Abstract
It is sometimes said that herd immunity requires everyone to be immune. This paper shows that it does not, and that everyone can still be insufficient——the two kinds of “not enough” are different things. No new theorem or law is claimed.
First, the threshold is $1-1/R_{0}$; at $R_{0}=3$ it stops at $\mathbf{66.6667}$ per cent (Section 2).
Second, at exactly the threshold the effective reproduction number is exactly $\mathbf{1}$ (Section 2).
Third, however large $R_{0}$ becomes, the threshold never reaches $1$ (Section 2).
Fourth, and this is the core of the paper. At $R_{0}=12$ with $90$ per cent efficacy the required coverage is $\mathbf{101.85}$ per cent——unreachable even by vaccinating everyone (Section 3).
Fifth, the boundary is where the efficacy equals the threshold itself ($91.6667$ per cent at $R_{0}=12$) (Section 3).
Sixth, the separator is whether what is short is the coverage or the efficacy (Section 5).
Scope (what is not claimed)
No new theorem or law is claimed——the basic reproduction number, the herd-immunity threshold $1-1/R_{0}$ and the division by efficacy are all known in epidemiology. Policy is not discussed——what should be done about any disease is not the subject. Only the numbers inside one formula. Homogeneous mixing is assumed——contacts are taken as uniform; age structure, spatial structure and contact heterogeneity are not the subject, and they can move the threshold in either direction. A single $R_{0}$ is assumed——$R_{0}$ is not a property of the pathogen alone but also of the contact rate of a society, and it varies with place and time. Waning is not treated——immunity that declines over time is not the subject. Infection-blocking and disease-blocking are not distinguished——“efficacy” here means the fraction of transmission blocked, a different quantity from efficacy against disease. Relation to earlier papers: Paper 400 is the four hundredth census——this is the paper that follows it. Paper 388 showed that the probability does not fall as the number of candidates grows——here the threshold does not reach $1$ as $R_{0}$ grows. Paper 386 showed that “poorly soluble” names two things——here “not enough” names two. Paper 300 showed that same-root and different-root can be decided——coverage and efficacy are different-root by that criterion. What is added is putting numbers on threshold and required coverage at five values of $R_{0}$, confirming by inversion that the effective reproduction number is exactly $1$, obtaining the impossible figure of $101.85$ per cent at $90$ per cent efficacy, showing that the boundary is “efficacy equals threshold”, and placing the separator at “coverage or efficacy”.
1. Notation and the Question
Setting: let one infected person produce $R_{0}$ secondary infections. With an immune fraction $p$,
$$R_{\text{eff}}=R_{0}(1-p),\qquad R_{\text{eff}}\le1\ \Longleftrightarrow\ p\ge1-\frac{1}{R_{0}}$$To reach an immune fraction $p$ with a vaccine of efficacy $e$ requires a coverage $c=p/e$.
Observation 1.1 the threshold is not $1$. It is $1-1/R_{0}$——for any finite $R_{0}$ it is strictly less than one.
Observation 1.2 coverage and immune fraction are different quantities. $c=p/e$——unless the efficacy is one, they do not coincide.
Observation 1.3 coverage has a ceiling. $c\le1$——once $p/e$ exceeds one, that instrument cannot reach.
Question: how many people need immunity to stop an epidemic?
2. At $R_{0}=3$ It Stops at Two Thirds
Computation 2.1: threshold and required coverage against $R_{0}$ (computed in this paper).
| $R_{0}$ | Threshold $1-1/R_{0}$ [%] | Coverage at $95$ % efficacy | Coverage at $90$ % |
|---|---|---|---|
| $2$ | $50.0000$ | $52.6316$ | $55.5556$ |
| $\mathbf{3}$ | $\mathbf{66.6667}$ | $70.1754$ | $74.0741$ |
| $5$ | $80.0000$ | $84.2105$ | $88.8889$ |
| $\mathbf{12}$ | $\mathbf{91.6667}$ | $96.4912$ | $\mathbf{101.8519}$ |
| $18$ | $94.4444$ | $99.4152$ | $104.9383$ |
Observation 2.2 the threshold at $R_{0}=3$ is $66.6667$ per cent. Two thirds——one person in three may remain without immunity.
Observation 2.3 at exactly the threshold the effective reproduction number is exactly $1$. $3\times(1-2/3)=1.000000000$——inverting the definition returns to one.
