Abstract
The threshold and the final size, both fixed by $R_{0}$, are readily taken to say one thing. This paper shows that their ratio changes with $R_{0}$—running from $\mathbf{1.7484}$ to $\mathbf{1.2413}$. No new theorem or law is claimed.
First, at $R_{0}=2$ the threshold is $\mathbf{0.500000}$ and the final size $\mathbf{0.796812}$ (Section 2).
Second, inverting the final size returns $R_{0}$—$2.000000000$ against $2.0$ (Section 2).
Third, at $R_{0}=1$ and at $R_{0}=0.9$ the final size is $\mathbf{0}$—both sides of the boundary were checked (Section 2).
Fourth, this is the core of the paper. Their ratio is $\mathbf{1.7484}$ at $R_{0}=1.5$ and $\mathbf{1.2413}$ at $R_{0}=5$ (Section 3).
Fifth, the larger $R_{0}$, the nearer the ratio to one—both approach one, at different rates (Section 3).
Sixth, the separator is whether the quantity came from a linearisation or from a nonlinear equation (Section 4).
Keywords: basic reproduction number R0, threshold theorem, final size equation, SIR model, mathematical epidemiology.
Scope (what is not claimed)
No new theorem or law is claimed—the threshold theorem and the final-size equation are both standard. No epidemiological advice—vaccination policy, estimating a real $R_{0}$ and the effect of interventions are not the subject; only the relation between two quantities. One model only—a closed, homogeneously mixing SIR model; age structure, space and time-varying contact rates are excluded. No adjudication of priority—to whom, and when, the threshold idea belongs is not treated. Efficacy is excluded—the vaccine efficacy of Paper 401 does not enter; only the immunised fraction itself. Relation to earlier papers: Paper 401 showed that not everyone needs immunity and yet everyone may not be enough—the threshold used there and the final size here are different quantities. Paper 300 showed that a shared root can be decided—two quantities from one $R_{0}$ can still be distinct roots. Paper 486 showed that optima in three fields are one vertex—there, separate-looking and one-rooted; here, same-looking and distinct-rooted. Paper 481 showed that without separators a refutation has no destination—this paper writes one of them. What is added is tabulating the two by $R_{0}$, solving the final size as $0.796812$ and confirming it by inversion, checking both sides of the boundary at $R_{0}=1$ and $0.9$, giving the ratio’s movement from $1.7484$ to $1.2413$, and placing the separator at linear versus nonlinear.
1. Notation and the Question
Setting: In a closed, homogeneously mixing population, write $R_{0}$ for the basic reproduction number. From the condition that no epidemic starts, the fraction that must be immune is
$$p_{c}=1-\frac{1}{R_{0}}$$while the fraction $z$ that has been infected once the epidemic ends is the positive solution of
$$z=1-e^{-R_{0}z}$$The first comes from a linearisation valid when infections are few; the second from a balance across the whole epidemic. Below, $R_{0}$ is varied and the two set side by side.
Observation 1.1 The threshold and the final size are both fixed by $R_{0}$. Both from one number—so they say one thing, runs the picture.
Observation 1.2 But the expressions differ in form. $1-1/R_{0}$ against $z=1-e^{-R_{0}z}$—one closed, the other containing itself.
Question: How far apart are they, and is that distance constant.
2. Setting the Two Side by Side
Computation 2.1: Evaluate both by $R_{0}$.
| $R_{0}$ | Threshold $1-1/R_{0}$ | Final size $z$ |
|---|---|---|
| $0.9$ | $0.000000$ | $\mathbf{0.000000}$ |
| $1.0$ | $0.000000$ | $\mathbf{0.000000}$ |
| $1.5$ | $0.333333$ | $0.582812$ |
| $2.0$ | $\mathbf{0.500000}$ | $\mathbf{0.796812}$ |
| $3.0$ | $0.666667$ | $0.940480$ |
Observation 2.2 The final size at $R_{0}=2$ matches the known value. $0.796812$—agreeing with the frequently quoted eighty per cent.
Observation 2.3 Inverting returns $R_{0}$. $-\ln(1-z)/z=2.000000000$ against $2.0$—an independent check.
Observation 2.4 At $R_{0}=1$ the final size is zero. The boundary—where the threshold is zero too, so the two coincide.
Observation 2.5 At $R_{0}=0.9$ it is zero as well. The other side—the threshold formula returns a negative value, cut at $0$, and the final-size equation has no positive solution at all.
Observation 2.6 The final size exceeds the threshold. $0.796812 > 0.500000$—because the epidemic stops only after the threshold is crossed.
3. The Ratio Is Not Constant
Computation 3.1: Divide the final size by the threshold.
| $R_{0}$ | Final size / threshold |
|---|---|
| $1.5$ | $\mathbf{1.7484}$ |
| $2.0$ | $1.5936$ |
| $3.0$ | $1.4107$ |
| $5.0$ | $\mathbf{1.2413}$ |
Observation 3.2 The ratio changes with $R_{0}$. From $1.7484$ to $1.2413$—were the two expressions of one quantity, it would be constant.
Observation 3.3 The larger $R_{0}$, the nearer to one. $1.2413$—both approach one, the threshold more slowly.
Observation 3.4 So “know the threshold and you know the final size” cannot be written as a proportion. $R_{0}$ must be passed through—no relation independent of $R_{0}$ joins them.
Observation 3.5 The difference is called the overshoot. $0.796812-0.500000=0.296812$ at $R_{0}=2$—the epidemic does not halt on reaching the threshold, and this is how far it runs past.
4. The Separator
The separator: did the quantity come from a linearisation, or from a nonlinear equation.
