Case Study 1 — Modeling an Outbreak with an Exponential Function

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Case Study 1 — Modeling an Outbreak with an Exponential Function

Field: Epidemiology / public health (Biology track) Calculus used: Exponential and logarithmic functions (§2.2), the modeling viewpoint (§2.7) Forward references: Chapter 3 (limits make "long-run behavior" precise), Chapter 5 (rate of change gives $N'(t)$), Chapter 19 (the logistic and SIR models repair this one)

The setup

In late February 2020, a novel coronavirus was spreading through cities in northern Italy. By the first week of March, hospitals in Lombardy were overflowing; by mid-March, much of Europe and North America was under lockdown. In those early weeks — before interventions took effect — the count of confirmed cases worldwide grew in a very particular way: it doubled every few days. That pattern is the unmistakable signature of exponential growth, and recognizing it early is what justified an unprecedented response.

This case study walks through the calculus-adjacent reasoning epidemiologists actually used: build an exponential model from the doubling time, use it to forecast when hospital capacity is exceeded, and — just as importantly — understand exactly where and why the model stops being trustworthy. Everything here uses only the function families of §2.2 and the modeling discipline of §2.7. The derivatives and differential equations come later; the function comes first.

What exponential growth says

If a quantity is multiplied by a fixed factor over each fixed time interval, it grows exponentially:

$$N(t) = N_0 \cdot r^{t/T},$$

where $N_0$ is the initial count, $r$ is the multiplier per period, and $T$ is the length of one period. Using the base-conversion of §2.2, this is equivalently

$$N(t) = N_0\, e^{kt}, \qquad k = \frac{\ln r}{T}.$$

The two forms are the same function dressed differently — exactly the representation point of §2.1. For COVID-19 in early March 2020, confirmed cases were doubling roughly every $3$ days, so $r = 2$ and $T = 3$ days:

$$N(t) = N_0 \cdot 2^{t/3} = N_0 \, e^{(\ln 2 / 3)\,t} \approx N_0\, e^{0.231\,t}.$$

The rate constant $k = \ln 2 / 3 \approx 0.231$ per day means about $23\%$ growth per day. The two descriptions — "$23\%$ per day" and "doubles every $3$ days" — carry identical information, but as Discussion Question 3 explores, they do not land on a listener the same way.

The hospital-capacity problem

Exponential growth is alarming precisely because the doubling time is short compared to the time needed to add hospital capacity. Suppose a city has $100$ confirmed cases today, the doubling time is $3$ days, and its hospitals can handle $10{,}000$ COVID-19 patients before being overwhelmed. When do the hospitals overflow?

Set $N(t) = 10{,}000$ with $N_0 = 100$, $T = 3$:

$$10{,}000 = 100 \cdot 2^{t/3} \;\Longrightarrow\; 100 = 2^{t/3}.$$

Take $\log_2$ of both sides (the inverse of the base-$2$ exponential, §2.5):

$$\log_2 100 = \frac{t}{3} \;\Longrightarrow\; t = 3\log_2 100.$$

Now $\log_2 100 = \dfrac{\ln 100}{\ln 2} \approx \dfrac{4.605}{0.693} \approx 6.64$, so

$$t \approx 3 \times 6.64 \approx 19.9 \text{ days}.$$

Just under three weeks. That is not enough time to build hospitals; it is barely enough to convert convention centers into emergency ICUs. That is why March 2020 felt like an emergency — the arithmetic of doubling left almost no runway.

# Exponential case growth, doubling every 3 days from 100 cases,
# against a hospital capacity of 10,000. Mark the overflow time.
import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(0, 30, 300)          # days
N0, T, capacity = 100, 3, 10_000
N = N0 * 2**(t / T)                   # vectorized exponential

t_overflow = T * np.log2(capacity / N0)   # exact: 3*log2(100) ≈ 19.9

plt.figure(figsize=(9, 5))
plt.plot(t, N, 'b-', lw=2, label='Cases  $N(t)=100\\cdot 2^{t/3}$')
plt.axhline(capacity, color='red', ls='--', label='Hospital capacity')
plt.axvline(t_overflow, color='green', ls=':', label=f'Overflow ≈ {t_overflow:.1f} d')
plt.xlabel('Days from today'); plt.ylabel('Confirmed cases')
plt.title('Exponential growth, doubling every 3 days')
plt.legend(); plt.grid(True, alpha=0.3); plt.show()
# Output: a curve crossing the capacity line at about t = 19.9 days.
print(f"Overflow at t = {t_overflow:.1f} days")   # ≈ 19.9 days

The power and the limits of the model

Three observations turn this from arithmetic into a lesson about modeling.

