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Module 2 · Lesson 2

Differential Equations: Weather, epidemics, chaos, and the Navier–Stokes problem

You'll see how differential equations model epidemics and the weather, and why chaos puts a horizon on even perfect models. You'll then follow a live case: a claimed solution to the Navier–Stokes problem, produced by an AI system, and how mathematicians are deciding whether it holds and who deserves credit.

What you will be able to do

  • Explain how differential equations model weather and epidemics, why chaos limits prediction, and what the Navier–Stokes problem asks.

0% of this lesson · 10 items · 1h 33m total · 1h 18m without the optional activity

Contents of this lesson10 items
  1. ReadingWhy Weather Forecasts Stop at About Ten Days3 min
  2. ReadingModeling Epidemics: The SIR Model4 min
  3. ReadingNumerical Weather Prediction4 min
  4. ReadingChaos: Sensitive Dependence and the Limits of Prediction4 min
  5. ReadingThe Navier–Stokes Problem: An AI-Generated Proof Under Review4 min
  6. Guided ReadingGuided Walkthrough: Stepping an Epidemic Forward Day by Day7 min
  7. Guided ConversationWhat Would It Take to Trust a Proof?12 min
  8. Hands-on Activity · optionalRun an Epidemic Model in a Spreadsheet15 min
  9. Knowledge CheckWeather, epidemics, chaos, and the Navier–Stokes problem10 min
  10. Graded QuizDifferential Equations and Dynamical Systems30 min

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Reading 3 min Done

Why Weather Forecasts Stop at About Ten Days

Look at a weather forecast and compare its days. Tomorrow's high temperature is usually close. The outlook for five days ahead is still useful for planning. By ten days ahead, the numbers are much shakier, and a forecast for a month from now, day by day, isn't worth much at all.

That's puzzling once you know how forecasts are made. The atmosphere is a fluid, and the equations that govern fluid flow and heat are well known. Weather centers solve them every day on powerful supercomputers, a task Bauer, Thorpe, and Brunet compare in scale to simulating the human brain (2015). So why can't forecasters push on to thirty days? Before you read on, decide what you think the obstacle is: missing data, slow computers, or something else.

Progress has been real. Bauer and colleagues report that forecast skill for three to ten days ahead has improved by about a day per decade, so that a six-day forecast in 2015 was about as good as a five-day forecast ten years before (Bauer, Thorpe, and Brunet 2015). Better physics, more observations, and faster machines all contributed. Yet the same review puts the horizon for accurate, reliable prediction of high-impact weather at about one to two weeks. More computing power has pushed forecasts closer to that horizon without removing it.

The reason was found in 1963 by the meteorologist Edward Lorenz. He studied a stripped-down set of differential equations, rules that link quantities to their own rates of change, meant to represent convection: warm fluid rising and cool fluid sinking. The rules involved no chance at all. Yet he found that the solutions were unstable and almost all of them never settled into a repeating pattern (Lorenz 1963). Two runs that start from almost the same state drift apart until they have nothing in common. Lorenz then asked what this meant for very-long-range weather prediction.

Measurements of the atmosphere are never perfect. If tiny differences in the starting state grow until they swamp the forecast, then better measurements and faster computers buy extra days, but not unlimited ones. The limit lies in the atmosphere itself, not in the machines.

That's one of the things the mathematics of differential equations and dynamical systems offers. It can tell you:

  • which systems can be predicted far ahead, such as the positions of the planets, and which can't, such as the weather
  • roughly how far ahead each can be predicted
  • what can still be said beyond that horizon, as odds and averages instead of exact states

This content reflects the field as of October 2026.

References

  • Bauer, Peter, Alan Thorpe, and Gilbert Brunet. 2015. "The Quiet Revolution of Numerical Weather Prediction." Nature 525 (7567): 47–55. doi:10.1038/nature14956.
  • Lorenz, Edward N. 1963. "Deterministic Nonperiodic Flow." Journal of the Atmospheric Sciences 20: 130–141.

Reading 4 min Done

Modeling Epidemics: The SIR Model

Introduction

Epidemics don't keep spreading until everyone has caught the disease. They rise, peak, and fade while many people are still unaffected. This reading explains the SIR model, the differential equations behind that pattern: where it came from, the rates that drive it, the reproduction number and threshold, and what the model is used for.

Kermack and McKendrick's 1927 Model

In 1927 W. O. Kermack and A. G. McKendrick published "A Contribution to the Mathematical Theory of Epidemics." They divided a population into the unaffected, the sick, and those removed from the outbreak by recovery or death. They found that, for given rates of infection, recovery, and death, there is a threshold population density: below it, an infected person doesn't set off an epidemic. They also found that an epidemic usually ends before every susceptible person is infected (Kermack and McKendrick 1927; O'Connor and Robertson 2020).

The simplest form of their approach is now called the SIR model: a model that divides a population into susceptible, infected, and recovered groups and describes the rates at which people move between them.

