What is the reproduction number in a corona pandemic?

coronavirus, sars-cov-2, lung

Governments talk about the reproduction number (R0) in pandemic models with the over familiarity of an Australian cricket commentator. Governments are effectively reactive to short term political soundings and by implication pushing an erroneous view that we can avoid a relatively significant number of COVID deaths by continuing the economic free fall.

But what is the reproduction number (R0)

The basic reproduction number (R0) is a known measure the transmission potential of a disease. It is the average number of secondary infections produced by a typical case of an infection in a population where everyone is susceptible. For example, if the R0 for measles in a population is 15, then we would expect each new case of measles to produce 15 new secondary cases (assuming everyone around the case was susceptible). R0 excludes new cases produced by the secondary cases.

The basic reproductive number is affected by several factors:

  • The rate of contacts in the host population
  • The probability of infection being transmitted during contact
  • The duration of infectiousness.

In general, for an epidemic to occur in a susceptible population R0 must be >1, so the number of cases is increasing. In many circumstances not all contacts will be susceptible to infection. This is measured by the effective reproductive rate (R).

Values of {\displaystyle R_{0}} of well-known infectious diseases (assayed before any social interventions). (Source)
Disease Transmission
Measles Aerosol 12–18
Chickenpox (varicella) Aerosol 10–12
Mumps Respiratory droplets 10–12
Polio Fecal–oral route 5–7
Rubella Respiratory droplets 5–7
Pertussis Respiratory droplets 5.5
Smallpox Respiratory droplets 3.5–6
HIV/AIDS Body fluids 2–5
SARS Respiratory droplets 0.19–1.08
COVID-19 Respiratory droplets and aerosol 2–6 
Common cold Respiratory droplets 2–3
Diphtheria Saliva 1.7–4.3
Influenza
(1918 pandemic strain)
Respiratory droplets 1.4–2.8
Ebola
(2014 Ebola outbreak)
Body fluids 1.5–1.9
Influenza
(2009 pandemic strain)
Respiratory droplets 1.4–1.6
Influenza
(seasonal strains)
Respiratory droplets 0.9–2.1
MERS Respiratory droplets 0.3–0.8
Nipah virus Body fluids 0.48

Effective reproductive number (R)

A population will rarely be totally susceptible to an infection in the real world. Some contacts will be immune, for example due to prior infection which has conferred life-long immunity, or as a result of previous immunisation. Consequently, not all contacts get infected and the average number of secondary cases per infectious case will be lower than the basic reproduction number. The effective reproductive number (R) is the average number of secondary cases per infectious case in a population made up of both susceptible and non-susceptible hosts. If R>1, the number of cases will increase, such as at the start of an epidemic. Where R=1, the disease is endemic, and where R<1 there will be a decline in the number of cases.

The effective reproduction number can be estimated by the product of the basic reproductive number and the fraction of the host population that is susceptible (x). So:

R = R0x
For example, if R0 for influenza is 12 in a population where half of the population is immune, the effective reproductive number for influenza is 12 x 0.5 = 6. Under these circumstances, a single case of influenza would produce an average of 6 new secondary cases.

To eliminate a disease from a population, R needs to be less than 1.

Herd immunity

Herd immunity occurs when a significant proportion of the population (or the herd) have been vaccinated (or are immune by some other mechanism), resulting in protection for susceptible (e.g. unvaccinated) individuals. The larger the number of people who are immune in a population, the lower the likelihood that a susceptible person will come into contact with the infection. It is more difficult for diseases to spread between individuals if large numbers are already immune as the chain of infection is broken.

The herd immunity threshold is the proportion of a population that need to be immune in order for an infectious disease to become stable in that community. If this is reached, for example through immunisation, then each case leads to a single new case (R=1) and the infection will become stable within the population.

Science and reality

The epidemic theory describes basic reproduction number, effective reproduction number, herd immunity, epidemics, epidemic curves, index cases, generation times, exceptions, clusters, significant clusters, aggregation of cases, etc. are all part of studies in science, but they are theories, not facts.

Calculating the reproduction number is possible, but does the reproduction number also indicate reality? No. It presents possible statistical trends, which are/were all wrong, as we have seen at the beginning of he pandemic. It can't even be used as an indicator.

Remember, it's theoretical, nothing to do with reality. The reproduction number is used for scientific research. It's an interesting study how the reproduction number compares with the reality of a pandemic, after the pandemic is gone. It's foolish to use the reproduction number during the pandemic and to build the nation's policy on a theory, which does not relate to the actual facts and reality. The reproduction number is mainly used in statistical modeling, which is - as we saw in the beginning of the pandemic outbreak - always wrong.

And that is exactly what is happening. For example, the Israeli government is using the reproduction number to determine if a (closed) sector of the society and economy can be (re-)opened or not. With as result, that their (promised three weeks) lockdown of the country is taking 63 days (and currently counting). The reality learns that they are wrong.

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