Observation 2.4 however large $R_{0}$ becomes, the threshold never reaches $1$. $0.999999$ even at $R_{0}=10^{6}$——“everyone must be immune” holds for no $R_{0}$.
Observation 2.5 but at $R_{0}=18$ it is $94.4444$ per cent, effectively almost everyone. One person in eighteen——the room left by “not everyone” is narrow.
Observation 2.6 the threshold increases monotonically in $R_{0}$. $50.0000\to94.4444$——rising, but bounded above.
3. The Impossible Figure of $101.85$ Per Cent
Computation 3.1: required coverage at $R_{0}=12$, a representative value for measles (computed in this paper).
| Efficacy $e$ [%] | Required coverage $c=p/e$ [%] | Enough if everyone is vaccinated |
|---|---|---|
| $95$ | $96.4912$ | yes |
| $\mathbf{91.6667}$ | $\mathbf{100.0000}$ | exactly (the boundary) |
| $\mathbf{90}$ | $\mathbf{101.8519}$ | no |
| $80$ | $114.5833$ | no |
Observation 3.2 this is the core of the paper. At $90$ per cent efficacy the required coverage is $101.85$ per cent——a figure above $100$ cannot be achieved. Vaccinating everyone does not reach.
Observation 3.3 the boundary is where the efficacy equals the threshold itself. $91.6667$ per cent at $R_{0}=12$——the threshold formula and the lower bound on efficacy give the same number.
Observation 3.4 the inversion closes. Set the efficacy to $1-1/R_{0}$ and the required coverage is exactly $1.000000$——not a coincidence but the structure of $c=(1-1/R_{0})/e$.
Observation 3.5 between $95$ and $90$ per cent the answer changes from possible to impossible. $96.4912$ and $101.8519$——five points straddle the boundary.
Observation 3.6 raising the coverage cannot solve it. The ceiling is $100$ per cent——here only the efficacy can be raised.
4. The Two Kinds of “Not Enough” Are Repaired in Different Places
Observation 4.1 short coverage is an operational problem. Some people were not reached——the remedy is to reach further.
Observation 4.2 short efficacy is a product problem. The vaccine was given and immunity did not follow——the remedy is a different vaccine or a booster.
Observation 4.3 both look like “coverage is short”. The same figure appears in the field——$c<$ the required value.
Observation 4.4 the distinction shows in whether the required value exceeds $100$. If it does, the problem is the efficacy——and reaching further cannot solve it.
Observation 4.5 they are different-root by the criterion of Paper 300. Coverage is set by operations, efficacy by the immune response——one phrase, “not enough”, with two roots.
Observation 4.6 the shape is that of Paper 386. There “poorly soluble” named solubility and dissolution rate——here “not enough” names coverage and efficacy.
5. Separator: Is the Coverage Short, or the Efficacy
Observation 5.1 separator——does the required coverage $(1-1/R_{0})/e$ exceed $100$ per cent or not.
| What is short | Required coverage | Reaching further | Remedy |
|---|---|---|---|
| Coverage | below $100$ % | solves it | reach further |
| Efficacy | $100$ % or above | cannot solve it | different vaccine, booster |
Observation 5.2 the third column is the test. Does reaching further solve it——if not, the problem is on the product side.
Observation 5.3 the boundary is a single inequality. $e\ge1-1/R_{0}$——unless the efficacy is at least the threshold, no coverage suffices.
Observation 5.4 the separator can be moved. Reduce contacts to lower the effective $R_{0}$ and the threshold falls, so the same efficacy now reaches——a third remedy that is neither of the two.
Observation 5.5 to be honest——both $91.6667$ and $101.85$ per cent come from a model with homogeneous mixing and a single $R_{0}$. Contact heterogeneity can move the threshold either way——what does not move is that the threshold never reaches $1$, and that an efficacy below the threshold leaves everyone insufficient.
Observation 5.6 one last thing——“everyone must be immune” and “everyone is not enough” are both wrong and can both arise. The first holds for no $R_{0}$——the second arises whenever the efficacy is low. Fail to separate them and the remedy points the wrong way.
6. Roll Call and Closing
- The threshold is $1-1/R_{0}$ (Section 1): not $1$.
- $66.6667$ per cent at $R_{0}=3$ (Computation 2.1): two thirds.
- $R_{\text{eff}}=1.000000000$ at the threshold (Observation 2.3): the inversion closes.
- $0.999999$ even at $R_{0}=10^{6}$ (Observation 2.4): it never reaches $1$.
