Reproduction number

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Introduction

The reproduction number is a concept in the epidemiology of infectious diseases. It is a measure of how infectious a disease is, and is required if you wish to calculate how many people you need to vaccinate if you are to achieve herd immunity.

When somebody gets an infectious disease, they may pass it on to nobody else, or they may infect 1, 2, or more other people (who become secondary cases). This can be displayed in a number of ways, including graphically.[1] The reproduction number, R, is the average (mean) number of secondary cases caused by each case of an infectious disease, during the infectious period.

The R number will, of course, depend on a large number of factors, including:

  • How the infectious organism is spread;
  • Behaviours which affect the likelihood of spread (social mixing, sexual and feeding practices...);
  • The level of susceptibility within the population - which will depend on factors such as:
  • prior immunity;
  • levels of nutrition and immune suppressions;
  • age

R0, R, and RC

The basic reproduction number - R0

Basic reproductive rate(R0, basic reproduction number, basic reproductive ratio) is the expected number of secondary cases produced by a typical primary case in an entirely susceptible population. When R0 < 1 the infection will die out but any value for R0 ≥ 1 implies it will spread (without control measures) and higher numbers are more likely to cause epidemics. When control measures are possible epidemiologists are more interested in the effective reproduction number (R).

The R number which would apply if nobody in the population had any immunity to the disease at all, in the absence of any control measures (such as when smallpox was first introduced to Pacific islands, or the the American continents) is referred to as the basic reproduction number or R0 (that's a zero or nought, not a letter "O"). R0 gives a measure of the infectiousness of the organism per se, which tends to be relatively fixed, as it is not affected by e.g. the uptake of vaccine or immunity from previous epidemics of the disease.

R0 is proportionate to:

  • The length of time that the case remains infectious (duration of infectiousness)
  • The number of contacts a case has with susceptible hosts per unit time (the contact rate)
  • The chance of transmitting the infection during an encounter with a susceptible host (the transmission probability).

This is plain common sense, and can be expressed mathematically as:

R = c p d

where:

  • c is the number of contacts per unit time,
  • p is the transmission probability per contact, and
  • d is the duration of infectiousness.

The effective reproduction number - R


Effective reproductive number (R) is the actual average number of secondary cases per primary case observed in a population with an infective disease. The value of R is typically smaller than the value of basic reproductive rate(R0), and it reflects the impact of control measures and depletion of susceptible persons by the infection.

Examples

Early in an new infectious disease R will be close to R0

  • SARS
    • R0=3.6 (95% CI 3.1-4.2) which was the same as R in early stages as this condition had no specific treatment[2]
    • R=0.7 (95% CI: 0.7-0.8) obtained by intense control measures and allowed fairly rapid control once recognised as a highly infectious disease with respiratory transmission
  • Swine influenza 2009
    • R0 northern hemisphere summer 1.4 – 1.5 with delay strategy
    • Initial R from southern hemisphere winter 1.8 to 2.3 in community/school winter outbreaks before disease recognised and control measures emplaced[3]

The effective reproduction number will change as, for example, people become immune to the disease. During an epidemic R will typically start as >1, fall to about 1 (at which stage the incidence of the disease will remain approximately static), or fall below 1, at which point the level the epidemic will cease - at least until the proportion of the population that is susceptible increases to levels at which another epidemic may arise. This explains the regular peaks and troughs in the incidence of e.g. Parvovirus B19 infection, or of most childhood illnesses prior to the introduction of vaccination.

The effective reproduction number - R is the basic reproduction number (R0) times the fraction of the population that is susceptible to infection (x):

R = R0 x

As an epidemic spreads, and people die or become immune to the disease, x decreases, and eventually becomes small enough that R drops below 1.

The control reproduction number - RC

The control reproduction number (RC) is the average number of secondary cases due to each case in the presence of control measures such as vaccination. The aim of control measures is to ensure that the disease is eliminated from a population, which will happen if RC is less than 1

In the case of vaccination, the control reproduction number is given by the following equation:

RC = R0 (1-h f)

where

  • h is the vaccine efficacy (the proportion of people vaccinated who will have complete protection), and
  • f is the vaccine coverage (proportion of the population that has been vaccinated).

(It's actually usually rather more complicated, as some of the population will have natural immunity.)

So, if the goal of a vaccination programme is to eliminate a disease (or, to put it another way, to ensure that there is herd immunity), the vaccine coverage that is required is given by the following equation:

  • Goal: RC < 1

Vaccine coverage required:

f > \frac{1-(1/R_{0})}{h}

R0 values and vaccine coverage levels for particular infectious diseases

R0 values, and the vaccine coverage required to prevent them are given for selected disease in the following table:[4]

Disease R0 Vaccine coverage (course completed) required for herd immunity
Diphtheria 6-7 85%
Measles 12-18 83%-94%
Mumps 4-7 75%-86%
Pertussis 12-17 92%-94%
Polio 5-7 80%-86%
Rubella 6-7 83%-94%
Smallpox 5-7 80%-85%

References


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