Description

• Your hand-in is to be uploaded as a bundle consisting of a) scanned copy of your signed Confirmation.pdf,
b) PDF file <lastname>.pdf containing your report c) your commented R/Rmd, Python, Python Notebook, Matlab, Julia, … file containing the code of your analysis.
A total of 25 points can be reached for the answers in the report. Note: A penalty is imposed on reports longer than 20 pages. Your final grade for the project module of the course is determined by your sum of points in the two project works – see the grading criteria of the course for further details.
Lycka till!
Exercise 1 (4 points)
This exercise deals with final size introduced in Lecture 1.
(a) Solve the final size equation 1 − τ = exp(−R0τ) numerically (the largest solution in [0,1]!) as a function of R0 and create a plot of it for R0 from 0 to 5.
(b) Do the same thing when a fraction r are immune. However, in the latter case you should plot the overall fraction infected, and not the fraction infected among those who were initially immune.
Exercise 2 (8 points)
In this exercise we consider the Susceptible-Exposed-Infectious-Recovered (SEIR) model in a closed population with parameters N = 100, β = 0.004, γ = 1/7 and a latency period with mean duration 5 days, i.e. the rate for the E → I transition is ρ = 1/5.
(a) Write up the ordinary differential equation system for the above SEIR model.
(c) Modify the SEIR equations such that β(t) becomes a time dependent function, which is β0 until time t1−w, is β1 after time t1 + w and changes linearly from β0 to β1 between t1 − w and t1 + w. Write the full β(t) as part of your report. Hint: Determine what ? should be in the equation below.
 β0 if t ≤ t1 − w,

β(t) = ? if t1 − w < t ≤ t1 + w,
 β1 if t1 + w < t
(d) Let t1 = 30, w = 5, β0 = 0.004 and β1 = 0.0012. Use the Euler scheme or your favorite ODE solver to solve the ODE system numerically and plot I(t) for t ∈ [0,100].
Exercise 3 (6 points)
Hint: Parametrise your optimization function using the log of the parameters to ensure valid parameter values at all times. Furthermore, hand-tuning the starting values of your optimization might be necessary in order to get a reasonable fit.
Exercise 4 (7 points)
(a) How many deaths are available in the data?
Hint: Let D be a random variable representing the reporting delay of a case. The support of D is 0,1,…,20. Consider the probability g(d) = P(D = d|D ≤ d), i.e. given that we know that D ≤ d, what is the probability that D is exactly equal to D days. How can we estimate g(d) in the reporting triangle? In your calculations you can use without proof that
.
(d) Calculate F(d) for d = 0,1,…,20 and state the resulting vector.
(f) Compare your results to the graph available from https://github.com/adamaltmejd/covid/blob/master/ docs/archive/deaths_lag_sweden_2020-06-29.png. Comment the differences.
Homepage: https://kurser.math.su.se/course/view.php?id=911