Canada – and variously, the world – is in the midst of a drive to vaccinate against the COVID-19 virus (COVID) and all of its variants. Eradication of the disease, it seems, will only be made possible when enough of the worldwide population is immunized, either through vaccination or through infection and survival of COVID. At that point, the world will have reached so-called herd immunity. Many parts of the world currently lack vaccine doses, while others, including Canada and the United States, are entering a phase at which supply is no longer the most pressing issue. The most compelling issues in the minds of many North Americans are whether or not to get vaccinated, and how high the vaccination level must be to achieve herd immunity.
This question is practical rather than academic. The social, psychological, economic, and medical health of people and nations are affected by our ability locally as a nation, and globally as a people, to control COVID. Freedom and liberty are also at stake. In the absence of vaccines, the intervention against COVID has taken the form of restrictions on, or alterations of, behaviour. This has included (at times) the wearing of masks and other personal protective devices, the banning of social activities, of education, of free movement, of the normal operation of many businesses, of religious gatherings, of sports, of expressions of love and affection, and even of the observance of death rites. Until COVID is controlled, and possibly eradicated, freedom- and health-loving Canadians may retain reasonable concern regarding the return of some elements of these restrictions.
As Canada and the world’s vaccination levels rise, some may view the question of herd immunity as moot. We may hope this is true, but we cannot be sure of it. The natural world is an uncertain one. Theories and hypotheses are subject to change, as are our collective behaviors.[1] New variants of concern, such as the Delta variant have apparent higher reproduction[2] numbers and may require a redefinition of the required level of herd immunity.
In order to reduce the uncertain problem of herd immunity vaccination rates, we must consider the following ideas:
- What is the reproduction rate (Ro) of COVID?
- What affects the effective reproduction rate (Re or Rt)?
- What is the relationship between Ro and the vaccination rates required to reach herd immunity?
- What is the relationship between liberty and vaccination?
Terms
Understanding many terms is required to adequately discuss COVID, herd immunity, and the spread of the disease. Here is a summary of some of the key terms related to this article.[3]Greater explanation for these terms and their relevance follows.
Summary of terms. If these values are accurately known,
decision processes regarding herd immunity can be more reasonably attempted.
Reproductive rate
Ro is the initial, basic, or pristine reproductive rate. It defines the disease’s infectious potential, though not its severity, and is the key variable to understand in determining the herd immunity vaccination rate. Ro is the number of secondary infections that are generated from an initial case prior to any element of the population becoming immune and prior to any mitigation strategies or non-pharmaceutical interventions (NPIs). NPIs are widely known as lockdowns or restrictions.[4]
Once the pandemic leaves the initial stage and intervention strategies are invoked, the reproductive rate varies and becomes an effective and time variant factor. Terms such as effective reproductive rate (Re) or time variant reproductive rate (Rt) are used. Ro is always greater than Re.[5]
It should be noted that Ro is not an absolute number. It can only be estimated, and the best estimate may vary by region. Ro is a function of the disease, but it is also affected by population, population density, social behaviours, how people live, shop, observe religious rites, and travel. The Royal Society, 2020, Reproduction number (R) and growth rate (r) of the COVID-19 epidemic in the UK: methods of estimation, data sources, causes of heterogeneity, and use as a guide in policy formulation, August 24, 2020[6]
Reproductive rate is difficult to estimate, and estimates will have a high variance
Ro is difficult to estimate, and can only be done through mathematical modeling that uses input data related to diagnosed cases such as from contact tracing, estimates of infected non-symptomatic persons, deaths, and hospitalizations, and seropositivity. The accuracy of some of this data will vary depending on location and even on the time variant diagnostic capability of medical professionals.[7]Ro may be directly estimated from the initial growth rate of the epidemic, though to do this, the generation time, or average time between infection events, must also be known or estimated.
