Basic Reproduction Number

Basic Reproduction Number R0, Herd Immunity and the Economics of Vaccination

R0 measures how many new infections one infectious person would cause, on average, if everyone around them could catch the infection. A value above 1 means an introduced infection can spread, a value below 1 means chains of transmission die out, and the value sets the share of the population that must be immune to stop sustained spread. In health economics R0 matters because it determines how far a vaccine or screening programme protects people it never reaches, and the ISPOR-SMDM task force on dynamic transmission models calls it a fundamental metric in infectious disease epidemiology. Once immunity builds up, the relevant quantity becomes the effective reproduction number, and the basic reproduction number fixes the herd immunity threshold $H = 1 - 1/R_0$ that vaccine coverage has to reach.

What R0 counts and what "basic" assumes

Diekmann, Heesterbeek and Roberts define R0 as the average number of new cases caused by one typical infected individual in a population consisting of susceptibles only. The word "basic" refers to that fully susceptible starting point, before immunity from infection or vaccination has built up. R0 is a count of cases per case, so it has no time unit; Delamater and colleagues stress that it is not a rate and does not show whether secondary cases appear within a day or months later.

R0 belongs to a pathogen in a particular population and setting. The WHO guide for economic evaluations of immunisation programmes writes it, for a homogeneously mixing population, as the product of the probability of transmission per contact $p$, the contact rate $c$ and the duration of infectiousness $D$, so contact patterns enter R0 alongside biology. Delamater and colleagues note a 2017 review that found feasible measles R0 values from 3.7 to 203.3, and that the often quoted measles range of 12 to 18 rests on data from 1912 to 1928 in the United States and 1944 to 1979 in England and Wales. Pitman and colleagues give a decision example in which malaria now has an R0 below 1 in northern Europe, so imported cases do not start epidemics even though most people are susceptible.

The threshold at one and the SIR formula

The value 1 matters because each generation of cases is R0 times the size of the one before. In the simplest SIR model, van den Driessche shows that the number of infectious people falls steadily to zero when R0 is below 1 and first rises when it is above 1, so a value of exactly 1 is a sharp threshold between die-out and an epidemic. Writing transmission in frequency-dependent form, R0 is the transmission rate multiplied by the mean infectious period.

$$R_0 = \frac{\beta}{\gamma}, \qquad r = \gamma (R_0 - 1)$$

where $\beta$ is the rate at which one infectious person causes new infections per unit of time in a fully susceptible population, $\gamma$ is the recovery rate (its inverse is the mean infectious period) and $r$ is the initial exponential growth rate of infections. Van den Driessche gives the growth result as $I(t) \approx I(0)\exp[\gamma(R_0 - 1)t]$ and notes that R0 does not depend on the fraction who die from the disease. Models with latent stages, age groups or vectors need a different derivation, even when the result is still called R0.

Several groups or stages and the next-generation matrix

When infected people differ by age, risk group or stage, R0 comes from the next-generation matrix $K$. Diekmann and colleagues define its element $K_{ij}$ as the expected number of new cases in state $i$ generated by one newly infected individual in state $j$, and R0 as the largest eigenvalue of $K$. Van den Driessche gives the standard compartmental recipe below.

$$R_0 = \rho(FV^{-1})$$

where $F$ holds the rates at which new infections arise, $V$ holds the other transitions between infected compartments (progression, recovery, death), $FV^{-1}$ is the next-generation matrix and $\rho$ is its spectral radius. For an illustrative two-group matrix with rows $(0.8, 1.2)$ and $(0.3, 0.5)$, the eigenvalues solve $\lambda^2 - 1.3\lambda + 0.04 = 0$, giving 1.268 and 0.032, so R0 is 1.268. Neither group sustains spread on its own, since both diagonal entries are below 1, yet transmission between groups takes the system above the threshold.

R0, the effective reproduction number and vaccination

Once some people are immune, fewer contacts lead to infection. The relevant quantity is then the effective reproduction number. Pitman and colleagues define it as R0 multiplied by the susceptible fraction of the population, and the WHO guide gives the same relation for a homogeneously mixing population.

$$R_t = R_0 \, x_t$$

where $R_t$ is the effective reproduction number at time $t$ and $x_t$ is the proportion of the population still susceptible. Vaccination therefore lowers $R_t$, not R0. Delamater and colleagues make the point directly. Removing susceptible people reduces the effective reproduction number, while R0 is unchanged because its definition assumes a completely susceptible population.

