R0 from growth rate and doubling time (simple approximations)
This calculator provides two practical mini-tools used in introductory microbiology and epidemiology:
(A) converting between an exponential growth rate r (per day) and doubling time (days), and
(B) estimating R0 from r using a clearly-labeled, assumption-based approximation.
Exponential growth model used
When cases (or another quantity) change approximately exponentially over a time window, a standard model is:
\[
C(t)=C_0\cdot \exp(r\cdot t)
\]
where C(t) is the relative number of cases at time t (days), C0 is the starting value, and
r is the exponential growth rate in day−1. If r > 0 the curve grows; if r < 0 it declines.
A) Growth rate r ↔ doubling time
The doubling time is the time required for an exponentially growing quantity to multiply by 2.
Starting from the exponential model, set the ratio
\(\frac{C(t+\Delta t)}{C(t)}=2\) and solve for \(\Delta t\).
Derivation.
\[
\begin{aligned}
\frac{C(t+\Delta t)}{C(t)}&=\frac{C_0\cdot \exp(r\cdot (t+\Delta t))}{C_0\cdot \exp(r\cdot t)}\\
&=\exp(r\cdot \Delta t)
\end{aligned}
\]
\[
\begin{aligned}
\exp(r\cdot \Delta t)&=2\\
r\cdot \Delta t&=\ln(2)\\
\text{doubling time}&=\Delta t=\frac{\ln(2)}{r}
\end{aligned}
\]
For decline (\(r<0\)), the same idea gives a halving time magnitude:
\[
\text{halving time}=\frac{\ln(2)}{|r|}
\]
In this calculator, you can explicitly choose “Growth” or “Decline” so the sign of r is handled consistently.
B) Simple R0 approximation from growth
A common introductory approximation links R0 to an exponential growth rate r using an assumed
serial interval / generation time T (days):
\[
R_0\approx \exp(r\cdot T)
\]
This is not a universal law—it is a simple bridge between “how fast cases grow” and “how many secondary cases occur on average,”
conditional on the serial interval assumption. That is why the calculator can show a mini-table of scenarios:
how the estimated R0 changes when T changes.
How to interpret the visuals
The doubling-time gauge summarizes speed: shorter times correspond to faster change (growth or decline).
The R0 meter highlights the threshold at R0 = 1:
values above 1 indicate growth in this simplified framing; values below 1 indicate decline.
The curve plot uses the exponential model
\(\,C(t)=C_0\cdot \exp(r\cdot t)\,\) to show relative cases over time and marks doubling (or halving) intervals of length
\(\,\ln(2)/|r|\,\).
Important limitations and cautions
-
These formulas assume a time window where growth is approximately exponential and conditions are relatively stable
(behavior, interventions, reporting, and susceptibility are not rapidly changing).
-
The link \(\,R_0\approx \exp(r\cdot T)\,\) depends strongly on the assumed T. Different diseases, variants,
settings, and measurement methods can imply different serial intervals.
-
Real R0/Rt estimation often uses more detailed models (e.g., renewal equations and generation-interval distributions),
and uncertainty intervals require statistical methods beyond this introductory calculator.
