What is Vaccines and herd immunity? — Biology

Definition 16.10 University Biology — Year 3 · Chapter 16 — Adaptive Immunity and Vaccination

A vaccine presents antigens of a pathogen, with an adjuvant or in a form that triggers innate receptors, so that the recipient acquires memory cells and antibodies without the disease (Chapter 13 lists the kinds). Its efficacy $E$ is the fraction of vaccinated people protected. Protection extends beyond the vaccinated: a pathogen spreading in a population where each case infects $R_{0}$ others on average when all are susceptible — the basic reproduction number — infects only $R_{0}$ times the susceptible fraction when some are immune, and dies out when that number falls below one. This is herd immunity: above a threshold of immune people the chain of transmission breaks and the unvaccinated — infants, the immunocompromised, those in whom the vaccine failed — are protected by the others.

Left: the threshold 1 - 1/R_0 — the more contagious the disease, the closer to everyone must be immune. Right: an SIR epidemic with R_0 = 3 in a naive population and in one with 60\,\% immune, which is not enough to stop it but flattens and slows it. — Left: the threshold $1 - 1/R_{0}$ — the more contagious the disease, the closer to everyone must be immune. Right: an SIR epidemic with $R_{0} = 3$ in a naive population and in one with $60\,\%$ immune, which is not enough to stop it but flattens and slows it.

Edward Jenner, who in 1796 protected a boy against smallpox with cowpox and founded vaccination (engraving of 1807, public domain). Right: the same act two centuries on — an intramuscular vaccine, whose antigen the deltoid’s dendritic cells will carry to the axillary lymph nodes.

Examples

Example 16.12 (Measles and smallpox)

Measles has $R_{0} \approx 15$ : $p_{c} = 1 - 1/15 = 93\,\%$ , and with a vaccine of efficacy $97\,\%$ after two doses the coverage needed is $96\,\%$ — which is why measles returns wherever vaccination slips a few per cent, and why an unvaccinated child in a well-vaccinated country is nonetheless safe. Smallpox had $R_{0} \approx 5$ – $7$ , a threshold near $80$ – $85\,\%$ , no animal reservoir, no asymptomatic carriers and a vaccine that worked in one dose: eradicable, and eradicated. Polio is nearly there. Influenza and the coronaviruses, whose antigens drift (Chapter 13) and whose immunity wanes, are not. In an unvaccinated population an epidemic with $R_{0} = 3$ infects, by the final-size relation, $1 - s_{\infty} = 94\,\%$ of people; with $R_{0} = 1.5$ , $58\,\%$ .

Example 16.7 (Kinetics of a response)

About one B cell in $10^{5}$ binds a given epitope, so a body’s $10^{11}$ B cells hold some $10^{6}$ precursors, of which perhaps a hundred meet the antigen in the draining node. Dividing every eight hours, a hundred cells become $10^{2}\times 2^{21} \approx 2\times 10^{8}$ in a week. Antibody appears in the serum after four to six days, IgM first, peaks at about two weeks and declines with a half-life of three weeks for IgG as the short-lived plasma cells die, settling at the level the long-lived marrow plasma cells maintain for years. A second exposure starts from $10^{4}$ – $10^{5}$ memory cells of high affinity already switched to IgG: antibody rises within two days, ten- to a hundredfold higher, and neutralises the pathogen before it can establish itself — which is what a vaccine buys.

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Examples

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