GroupWho's in itMoves in byMoves out by
Susceptible (S)People who can catch the diseaseNobody, in the simplest versionCatching it
Infected (I)People who have it and can pass it onCatching itRecovering
Recovered (R)People who can no longer catch or pass it onRecoveringNobody

Susceptible people are those who can still catch the disease. Infected people are those who currently have it and can pass it on. Recovered people are those who have had it and can no longer catch or pass it on; in Kermack and McKendrick's version, this group also includes those who died.

The Rates in Words

Each group's size changes at a rate given by a rule, so the model is a set of three linked differential equations. A differential equation is a rule that links a quantity to its own rate of change.

New infections need contact between a susceptible person and an infected one. So the number of new infections per day is proportional to the number of susceptible people times the number of infected people, divided by the population size. Recoveries are proportional to the number of infected people: if an illness lasts about ten days on average, about a tenth of the infected recover each day.

In a town of 1,000, for example, you might take new infections per day as 0.3 \times S \times I \div 1000 and recoveries as 0.1 \times I, where S and I are the numbers susceptible and infected. The 0.3 is the infection rate and the 0.1 is the recovery rate.

The Reproduction Number and the Threshold

The reproduction number is the average number of new infections that one case causes early in an outbreak, when nearly everyone is susceptible. In the SIR model it equals the infection rate divided by the recovery rate. In the example, 0.3 \div 0.1 = 3. When it's above 1, cases grow; when it's below 1, an outbreak fades out.

The threshold is the level of susceptible people above which infections can rise. In the SIR model, infections rise only while the fraction of people still susceptible is greater than 1 divided by the reproduction number. With a reproduction number of 3, that's one-third of the population.

Why Epidemics Peak, and What the Model Is For

As the outbreak spreads, the susceptible group shrinks, so each infected person meets fewer people who can catch it. Once the susceptible fraction falls below the threshold, recoveries outpace new infections, and the number infected peaks and starts to fall. In the example, carried forward by computer, infections peak around day 28 at about 316 people, and about 53 of the 1,000 have still never been infected by day 100. The epidemic burns out because it runs short of susceptible people, not because everyone has caught it.

Models descended from Kermack and McKendrick's are used, alongside other tools, to estimate when an outbreak might peak, how many people might need hospital care at once, and how measures that reduce contact might lower the peak. The simple version has clear limits. It assumes everyone mixes evenly, while real contacts cluster in households, schools, and workplaces, and people change their behavior as an outbreak unfolds. Practical models add more groups and more structure to deal with this.

Conclusion

The SIR model describes an epidemic with three linked rates of change, a structure that goes back to Kermack and McKendrick's 1927 paper. Its reproduction number and threshold explain why outbreaks grow, peak, and decline before everyone is infected. Real outbreaks are messier, so working models build on this core with more detail.

This content reflects the field as of October 2026.

Key Terms

  • SIR model: A model that divides a population into susceptible, infected, and recovered groups and describes the rates at which people move between them.
  • Susceptible: The group of people who can still catch the disease.
  • Infected: The group of people who currently have the disease and can pass it on.
  • Recovered: The group of people who have had the disease and can no longer catch or pass it on.
  • Reproduction number: The average number of new infections that one case causes early in an outbreak, when nearly everyone is susceptible.
  • Threshold: The level of susceptible people above which infections can rise.

References

  • Kermack, W. O., and A. G. McKendrick. 1927. "A Contribution to the Mathematical Theory of Epidemics." Proceedings of the Royal Society of London, Series A 115 (772): 700–721.
    • Free: None confirmed
    • Free companion: O'Connor and Robertson 2020
  • O'Connor, J. J., and E. F. Robertson. 2020. "Contributions to the Mathematical Theory of Epidemics." MacTutor History of Mathematics Archive, University of St Andrews. Last updated November 2020.

Reading 4 min Done

Numerical Weather Prediction

Introduction

Every forecast you see started as a very large calculation. Forecasters don't extend yesterday's trend; they compute the future state of the atmosphere from the laws of physics. This reading explains how that works: the equations, the grid, the starting state, the ensemble of forecasts, and how much forecasts have improved.

The Atmosphere as a Fluid

Numerical weather prediction is forecasting the weather by solving the equations of the atmosphere's motion and heat on a computer, stepping forward in time. The atmosphere is a fluid, so its motion obeys the equations of fluid flow. Bauer, Thorpe, and Brunet list the main ones: the Navier–Stokes and mass continuity equations, which describe how air moves and how mass is conserved on a rotating Earth, together with the first law of thermodynamics and the ideal gas law (2015). Between them, they govern changes in wind, pressure, density, and temperature.

All of these are differential equations, rules that link quantities to their own rates of change. Nobody can solve them with a formula for the whole atmosphere, so they're solved numerically: compute the current rates, take a short step forward in time, and repeat (Bauer, Thorpe, and Brunet 2015).

Gridding the Globe and Measuring the Start

A computer can't track every parcel of air, so models work on a grid: a network of points covering the globe, at several heights, where the model computes wind, temperature, pressure, and moisture. Finer grids capture more detail but cost more computing. In 2015 the European Centre for Medium-Range Weather Forecasts (ECMWF) ran its highest-resolution global model with points about 16 kilometers apart (Bauer, Thorpe, and Brunet 2015). By 2025 its physics-based model used a 9-kilometer spacing (European Centre for Medium-Range Weather Forecasts 2025).