Besides data type and quality, and variations in model and other variables, and in the effect of local cultural modalities, there is another reason that Ro is difficult to estimate: superspreader events. It has been noted for most infectious diseases that the majority of disease transmission comes from a small fraction of infected persons.[8] Probability distributions show that this is strongly the case for COVID. The term k, also known as dispersion, aggregation, or clumpiness, describes how a small number of spreaders affect Ro. K is an inverse term, where low values mean high degrees of superspreading. Due to this effect, the heterogeneity of estimates for Ro are quite high for COVID, giving high uncertainty to any assessment of its value.[9]
Our best estimate of the reproductive rate of COVID
There have been numerous attempts to estimate Ro for COVID around the world, many of them in China, but also in France, Spain, Germany, Italy, and elsewhere. Given the difficulties of estimating Ro and its importance in determining the herd immunity vaccination rate, there was a high level of motivation to conduct such studies.[10]A recent worldwide review of such studies validated 31 estimates for Ro, and produced an average estimate for the original, “wild” strain.[11]These results are summarized below in bullet and chart form. Estimates of Ro for the Alpha and Delta variants from another source are included.[12]
- Original or Wild COVID: 2.87 (2.39 to 3.34)
- Alpha variant: 4 to 5
- Delta variant: 5 to 8
Chart showing the estimate for Ro for COVID and two of its variants.
The source data for this chart is [13]and [14].
The chart is noteworthy because it illustrates the large variation in the estimates for Ro.
Effective reproduction rate and interventions
Prior to the availability of vaccines, non-pharmaceutical interventions (NPI) were used in attempting to control the effect of COVID. NPIs are more commonly known as restrictions, interventions, mitigations, or lockdowns. These NPIs include school and workplace closures, and restrictions on travel, public transportation, and gatherings. All such restrictions are intrusions on freedom and liberty. The NPIs were intended to help get Re below 1.
A study regarding the effectiveness of various key NPIs was recently conducted by Li et al using data from 131 countries.[15]There were challenges to determining the effectiveness of each NPI. First, estimating instantaneous values of Re is difficult and uncertain for the same reasons that estimating Ro is uncertain and difficult. The study aimed to isolate the effect of each NPI. The effect of certain confounding variables that may have affected Re, such as increases in hygiene, could not be removed. In cases of countries where NPIs were applied inconsistently by geography, additional error was introduced to this work. Another challenge was that the study can only account for the policy decision (NPI), not whether the population complied with the policy.[16]Despite these challenges, meaningful results appear to have been achieved.
This study shows that certain of these NPIs were successful. Banning public events and gatherings of greater than 100 people were determined to be the most effective NPIs. It also shows that there is a delay of 1 to 3 weeks after the NPI is introduced before the reduction in Re manifests. A similar delay was observed in the timing of the increase in Re after the ban was lifted.
This study and related studies in the future will be tools that governments have for future outbreaks of COVID or similar infectious diseases. This study demonstrates that a reduction in spread can be achieved even if vaccination-rate herd immunity is not achieved, although at the price of certain elements of freedom and liberty.
Figure from Li et al’s[17]global study on the effect of restrictions on Re. This is the change over time for Re following either the introduction (blue) or lifting (red) of particular restrictions (NPI). The error bars are the 95% confidence intervals for the estimate of Re.
Ro and the herd immunity vaccination rate
How is the herd immunity vaccination rate estimated? There are some assumptions to be considered, the first of which we know is that we know the Ro. The second, that we assume vaccination effectively suppresses the spread of the disease. The third assumption is that we wish to vaccinate to the Ro level of the disease and not to some lower Re, the choice of which would require infection avoidance, or interventions in freedom and behaviour.
Given these assumptions, Patrick Honner of Quanta Magazine shows how to derive the equation for herd immunity vaccination rates.[18]His derivation is reproduced below.
If N is the number of persons that each infected individual comes into contact with during their infectious period.
If V is the number of vaccinated persons out of N people contacted.
Then the number of new infections will be:
Since we are looking for the rate of new infections, and we want that rate to be 1 or less, we set the equation equal to 1:
We want to find the vaccination rate as a fraction (or percentage), which is V/N:
We use this relationship to create a chart defining the herd immunity vaccination rate as a function of Ro.
The estimated herd immunity vaccination rates of a few common diseases are noted in the chart. [19][20] [21][22]
Chart showing the relationship between the Ro of various diseases
and their corresponding herd immunity vaccination rates.
Polio requires very high immunization rates in the 80% range and measles requires even higher rates. Polio was eradicated from the United States and Canada (and all of the Americas) in 1994.[23][7]
Estimated herd immunity vaccination rate for COVID
The range of Ro estimates for COVID can also be labelled and considered. Let us lay out the key estimates of COVID’s Ro (which we defined earlier) and their corresponding herd immunity vaccination rate.
