1 Question 1 (8 marks)

Consider the closed compartmental model described by the following equations, with days as the time unit of the model, and where \(\beta,\,\gamma\) and \(\sigma\) have the same meanings as in lectures:

\[\begin{align*} \frac{dS}{dt} & = -\frac{\beta IS}{N}\\ \frac{dE}{dt} &= \frac{\beta IS}{N} - \sigma E\\ \frac{dI}{dt} &= \sigma E - \gamma I\\ \frac{dR}{dt} &= \alpha \gamma I\\ \frac{dD}{dt} &= \lambda\gamma I. \end{align*}\]

  1. Draw the associated diagram, showing the compartments and possible routes from one to another.
  2. Suggest a sensible meaning for the compartment \(D\).
  3. Following from your answer to 2, interpret the parameters \(\alpha\) and \(\lambda\), and identify a constraint on the relationship between them. Can you use this relationship to reduce the number of parameters?
  4. Suppose that, once recovered, people become susceptible again after an average of 6 weeks. Rewrite the equations that change as a result of this (use \(\eta\) to denote the rate at which immunity wanes) and suggest a value for \(\eta\).

2 Question 2 (12 marks)

The .Rdata file ‘assignment1_df.Rdata’ (which you can find under ‘Datasets’ in the ‘resources’ section on Ultra) loads a data frame named assignment1_df, which contains synthesized cumulative daily death counts for some infectious disease over 40 days. The disease spread through a population of 1080 people, all of whom were susceptible apart from the initial infectious person. You can assume that the pre-infectious period of this disease is negligible, and that once recovered, survivors have perfect immunity.

  1. Adapt the model in Question 1 to model this disease. Draw the new flow diagram and write the new set of ODEs. What parameters will the model have?
  2. Use the cumulative death count data to find values of the model parameters that minimize the RMSE.
  3. Estimate the value of \(R_0\) for this model, and explain what this means in terms of the disease being modelled.

Please submit your R code (you may have to screenshot it or convert it to a form that is uploadable to Gradescope) along with your answers. You may adapt the code in the R files used in workshops 1 and 2.