A forecast also needs a starting point. Initial conditions are the measured state of the atmosphere at the moment a forecast begins. They come from satellites, weather stations, balloons, aircraft, and ships, blended with a short recent forecast to fill the gaps. Bauer and colleagues put the number of observations used each day at around ten million, and in 2025 ECMWF described its initial conditions as combining a short-term forecast with about 60 million observations (European Centre for Medium-Range Weather Forecasts 2025).

Ensembles: Many Forecasts at Once

The measured starting state is never exact, and small errors in it grow as the forecast runs. So forecast centers don't rely on a single run. An ensemble forecast is a set of many forecasts run from slightly different starting states, and often slightly different model settings, whose spread estimates how uncertain the forecast is. In 2015 ECMWF's ensemble had 50 members, run on a coarser grid than its single best forecast (Bauer, Thorpe, and Brunet 2015).

When the members agree, forecasters can be confident. When they fan out, the forecast is uncertain, and the spread says by how much. Ensembles are one of the main sources of the probabilities in forecasts, such as a 70 percent chance of rain. Bauer and colleagues describe modern forecasts as giving "an inherently probabilistic assessment" (2015).

Steady Gains, and a New Kind of Model

Bauer, Thorpe, and Brunet's 2015 review describes a "quiet revolution": forecast skill for three to ten days ahead improved by about a day per decade, through gains in physics, observations, data handling, and computing. Their review also places the limit for accurate prediction of high-impact weather at about one to two weeks.

A newer approach now runs alongside the physics. In February 2025 ECMWF made operational a machine-learning forecast system, trained on how the weather has evolved in the past. It starts from the same initial conditions as the physics-based model, and ECMWF reported that it beats leading physics-based models on many measures while using about 1,000 times less energy per forecast. ECMWF describes the two kinds of model as complementary (European Centre for Medium-Range Weather Forecasts 2025).

Conclusion

Weather forecasts come from stepping the equations of fluid flow and heat forward on a grid, starting from measured initial conditions, and running ensembles to measure uncertainty. Skill has improved steadily for decades, and machine-learning models now run alongside physics-based ones. The horizon of one to two weeks for detailed forecasts hasn't moved away.

This content reflects the field as of October 2026.

Key Terms

  • Numerical weather prediction: Forecasting the weather by solving the equations of the atmosphere's motion and heat on a computer, stepping forward in time.
  • Grid: A network of points covering the globe, at several heights, where a weather model computes wind, temperature, pressure, and moisture.
  • Initial conditions: The measured state of a system, such as the atmosphere, at the moment a forecast begins.
  • Ensemble forecast: A set of many forecasts run from slightly different starting states, whose spread estimates how uncertain the forecast is.

References

  • Bauer, Peter, Alan Thorpe, and Gilbert Brunet. 2015. "The Quiet Revolution of Numerical Weather Prediction." Nature 525 (7567): 47–55. doi:10.1038/nature14956.
  • European Centre for Medium-Range Weather Forecasts. 2025. "ECMWF's AI Forecasts Become Operational." News, February 25, 2025.

Reading 4 min Done

Chaos: Sensitive Dependence and the Limits of Prediction

Introduction

If you know a system's rules exactly and its present state almost exactly, can you predict its future? For many systems the answer is no, and the reason isn't randomness. This reading explains chaos: how Lorenz found it in a weather model, why exact rules don't guarantee prediction, how Poincaré had glimpsed it, and what can still be predicted.

Lorenz's Rounded Numbers

In 1961 the meteorologist Edward Lorenz was running a weather model on a computer, a set of differential equations, rules that link quantities to their own rates of change. He restarted a run partway through, typing in values from a printout. The printout showed fewer decimal places than the machine stored, so the new start differed from the old one in the fourth decimal place. The rerun soon gave significantly different results, and Lorenz first suspected a fault in the computer (O'Connor and Robertson 2008).

There was no fault. The difference came from the equations. Lorenz had found sensitive dependence on initial conditions: the property that tiny differences in a system's starting state grow over time into large differences. In his 1963 paper he studied a much simpler system, three equations representing convection, warm fluid rising and cool fluid sinking. He found that its solutions were unstable and that almost all of them never settled into a repeating pattern (Lorenz 1963).

Figure: The path traced by the solution of Lorenz's three convection equations, drawn in three dimensions: two looping lobes, like a butterfly's wings, with the path circling one lobe several times, then switching to the other, never retracing itself. Alt text: A tangled curve forming two looping, wing-shaped lobes that the path switches between without ever repeating.

In 1972 Lorenz gave a talk titled "Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?" The phrase "butterfly effect" soon became a popular name for sensitive dependence (O'Connor and Robertson 2008).

Exact Rules, Limited Prediction

A system is deterministic when its present state and its rules fix its future completely, with no randomness. Lorenz's equations are deterministic: start them from the same values and they give the same result every time.

