Estimating Herd Immunity Gains from Mandatory Vaccination Policies — 2

December 17, 2014
Flu Shot

In the previous post, I sketched out a simple model to estimate the direct benefits associated with mandatory vaccination policies. Rough calculations suggested that a vaccine mandate could decrease the amount of time an unimmunized patient spent in direct contact (which I take to mean within 2 meters, the distance that aerosolized virus can travel)12 with unimmunized persons by 3.2 hours. This was based on the assumption that the patient spent 16 hours in direct contact with healthcare workers, a vaccine efficacy of 60%, and a vaccination rate of 40%. A mandate was presumed to improve healthcare worker vaccination rates to about 94%.3 Finally, I assumed that we could consider the cascading effect of herd immunity up to 3 degrees of separation from the patient. We can improve on the assumptions made to exposure time, vaccine efficacy, voluntary vaccination rates, and degrees of separation to provide a more realistic estimate.

Exposure Times

In a 16-hour day, a patient’s time may be spent alone or in direct contact with healthcare workers, other patients or residents, or visitors. van den Dool et al. estimate that approximately 52% of patient time in a longterm care facility is spent in contact with healthcare workers and 7% with other residents, and patients see 0.7 visitors per day.4 It’s important to note that van den Dool’s estimates are based on direct observation of nursing homes in the Netherlands. I could find no other studies to verify the proportion of time patients spend in close contact with others in a typical day.

It’s unclear if van den Dool’s estimates include time spent in contact with persons who are not healthcare workers; for example, a porter or administrative staff. That can be important when determining to whom mandates should be applied and what the literature (that is used to justify the use of mandates) characterizes as a healthcare worker when measuring vaccination rates and effects.5 It also doesn’t account for time a patient spends with objects which may have been exposed to a viral shedder (such as foods prepared by an employee who has no direct patient contact or that has been brought in from off-site), given the fact that transmission can occur through contact with contaminated surfaces. Owing to a lack of data, I’ll have to ignore these differences, but I point them out as limitations to the analysis. I’ll assume that visitors account for 6% of direct patient contact time. The remaining 35% of the patient’s time is therefore considered time not exposed to any viral sources.

Incorporating exposure times into the model means weighting the immunization rate (IR) at each level according to the proportion of time the patient is exposed to each group. Considering the exposure from healthcare workers (h), visitors (v), other patients (r), and the 35% component of non-exposure, then the fraction of a patient’s 16-hour day exposed to unimmunized contacts, µ, becomes:

µ = 1 - ((VEh × VRh × ph) + (VEv × VRv × pv) + (VEr × VRr × pr) + 0.35)

Vaccine Efficacy

Seasonal influenza vaccine efficacy varies widely across age groups. Osterholm estimated vaccine efficacy at 59% for adults 18–65 years, but questioned the efficacy in persons over 65.6 Other systematic reviews have come to similar conclusions.789 On the other hand, there is some evidence to suggest a benefit to elderly patients in institutional settings.1011 Let’s assume a vaccine efficacy for other residents in the facility at 30%.

Voluntary vaccination rates

In the previous post, I pointed out that voluntary vaccination rates among healthcare workers varies widely across jurisdictions and institutions.1213 The important parameter with respect to assessing the effect of mandates is the voluntary rate, with or without programs that help to encourage uptake without the use of mandates. Therefore, I’ll avoid the use of national measures that may represent a blending of strategies since they do not reflect the true voluntary baseline rate.

Following a failure to suppress a mandate through the Labour Relations Board, healthcare workers in British Columbia were subjected to a mandate in the 2013–2014 season (though strict enforcement was supposedly postponed until the 2014–2015 season). I’ll use BC as a case study for the purposes of demonstrating the model developed here.

In 2005, Canadian Community Health Survey found that people aged 18–64 with a high risk medical condition, 43.6% were vaccinated against seasonal influenza. National statistics gathered in 2012 from the Public Health Agency of Canada suggest that only 37.2% of the general population were vaccinated against influenza. I’ll use an estimate for the general BC public vaccination rate (VRv) at 40%.

Given that the best evidence for any benefit to patients comes from trials conducted in longterm care facilities,14 I’ll constrain the analysis to this subgroup. The vaccination rates of residents in longterm care (VRr) is reported by the BC CDC was 89% for the 2012/2013 season.15 Ksienski reports that vaccination rates of healthcare workers (VRh) rose from 57% to 75% in longterm care facilities and from 40% to 74% in acute care facilities.16 Perhaps with stricter enforcement vaccination rates may rise to the 94% levels reported elsewhere.3 Therefore, we’ll consider the change in unexposed times, ∆µ, in going from 57% to 75% vaccination rates and from 57% to a hypothetical 94% vaccination rate.

Degrees of separation

Pipon and Frenette suggested that an ethically workable mandate must be based on a principle of proximity.

Proximity with patients refers to the degree of separation between a person — either directly or indirectly — with a patient, with a patient’s health care provider, or with the patient’s environment. […] We argue that up to two degrees of separation of any kind (direct, indirect, etc.) presents a sufficient risk that justifies vaccination. It should be noted that the number of degrees are here set arbitrarily for the purpose of demonstration, and should be established by each institution’s infection control department or designated experts on the basis of studies demonstrating the relationship between distance and risk of transmission.