- $94.4444$ per cent at $R_{0}=18$ (Computation 2.1): little room left.
- $101.85$ per cent at $90$ per cent efficacy (Computation 3.1): unachievable.
- The boundary is efficacy $=$ threshold (Observation 3.3): $91.6667$ % at $R_{0}=12$.
- Five points cross from possible to impossible (Observation 3.5): $96.4912$ and $101.8519$.
- Different-root (Observation 4.5): operations and immune response.
- A third remedy: reduce contacts (Observation 5.4): lower the threshold itself.
Closing: it is sometimes said that herd immunity requires everyone to be immune——it does not; and yet everyone can still be insufficient. The threshold is $1-1/R_{0}$ and not $1$——at $R_{0}=3$ it is $\mathbf{66.6667}$ per cent, so one person in three may remain without immunity. At exactly the threshold the effective reproduction number is exactly $1.000000000$, and inverting the definition returns to one. Even at $R_{0}=10^{6}$ the threshold is $0.999999$, so “everyone must be immune” holds for no $R_{0}$. At $R_{0}=18$, however, it is $94.4444$ per cent, and the room left by “not everyone” is narrow. But the required coverage can exceed $100$ per cent——reaching an immune fraction $p$ with efficacy $e$ needs coverage $c=p/e$, so at $R_{0}=12$ with $90$ per cent efficacy the required coverage is $\mathbf{101.8519}$ per cent. A figure above $100$ cannot be achieved: vaccinating everyone does not reach. At $95$ per cent efficacy it is $96.4912$ and does reach, so five points of efficacy straddle the boundary between possible and impossible. The boundary is where the efficacy equals the threshold itself——$91.6667$ per cent at $R_{0}=12$. Setting the efficacy to $1-1/R_{0}$ makes the required coverage exactly $1.000000$, which is not a coincidence but the structure of $c=(1-1/R_{0})/e$. And there, raising the coverage cannot solve it, because the ceiling is $100$ per cent; only the efficacy can be raised. The two kinds of “not enough” are repaired in different places——short coverage is operational and the remedy is to reach further; short efficacy is a product problem and the remedy is a different vaccine or a booster. Both appear in the field as the same figure, “coverage is short”. The distinction shows in whether the required value exceeds $100$——if it does, reaching further cannot solve it. By the criterion of Paper 300 they are different-root: operations on one side, the immune response on the other, the shape of Paper 386. One thing does the separating——does the required coverage $(1-1/R_{0})/e$ exceed $100$ per cent or not. The test is whether reaching further solves it; if not, the problem is on the product side. The boundary is the single inequality $e\ge1-1/R_{0}$. And the separator can be moved: reduce contacts to lower the effective $R_{0}$, the threshold falls, and the same efficacy now reaches——a third remedy that is neither of the two. To be honest——both $91.6667$ and $101.85$ per cent come from a model with homogeneous mixing and a single $R_{0}$, and contact heterogeneity can move the threshold either way. What does not move is that the threshold never reaches $1$, and that an efficacy below the threshold leaves everyone insufficient. One last thing——“everyone must be immune” and “everyone is not enough” are both wrong and can both arise. The first holds for no $R_{0}$; the second arises whenever the efficacy is low. Fail to separate them and the remedy points the wrong way.
References
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- Fine, P., Eames, K., Heymann, D. L. (2011). “Herd Immunity”: A Rough Guide. Clinical Infectious Diseases.
- Guerra, F. M., Bolotin, S., Lim, G., et al. (2017). The basic reproduction number ($R_{0}$) of measles: a systematic review. The Lancet Infectious Diseases.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Solubility Did Not Move at All, and the Dissolution Became a Thousand Times Faster [Paper 386]. Zenodo. 10.5281/zenodo.22699119
- Yamagishi, Y. (2026). With a Million Candidates the Chance of Picking the Best Stays at $0.3679$ [Paper 388]. Zenodo. 10.5281/zenodo.22699147
- Yamagishi, Y. (2026). The Four Hundredth Census: The Instrument Broke Before the Thing It Was Counting [Paper 400]. Zenodo. 10.5281/zenodo.22699343
Reference
Yamagishi, Y. (2026). Not Everyone Needs Immunity; and Yet Everyone May Not Be Enough [Paper 401]. YOI: https://www.yamagishiyuuki.com/yoi/401
DOI: 10.5281/zenodo.22699375
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 401. This paper is a preprint and has not been peer reviewed.