Observation 4.1 From a linearisation, and it has a closed form. $1-1/R_{0}$—looking only at the moment when infections are few, and containing nothing of what follows.
Observation 4.2 From a nonlinear equation, and it contains itself. $z=1-e^{-R_{0}z}$—a balance across the whole epidemic, solvable only numerically.
Observation 4.3 The two are distinct roots by the criterion of Paper 300. Both come from $R_{0}$—but one is a condition for an epidemic to begin and the other a result after it has: they ask about different moments.
Observation 4.4 The moving ratio is the evidence. From $1.7484$ to $1.2413$—a shared root would admit a conversion independent of $R_{0}$.
Observation 4.5 Paper 401 used only the threshold side. Not everyone needs immunity—what did the work there was $1-1/R_{0}$, not the final size.
Observation 4.6 Paper 486 runs the other way. There, separate-looking and one-rooted—here, same-looking and distinct-rooted. The same test serves in both directions.
Observation 4.7 To be honest—the model is a closed, homogeneously mixing population. No age structure, no space, no varying contact rate—real epidemics usually end below this, so $0.796812$ is close to an upper bound. What is shown is a relation between two quantities, not a prediction.
5. Roll Call and Closing
- Threshold $0.500000$ at $R_{0}=2$ (Computation 2.1): final size $0.796812$.
- Matching the familiar eighty per cent (Observation 2.2): an independent check.
- Inversion gives $2.000000000$ (Observation 2.3): returning $R_{0}$.
- Zero at $R_{0}=1$ and at $0.9$ (Observation 2.5): both sides of the boundary.
- Ratio from $1.7484$ to $1.2413$ (Computation 3.1): the core.
- Nearer to one at larger $R_{0}$ (Observation 3.3): the threshold lags.
- An overshoot of $0.296812$ (Observation 3.5): how far it runs past.
- The separator is linear or nonlinear (Observation 4.3): distinct roots by Paper 300.
- The reverse of Paper 486 (Observation 4.6): the test serves both ways.
- A homogeneously mixing model (Observation 4.7): not a prediction.
Closing: The threshold and the final size are both fixed by $R_{0}$—both from one number, so they say one thing. But the expressions differ in form: $1-1/R_{0}$ against $z=1-e^{-R_{0}z}$, one closed and the other containing itself. Setting the two side by side—at $R_{0}=2$ the threshold is $\mathbf{0.500000}$ and the final size $\mathbf{0.796812}$, agreeing with the frequently quoted eighty per cent. Computing $-\ln(1-z)/z$ from the final size returns $2.000000000$. At $R_{0}=1$ both are zero and coincide. At $R_{0}=0.9$ both are zero as well: the threshold formula returns a negative value, cut at $0$, and the final-size equation has no positive solution. The final size exceeds the threshold, because the epidemic stops only after the threshold is crossed. The ratio is not constant—dividing gives $\mathbf{1.7484}$ at $R_{0}=1.5$, $1.5936$ at $2.0$, $1.4107$ at $3.0$ and $\mathbf{1.2413}$ at $5.0$. Were the two expressions of one quantity, it would be constant. The larger $R_{0}$, the nearer to one, with the threshold approaching more slowly. So “know the threshold and you know the final size” cannot be written as a proportion: $R_{0}$ must be passed through, and no relation independent of it joins them. The difference, called the overshoot, is $0.296812$ at $R_{0}=2$—how far the epidemic runs past its own threshold. One thing separates them—did the quantity come from a linearisation, or from a nonlinear equation. From a linearisation, and it has a closed form: $1-1/R_{0}$ looks only at the moment when infections are few. From a nonlinear equation, and it contains itself: $z=1-e^{-R_{0}z}$ is a balance across the whole epidemic, solvable only numerically. The two are distinct roots by the criterion of Paper 300: both come from $R_{0}$, but one is a condition for an epidemic to begin and the other a result after it has. The moving ratio is the evidence, since a shared root would admit a conversion independent of $R_{0}$. Paper 401 used only the threshold side, where what did the work was $1-1/R_{0}$. Paper 486 runs the other way—separate-looking and one-rooted there, same-looking and distinct-rooted here—so the same test serves in both directions. To be honest—the model is a closed, homogeneously mixing population, with no age structure, no space and no varying contact rate. Real epidemics usually end below this, so $0.796812$ is close to an upper bound. What is shown is a relation between two quantities, not a prediction.
References
- Diekmann, O., Heesterbeek, H., & Britton, T. (2013). Mathematical Tools for Understanding Infectious Disease Dynamics. Princeton University Press.
- Kermack, W. O., & McKendrick, A. G. (1927). A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society A.
- Ross, R. (1911). The Prevention of Malaria (2nd ed.). John Murray.
- 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). Not Everyone Needs Immunity; and Yet Everyone May Not Be Enough [Paper 401]. Zenodo. 10.5281/zenodo.22699375
- Yamagishi, Y. (2026). A Broken Law Can Be Blamed on Any of the Assumptions [Paper 481]. Zenodo. 10.5281/zenodo.22733927
- Yamagishi, Y. (2026). The “Exactly Half” in Three Fields Is the Vertex of a Parabola, Not Physics [Paper 486]. Zenodo. 10.5281/zenodo.22759952
Reference
Yamagishi, Y. (2026). The Threshold and the Final Size Both Come from $R_{0}$, and Their Ratio Is Not Constant [Paper 511]. YOI: https://www.yamagishiyuuki.com/yoi/511
DOI: 10.5281/zenodo.22862074
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 511. This paper is a preprint and has not been peer reviewed.