John M. Drake

Topics

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

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

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)

- 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