Herd immunity threshold and critical vaccination coverage

Spread stops when each case infects fewer than one other person. In a homogeneously mixing population that happens once the susceptible share $x$ falls below $x^{*} = 1/R_0$. The WHO guide defines the herd immunity threshold as follows.

$$H = 1 - \frac{1}{R_0}$$

where $H$ is the proportion of the population that must be immune. Pitman and colleagues state that in a homogeneously mixing population, vaccine coverage must exceed this value for herd immunity to occur. The WHO guide calls it the critical level of effective immunisation and adds that actual coverage has to be higher, because vaccine protection is usually imperfect and not always given at the age the simple formula assumes. Its table puts measles R0 at 9.3 to 15.6 with a critical proportion of 89 to 94%, and warns that these properties can differ substantially for the same disease between populations. If a vaccine fully protects a fraction $E$ of those vaccinated and gives nothing to the rest, coverage $c$ immunises $cE$ of the population, so the critical coverage is $H/E$. If $E$ is below $H$, this exceeds 100% and such a vaccine cannot stop sustained spread on its own.

Estimating R0 from data

R0 is rarely observed directly. Delamater and colleagues report that it is nearly always estimated after the event from serological data or with mathematical models, and that estimates from models with small differences in structure can differ even on the same data. Three routes are common in vaccine evaluation.

• Early growth. The growth rate $r$ of an epidemic is converted to R0 through the generation interval, the time between infection of a case and infection of the people it infects. Wallinga and Lipsitch show that the different formulas differ only in the assumed shape of that distribution. An exponential distribution gives $R = 1 + rT_c$, with $T_c$ the mean generation interval, while a fixed interval gives $R = e^{rT_c}$, an upper bound for a given growth rate.
• Final size. Van den Driessche shows that after an SIR epidemic, R0 can be recovered from the fraction left uninfected, $s$, as $R_0 = -\ln s / (1 - s)$.
• Age at infection. For endemic infections before vaccination, the WHO guide gives the approximation $R_0 = L/A$, with $L$ life expectancy and $A$ mean age at infection, under strong assumptions about the age distribution.

The WHO guide asks that, where R0 is a direct model input, the basis for its estimation be clearly described, including assumptions on mixing, survival and heterogeneity. In a model fitted by calibration, different combinations of transmission rate and infectious period can reproduce the same early growth while implying different R0 values, so the estimate carries structural uncertainty as well as sampling error.

Role in dynamic transmission models and vaccine evaluation

A vaccine that prevents infection also prevents the onward cases that infection would have caused. Pitman and colleagues describe how static models assume a constant force of infection, while dynamic transmission models let infection risk depend on how many people are infectious, so unvaccinated people benefit from a lower risk. They report that only 11% of 208 cost-effectiveness studies of vaccination programmes used an approach that could capture these indirect effects. R0 is a main determinant of how large those effects are, which is why it appears in model calibration and scenario design for vaccination externalities.

The task force sets out when the choice matters. A static model is acceptable when the target group contributes little to transmission, when protection is almost entirely direct (for example vaccinating older people against influenza), or when the static result is already cost effective and indirect effects would only improve it. Where a static model shows a programme as unattractive or close to the willingness-to-pay threshold, supplementary dynamic modelling should test whether herd effects, replacement or a shift in the age at infection change the result. The age shift can work against the programme, and partial rubella coverage raised the age at first infection in several countries and was followed by paradoxical increases in congenital rubella.

Worked example with static and dynamic views of one vaccine programme

All numbers are illustrative. An SIR infection has $\beta = 0.30$ per day and $\gamma = 0.10$ per day in a population of 100,000, and a vaccine fully protects 90% of those vaccinated. A static model holds each unprotected person's risk at the level seen without the programme; a dynamic model lets the risk fall as fewer people transmit.

Step 1. R0 and early growth. R0 is 0.30/0.10 = 3, with a mean infectious period of 10 days. The growth rate is $r = 0.10 \times (3 - 1) = 0.20$ per day, so cases double every $\ln 2 / 0.20 = 3.47$ days. If the same growth were observed for an infection with a fixed 10-day generation interval, Wallinga and Lipsitch's upper bound gives $e^{0.20 \times 10} = 7.39$, so the generation interval assumption alone can more than double the estimate.