The model captured what mattered. In early March 2020, models of exactly this shape were used at Imperial College London, the CDC, and many national agencies. They predicted catastrophic hospital overload within weeks absent drastic reductions in contact rates. Where interventions arrived in time, the predicted overload was often averted; where they arrived late (Lombardy, New York City), it happened. The exponential model was correct enough to drive policy that saved lives.

The model is also wrong. Exponential growth cannot continue forever. Once infected people recover and gain immunity, once behavior changes, once public-health measures take hold, the per-day multiplier drops. The honest range of $N(t) = N_0 e^{kt}$ is the early phase, while susceptibles are effectively unlimited. As $N(t)$ becomes a real fraction of the population, the constant-$k$ assumption fails — which is Discussion Question 2 made quantitative. In Chapter 19 we replace it with the logistic model (growth that slows toward a carrying capacity) and the SIR model (a system of differential equations tracking Susceptible, Infected, and Recovered groups) — the SIR anchor that recurs through the rest of this book and culminates in the Chapter 39 capstone.

Even the wrong model was useful. A model need not be exact to earn its keep. This one captured the one qualitative fact that drove every decision: this is growing fast, and you have weeks, not months. By the time interventions made the model quantitatively wrong, it had already done its job.

The Key Insight. A mathematical model is a tool, not a description of reality. The exponential model of outbreak spread was wrong in detail yet decisive in the qualitative forecast that justified urgent action. Good modeling means choosing a function that is good enough for the question at hand — the §2.7 discipline in its purest form.

Why this is the gateway to calculus

So far we used only the exponential function and its inverse — no derivatives, no integrals. So is this calculus? It is the doorway. Here is what the coming chapters add to this exact example:

Chapter 3 (limits) gives precise meaning to "$N(t)$ as $t \to \infty$," letting us state long-run behavior rigorously.
Chapter 5 (rate of change) gives $N'(t)$, the new infections per day. For exponential models $N'(t) = k\,N(t)$: the growth rate is proportional to the current count. That proportionality is the defining property of exponential growth, and it is why $e^{kt}$ shows up — a thread we pick up the moment derivatives arrive.
Chapter 19 (differential equations) turns "$N' = kN$" into the launching point for the logistic and SIR models.
Chapter 23 (Taylor series) explains why $e^{kt} \approx 1 + kt$ for small $t$ — why an exponential "looks linear at first" before it takes off.

The exponential model, in other words, is the first paragraph of a long story that calculus spends the rest of the book telling.

Discussion questions

The model assumes a constant doubling time. What real-world factors make the doubling time change over an outbreak, and which of them shorten it versus lengthen it?
Make the breakdown precise: roughly what value of $N(t)$, as a fraction of a city's population, should make you distrust the constant-$k$ assumption? Why?
Why is "doubles every 3 days" more useful for public communication than "the daily growth rate is $0.231$," even though they encode the same model?
Confirmed cases undercount true cases (testing latency, asymptomatic spread). How would you adjust $N_0$ or $k$ to reflect that, and which forecast — overflow date or peak height — is more sensitive to the adjustment?
Some critics in March 2020 called exponential models "too pessimistic" because they ignored adaptive behavior. Looking back at your own country's first-wave trajectory, were the critics right, wrong, or both — and what does that say about the range over which the model is valid?

Your turn — mini-project

Pull real first-wave data (e.g. Our World in Data, https://ourworldindata.org/covid-cases).

Choose a two-week window from the first wave (Feb–Apr 2020 in most countries).
Fit $N(t) = N_0 e^{kt}$ by linear regression on $\ln N(t) = \ln N_0 + kt$ — taking the log turns the exponential into a straight line whose slope is $k$ (a §2.2/§2.5 trick).
Compute the doubling time $T_{\text{double}} = \ln 2 / k$.
Plot the data and the fitted curve together.
Extrapolate the fit two weeks past your window and overlay the actual data. Mark where the model and reality part ways — that gap is the model's range of validity, made visible.

You will return to this dataset in Chapter 19 with differential equations in hand.

Case Study 1 — Modeling an Outbreak with an Exponential Function

The setup

What exponential growth says

The hospital-capacity problem

The power and the limits of the model

Why this is the gateway to calculus

Discussion questions

Your turn — mini-project

Further reading (annotated)