$R_0 > 1$

$R_0 > 1$

$R_0 > 1$

$R_0 > 1$

$R_0 > 1$

$R_0 > 1$

$R_0 < 1$

$R_0 < 1$

$R_0 < 1$

$R_0 < 1$

Outbreak behaviour depends on $R_0$

• If $R_0 > 1$ we expect an outbreak to take off
• If $R_0 < 1$ if we expect a just a few cases with no sustained outbreak

Epidemic dynamics

Terminology

• $S(t)$, $S_t$ or $S$ is the number of susceptibles

• $I(t)$, $I_t$ or $I$ is the number of infecteds

• $R(t)$, $R_t$ or $R$ is the number of recovereds

• $N$ is the total population size

  • i.e. $N = S + I + R$

Modelling transmission

• Total rate at which new infections currently arise in the population

$$\beta \times I \times \frac{S}{N}$$

• $\beta$ is the transmission rate
• $I$ is the current number of infected individuals
• $\frac{S}{N}$ is the proportion of population that is still susceptible

Modelling recovery

• Total rate at which individuals are recovering

$$\sigma \times I$$

• $\sigma$ is the recovery rate
  
• $I$ is the current number of infected individuals

SIR model in difference equation form

$$S_{t+1} = S_t - \beta \times I_t \times (\frac{S_t}{N})$$

SIR model in difference equation form

$$S_{t+1} = S_t - \beta \times I_t \times (\frac{S_t}{N})$$

$$I_{t+1} = I_t + \beta \times I_t \times (\frac{S_t}{N}) - \sigma \times I_t$$

SIR model in difference equation form

$$S_{t+1} = S_t - \beta \times I_t \times (\frac{S_t}{N})$$

$$I_{t+1} = I_t + \beta \times I_t \times (\frac{S_t}{N}) - \sigma \times I_t$$

$$R_{t+1} = R_t + \sigma \times I_t$$

SIS process

SIS model in difference equation form

$$S_{t+1} = S_t – \beta \times I_t \times \frac{S_t}{N} + \sigma \times I_t$$

$$I_{t+1} = I_t + \beta \times I_t \times \frac{S_t}{N} – \sigma \times I_t$$

Confusing notation…

$$\beta \times I \times (\frac{S}{N})$$

is the same as…

$$\beta I(\frac{S}{N})$$

is the same as

$$\frac{\beta I S}{N}$$

Basic reproduction number, $R_0$

• This is a key concept in epidemiology and is defined as:

  “the average number of secondary cases arising from an infected individual introduced into a fully susceptible population”

Calculating $R_0$

$$\begin{align} R_0 = {} & \textrm{number of new infections per day} \\ & \textrm{arising from 1 infected in a fully} \\ & \textrm{susceptible population}\; \times \\ & \textrm{number of days infectious} \end{align}$$

Calculating $R_0$

$$\begin{align} R_0 = {} & \textrm{number of new infections per day} \\ & \textrm{arising from 1 infected in a fully} \\ & \textrm{susceptible population}\; \times \\ & \textrm{number of days infectious} \\ = {} & \beta \times \textrm{number of days infectious} \end{align}$$

Note on rates and duration

• We convert between a rate and a duration using the reciprocal
  • $\textrm{duration} = \frac{1}{\textrm{rate}}$
  • $\textrm{rate} = \frac{1}{\textrm{duration}}$
• Examples:

  • Recovery rate = 0.1 per day

  • Infectious period = $\frac{1}{0.1}$ = 10 days

  • Infectious period = 3 weeks

  • Recovery rate = $\frac{1}{3}$ per week

Calculating $R_0$

$$\begin{align} R_0 = {} & \textrm{number of new infections per day} \\ & \textrm{arising from 1 infected in a fully} \\ & \textrm{susceptible population}\; \times \\ & \textrm{number of days infectious} \\ = {} & \beta \times \textrm{number of days infectious} \end{align}$$

Calculating $R_0$

$$\begin{align} R_0 = {} & \textrm{number of new infections per day} \\ & \textrm{arising from 1 infected in a fully} \\ & \textrm{susceptible population}\; \times \\ & \textrm{number of days infectious} \\ = {} & \beta \times \textrm{number of days infectious} \\ = {} & \beta \times \frac{1}{\textrm{recovery rate}} \end{align}$$

Calculating $R_0$

$$\begin{align} R_0 = {} & \textrm{number of new infections per day} \\ & \textrm{arising from 1 infected in a fully} \\ & \textrm{susceptible population}\; \times \\ & \textrm{number of days infectious} \\ = {} & \beta \times \textrm{number of days infectious} \\ = {} & \beta \times \frac{1}{\textrm{recovery rate}} \\ = {} & \beta \times \frac{1}{\sigma} = \frac{\beta}{\sigma} \end{align}$$

Calculating $R_0$

$$\begin{align} R_0 = {} & \textrm{number of new infections per day} \\ & \textrm{arising from 1 infected in a fully} \\ & \textrm{susceptible population}\; \times \\ & \textrm{number of days infectious} \\ = {} & \beta \times \textrm{number of days infectious} \\ = {} & \beta \times \frac{1}{\textrm{recovery rate}} \\ = {} & \beta \times \frac{1}{\sigma} = \frac{\beta}{\sigma} \\ = {} & \frac{transmission.rate}{recovery.rate} \end{align}$$

Epidemic dynamics for SIR, $R_0 = 2$

Epidemic dynamics for SIR, $R_0 = 2$

Epidemic dynamics for SIR, $R_0 = 2$

Epidemic dynamics for SIR, $R_0 = 2$

Epidemic dynamics for SIR, $R_0 = 2$

Epidemic dynamics for SIR, $R_0 = 2$

Epidemic dynamics for SIR, $R_0 = 2$

Key features of dynamics

• Infection burns itself out
• Not all individuals become infected

Key features of dynamics

• Infection burns itself out

• Not all individuals become infected

• Chain of transmission eventually halts due to insufficient susceptibles, not a complete lack of susceptibles

$R_0 < 1$

$R_0 < 1$

$R_0 < 1$

$R_0 < 1$

Dynamics for $R_0 = 0.5$

SIS process

SIS model in difference equation form

$$S_{t+1} = S_t – \beta \times I_t \times \frac{S_t}{N} + \sigma \times I_t$$

$$I_{t+1} = I_t + \beta \times I_t \times \frac{S_t}{N} – \sigma \times I_t$$

SIS simulation for $R_0 = 3$

SIS simulation for $R_0 = 3$

Practicals

• Programming in R Practical:
  • Extending your programming skills
  • Building simple disease dynamics models in R