- Original or Wild COVID: 2.87 (2.39 to 3.34), corresponding to: 65.2% (58.1% to 70.0%) vaccination rate.
- Alpha variant: 4 to 5, corresponding to 75% to 80% vaccination rate.
- Delta variant: 5 to 8, corresponding to 80% to 87.5% vaccination rate.
Chart showing the relationship between the Ro of various COVID variants and the herd immunity vaccination rate. The Delta variant could require rates as high as about 87.5%.
Interpretation
Is this data suggesting that we are doomed unless our vaccination rates climb into the mid-80s, or perhaps higher? In this case, it appears that fear is not warranted. Careful and informed decisions will be more useful in seeing that Canadians enjoy the outcome of their choosing.
According to Our World in Data, as of July 21, 2021, Canada enjoys a vaccination rate for eligible citizens of 70.8% for one dose and 52.6% for two doses.[24]These rates of vaccination will continue to climb, eventually finding some unknown maximum number above 71%.
The current vaccination rate, assuming full two-dose vaccination eventually matches single-dose vaccination, would likely be sufficient to meet herd immunity for the original or “wild” strain of COVID.
Concerned citizens will note that the Alpha and in particular, Delta variants of concern could require vaccination rates as high as 87.5% for herd immunity to be reached. This is rationally concerning but should be viewed carefully. The estimates for the Ro, and corresponding required vaccination rate for herd immunity are uncertain. The Delta variant vaccination requirement could be as low as 80%. Or it could be higher. One could argue that we should reach for the highest vaccination rate possible, because there could be future variants with higher Ro values. However, becoming fearfully attached to a particular number is not warranted. Our level of certainty is insufficient to look at this question too finely.
The other reason that we should not feel great angst over where the vaccination number peaks, given that it exceeds, or will exceed 71%, is that we have a choice – a tradeoff – that we are in position to make.
The use of Re, rather than Ro has been argued as being more appropriate for determining the herd immunity vaccination rate.[25]It has been demonstrated that the curtailment of freedom and liberty in the form of restrictions on activities can lower the effective reproduction rate, Re, of COVID.[26]If a COVID variant such as Delta causes another spike of cases in some element of our population due to an insufficient vaccination rate, we may be choosing to endure whatever restriction in freedom required to create some Re (which will always be lower than Ro) that our herd immunity level can accommodate.
Argument that the effective reproductive rate is inversely proportional in some manner to the severity of the non-pharmaceutical interventions (restrictions) being imposed on the population.
Having taken a close look at COVID reproduction rates and the effectiveness of NPIs, we should better understand the challenges that North American politicians and medical officials have grappled with over the last 18 months. Ideally, such analysis will also help our leaders make well-informed decisions in the future.
References
[1] Weston, Laurie, 2021, Science—there is method to the madness: BIG Media, April 12, 2021
[2] Gallagher, James, 2021, Covid: is there a limit to how much worse variants can get?, BBC News, June 12, 2021
[3] The Royal Society, 2020, Reproduction number (R) and growth rate (r) of the COVID-19 epidemic in the UK: methods of estimation, data sources, causes of heterogeneity, and use as a guide in policy formulation, August 24, 2020
[4] The Royal Society, 2020, Reproduction number (R) and growth rate (r) of the COVID-19 epidemic in the UK: methods of estimation, data sources, causes of heterogeneity, and use as a guide in policy formulation, August 24, 2020
[5] The Royal Society, 2020, Reproduction number (R) and growth rate (r) of the COVID-19 epidemic in the UK: methods of estimation, data sources, causes of heterogeneity, and use as a guide in policy formulation, August 24, 2020
[6] The Royal Society, 2020, Reproduction number (R) and growth rate (r) of the COVID-19 epidemic in the UK: methods of estimation, data sources, causes of heterogeneity, and use as a guide in policy formulation, August 24, 2020
[7] The Royal Society, 2020, Reproduction number (R) and growth rate (r) of the COVID-19 epidemic in the UK: methods of estimation, data sources, causes of heterogeneity, and use as a guide in policy formulation, August 24, 2020
[8] The Royal Society, 2020, Reproduction number (R) and growth rate (r) of the COVID-19 epidemic in the UK: methods of estimation, data sources, causes of heterogeneity, and use as a guide in policy formulation, August 24, 2020
[9] The Royal Society, 2020, Reproduction number (R) and growth rate (r) of the COVID-19 epidemic in the UK: methods of estimation, data sources, causes of heterogeneity, and use as a guide in policy formulation, August 24, 2020
[10] Billah Arif, Mamum Miah, Nuruzzaman Khan, 2020, Reproductive number of coronavirus: A systematic review and meta-analysis based on global level evidence, PLOS ONE. 15 (11): e0242128.