Chaos is behavior of a system that follows exact rules, yet in which tiny differences in the starting state grow until long-range prediction fails. Determinism isn't predictability. To predict a chaotic system far ahead, you'd need its starting state with impossible precision, because any measurement error, however small, eventually grows as large as the quantities being predicted. Improving the measurements buys extra time, but in a chaotic system each improvement buys less than you might hope, because errors tend to grow quickly once they start.

Poincaré Saw It First

Lorenz wasn't the first to meet this behavior. In 1889–1890 Henri Poincaré, studying three bodies that attract one another by gravity, gave the first mathematical description of chaotic motion (O'Connor and Robertson 2003). Bauer, Thorpe, and Brunet note that Poincaré recognized that small changes to a starting state could produce very different outcomes (2015). What Lorenz added, with a computer, was a concrete case in a weather-like system, and a clear statement of what it meant for forecasting.

What Can Still Be Predicted

The predictability horizon is the time beyond which forecasts of a system's exact state stop being useful. For high-impact weather, Bauer, Thorpe, and Brunet put it at about one to two weeks (2015). Different systems have very different horizons: the planets' positions can be computed far ahead, while a single day's weather can't.

Beyond the horizon, some things can still be said. Forecasters run many simulations from slightly different starts and report the odds of different outcomes. Averages and patterns over long periods, which is what climate describes, can be studied even when the weather on a particular day can't be predicted. Chaos limits forecasts of exact states, not every statement about a system.

Conclusion

Chaos is a property of exact rules, not a lack of them: tiny differences in the starting state grow until exact prediction fails. Poincaré met it in the three-body problem and Lorenz in a weather model. Forecasters now work within the predictability horizon it sets, and use probabilities and averages beyond it.

This content reflects the field as of October 2026.

Key Terms

  • Sensitive dependence on initial conditions: The property that tiny differences in a system's starting state grow over time into large differences.
  • Deterministic: Describes a system whose present state and rules fix its future completely, with no randomness.
  • Chaos: Behavior of a system that follows exact rules, yet in which tiny differences in the starting state grow until long-range prediction fails.
  • Predictability horizon: The time beyond which forecasts of a system's exact state stop being useful.

References

  • Bauer, Peter, Alan Thorpe, and Gilbert Brunet. 2015. "The Quiet Revolution of Numerical Weather Prediction." Nature 525 (7567): 47–55. doi:10.1038/nature14956.
  • Lorenz, Edward N. 1963. "Deterministic Nonperiodic Flow." Journal of the Atmospheric Sciences 20: 130–141.
  • O'Connor, J. J., and E. F. Robertson. 2003. "Jules Henri Poincaré." MacTutor History of Mathematics Archive, University of St Andrews. Last updated October 2003.
  • O'Connor, J. J., and E. F. Robertson. 2008. "Edward Norton Lorenz." MacTutor History of Mathematics Archive, University of St Andrews. Last updated July 2008.

Reading 4 min Done

The Navier–Stokes Problem: An AI-Generated Proof Under Review

Introduction

Water from a tap, smoke from a candle, and air over a wing all obey the same equations of fluid flow. Whether those equations can break down is one of the seven Millennium Prize Problems, and in September 2026 an AI developer claimed that one of its systems had answered it. This reading explains what the problem asks, what has been claimed, and what is still being checked, as of mid-October 2026.

What the Problem Asks

The Navier–Stokes equations are differential equations that describe how a fluid such as water or air moves, accounting for pressure, internal friction, and outside forces. The Clay Mathematics Institute calls them "the equation which governs the flow of fluids such as water and air" (Clay Mathematics Institute n.d.). The closely related Euler equations describe a fluid with no internal friction.

In 2000 the Institute named the problem one of its Millennium Prize Problems, each carrying a $1 million prize (Clay Mathematics Institute 2026). It asks, roughly, whether a smooth flow in three dimensions always stays smooth, or whether it can develop a blowup: a singularity in finite time, where the fluid's speed or how sharply it changes becomes infinite at some point.

The official statement accepts either a proof that smooth flows stay smooth or an example of breakdown, and the breakdown option allows forcing: an external force applied to the fluid, such as stirring. Córdoba and Martínez-Zoroa write that a forced blowup for Navier–Stokes, "if verified under the stated hypotheses," would establish one of the breakdown alternatives the prize allows (2026).

A Month of Claims

Date in 2026SourceWhat was reported
September 7Terence Tao's blogWork by Levent Alpöge and Tristan Buckmaster showing blowup with a smooth external force for three simpler fluid equations, including the three-dimensional Euler equations, building on methods of Diego Córdoba and Luis Martínez-Zoroa. Not yet Navier–Stokes (Tao 2026)
September 8OpenAIAn internal AI system produced a proof of blowup for Navier–Stokes with a smooth external force, starting from fluid at rest, checked in the Lean proof software. OpenAI recognized Alpöge and Buckmaster's priority on related work and said it won't claim the prize (OpenAI 2026)
September 11Clay Mathematics InstituteThe problem has "apparently been settled"; its review is "deliberately unhurried" (Clay Mathematics Institute 2026)
October 4Córdoba and Martínez-ZoroaAssessing the announced proofs "requires a detailed mathematical analysis" (Córdoba and Martínez-Zoroa 2026)

Tao described Alpöge and Buckmaster's arguments as "heavily AI-assisted" (Tao 2026).

