Building Epidemiological Models
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”
Basic reproduction number \(R_0 > 1\)
\(R_0 > 1\)
\(R_0 > 1\)
\(R_0 > 1\)
\(R_0 > 1\)
\(R_0 > 1\)
\(R_0 > 1\)
Basic reproduction number \(R_0 < 1\)
\(R_0 < 1\)
\(R_0 < 1\)
\(R_0 < 1\)
\(R_0 < 1\)
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
Model structure
- Determined by the biology of the infection
- do infected individuals die?
- do infected individuals recover?
- are recovered individuals immune?
- are there latently infected individuals?
SIR process
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
Basic reproduction number \(R_0 < 1\)
\(R_0 < 1\)
\(R_0 < 1\)
\(R_0 < 1\)
\(R_0 < 1\)
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