Step 2. Thresholds. $H = 1 - 1/3 = 0.667$, so 66.7% of the population must be immune, and critical coverage is $H/E = 0.667/0.90 = 0.741$, or 74.1%.

Step 3. No programme. Van den Driessche's SIR final size equation, written for the share infected $z$ as $\ln(1 - z) = -R z$ and applied with $R = 3$, gives an attack rate of $z = 0.94048$, or 94,048 infections.

Step 4. Fifty per cent coverage. 50,000 people are vaccinated and 45,000 protected, leaving $x = 0.55$ susceptible and $R_t = 3 \times 0.55 = 1.65$. The static model applies the old attack rate to the 55,000 unprotected people, giving 55,000 × 0.94048 = 51,726 infections, so 42,322 are averted, all directly. The dynamic model solves the final size equation for the unprotected group with $R = 1.65$, giving $z = 0.66769$ among the unprotected and 55,000 × 0.66769 = 36,723 infections, so 57,325 are averted, of which 15,003 are indirect.

Step 5. Cost per infection averted. At an illustrative GBP 20 per vaccinee the programme costs GBP 1,000,000 in total.

$$\frac{1{,}000{,}000}{42{,}322} = 23.63, \qquad \frac{1{,}000{,}000}{57{,}325} = 17.44$$

where the numerators are programme cost in GBP and the denominators are infections averted under the static and dynamic models, so cost per infection averted is GBP 23.63 and GBP 17.44.

Step 6. Interpretation. At 80% coverage, $R_t = 3 \times (1 - 0.72) = 0.84$, so a major epidemic cannot occur in the dynamic model, while the static model still predicts 26,333 infections. The static model understates benefit by about a quarter at 50% coverage, and it cannot show that coverage above the threshold prevents the epidemic altogether. With a larger R0 the threshold moves up and the indirect share changes, which is why the cost-effectiveness result of a vaccine depends on how R0 was estimated.

Limitations and common misreadings

R0 is a compact summary of transmission potential. Much of its misuse comes from reading more into it than the model behind it supports.

• A constant of the pathogen. R0 depends on contact patterns and setting; a value from another country or decade may not apply.
• A measure of severity. R0 contains nothing about symptoms, hospital use or death, so two infections with similar R0 can have very different costs and QALY losses.
• Lowered by vaccination. Programmes reduce the effective reproduction number; quoting a current estimate as R0 confuses the two.
• The herd immunity formula as a coverage target. The formula assumes homogeneous mixing and durable, complete protection; imperfect vaccines raise the coverage needed, and the WHO guide notes that age-dependent transmission can make the simple threshold too high or too low.
• Below 1 means no cases. Imported cases and short chains of transmission still occur.
• A threshold as a decision rule. A programme that leaves $R_t$ above 1 may still avert many infections at good value, and one that pushes it below 1 may cost more than it gains; the comparison belongs in a full cost-effectiveness analysis.

Sources

• Delamater PL, Street EJ, Leslie TF, Yang YT, Jacobsen KH. Complexity of the basic reproduction number (R0). Emerging Infectious Diseases 2019;25(1):1-4. https://doi.org/10.3201/eid2501.171901
• Diekmann O, Heesterbeek JAP, Roberts MG. The construction of next-generation matrices for compartmental epidemic models. Journal of the Royal Society Interface 2010;7(47):873-885. https://doi.org/10.1098/rsif.2009.0386
• Pitman R, Fisman D, Zaric GS, Postma M, Kretzschmar M, Edmunds J, Brisson M. Dynamic transmission modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-5. Value in Health 2012;15(6):828-834. https://doi.org/10.1016/j.jval.2012.06.011
• van den Driessche P. Reproduction numbers of infectious disease models. Infectious Disease Modelling 2017;2(3):288-303. https://doi.org/10.1016/j.idm.2017.06.002
• Wallinga J, Lipsitch M. How generation intervals shape the relationship between growth rates and reproductive numbers. Proceedings of the Royal Society B 2007;274(1609):599-604. https://doi.org/10.1098/rspb.2006.3754
• World Health Organization. WHO guide for standardization of economic evaluations of immunization programmes, 2nd edition (WHO/IVB/19.10). Geneva: WHO; 2019. Sections 6.1.1 and 6.2. https://iris.who.int/handle/10665/329389