What Is Still Being Checked

OpenAI's proof was checked by formal verification: checking every step of a proof by computer, in a proof assistant such as Lean. OpenAI says the Lean version was itself produced by an AI model, in about 17 hours of additional work (OpenAI 2026). Such a check confirms that the formal statement follows from the formal steps. It doesn't confirm that the formal statement matches the problem as Clay posed it, or explain the argument to the people who must judge it. That work falls to experts. On October 6, Alexander Bastounis, Fabian Circelli, and Anders Hansen posted a preprint, a paper not yet peer reviewed, arguing that parts of OpenAI's Lean proof don't correspond to its written proof. They make no claim about whether the written proof is correct (Bastounis, Circelli, and Hansen 2026). As of October 13, 2026, the Clay Institute's page lists the problem as "Active" (Clay Mathematics Institute n.d.).

Credit is disputed too. Priority is credit for being first to a result. OpenAI says it began after hearing a rumor that two Millennium Prize Problems had been resolved, which it later linked to Alpöge and Buckmaster, and on September 10 it updated its announcement with an investigation into whether Buckmaster's prior use of OpenAI's tools could have influenced its system, concluding that it couldn't (OpenAI 2026). Public disagreement over credit, and over how the work was done, hasn't been resolved.

None of the announced results claims blowup for Navier–Stokes with no external force at all. That question remains open.

Conclusion

As of mid-October 2026, a claimed AI-generated proof of Navier–Stokes blowup with a smooth external force is under expert review, and the Clay Institute lists the problem as active. The verdict depends on detailed checking that, by the Institute's own account, won't be rushed.

This content reflects the field as of October 2026.

Key Terms

  • Navier–Stokes equations: Differential equations that describe how a fluid such as water or air moves, accounting for pressure, internal friction, and outside forces.
  • Blowup: The development of a singularity in finite time, where a fluid's speed or how sharply it changes becomes infinite at some point.
  • Forcing: An external force applied to a fluid, such as stirring.
  • Formal verification: Checking every step of a proof by computer, in a proof assistant such as Lean.
  • Priority: Credit for being first to a result.

References

  • Bastounis, Alexander, Fabian Circelli, and Anders C. Hansen. 2026. "Navier–Stokes Lost in Translation: Why Lean Verification of AI Autoformalisation Does Not Guarantee Correct Natural Language Proofs." arXiv preprint arXiv:2610.08144, submitted October 6, 2026.
  • Clay Mathematics Institute. n.d. "Navier–Stokes Equation." Millennium Prize Problems. Accessed October 13, 2026.
  • Clay Mathematics Institute. 2026. "Navier–Stokes Announcement." September 11, 2026.
  • Córdoba, Diego, and Luis Martínez-Zoroa. 2026. "On Classical Solutions and Singularity Formation in Incompressible Fluids." Guest post on Terence Tao's blog, What's New, October 4, 2026.
  • OpenAI. 2026. "On the Navier–Stokes Millennium Prize Problem." September 8, 2026; updated September 10, 2026.
  • Tao, Terence. 2026. "Finite Time Blowup with Smooth Forcing Term for the Incompressible Porous Medium, Boussinesq, and Incompressible Euler Equations." What's New, September 7, 2026.

Guided Reading 7 min Done

Guided Walkthrough: Stepping an Epidemic Forward Day by Day

Introduction

Epidemic models can sound like something only a supercomputer could run. The simplest one, the SIR model, can be stepped forward with a calculator. This walkthrough follows an outbreak in a small town for two days by hand, then uses the model's numbers to explain why the outbreak grows, where it turns, and why it stops before everyone has been infected.

The SIR model goes back to W. O. Kermack and A. G. McKendrick's 1927 paper on the mathematical theory of epidemics. They showed that an epidemic needs a threshold density of susceptible people to take off, and that it usually ends before every susceptible person is infected (Kermack and McKendrick 1927; O'Connor and Robertson 2020).

The Starting Point

A town has 1,000 people. On day 0, 10 of them are infected, 990 are susceptible, and none have recovered. The SIR model splits the town into three groups:

  • S, susceptible: people who can still catch the disease
  • I, infected: people who have it and can pass it on
  • R, recovered: people who have had it and can no longer catch or pass it on

Two rules say how people move between the groups each day:

\text{new infections} = 0.3 \times S \times I \div 1000
\text{recoveries} = 0.1 \times I

The first rule says that infections come from contact between susceptible and infected people, so they're proportional to both. The 0.3 is the infection rate, and dividing by 1,000, the town's population, turns the count of possible contacts into a share of the town. The second rule says that about a tenth of the infected recover each day, which fits an illness lasting about ten days on average.

