Basic reproduction number

John M. Drake

Overview

Topics

  • Basic reproduction number, \( R_0 \)
  • Invasion threshold
  • Next generation method

Epidemic trajactories differ according to parameters


plot of chunk unnamed-chunk-2


plot of chunk unnamed-chunk-3

Generalization

  • Can we make general statements about epidemics without resorting to extensive numerical integration?
  • Under what conditions will an infectious disease invade a system

The invasion threshold

Assume an infectious individual arises in a Wholly Susceptible and otherwise Disease Free population

Initial conditions: \( X(0)=N-1 \approx N \), \( Y(0)=1 \), and \( Z(0)=0 \)

Invasion occurs only if \( dY/dt > 0 \)

\[ \begin{align} \frac{dY}{dt} = \beta X Y/N - \gamma Y &> 0 \\ Y(\beta X/N - \gamma) &>0 \\ X/N &> \gamma/\beta \end{align} \]

Since \( X \approx N \), this is satisfied when \( 1>\gamma / \beta \), which is equivalent to \( \beta/\gamma > 1 \)

Kermack & McKendrick (1927)

Basic reproduction number

  • For the \( SIR \) model, the ration \( \beta/\gamma \) gives te number of secondary cases that will be infected before the index case recovers
  • This quantity is universally referred to as \( R_0 \) and called the Basic Reproduction Number or Basic Reproductive Ratio
  • We can extend the concept of \( R_0 \) to other models with the following definition
  • The basic reproduction number is the number of secondary cases generated a single typical infected case in an entirely suspectible population
  • Hence, \( R_0 \) depends on both properties of the pathogen and properties of the population into which it is introduced

Basic reproduction number vs model parameters