Dynamical Systems in Biology

Tutorial 3

Integrating ODEs with deSolve

Install and load the package deSolve. Take a look at the first example of the documentation of ode().

Exercise 1: SIR model

The ODEs of the SIR model, \(\dot{\vec x} = f(\vec x)\), are given by: \[ \begin{align} \dot{S}(t) & = - r \cdot S(t) \cdot I(t), \\ \dot{I}(t) & = r \cdot S(t) \cdot I(t) - a \cdot I(t), \\ \dot{R}(t) & = a \cdot I(t), \end{align} \] where \(\mathbb{R} \ni t \mapsto S(t),I(t),R(t) \in \mathbb{R}\) and the parameters \((a, r) \in \mathbb{R}_{+}^2\) are positive.

• Write a function SIR <- function(time, ini, pars) that can be passed as func argument to ode(). Add two functions out2config <- function(out, parameter) and out2phase <- function(out, parameter) that arrange the output of ode() in a config or phase space data.frame with columns:
  • time, value, species, parameter, where species \(\in \{S,I,R\}\) and parameter, labelling the parameter choice, are character variables.
  • S, I, R, parameter.
• Use ode() to integrate the system for \(r = 0.1\); \(a=50\) and the initial values \(R(t=0) = 0\), \(S(t=0) = 450\), \(N = R(t=0) + S(t=0) + I(t=0) = 1000\).
  • Plot solutions for \(S(t=0) = \{450,950\}\) in configuration space.
  • Plot solutions for \(r\in [0.1,0.5]\) in phase space.
  • Plot solutions for \(r\in [0.1,0.5]\) but \(I(t=0)=1\) in phase space and compare it to the previous plot.
• How many individuals need to be infected at \(t=0\) such that an epidemic occurs?
  • Plot the derivative dI as a function of \(I(t=0)\).
  • Find the root of dI using uniroot.
• Evaluate the severity of the disease as a function of the parameters a and r for \(N = 1000\), \(S(t=0)=950\), \(R(t=0)=0\).
  • plot Imax, the maximal value of \(I(t)\), in a 2d grid plot via geom_tile()
  • mark the non-epidemic region by adding a line via annotate("line",...)
• Simulate the SIR model in configuration space for \(I(t=0) = 50\), \(S(t=0) \in [100,1000]\) and \(r=0.1, a=10\). What difference would you expect for an epidemic in Manhattan (27.500 citizens per km\(^2\)) and Mongolia (1,7 citizens per km\(^2\))?

Cathedral exercise

Close to the outside staircase leading to the top panorama platform there are seven huge figures. During the renovation around 1900 one of the figures was replaced by a figure of the Canon. What about this fact was so funny for a church representative from Konstanz when he visited the inauguration ceremony after the renovation?