Basic reproduction number

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

• As before, pathogen invasion occurs if \( dY/dt > 0 \)
• Assuming nearly everyone initially susceptible \( X \approx N \), then \( \beta NY - \gamma Y > 0 \implies Y(\beta N- \gamma) > 0 \)
• Now, we define \( R_0 = \beta N / \gamma \)
• Conclusion: For density dependent transmission there is a threshold population density required for invasion

We assume infectious individuals die at rate \( \alpha \)

\[ \frac{dY}{dt} = ... -\gamma Y - \alpha Y \]

• If the population size doesn't change substantially theb frequency- and density-dependent transmission are equivalent (\( \beta_{FD} = N \times \beta_{DD} \))
• Otherwise
  • Frequency dependent transmission is typically assumed to be more appropriate for large populations with heterogeneous mixing, sexually transmitted diseases, and vector-borne pathogens
  • Density dependent transmission is assumed to represent wildlife and livestock diseases, especially those with smaller population sizes
  • Empirical evidence suggests that many disease systems are intermediate

Our informal derivations of \( R_0 \) were possible because there was only one class of infected individual (i.e. \( Y \))

The Next Generation Method can be used to find expressions for \( R_0 \) in more complicated situations where the population can be divided into disjoint categories, for instance when we have latent infections, asymptomatic infections, etc.

Diekmann et al. 1990. Journal of Mathematical Biology 28:365–382.

Heffernan et al. 2005. Journal of the Royal Society Interface 2:281-293.

Let \( x= \left\{x_1, x_2, ..., x_n \right\} \) represent the set of \( n \) infected host compartments and \( y= \left\{y_1, y_2, ..., y_m \right\} \) the set of \( m \) other host compartments.

Write the model in terms of two subsystems

\[ \frac{dx_i}{dt} = \mathcal{F}_i(x,y) - \mathcal{V}_i(x,y) \\ \frac{dy_j}{dt} = \mathcal{G}_j(x,y) \]

where \( \mathcal{F} \) is the rate at which new infected individuals enter compartment \( i \), \( \mathcal{V} \) is the rate at which already infected individuals move among compartments and \( \mathcal{G} \) are the flows involving uninfected individuals.

1. There are no infections if there are no existing infections (no importation of infected individuals)
2. The function \( \mathcal{F} \) cannot be negative
3. If a compartment is empty it can only have inflow
4. The net outflow of infected classes is nonnegative
5. The uninfected subsystem has a unique asymptotically stable equilibrium

(Violations of these assumptions are unusual, but sometimes possible)

Van den Dreissche & Watmough. 2008. Further Notes on the Basic Reproduction Number. (book chapter)

The two subsystems are effectively decoupled when close to the disease free equilibrium, \( y^* \)

(Linear approximation) Define matrices \( F \) and \( G \)

\[ F_{ij} = \frac{\partial F_i}{\partial x_j}(0,y^*) \\ V_{ij} = \frac{\partial V_i}{\partial x_j}(0,y^*) \]

and next generation matrix

\[ K = FV^{-1} \]

Entry \( K_{ij} \) represents the average number of secondary cases induced in compartment \( i \) by an individual in compartment \( j \). \( R_0 \) is given by the dominant eigenvalue of \( K \).

Using the notation defined above

\( x = \left\{ E, I \right\} \) and \( y = \left\{ S, R \right\} \)

We find matrix \( F \) from

\( \mathcal{F}_1 = \beta S I \) and \( \mathcal{F}_2 = 0 \)

from

\[ F = \begin{pmatrix} \frac{\partial(\beta SI)}{\partial E} & \frac{\partial(\beta SI)}{\partial I} \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & \beta \\ 0 & 0 \end{pmatrix} \]

We find matrix \( V \) from

\( \mathcal{V}_1 = (\mu + \sigma) E \) and \( \mathcal{V}_2 = (\mu +\gamma) I - \sigma E \)

from

\[ V = \begin{pmatrix} \frac{\partial (\mu + \sigma) E}{\partial E} & \frac{\partial (\mu + \sigma) E}{\partial I} \\ \frac{\partial (\mu +\gamma) I - \sigma E}{\partial E} & \frac{\partial (\mu +\gamma) I - \sigma E}{\partial I} \end{pmatrix} = \begin{pmatrix} \mu + \sigma & 0 \\ -\sigma & \mu + \gamma \end{pmatrix} \]

From linear algebra, we recall that the inverse of a 2x2 square matrix

\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

So, we get

\[ FV^{-1} = \begin{pmatrix} 0 & \beta \\ 0 & 0 \end{pmatrix} \begin{pmatrix} \frac{\mu+\gamma}{(\mu + \gamma)(\mu+\sigma)} & 0 \\ \frac{\sigma}{(\mu + \gamma)(\mu+\sigma)} & \frac{\mu+\sigma}{(\mu + \gamma)(\mu+\sigma)} \end{pmatrix} \]

\[ FV^{-1} = \begin{pmatrix} \frac{\beta \sigma}{(\mu + \gamma)(\mu+\sigma)} & \frac{\beta (\mu +\sigma)}{(\mu + \gamma)(\mu+\sigma)} \\ 0 & 0 \end{pmatrix} \]

The basic reproduction number is given by the dominant eigenvalue of this matrix (which can be found using the quadratic formula), i.e.

\[ R_0 = \frac{\beta \sigma}{(\mu+\gamma)(\mu + \sigma)} \]

Discussion question: What happens as \( \sigma \) gets large, i.e. \( \sigma \rightarrow \infty \)?

Although cumbersome to use, the next generation method can be applied to obtain \( R_0 \) for many significantly more complicated models.

This is useful because \( R_0 > 1 \) is an invasion criterion and has applications in control, such as this model from van den Driessche & Watmough for infection with treatment.

Presentations and exercises draw significantly from materials developed with Pej Rohani, Ben Bolker, Matt Ferrari, Aaron King, and Dave Smith used during the 2009-2011 Ecology and Evolution of Infectious Diseases workshops and the 2009-2019 Summer Institutes in Statistics and Modeling of Infectious Diseases.

Licensed under the Creative Commons attribution-noncommercial license, http://creativecommons.org/licenses/bync/3.0/. Please share and remix noncommercially, mentioning its origin.