Basic reproduction number

${\displaystyle R_{0}}$ is the average number of people infected from one other person. For example, Ebola has an ${\displaystyle R_{0}}$ of two, so on average, a person who has Ebola will pass it on to two other people.

Suppose that infectious individuals make an average of ${\displaystyle \beta }$ infection-producing contacts per unit time, with a mean infectious period of ${\displaystyle \tau }$. Then the basic reproduction number is:

$${\displaystyle R_{0}=\beta \,\tau }$$

This simple formula suggests different ways of reducing ${\displaystyle R_{0}}$ and ultimately infection propagation. It is possible to decrease the number of infection-producing contacts per unit time ${\displaystyle \beta }$ by reducing the number of contacts per unit time (for example staying at home if the infection requires contact with others to propagate) or the proportion of contacts that produces infection (for example wearing some sort of protective equipment). Hence, it can also be written as[11]

${\displaystyle R_{0}={\overline {c}}\,T\,\tau ,}$

where ${\displaystyle {\overline {c}}}$ is the rate of contact between susceptible and infected individuals and ${\displaystyle T}$ is the transmissibility, i.e, the probability of infection given a contact. It is also possible to decrease the infectious period ${\displaystyle \tau }$ by finding and then isolating, treating or eliminating (as is often the case with animals) infectious individuals as soon as possible.

Latent period is the transition time between contagion event and disease manifestation. In cases of diseases with varying latent periods, the basic reproduction number can be calculated as the sum of the reproduction numbers for each transition time into the disease. An example of this is tuberculosis (TB). Blower and coauthors calculated from a simple model of TB the following reproduction number:[12]

$${\displaystyle R_{0}=R_{0}^{\text{FAST}}+R_{0}^{\text{SLOW}}}$$

In their model, it is assumed that the infected individuals can develop active TB by either direct progression (the disease develops immediately after infection) considered above as FAST tuberculosis or endogenous reactivation (the disease develops years after the infection) considered above as SLOW tuberculosis.[13]

In populations that are not homogeneous, the definition of ${\displaystyle R_{0}}$ is more subtle. The definition must account for the fact that a typical infected individual may not be an average individual. As an extreme example, consider a population in which a small portion of the individuals mix fully with one another while the remaining individuals are all isolated. A disease may be able to spread in the fully mixed portion even though a randomly selected individual would lead to fewer than one secondary case. This is because the typical infected individual is in the fully mixed portion and thus is able to successfully cause infections. In general, if the individuals infected early in an epidemic are on average either more likely or less likely to transmit the infection than individuals infected late in the epidemic, then the computation of ${\displaystyle R_{0}}$ must account for this difference. An appropriate definition for ${\displaystyle R_{0}}$ in this case is "the expected number of secondary cases produced, in a completely susceptible population, produced by a typical infected individual".[14]

The basic reproduction number can be computed as a ratio of known rates over time: if an infectious individual contacts ${\displaystyle \beta }$ other people per unit time, if all of those people are assumed to contract the disease, and if the disease has a mean infectious period of ${\displaystyle {\dfrac {1}{\gamma }}}$, then the basic reproduction number is just ${\displaystyle R_{0}={\dfrac {\beta }{\gamma }}}$. Some diseases have multiple possible latency periods, in which case the reproduction number for the disease overall is the sum of the reproduction number for each transition time into the disease. For example, Blower et al.[12] model two forms of tuberculosis infection: in the fast case, the symptoms show up immediately after exposure; in the slow case, the symptoms develop years after the initial exposure (endogenous reactivation). The overall reproduction number is the sum of the two forms of contraction: ${\displaystyle R_{0}=R_{0}^{FAST}+R_{0}^{SLOW}}$.

If an individual, after getting infected, infects exactly ${\displaystyle R_{0}}$ new individuals only after exactly a time ${\displaystyle \tau }$ (the serial interval) has passed, then the number of infectious individuals over time grows as

$${\displaystyle n_{E}(t)=n_{E}(0)\,R_{0}^{t/\tau }=n_{E}(0)\,e^{Kt}}$$

or

$${\displaystyle \ln(n_{E}(t))=\ln(n_{E}(0))+\ln(R_{0})t/\tau .}$$

The underlying matching differential equation is

$${\displaystyle {\frac {dn_{E}(t)}{dt}}=n_{E}(t){\frac {\ln(R_{0})}{\tau }}.}$$

or

$${\displaystyle {\frac {d\ln(n_{E}(t))}{dt}}={\frac {\ln(R_{0})}{\tau }}.}$$

In this case, ${\displaystyle R_{0}=e^{K\tau }}$ or ${\displaystyle K={\frac {\ln R_{0}}{\tau }}}$.

For example, with ${\displaystyle \tau =5~\mathrm {d} }$ and ${\displaystyle K=0.183~\mathrm {d} ^{-1}}$, we would find ${\displaystyle R_{0}=2.5}$.

If ${\displaystyle R_{0}}$ is time dependent

$${\displaystyle \ln(n_{E}(t))=\ln(n_{E}(0))+{\frac {1}{\tau }}\int \limits _{0}^{t}\ln(R_{0}(t))dt}$$

showing that it may be important to keep ${\displaystyle \ln(R_{0})}$ below 0, time-averaged, to avoid exponential growth.

In this model, an individual infection has the following stages:

1. Exposed: an individual is infected, but has no symptoms and does not yet infect others. The average duration of the exposed state is ${\displaystyle \tau _{E}}$.
2. Latent infectious: an individual is infected, has no symptoms, but does infect others. The average duration of the latent infectious state is ${\displaystyle \tau _{I}}$. The individual infects ${\displaystyle R_{0}}$ other individuals during this period.
3. Isolation after diagnosis: measures are taken to prevent further infections, for example by isolating the infected person.

This is a SEIR model and ${\displaystyle R_{0}}$ may be written in the following form[16]

$${\displaystyle R_{0}=1+K(\tau _{E}+\tau _{I})+K^{2}\tau _{E}\tau _{I}.}$$

This estimation method has been applied to COVID-19 and SARS. It follows from the differential equation for the number of exposed individuals ${\displaystyle n_{E}}$ and the number of latent infectious individuals ${\displaystyle n_{I}}$,

$${\displaystyle {\frac {d}{dt}}{\begin{pmatrix}n_{E}\\n_{I}\end{pmatrix}}={\begin{pmatrix}-1/\tau _{E}&R_{0}/\tau _{I}\\1/\tau _{E}&-1/\tau _{I}\end{pmatrix}}{\begin{pmatrix}n_{E}\\n_{I}\end{pmatrix}}.}$$

The largest eigenvalue of the matrix is the logarithmic growth rate ${\displaystyle K}$, which can be solved for ${\displaystyle R_{0}}$.

In the special case ${\displaystyle \tau _{I}=0}$, this model results in ${\displaystyle R_{0}=1+K\tau _{E}}$, which is different from the simple model above (${\displaystyle R_{0}=\exp(K\tau _{E})}$). For example, with the same values ${\displaystyle \tau =5~\mathrm {d} }$ and ${\displaystyle K=0.183~\mathrm {d} ^{-1}}$, we would find ${\displaystyle R_{0}=1.9}$, rather than the true value of ${\displaystyle 2.5}$. The difference is due to a subtle difference in the underlying growth model; the matrix equation above assumes that newly infected patients are currently already contributing to infections, while in fact infections only occur due to the number infected at ${\displaystyle \tau _{E}}$ ago. A more correct treatment would require the use of delay differential equations.[17]