From: 3blue1brown
The phrase exponential growth is familiar to most people, yet human intuition often struggles to grasp its true implications [00:00:06]. Small initial numbers can lead to surprising large outcomes, even when the overall trend is perfectly consistent with exponential patterns [00:00:17].
Defining Exponential Growth
Exponential growth means that moving from one day to the next involves multiplying by a constant factor [00:00:47]. For example, in data for recorded COVID-19 cases, the number of cases each day tended to be a multiple of about 1.15 to 1.25 of the previous day’s cases [00:00:55]. Viruses are a textbook example of this growth type because new cases are caused by existing ones [00:01:03].
The Mechanism of Growth
If ‘n’ is the number of cases on a given day, ‘e’ is the average number of people each infected individual exposes, and ‘p’ is the probability of an exposure leading to a new infection, then the number of new cases is e * p * n [00:01:24]. The fact that ‘n’ (the current number of cases) is a factor in its own change is what causes rapid acceleration [00:01:29]. As ‘n’ grows, the rate of growth itself increases [00:01:35]. This is equivalent to multiplying the total cases by a constant greater than 1 each day [00:01:47].
Visualizing Exponential Growth
To better visualize exponential growth, a logarithmic scale for the y-axis can be used [00:01:52]. On this scale, each fixed distance step represents multiplication by a certain factor (e.g., a power of 10) [00:02:01]. When plotted on a logarithmic scale, exponential growth appears as a straight line [00:02:05].
Using data, a linear regression can determine the best fit line. For instance, if cases multiply by 10 every 16 days on average, this indicates a strong exponential fit [00:02:22].
Counterintuitive Implications
One counterintuitive aspect of exponential growth is how different initial states can quickly converge. A country with 60 cases versus one with 6,000 might seem vastly different, but if numbers multiply by 10 every 16 days, the second country is only about a month behind the first [00:02:55].
If an exponential trend continues, numbers can escalate rapidly:
- From 10,000 cases to 1 million in 30 days [00:03:09].
- To 10 million in 47 days [00:03:15].
- To 100 million in 64 days [00:03:15].
- To 1 billion in 81 days [00:03:19].
The Limits of Exponential Growth: The Logistic Curve
True exponentials essentially never exist in the real world [00:05:03]; every observed instance is the start of a logistic curve [00:05:06].
A logistic curve initially mimics exponential growth but eventually levels out as it approaches a total population size [00:04:57]. This leveling occurs because factors like ‘e’ (exposure) or ‘p’ (infection probability) must decrease [00:04:07]. For example, if a large proportion of the population is already infected, the probability of exposure leading to a new infection decreases [00:04:21].
Inflection Point
The point where a logistic curve transitions from curving upward to curving downward is called the inflection point [00:05:13]. At this point, the number of new cases each day stops increasing and remains roughly constant before starting to decrease [00:05:22].
The Growth Factor
The growth factor is the ratio between the number of new cases on one day and the number of new cases the previous day [00:05:37].
- During the exponential part of a curve, this factor consistently stays above one [00:05:59].
- When the growth factor approaches one, it signals that the inflection point has been reached [00:06:02].
A subtle difference in the growth factor can have vastly different implications:
- A growth factor of one might mean the total number of cases will max out at about twice the current total [00:06:33].
- A growth factor slightly greater than one, however, means the system is still in the exponential part, implying orders of magnitude of growth are still possible [00:06:43].
Halting Exponential Growth
While an idealized model might suggest saturation at the total population, real-world factors like people clustering in local communities and less random shuffling of individuals modify the spread [00:07:00]. However, even with slight travel between clusters, the growth follows similar exponential-inducing laws [00:07:06].
Fortunately, the two key factors contributing to growth (‘E’ for exposure and ‘P’ for infection probability) can be reduced [00:07:45]:
- Exposure can decrease when people stop gathering and traveling [00:07:49].
- Infection rates can go down with practices like increased hand washing [00:07:53].
Exponential growth is highly sensitive to the constant multiplier [00:08:00]. For instance, a 15% daily growth rate could lead to over 100 million cases in 61 days from 21,000 [00:08:13]. But if that rate drops to 5%, the projection for the same period falls drastically to around 400,000 cases [00:08:26]. This highlights that proactive measures can significantly mitigate potential outcomes [00:08:26].