Basic reproduction number
Encyclopedia
In epidemiology
, the basic reproduction number (sometimes called basic reproductive rate, basic reproductive ratio and denoted R0, r nought) of an infection
is the mean number of secondary cases caused by an individual infected soon after disease introduction into a population with no pre-existing immunity
to the disease in the absence of interventions to control the infection. This metric is useful because it helps determine whether or not an infectious disease
can spread through a population. The roots of the basic reproduction concept can be traced through the work of Alfred Lotka, Ronald Ross
, and others, but its first modern application in epidemiology was by George MacDonald in 1952, who constructed population models of the spread of malaria
.
When
the infection will die out in the long run (provided infection rates are constant). But if
the infection will be able to spread in a population.
Generally, the larger the value of R0, the harder it is to control the epidemic. For simple models, the proportion of the population that needs to be vaccinated to prevent sustained spread of the infection is given by 1 − 1/R0. The basic reproductive rate is affected by several factors including the duration of infectivity
of affected patients, the infectiousness of the organism, and the number of susceptible people in the population that the affected patients are in contact with.
, evolutionary invasion analysis
and life history theory
. It represents the average number of offspring produced over the lifetime of an individual (under ideal conditions).
For simple population models
, R0 can be calculated, provided an explicit decay rate (or "death rate") is given. In this case, the reciprocal of the decay rate (usually 1/d) gives the average lifetime of an individual. When multiplied by the average number of offspring per individual per timestep (the "birth rate" b), this gives R0 = b / d. For more complicated models that have variable growth rates (e.g. because of self-limitation or dependence on food densities), the maximum growth rate should be used.
s, what is often claimed to be R0 is, in fact, simply a threshold, not the average number of secondary infections. There are many methods used to derive such a threshold from a mathematical model, but few of them always give the true value of R0. This is particularly problematic if there are intermediate vectors between hosts, such as malaria
.
What these thresholds will do is determine whether a disease will die out (if R0 < 1) or whether it may become endemic (if R0 > 1), but they generally can not compare different diseases. Therefore, the values from the table above should be used with caution, especially if the values were calculated from mathematical models.
Methods include the survival function
, rearranging the largest eigenvalue of the Jacobian matrix, the next-generation method, calculations from the intrinsic growth rate, existence of the endemic equilibrium, the number of susceptibles at the endemic equilibrium, the average age of infection and the final size equation. Few of these methods agree with one another, even when starting with the same system of differential equations. Even fewer actually calculate the average number of secondary infections. Since R0 is rarely observed in the field and is usually calculated via a mathematical model, this severely limits its usefulness.
Epidemiology
Epidemiology is the study of health-event, health-characteristic, or health-determinant patterns in a population. It is the cornerstone method of public health research, and helps inform policy decisions and evidence-based medicine by identifying risk factors for disease and targets for preventive...
, the basic reproduction number (sometimes called basic reproductive rate, basic reproductive ratio and denoted R0, r nought) of an infection
Infection
An infection is the colonization of a host organism by parasite species. Infecting parasites seek to use the host's resources to reproduce, often resulting in disease...
is the mean number of secondary cases caused by an individual infected soon after disease introduction into a population with no pre-existing immunity
Immunity (medical)
Immunity is a biological term that describes a state of having sufficient biological defenses to avoid infection, disease, or other unwanted biological invasion. Immunity involves both specific and non-specific components. The non-specific components act either as barriers or as eliminators of wide...
to the disease in the absence of interventions to control the infection. This metric is useful because it helps determine whether or not an infectious disease
Infectious disease
Infectious diseases, also known as communicable diseases, contagious diseases or transmissible diseases comprise clinically evident illness resulting from the infection, presence and growth of pathogenic biological agents in an individual host organism...
can spread through a population. The roots of the basic reproduction concept can be traced through the work of Alfred Lotka, Ronald Ross
Ronald Ross
Sir Ronald Ross KCB FRS was a British doctor who received the Nobel Prize for Physiology or Medicine in 1902 for his work on malaria. He was the first Indian-born person to win a Nobel Prize...
, and others, but its first modern application in epidemiology was by George MacDonald in 1952, who constructed population models of the spread of malaria
Malaria
Malaria is a mosquito-borne infectious disease of humans and other animals caused by eukaryotic protists of the genus Plasmodium. The disease results from the multiplication of Plasmodium parasites within red blood cells, causing symptoms that typically include fever and headache, in severe cases...
.
