasing the spread of the flu during the summer.) Find the best value for Rt 0 for the summer period. 8. Goal: how to choose R 0 t throughout the year to control the epidemic? Assume i 1 = 0.0 and Rt 0 varies through the year as in Question 7. Also assume that Rt 0 can be changed from its base value to R t at a cost of R t Rt 0 (for the day) and that a total of 10 units is available over the year to make these changes. What is the best set of values for R t to minimize the total number catching the flu in the year. (An exact minimum is not required here. As long as your answer has got the right properties and is approximately optimal that s fine.) 9. Can the the optimization problem in Question 8 be formulated as an LP?