[…]

For example, delivery or repair personnel may be employees of a third party that is contractually bound to the institution, but are nonetheless part of the environmental aspect of patient care. Here, proximity plays an important role. The delivery of medical supplies or food to the hospital will in most cases be too distant to be problematic; these personnel do not fulfill the minimal conditions unless the services are provided directly on the units (e.g.,  repairs conducted to unit resources), near patients, or even at their bedside. By contrast, even if caterers work at a distance from patients, they enter the patient environment directly through food trays, and serve HCP in the institution’s cafeteria.17

In the previous post, I used n = 3 to demonstrate the cascading effect of herd immunity on a patient’s primary contacts. In effect, it means the analysis relies on the uninmmunized secondary contacts who are protected by immunized tertiary contacts to protect a portion of the patient’s unimmunized direct contacts. This corresponds to Pipon & Frenette’s recommendation of vaccinating up to second degree contacts. If secondary contacts are subjected to a mandate, then it is reasonable to consider the herd effects afforded by their direct contacts: namely, the patient’s tertiary contacts.

Application and analysis

We can now provide an estimate of the fraction of time a patient spend in direct contact with people who are neither immunized nor protected from influenza.

µ = (1 - (IRh + IRv + IRr + 0.35))(1 - IRv)2
µ = (1 - ((VEh × VRh × ph) + (VEv × VRv × pv) + (VEr × VRr × pr) + 0.35))(1 - (VEv × VRv))2

In going from VRh = 0.57 to 0.75, we find ∆µ = 0.032, or about 31 minutes.18 If the higher VRh = 0.94 is achieved, then the savings is doubled to about 63 minutes. It’s interesting to note that when all other parameters are equal, Δµ is linearly proportional to ΔVRh, in line with what van den Dool et al. predicted.4

Herd immunity through three degrees of separation from an unimmunized patient for different categories of primary contacts.

Herd immunity through three degrees of separation from an unimmunized patient for different categories of primary contacts.

Framing the benefits of herd immunity in terms of exposure minutes saved per patient (per day) can be more easily understood by persons who are not familiar with interpreting epidemiological studies, such arbitrators and judges who may be called on to resolve legal disputes regarding vaccination mandates. In the next post, I’ll try and show how ∆µ can be used to inform how a mandate should be constructed in order to balance the legal rights and interests of healthcare workers from a principled basis.


  1. WE Bischoff et al. Exposure to Influenza Virus Aerosols During Routine Patient Care. Journal of Infectious Diseases 207(7): 1037–1046 (2013). 
  2. P Fabian et al. Influenza Virus in Human Exhaled Breath: An Observational Study. PLoS ONE 3(7): e2691 (2008). 
  3. SI Pitts et al. A Systematic Review of Mandatory Influenza Vaccination in Healthcare Personnel. American Journal of Preventative Medicine 47(3): 330–340 (2014). 
  4. C van den Dool et al. The effects of influenza vaccination of health care workers in nursing homes: Insights from a mathematical model. PLoS Medicine 5(10): e200 (2008). 
  5. MC Lindley et al. Evaluating a standardized measure of healthcare personnel influenza vaccination. American Journal of Preventative Medicine 45(3):297–303 (2013). 
  6. MT Osterholm et al. Efficacy and effectiveness of influenza vaccines: a systematic review and meta-analysis. Lancet Infectious Diseases 12: 36–44 (2012). 
  7. T Jefferson et al. Vaccines for preventing influenza in the elderly. y. Cochrane Database of Systematic Reviews, Art. No. CD004876 (2010). 
  8. T Jefferson et al. Efficacy and effectiveness of influenza vaccines in elderly people: a systematic review. Lancet 366: 1165–1174 (2005). 
  9. B Michiels et al. A systematic review of the evidence on the effectiveness and risks of inactivated influenza vaccines in different target groups. Vaccine. 29(49):9159–70 (2011). 
  10. JC Kwong et al. Vaccine Effectiveness Against Laboratory-Confirmed Influenza Hospitalizations Among Elderly Adults During the 2010–2011 Season. Clinical Infectious Diseases 57(6): 820–827 (2013). 
  11. B Ridenhour et al. Effectiveness of Inactivated Influenza Vaccines in
    Preventing Influenza-Associated Deaths and Hospitalizations among Ontario Residents Aged > 65 Years: Estimates with Generalized Linear Models Accounting for Healthy Vaccinee Effects. PLoS ONE 8(10): e76318 (2013). 
  12. PR Blank et al. Vaccination coverage rates in eleven European countries during two consecutive influenza seasons. Journal of Infection 58(6): 1–13 (2009). 
  13. H Hollmeyer et al. Review: interventions to increase influenza vaccination among healthcare workers in hospitals. Influenza and Other Respiratory Viruses 7(4): 604–621 (2012). 
  14. F Ahmed et al. Effect of Influenza Vaccination of Healthcare Personnel on Morbidity and Mortality Among Patients: Systematic Review and Grading of Evidence. Clinical Infectious Diseases 58(1): 50–57 (2014). 
  15. See similar statistics from Public Health Ontario 
  16. DS Ksienski. Mandatory seasonal influenza vaccination or masking of British Columbia health care workers: Year 1. Canadian Journal of Public Health 105(4):e312-6 (2014) Abstract only. 
  17. JCB Pipon & M Frenette. Mandatory Influenza Vaccination: How Far to Go and Whom to Target Without Evidence? American Journal of Bioethics 13(9): 48–50 (2013). 
  18. Showing my work, µ = (1 - ((0.59 × 0.57 × 0.52) + (0.59 × 0.4 × 0.06) + (0.3 × 0.89 × 0.07) + 0.35))(1 - (0.59 × 0.4))2