PU650 Unit 8 Vaccination and Immunity Analysis Example

Ninety-Three Percent Fully Vaccinated and Still Fourteen Cases: A Unit 8 Vaccination and Immunity Analysis Computing Herd Immunity Thresholds for Varicella at a Composite Elementary School

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Purdue University Global

PU650: Infectious Disease Epidemiology

Unit 8 Vaccination and Immunity Analysis

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[Date]

Ashby Elementary, its records and its cases are composite; the formulas, the vaccine effectiveness estimates, the contagiousness findings and the research cited are real.

The Situation

Ashby Elementary enrolls 640 children from kindergarten to fifth grade. School records show 595 children, 93 percent, with two documented doses of varicella vaccine, 26 children, 4 percent, with one dose, and 19 children, 3 percent, with no doses because of medical or nonmedical exemptions. Between January 12 and February 23, fourteen children developed varicella. Ten had two doses, one had one dose and three were unvaccinated. Eleven of the fourteen had fewer than fifty lesions and no fever above 101 degrees. Parents and teachers asked the same question: how can a school this well vaccinated have an outbreak?

The Formulas

The herd immunity threshold is the share of a population that must be immune for each case to cause, on average, fewer than one new case. It equals one minus the reciprocal of the basic reproduction number. Because vaccines do not protect everyone who receives them, the coverage needed to reach that threshold is the threshold divided by vaccine effectiveness (Fine et al., 2011). The effective reproduction number in a partly immune population is the basic reproduction number multiplied by the share still susceptible.

The Inputs

Basic reproduction number: varicella is among the most transmissible childhood infections, and estimates commonly cited for school-age settings run from about 8 to 12. The script uses 8, 10 and 12. Vaccine effectiveness: a meta-analysis of postlicensure studies in healthy children pooled one-dose effectiveness at 81 percent against all varicella and 98 percent against moderate or severe disease, and two-dose effectiveness at 92 percent against all varicella (Marin et al., 2016).

What the Script Returns

1. Basic reproduction number 8: threshold 87.5 percent; coverage needed with two doses 95.1 percent; with one dose, 108 percent, which is impossible.

2. Basic reproduction number 10: threshold 90.0 percent; two-dose coverage needed 97.8 percent; one dose impossible.

3. Basic reproduction number 12: threshold 91.7 percent; two-dose coverage needed 99.6 percent; one dose impossible.

At Ashby, the share protected is 0.93 times 0.92 plus 0.04 times 0.81, or 88.8 percent, ignoring the few exempt children who may have had chickenpox already. The effective reproduction number is therefore 0.90 at a basic number of 8, 1.12 at 10 and 1.34 at 12. A school can be 93 percent fully vaccinated and still sit just above the line at which each case replaces itself, because the line for chickenpox is set so high.

Reading the Result

At the middle of the range, Ashby's effective reproduction number sits slightly above one. That predicts exactly what happened: not an explosive outbreak, but a slow chain of cases, each producing about one more, over several weeks. The answer to the parents' question is arithmetic. Two doses at 92 percent effectiveness leave about eight in every hundred vaccinated children susceptible, and in a school of 640 with 595 two-dose children, that is roughly 48 children, plus the unvaccinated and the one-dose failures. Most of the susceptible children at Ashby are vaccinated children, which is why most of the cases were too.

A Complication That Helps: Breakthrough Cases Spread Less

The calculation above treats every case as equally contagious, and that is not true for varicella. In a large household contact study, vaccinated cases overall were about half as contagious as unvaccinated cases, and vaccinated cases with fewer than fifty lesions were about one third as contagious, while those with fifty or more lesions were about as contagious as unvaccinated cases (Seward et al., 2004). If breakthrough cases at Ashby transmit at half the rate of unvaccinated cases, the infectiousness-weighted share of susceptible children falls from 11.2 percent to about 7.1 percent, and the effective reproduction number at a basic number of 10 falls from 1.12 to about 0.71. The milder disease in eleven of Ashby's fourteen children is one reason the outbreak stayed small.

A Complication That Hurts: Waning

If two-dose effectiveness had waned to 85 percent in older children, the protected share would fall to 82.3 percent and the effective reproduction number at a basic number of 10 would rise to 1.77; at 80 percent effectiveness it would reach 2.24. Studies of waning after two doses are fewer and less consistent than the pooled estimates, so this is a sensitivity analysis, not a finding. It says that the school's margin is thin and would disappear with even modest waning.

Control Measures the Analysis Supports

1. A second dose now for the 26 one-dose children, which moves each from 81 to about 92 percent protection.

2. Vaccination offered to the 19 unvaccinated children who have no history of disease, with nonmedical exemptions reviewed under the state's outbreak rules.

3. Exclusion of cases until all lesions have crusted, or, for vaccinated children whose rash never blisters, until no new lesions appear for 24 hours, following national recommendations (Marin et al., 2007).

4. Notification letters to families of children who are pregnant, immunocompromised or caring for infants.

What to Tell Parents

The parents' question deserves an answer in plain words, and the analysis supplies one. Vaccination worked at Ashby in the way the evidence says it should. Of the fourteen children who became ill, eleven had mild cases with fewer than fifty spots, and none was hospitalized, which matches the near-complete protection two doses give against moderate and severe disease. Chickenpox spreads so easily that even a very well vaccinated school can carry a slow chain of mild cases, because a small share of vaccinated children will always remain susceptible. Illness struck 3 of the 19 unvaccinated children, about 16 percent, against 11 of the 621 vaccinated ones, under 2 percent, a risk close to nine times higher. The school letter therefore says three things: the vaccine is doing its job, a second dose matters for the children who have had only one, and children with new spots should stay home until the rash has crusted. It avoids blaming the families of unvaccinated children, whose children are named nowhere and whose exemptions the health department handles directly.

Limits

The analysis assumes random mixing, while classrooms and grade levels mix far more within themselves than across the school, which can let an outbreak run within one grade even when the school-wide number is below one. It treats vaccine protection as all-or-nothing rather than partial, ignores children who had chickenpox before entering school and uses a range for the basic reproduction number rather than an estimate from Ashby's own data. Fourteen cases are too few to estimate the effective reproduction number directly.

References

Fine, P., Eames, K., & Heymann, D. L. (2011). "Herd immunity": A rough guide. Clinical Infectious Diseases, 52(7), 911-916. https://doi.org/10.1093/cid/cir007

Marin, M., Guris, D., Chaves, S. S., Schmid, S., & Seward, J. F. (2007). Prevention of varicella: Recommendations of the Advisory Committee on Immunization Practices (ACIP). MMWR Recommendations and Reports, 56(RR-4), 1-40.

Marin, M., Marti, M., Kambhampati, A., Jeram, S. M., & Seward, J. F. (2016). Global varicella vaccine effectiveness: A meta-analysis. Pediatrics, 137(3), Article e20153741. https://doi.org/10.1542/peds.2015-3741

Seward, J. F., Zhang, J. X., Maupin, T. J., Mascola, L., & Jumaan, A. O. (2004). Contagiousness of varicella in vaccinated cases: A household contact study. JAMA, 292(6), 704-708. https://doi.org/10.1001/jama.292.6.704