| Disease | Transmission | R0 |
|---|---|---|
| Measles Measles Measles, also known as rubeola or morbilli, is an infection of the respiratory system caused by a virus, specifically a paramyxovirus of the genus Morbillivirus. Morbilliviruses, like other paramyxoviruses, are enveloped, single-stranded, negative-sense RNA viruses... |
Airborne | 12–18 |
| Pertussis Pertussis Pertussis, also known as whooping cough , is a highly contagious bacterial disease caused by Bordetella pertussis. Symptoms are initially mild, and then develop into severe coughing fits, which produce the namesake high-pitched "whoop" sound in infected babies and children when they inhale air... |
Airborne droplet | 12–17 |
| Diphtheria Diphtheria Diphtheria is an upper respiratory tract illness caused by Corynebacterium diphtheriae, a facultative anaerobic, Gram-positive bacterium. It is characterized by sore throat, low fever, and an adherent membrane on the tonsils, pharynx, and/or nasal cavity... |
Saliva | 6–7 |
| Smallpox Smallpox Smallpox was an infectious disease unique to humans, caused by either of two virus variants, Variola major and Variola minor. The disease is also known by the Latin names Variola or Variola vera, which is a derivative of the Latin varius, meaning "spotted", or varus, meaning "pimple"... |
Social contact | 5–7 |
| Polio | Fecal-oral route | 5–7 |
| Rubella Rubella Rubella, commonly known as German measles, is a disease caused by the rubella virus. The name "rubella" is derived from the Latin, meaning little red. Rubella is also known as German measles because the disease was first described by German physicians in the mid-eighteenth century. This disease is... |
Airborne droplet | 5–7 |
| Mumps Mumps Mumps is a viral disease of the human species, caused by the mumps virus. Before the development of vaccination and the introduction of a vaccine, it was a common childhood disease worldwide... |
Airborne droplet | 4–7 |
| HIV HIV Human immunodeficiency virus is a lentivirus that causes acquired immunodeficiency syndrome , a condition in humans in which progressive failure of the immune system allows life-threatening opportunistic infections and cancers to thrive... /AIDS AIDS Acquired immune deficiency syndrome or acquired immunodeficiency syndrome is a disease of the human immune system caused by the human immunodeficiency virus... |
Sexual contact | 2–5 |
| SARS | Airborne droplet | 2–5 |
| Influenza Influenza Influenza, commonly referred to as the flu, is an infectious disease caused by RNA viruses of the family Orthomyxoviridae , that affects birds and mammals... (1918 pandemic Spanish flu The 1918 flu pandemic was an influenza pandemic, and the first of the two pandemics involving H1N1 influenza virus . It was an unusually severe and deadly pandemic that spread across the world. Historical and epidemiological data are inadequate to identify the geographic origin... strain) |
Airborne droplet | 2–3 |
When
- R0 < 1
the infection will die out in the long run (provided infection rates are constant). But if
- R0 > 1
the infection will be able to spread in a population.
Generally, the larger the value of R0, the harder it is to control the epidemic. For simple models, the proportion of the population that needs to be vaccinated to prevent sustained spread of the infection is given by 1 − 1/R0. The basic reproductive rate is affected by several factors including the duration of infectivity
Infectivity
In epidemiology, infectivity refers to the ability of a pathogen to establish an infection. More specifically, infectivity is a pathogen's capacity for horizontal transmission that is, how frequently it spreads among hosts that are not in a parent-child relationship...
of affected patients, the infectiousness of the organism, and the number of susceptible people in the population that the affected patients are in contact with.
Other uses
R0 is also used as a measure of individual reproductive success in population ecologyPopulation ecology
Population ecology is a sub-field of ecology that deals with the dynamics of species populations and how these populations interact with the environment. It is the study of how the population sizes of species living together in groups change over time and space....
, evolutionary invasion analysis
Evolutionary invasion analysis
Evolutionary invasion analysis, also known as adaptive dynamics, is a set of techniques for studying long-term phenotypical evolution developed during the 1990s...
and life history theory
Life history theory
Life history theory posits that the schedule and duration of key events in an organism's lifetime are shaped by natural selection to produce the largest possible number of surviving offspring...
. It represents the average number of offspring produced over the lifetime of an individual (under ideal conditions).
For simple population models
Population modeling
A population model is a type of mathematical model that is applied to the study of population dynamics.Models allow a better understanding of how complex interactions and processes work. Modeling of dynamic interactions in nature can provide a manageable way of understanding how numbers change over...
, R0 can be calculated, provided an explicit decay rate (or "death rate") is given. In this case, the reciprocal of the decay rate (usually 1/d) gives the average lifetime of an individual. When multiplied by the average number of offspring per individual per timestep (the "birth rate" b), this gives R0 = b / d. For more complicated models that have variable growth rates (e.g. because of self-limitation or dependence on food densities), the maximum growth rate should be used.
Limitations of R0
When calculated from mathematical models, particularly ordinary differential equationOrdinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s, what is often claimed to be R0 is, in fact, simply a threshold, not the average number of secondary infections. There are many methods used to derive such a threshold from a mathematical model, but few of them always give the true value of R0. This is particularly problematic if there are intermediate vectors between hosts, such as malaria
Malaria
Malaria is a mosquito-borne infectious disease of humans and other animals caused by eukaryotic protists of the genus Plasmodium. The disease results from the multiplication of Plasmodium parasites within red blood cells, causing symptoms that typically include fever and headache, in severe cases...
.
What these thresholds will do is determine whether a disease will die out (if R0 < 1) or whether it may become endemic (if R0 > 1), but they generally can not compare different diseases. Therefore, the values from the table above should be used with caution, especially if the values were calculated from mathematical models.
Methods include the survival function
Survival function
The survival function, also known as a survivor function or reliability function, is a property of any random variable that maps a set of events, usually associated with mortality or failure of some system, onto time. It captures the probability that the system will survive beyond a specified time...
, rearranging the largest eigenvalue of the Jacobian matrix, the next-generation method, calculations from the intrinsic growth rate, existence of the endemic equilibrium, the number of susceptibles at the endemic equilibrium, the average age of infection and the final size equation. Few of these methods agree with one another, even when starting with the same system of differential equations. Even fewer actually calculate the average number of secondary infections. Since R0 is rarely observed in the field and is usually calculated via a mathematical model, this severely limits its usefulness.