# Basic reproduction number in epidemiology

The basic reproduction number $R_0$ of diseases depends on the population size (or the number of susceptible individuals). The commonly reported $R_0$ is based on what population size (when they do not mention about it)? Is there any standard population size for reporting $R_0$?

• I'd think there'd be a required sample size to get your data within the necessary margin of error, but agencies like WHO actually provide standard population data I believe. Someone let me know if i'm mistaken! – CKM Nov 16 '15 at 1:34
• Actually, population density and structure is far more important than population size. – Corvus Nov 16 '15 at 18:52
• OK, so $R_0$ depends on the environment. For example, different countries experience different $R_0$ for the same pathogen. So the commonly reported values correspond with what environmental conditions? – quibble Nov 17 '15 at 0:28

From what I understand (as an ecologist/population modeller), $R_0$ in epidemiology is not dependent on host population size per se, at least not in its basic form. It is also not dependent on the number of susceptible individuals, since it is defined as the number of secondary infections in a fully susceptible population, see e.g. this section in Farrington et al (2001):

The basic reproduction number of an infectious agent in a given population is the average number of secondary infections which one typical infected individual would generate if the population were completely susceptible.

This is also consistent with the Wikipedia description ("...in an otherwise uninfected population"). $R_0$ is however dependent on the environment (dispersal routes, host-host interactions etc), which also why it is used to evaluate and compare the effect of control measures. Host-host interactions is also dependent on population density, so $R_0$ is indirectly influenced by host population density, and this is partially why estimated $R_0$ values for the same disease will differ between countries and populations.

Fundamentally and conceptually, $R_0$ is the same thing as the Net reproductive rate (often also labelled $R_0$) in demography and population modelling, and it was originally borrowed from demography to epidemiology. As such, the net reproductive rate is often used in for instance pest control to evaluate the potential impact of pest species on agricultural crops, in direct analog to the basic reproduction number in epidemiology (see Emiljanowicz et al, 2014 for a randomly picked, recent example). Most commonly, $R_0$ (both in population dynamics and epidemiology) is calculated from static demographic rates, but nothing prevents you from considering stochastic effects, density dependence or e.g. ways that demographic rates are functions of host density or population size. This could lead to formulations where $R_0$ is an explicit function of host population density, but that is not the standard use of $R_0$.

To me, it seems like you are asking for the effective reproduction number ($R(t)$), which can be defined by:

...the average number of secondary cases that result from an infectious case in a particular population (Box 1). R depends on the level of susceptibility in the population, in contrast to the basic reproduction number (R0), which is the average number of secondary cases arising from one infectious case in a totally susceptible population."

This section is from a WHO paper on meases (Chiew et al, 2013). So basically, $R(t)$ is the average number of secondary cases per primary case at time $t$ (including the effects of immunity and/or control measures). As you can see, $R(t)$ is a function of time, and it can e.g. be used to describe the temporal dynamics of SIR-type models.

• But a well established result in epidemiology models (e.g., the standard SIR model) is that a disease will die out before infecting all susceptible individuals in the population. Also an epidemic will not even take place if the population size is small. This is because $R_0$ itself is a dynamic quantity. – quibble Dec 1 '15 at 1:12
• @quibble That diseases will die out is not a function of the $R_0$, but of the buildup of immunity described by the dynamics of SIR models. $R_0$ is not supposed to describe this, and bringing it into discussion is just confusing things.The dynamics in small populations is also another thing (related to e.g. low host density or stochasticity). Remember that $R_0$ is mainly a measure of maximum growth, similar to e.g. $r_max$ in population models, which is describing exponential growth under "ideal" conditions (e.g. a completely susceptible population). – fileunderwater Dec 1 '15 at 9:29
• @quibble What you seem to be after is the effective reproduction number ($R_t$), which can be described as "...the average number of secondary cases that result from an infectious case in a particular population (Box 1). R depends on the level of susceptibility in the population, in contrast to the basic reproduction number (R0), which is the average number of secondary cases arising from one infectious case in a totally susceptible population." (Chiew et al, 2013) – fileunderwater Dec 1 '15 at 9:32