Understanding the Reproduction Number, Lock-downs and Herd Immunity

During a disease epidemic, lock-downs cut the reproduction number -- that is the number of people infected by each infected person -- by reducing opportunities for viral spread. This can be seen in the Canadian data, which show that the number of newly reported cases turned down within two week of the onset of a national lock-down (see figure 2). As the incubation period for the disease is approximately two weeks, that indicates that the lock-down had a more or less immediate effect in reducing the spread of the virus.

Figure 2. New reported COVID-19 cases (n=5,590

Footnote1) in Canada by episode dateFootnote2 as of April 1, 2020, 11 am EDT

When R0 is less than one, the epidemic dies down and is ultimately extinguished. Problem is, if R0 is driven down by means of a lock-down, any remaining cases or new cases from whatever source are likely to re-ignite the epidemic when the lock-down ends. That is the danger China now faces. China is dealing with this risk by testing people entering the country so that imported cases can be quarantined, and so that contacts of new cases of any source, can be tracked down and quarantined.

In time, if the proportion of the population that has been infected rises high enough, R0 will fall below 1 without a lock-down because most people are immune and cannot therefore be (re-)infected. That is why, in Britain, epidemiologists are now conducting surveys to determine how many people have already acquired immunity. If the proportion is high enough, probably over 60%, herd immunity will have been achieved. That means that even without control over human interactions, R0 will be less than 1 and the virus will be unable to spread. At that point, any new outbreak will die out spontaneously and the national lock-down can be ended.

A further consequence of making a reliable estimate of the population-wide infection rate is that it will provide a basis for an accurate estimate of the mortality rate for the disease. At present, mortality rate estimates are hopelessly skewed because the infection rate includes only those with disease symptoms who are known to medical authorities. Those who suffer the illness without reporting to a doctor, or who are infected without experiencing disease symptoms, are currently excluded from mortality rate estimates, thus elevating the apparent death rate, probably many times in excess of the actual rate.

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