[11] Billah Arif, Mamum Miah, Nuruzzaman Khan, 2020, Reproductive number of coronavirus: A systematic review and meta-analysis based on global level evidence, PLOS ONE. 15 (11): e0242128.
[12] Gallagher, James, 2021, Covid: is there a limit to how much worse variants can get?, BBC News, June 12, 2021
[13] Gallagher, James, 2021, Covid: is there a limit to how much worse variants can get?, BBC News, June 12, 2021
[14] Billah Arif, Mamum Miah, Nuruzzaman Khan, 2020, Reproductive number of coronavirus: A systematic review and meta-analysis based on global level evidence, PLOS ONE. 15 (11): e0242128
[15] You Li, Harry Campbell, Durga Kulkarni, Alice Harpur, Madhurima Nundy, Xin Wang, et al, 2020, The temporal association and lifting non-pharmaceutical interventions with the time-varying reproduction number ® of SARS-CoV02: A modelling study across 131 countries: The Lancet Infectious Diseases, Vol. 21, No. 2. P193-2020
[16] You Li, Harry Campbell, Durga Kulkarni, Alice Harpur, Madhurima Nundy, Xin Wang, et al, 2020, The temporal association and lifting non-pharmaceutical interventions with the time-varying reproduction number ® of SARS-CoV02: A modelling study across 131 countries: The Lancet Infectious Diseases, Vol. 21, No. 2. P193-2020
[17] You Li, Harry Campbell, Durga Kulkarni, Alice Harpur, Madhurima Nundy, Xin Wang, et al, 2020, The temporal association and lifting non-pharmaceutical interventions with the time-varying reproduction number ® of SARS-CoV02: A modelling study across 131 countries: The Lancet Infectious Diseases, Vol. 21, No. 2. P193-2020
[18] Patrick Honner, 2018, How Math (and Vaccines) Keep You Safe From the Flu, Quanta Magazine
[19] Centers for Disease Control Prevention (CDC), 1994, Certification of poliomyelitis eradication-the Americas, 1994, MMWR Morbidity and Mortality Weekly Report. 43 (39): 720-722
[20] Guerra Fiona, Shelly Bolotin, Gillian Lim, Jane Heffernan, Shelley Deeks, Li Ye, Natasha Crowcroft, 2017, The basic reproduction number (R0) of measles: a systematic review, The Lancet. Infectious Diseases. 17 (12): e420–e428. doi:10.1016/S1473-3099(17)30307-9. PMID 28757186
[21] Fraser Christophe, Christl Donnelly, Simon Cauchemez, William Hanage, Maria Van Kerkhove, Dierdre Hollingsworth, et al., June 2009, Pandemic potential of a strain of influenza A (H1N1): early findings. Science. 324 (5934): 1557 61. Bibcode:2009Sci…324.1557F. doi:10.1126/science.1176062. PMC 3735127. PMID 19433588
[22] Freeman, Colin., 2020, Magic Formula that will determine whether Ebola is beaten, The Telegraph, November 6, 2014
[23] Centers for Disease Control Prevention (CDC), 1994, Certification of poliomyelitis eradication-the Americas, 1994, MMWR Morbidity and Mortality Weekly Report. 43 (39): 720-722
[24] Coronavirus (COVID-19) Vaccinations, Our World in Data, July 21, 2021
[25] Delamater, Paul, Erica Street, Timothy Leslie, Tony Yang, Kathryn Jacobsen, 2019, Complexity of the Basic Reproduction Number (Ro), Centers for Disease Control and Prevention: Emerging Infectious Diseases, 25, 1 , DOI: 10.3201/eid2501.171901
[26] You Li, Harry Campbell, Durga Kulkarni, Alice Harpur, Madhurima Nundy, Xin Wang, et al, 2020, The temporal association and lifting non-pharmaceutical interventions with the time-varying reproduction number ® of SARS-CoV02: A modelling study across 131 countries: The Lancet Infectious Diseases, Vol. 21, No. 2. P193-2020