Each day, new infections leave S and join I, and recoveries leave I and join R. These two rules are a daily-step version of the model's differential equations, rules that link quantities to their own rates of change. Stepping them forward one day at a time is the same move as any numerical method: compute today's rates from today's numbers, then update.

Walking Through the Daily Steps

Step 1: Compute day 1

Start from S = 990 and I = 10. New infections are

0.3 \times 990 \times 10 \div 1000 = 2.97

Recoveries are 0.1 \times 10 = 1. Now update each group. The susceptible group loses the new infections: 990 - 2.97 = 987.03. The infected group gains the new infections and loses the recoveries: 10 + 2.97 - 1 = 11.97. The recovered group gains the recoveries: 0 + 1 = 1.

So after day 1, S = 987.03, I = 11.97, and R = 1. As a check, the three still add up to 1,000. Nobody enters or leaves the town in this model; people only move between groups.

Step 2: Compute day 2

Use the day-1 numbers. New infections are 0.3 \times 987.03 \times 11.97 \div 1000, which is about 3.54. Recoveries are 0.1 \times 11.97, which is about 1.20. Update again:

  • S: 987.03 - 3.54 \approx 983.49
  • I: 11.97 + 3.54 - 1.20 \approx 14.32
  • R: 1 + 1.20 = 2.20

The infected group grew from 10 to 11.97 to 14.32. Each day's new infections outnumber the day's recoveries, so the outbreak is growing, and growing faster each day, because more infected people means more contacts.

Step 3: Find the reproduction number

The reproduction number is the average number of new infections one case causes early on, when nearly everyone is susceptible. In the SIR model it's the infection rate divided by the recovery rate:

0.3 \div 0.1 = 3

Here's why. Early on, almost everyone is susceptible, so each infected person causes about 0.3 new infections a day. An infected person stays infected for about ten days on average, which is 1 divided by the recovery rate of 0.1. Over those ten days, at 0.3 per day, each case causes about 3 more. A reproduction number above 1 means the outbreak grows.

Step 4: Find the threshold

Infections rise on any day when new infections exceed recoveries. Compare the two rules: new infections are 0.3 \times S \times I \div 1000 and recoveries are 0.1 \times I. Both contain I, so the comparison depends only on the share of the town still susceptible. Infections rise when

0.3 \times S \div 1000 > 0.1

which is the same as S \div 1000 > 1/3. So infections keep rising only while more than a third of the town is susceptible, about 333 people. That's the threshold, and it's 1 divided by the reproduction number.

On day 2, 983.49 people are susceptible, far above 333, so the outbreak has a long way to grow.

Step 5: Explain the peak and decline

You don't need to step every day to see what happens next. Each new infection moves someone out of S, so the susceptible group keeps shrinking. As it shrinks, each infected person meets fewer people who can catch the disease, and new infections per case fall. When S drops below about 333, recoveries outpace new infections, and the number infected starts to fall. That turning point is the peak.

If you carry the same arithmetic forward, as a spreadsheet can, the infected count peaks around day 28 at about 316 people, just as S passes through about 333. After the peak, the infected count falls steadily. By day 100, about 53 people have never been infected. The outbreak ends because it runs out of susceptible people to reach, not because everyone has caught the disease.

Key Considerations

Fractions of people are a feature, not a mistake. A count of 11.97 infected people can't happen in a real town, but the model treats each group as a smoothly varying quantity, the way a population model does. Rounding at every step would add errors that build up over many days. Keep the decimals while stepping, and round only when you report a result.

The order of updates matters. Compute both new infections and recoveries from the same day's numbers before changing any group. If you update S first and then use the new S to compute infections, you're mixing two days, and the totals won't add up the same way.

The common mistake: assuming the epidemic stops only when everyone has been infected. It's natural to picture an outbreak burning until there's no one left. The threshold shows why that's wrong. Infections fall as soon as the susceptible share drops below 1 divided by the reproduction number, here one-third. After that, each case leads to fewer than one new case on average, and the outbreak fades while some people have never been infected. Kermack and McKendrick noted this pattern in 1927 (O'Connor and Robertson 2020).

This is a model, not a forecast. The simple SIR model assumes everyone mixes evenly and that the rates stay fixed. Real contacts cluster, and people change their behavior during an outbreak. The model explains the shape of an epidemic; practical forecasts need more detail.

Summary

Stepping the SIR model forward means computing each day's new infections and recoveries from that day's numbers, then moving people between groups. For a town of 1,000 starting with 10 infected, an infection rate of 0.3, and a recovery rate of 0.1:

DayNew infectionsRecoveriesSIR
0——990100
12.971987.0311.971
23.541.20983.4914.322.20

Reproduction number: 0.3 \div 0.1 = 3. Threshold: infections rise only while more than one-third of the town, about 333 people, is susceptible.

  1. Each row adds up to 1,000, which checks the arithmetic.
  2. Infections rise on both days because S is far above the threshold.
  3. Carried further, infections peak around day 28, when S falls to about the threshold, and the outbreak ends with some people never infected.

