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Microsoft Word - NATS1920Project2.docx 1 NATS 1920 Modelling Infectious Disease Project Complete this worksheet by April 12th, 11:59pm (ET). Answers are to be submitted online, on eClass. You may print this sheet and write your answers directly on it, or you can write your answers on a blank piece of paper. If you do the latter, please label your answers using the question number and section. You must write your name on your report. If you are working in a pair, please include both of your names on your submission. If working in a pair, only one of you needs to submit the report. Understanding - and recognizing- the differences between otherwise very similar models is a great way to build understanding of how modeling works as a whole. Adding new parts to a model to address specific parts of disease biology is all you really need to know to start making your own! Within a model, we look at differences in how the system behaves. For diseases, there are obvious outcomes we might want to examine: peak infection during an outbreak, or how much a population size is being suppressed by disease. We’re going to examine the outcomes of different scenarios across a family of SIR-type models. Please go to: https://rozins.shinyapps.io/model/ to access the modelling software. In the following models, we will use a few Greek letters. It is convention to use Greek letters to represent pathogen-related parameters. The following letters are almost always used for these terms. ● ? - beta - transmission term ● ? - gamma - recovery rate ● ⍺ - alpha - disease-induced mortality We have also used the following Greek letters for today’s simulations, although they may represent different parameters in other contexts. ● ? - sigma - advancement rate from exposed to infectious ● ? - omega - rate at which immunity wanes (R to S) Additionally, the following definitions might be useful to you: ● Epidemic: The infection spreads rapidly to a large portion of the population of interest in a short period of time (e.g., novel coronavirus in China) 2 ● Endemic: The infection is constantly maintained at a baseline level in the population of interest (e.g., chickenpox in the UK) Primer on R0 R0 (“R nought”, or the basic reproductive number) is one of the most important concepts in epidemiological modelling, and is central to understanding disease dynamics. R0 is defined as the number of new cases, on average, produced by a single infectious individual in a completely susceptible (“naive”) population. Remember: for a disease to be able to spread in a population, the R0 must be greater than 1- i.e., each new case must produce more than one additional cases. In simple terms, we can think about R0 as being the result of two processes: the rate at which new cases are generated by the individual, and the duration of infection: ??????????????????????????? ∗ ??????????????????? This can be re-written as: ??????????????????????????? ???????ℎ??ℎ??????????????????????????????? If you think about R0 in these terms, it can be easier to come up with a formula for R0 that is appropriate for the disease of interest. For all diseases, the transmission parameter ? will be involved in the generation of new cases per unit time (i.e., in the numerator). For diseases that have density dependent transmission, the population size N will affect how many new cases arise from a single case, and for diseases where transmission is density independent i.e. frequency dependent, the population size will not play a role in the same way. Similarly, the factors that determine the infectious period or duration of a disease will depend upon the system. For acute, mild diseases, like a cold, the infectious period is determined by the recovery rate. For chronic diseases or diseases from which hosts do not recover, the infectious period will be limited by death- either natural death, or disease-induced mortality. For any given disease, you should think about what factors lead to individuals leaving the infectious class. 3 Part 1: Basic SIR [33 marks total] The most basic S-I-R model looks at an outbreak of a pathogen in the short term. It doesn’t necessarily include demography- births, deaths, or immigration/emigration. ‘SIR’ comes from the three classes of individual host which are defined in the model: S - ‘Susceptible’ - individual hosts which can be infected by the pathogen. I - ‘Infectious’ - hosts which have been infected by the pathogen and are currently infectious (the difference between infected vs. infectious is important in later models). R - ‘Removed or Recovered’ - hosts which are no longer able to be infected and are not infectious N - Population Size - The total size of the population (S + I + R) is commonly called “N” This model also uses two parameters, which we call: ? - Transmission Rate - ? is the result of both how frequently contact between individuals occurs, and how often this contact results in transmission. Larger values of ? in the same model indicate more transmissibility. ? - Recovery Rate - ? reflects how quickly individuals recover from the infection. As discussed in class, ? is related to the average duration of infection where ???????? = ! " The system of differential equations governing the basic SIR model are displayed in the app, see link on page 1. Here is an example of an S-I-R-type flow diagram for the system: 1. What is the equation that describes the R0 for this model? (Refer to R0 primer on page 2 for additional guidance) [1 mark] β*N/γ 2. In class we discussed the idea that, for diseases with density dependent transmission, there is a minimum host density or ‘critical community size’ which must be reached in order for an epidemic to take off. For the default parameters (?=0.001, ?=0.1), what is the critical community size? Using the app, what happens when the population is below this size? [4 marks] Hint: What is the value of R0 required for invasion of a population? Refer to the R0 primer. γ/β = 100 3. What is the relationship between R0 and the vaccination threshold? [2 marks] Herd Immunity Threshold = R0 increases so does the vaccination threshold 4 4. In the table below, we have given you values for various parameters. By rearranging your formula for R0, write the formula to calculate the value in that column based on the values in the other columns. Then, calculate the missing parameters and enter the calculated values into the app. Look at the model behaviour, and use this to complete the table. For % that contract the disease, estimate the value by inputting calculated parameters into the app and examining the resulting plot. [18 marks] N ? β R0 % contracted Herd immunity threshold Beta = R0*gamma/N R0 = beta*N/ gamma=beta* N*d Look at plot 1-1/R0 500 0.5 0.004 2 99%ish 1-1/2 = 50% 500 0.3 0.001 1.667 ~70% 1-1/1.666 ≈ 40% 1000 0.3 0.001 3.333 ~95% 1-1/3.333=70% 100 0.1 0.003 0.6 ~2% 1-1/0.6 = -66% 500 0.2 0.005 12.5 100% 1-1/12.5 = 92% 250 0.4 0.003 1.875 75% 1-1/1.875 = ~46% 5. Now, change the “Proportion Vaccinated” parameter. How are vaccinated individuals incorporated into this model? [2 marks] In the R class at time t=0 6. What is the effect of vaccination on the R0 of the disease? Remember the definition of R0. [2 marks] Vaccination has no effect on the value of R0 for a disease, as vaccination moves people into the recovered class but R0 is a function of ?, ?, and N. The definition of R0 includes the concept of an entirely naive population, so S ≈ N. 5 7. Now, change the ? (recovery rate) parameter. What happens to R0 and the epidemic as you increase ?? Decrease? Why? [4 marks] is a product of transmission rate, population size, and duration. The R0 increases as ? decreases. The epidemic peaks earlier and higher. This is because the duration increases as ? decreases, and Part 2: Basic SEIR [18 marks] We can extend the basic SIR model to include information about the incubation period- the time between infection of the host and the point at which they actually become infectious. We do this by adding another class (variable), and another parameter: E - Exposed - Individuals who are infected with a pathogen, but cannot yet infect other hosts ? - the Advancement Rate - how quickly individuals move from the exposed (E) to infectious (I) class; the incubation period is equal to 1/? 1. In the space below, draw a flow diagram of the SEIR Model. [4 marks] 2. Is the equation for R0 different for the SEIR model than the SIR model? Why or why not? [4 marks] No, R0 deals with time spent in I and the size of ?. Neither of which are changed by introducing the E compartment. 3. Compare the dynamics of the SIR and SEIR models by running both for the same set of parameters. Then, vary the advancement rate ?. How does the introduction of the E class and the length of the incubation period change the resulting epidemic? Consider the timing and magnitude of peak caseload, as well as the proportion of the population that is infected by the end of the epidemic.