© 2018 Aaron A. King.

A deadly disease outbreak

In the Challenge, you are concerned with a new infection that has been introduced into your city of 1 million inhabitants. The infection spreads through direct contact and can be fatal.

Diagram of the model. S represents susceptible hosts; I, infected hosts; R, recovered hosts; and D, hosts killed by the disease.

Diagram of the model. S represents susceptible hosts; I, infected hosts; R, recovered hosts; and D, hosts killed by the disease.

We suppose that the rate at which each hosts makes contact with other hosts is roughly constant. Accordingly, we assume that the the infection probability, \(\lambda\), is given by \[\lambda_t = 1-e^{-b\,(I_{t}+h)/N_t}.\] The parameter \(h\) is the average number of infectious individuals visiting the population from elsewhere. \(N_t=S_t+I_t+R_t\) is the population size at time \(t\).

With a timestep of 1 d, the model equations are: \[\begin{pmatrix}S_{t+1}\\I_{t+1}\\R_{t+1}\\D_{t+1}\end{pmatrix}= \begin{pmatrix} (1-\mu)\,(1-\lambda_t) & 0 & 0 & 0 \\ (1-\mu)\,\lambda_t & (1-\nu)\,(1-\gamma) & 0 & 0\\ 0 & (1-\nu)\,\gamma & 1-\mu & 0\\ 0 & \nu & 0 & 1 \end{pmatrix}\,\begin{pmatrix}S_{t}\\I_{t}\\R_{t}\\D_{t}\end{pmatrix}+\begin{pmatrix}B\\0\\0\\0\end{pmatrix} \]

We will assume \(\gamma=1/10\), \(\nu=2/10\), \(b=1\,\mathrm{d}^{-1}\), \(B=50\,\mathrm{d}^{-1}\), and \(\mu=1/50/365\). At time \(t=0\), the population is fully susceptible. With these values, and assuming \(h=0.1\), the first 100 d of the outbreak look like the following.

Specific Tasks

  1. Explain the meanings of the symbols in the model equation. What do their values say about the nature of the infection and the host?
  2. What is the case fatality rate (i.e., the chance that an infection results in host death)?
  3. Write code to iterate the model equations. Check your code using the plot above. How many die from the disease within the first year?
  4. Determine the equilibrium fraction of susceptibles and infecteds. Make plots to show how these depend on the transmission rate, \(b\).
  5. At equilibrium, how many are dying each year from the disease?
  6. Suppose you are able to develop an vaccine that is 100% effective in preventing infection. What fraction of the population must you vaccinate to eradicate the disease?

Important Note: You will turn in each draft of your report via the course Canvas site. Upload both the Rmarkdown document and a PDF version of your report. Choose the filenames according to the following formula: Challenge_X-Y.Z where X is the challenge problem number, Y is the draft number (1, 2, or 3), and Z is the appropriate extension (.Rmd or .pdf).