References

  • Kermack, W. O., and A. G. McKendrick. 1927. "A Contribution to the Mathematical Theory of Epidemics." Proceedings of the Royal Society of London, Series A 115 (772): 700–721.
    • Free: None confirmed
    • Free companion: O'Connor and Robertson 2020
  • O'Connor, J. J., and E. F. Robertson. 2020. "Contributions to the Mathematical Theory of Epidemics." MacTutor History of Mathematics Archive, University of St Andrews. Last updated November 2020.

Guided Conversation 12 min Done

What Would It Take to Trust a Proof?

In this conversation you'll start from a forecast you rely on and think through why some predictions hold far ahead while others fade within days. Then you'll look at a live case, a claimed proof about the equations of fluid flow produced by an AI system, and decide what you'd want to know before calling it correct. You'll leave with two tests of your own for trusting a claimed result.

You'll have this conversation with an AI assistant, using your own account. Choose a button to open a new chat with the prompt already filled in, then press send to start. If the chat opens empty, copy the prompt and paste it in.

Run this conversation in whichever assistant you already use:

Open in ClaudeOpen in ChatGPT
Claude desktop app

To use another LLM, simply copy and paste the prompt into its chat window.

Show the full prompt (it lists misreadings to watch for, so skip it if you would rather come to the conversation fresh) 
Guided Conversation: What Would It Take to Trust a Proof? (about 12 minutes)

Note to the learner: press send to start. Everything below is facilitator guidance for the AI. It lists misconceptions to watch for, so skip it if you'd rather come to the conversation fresh.

Please facilitate a reflective dialogue with me. I'm an adult who is comfortable with basic high-school algebra but may be rusty or uneasy with math, exploring what differential equations and dynamical systems are and whether to study them further. I'm studying weather, epidemics, chaos, and the Navier-Stokes problem. Follow this guidance for the whole conversation.

GOAL
Explain how differential equations model weather and epidemics, why chaos limits prediction, and what the Navier-Stokes problem asks, ending on how mathematicians accept a result.

HOW TO RUN THE CONVERSATION
- Ask one question at a time, then wait for my reply. Keep each of your turns under about 120 words.
- Don't lecture. Explain a point only when I need it to continue, then return to my examples.
- Be curious, collegial, and patient. Define terms briefly on first use (chaos, reproduction number, formal verification). Welcome disagreement when I give a reason.
- Plain conversation only: don't search the web or create files or documents. On Navier-Stokes, use only the facts below and note that its status may have changed since October 2026.
- Aim for about 12 minutes, mostly on topics 2 and 3. If my replies are brief, offer one concrete prompt, such as "would you plan an outdoor event around a ten-day forecast?", and move on. If I seem uncertain, shorten the conversation to 5-7 minutes. Always reach the final topic.
- Start now. Open with one or two warm sentences: this is a conversation, not a quiz; my reasons matter more than being right; I can ask you to clarify anything. Then ask me to name a forecast I rely on (weather, traffic, a health projection) and how far ahead I trust it.

TOPICS, IN ORDER
1. A forecast I rely on. Ask why my trust fades with distance: the model, the data, or something else? Follow-up: has it improved in my lifetime?
2. Chaos and the horizon. Ask why a perfect model of a chaotic system, such as the atmosphere, still can't predict far ahead. Help me connect it to tiny starting errors that grow. Follow-up: what can still be predicted beyond the horizon?
3. The Navier-Stokes case. Describe it briefly from the key points, then ask what I'd want to know before calling it solved. Draw out the difference between a computer checking each step and experts agreeing the right statement was proved. Follow-up: does it matter that an AI system produced the proof?
4. Closing. Ask me to name two things that would convince me a claimed proof is correct. This gives me something to take into hands-on practice with a model, where I'll also judge what makes a result trustworthy.

KEY POINTS TO KEEP ACCURATE
- Weather models step the equations of fluid flow and heat forward on a grid; ensembles of runs estimate uncertainty. Detailed forecasts have a horizon of about one to two weeks.
- Chaos is sensitive dependence on initial conditions in a deterministic system, not randomness.
- In the SIR epidemic model, infections rise only while the susceptible share is above 1 divided by the reproduction number, so outbreaks fade before everyone is infected.
- The Navier-Stokes problem, a Clay Millennium Prize Problem since 2000, asks whether smooth 3D fluid flows always stay smooth or can blow up in finite time. The official statement allows a breakdown example with a smooth external force.
- Sept 7, 2026: Terence Tao reported heavily AI-assisted work by Levent Alpoge and Tristan Buckmaster proving forced blowup for three simpler fluid equations, not yet Navier-Stokes.
- Sept 8, 2026: OpenAI announced that an internal AI system proved blowup for Navier-Stokes with a smooth external force, with a formal version checked in the Lean proof software. It recognized Alpoge and Buckmaster's priority on related work and said it won't claim the prize.
- Sept 11, 2026: Clay said the problem had apparently been settled and its review would be deliberately unhurried. As of October 13, 2026, the claim is under review and Clay's page lists the problem as active. In October, Diego Cordoba and Luis Martinez-Zoroa wrote that assessing the proofs requires detailed mathematical analysis. A public dispute over credit and over how the work was done continues.
- Oct 6, 2026: a preprint by Alexander Bastounis, Fabian Circelli, and Anders Hansen argued that parts of OpenAI's Lean proof don't match its written proof. It makes no claim about whether the written proof is correct.
- No announced result claims blowup with no external force at all.

MISCONCEPTIONS TO CORRECT GENTLY
When one appears, name the accurate version briefly, then return to my examples.
- "Chaos means the system is random": it's deterministic; the same start gives the same result, but tiny differences grow.
- "Better computers will remove the forecast horizon": they add days of skill, but chaos sets a limit computing can't remove.
- "A machine-checked proof settles everything": experts still need to check that the formal statement matches the problem and to understand the argument.

LIMITS
- Don't state that Navier-Stokes is solved or unsolved beyond the reported status. Don't take sides in the credit dispute.
- No health advice; if I mention a real outbreak, keep to how models work. No equations.

TO FINISH
After I name my two tests, close in one short turn:
- Affirm one specific thing I worked out, in my own words where possible.
- Suggest one or two next steps that fit how the conversation went. Possible steps: check the Clay Mathematics Institute's page for the problem's current status; compare a three-day and a ten-day forecast for the same day; read what a proof assistant checks.
- Restate my two tests on their own line, labeled "My tests for trusting a proof", so I can copy them.

Hands-on Activity 15 minOptional Done

Run an Epidemic Model in a Spreadsheet

Overview

This optional activity has you build the SIR epidemic model in a spreadsheet and watch an outbreak rise and fall over 100 days. The SIR model splits a population into susceptible (S), infected (I), and recovered (R) groups and moves people between them each day. You'll change the infection rate and see what happens to the size and timing of the peak. No formulas beyond multiplication, addition, and subtraction are needed.

What You'll Need

  • A spreadsheet program. Free online ones work.

Your Task

Build an SIR model for a town of 1,000 people, step it forward 100 days, and record the peak number infected and the day it happens. Then rerun it with two lower infection rates and write two sentences on what lowering the rate does to the peak and its timing.

Steps

  1. Set up the columns. In row 1, type the headings Day, S, I, and R in columns A to D. In row 2, enter 0, 990, 10, and 0.
  2. Write the update rules in row 3. Each day, new infections are 0.3 × S × I ÷ 1000 and recoveries are 0.1 × I, both computed from the row above. Enter these formulas:
    • A3: =A2+1
    • B3: =B2-0.3*B2*C2/1000
    • C3: =C2+0.3*B2*C2/1000-0.1*C2
    • D3: =D2+0.1*C2
  3. Fill down 100 rows. Select A3 to D3 and fill down to row 102, which is day 100. Find the largest value in column I and the day it occurs. Many spreadsheets have a MAX function, such as =MAX(C2:C102), that finds it for you.
  4. Change the infection rate. Replace 0.3 with 0.2 in both B3 and C3, fill down again, and record the new peak and its day. Then do the same with 0.15.
  5. Write two sentences. Describe what lowering the infection rate did to the height of the peak and to when it came.

What to Expect

With an infection rate of 0.3, the number infected rises, peaks around day 28 at about 316 people, and then falls. With 0.2, the peak is lower and later, around day 45 at about 162. With 0.15, it's lower and later still, around day 66 at about 71. Your numbers may differ slightly in the last decimal places depending on rounding, but the pattern should match: a lower infection rate gives a lower, later peak.

You may also notice that with the lower rates, more people are still susceptible at day 100, and with 0.15 the outbreak is still going. In every row, S, I, and R should add up to 1,000.

Self-Check

When you're done, check that:

  • S + I + R equals 1,000 in every row.
  • The I column rises, reaches a peak, and then falls in your first run.
  • A lower infection rate gave a lower and later peak.
  • You recorded the peak and its day for at least three runs.
  • Your two sentences describe both the height and the timing of the peak.

Nothing is uploaded. Write in your own notebook or document and keep it.

Knowledge Check 10 min Done

Weather, epidemics, chaos, and the Navier–Stokes problem

This ungraded knowledge check assesses your understanding of how differential equations model weather and epidemics, why chaos limits prediction, and what the Navier–Stokes problem asks. You'll be asked about the SIR model, numerical weather prediction, chaos and sensitive dependence, and the Navier–Stokes problem and the status of the 2026 claim.

Note: Use this to test yourself, review the feedback on any questions you miss, and retry until you feel confident before moving forward.

5 questions · ungraded · retry as often as you like

Graded Quiz 30 min Done

Differential Equations and Dynamical Systems

This graded quiz assesses your understanding of how differential equations state laws of change and what they can and can't predict. You'll be asked about the history from Newton to Poincaré, what a differential equation is, numerical stepping and equilibrium, exponential growth and decay, the SIR model, numerical weather prediction, chaos, and the Navier–Stokes problem.

Note: Aim for a score of 80 percent or higher. If you score lower, use the feedback to review the topics you missed, then retake the quiz.

10 questions · target score 80% · 3 forms, rotated on each attempt