Epidemic Curve Simulator

An outbreak does not stop because it runs out of people. That is the most surprising thing the simplest epidemic model has to say, and it has been saying it since 1927. The curve of infections climbs, bends and falls while a share of the population has still never met the infection at all.

This page is about that model and nothing wider. It is a teaching model, a way to see how a few assumptions produce a familiar shape. It is not a forecast of any outbreak and it gives no advice about health. Its value lies in showing which conclusions follow from arithmetic alone, and where the arithmetic stops being a fair picture of the world.

A chemist, a colonel and three boxes

The two authors met in a laboratory in Edinburgh. Anderson McKendrick trained as a doctor in Glasgow, served in the Indian Medical Service, directed the Pasteur Institute at Kasauli and rose to lieutenant colonel before he came home in 1920 to run the research laboratory of the Royal College of Physicians. He had also taken time on leave to study mathematics. William Kermack arrived a year later to head the chemical section.

On the evening of 2 June 1924 an experiment exploded in Kermack’s laboratory and threw caustic alkali into his eyes. He left hospital two months later completely blind, at twenty-six. Bench chemistry was over for him. Colleagues read the journals aloud, and he turned to work that could be done in the head. Part of that work, with McKendrick, became a paper in the Proceedings of the Royal Society in 1927, the first of five on the mathematical theory of epidemics.

The version taught today is a special case of what they wrote. Sort everyone into three boxes. S holds the susceptible, who can catch the infection. I holds the infectious, who can pass it on. R holds the removed, who have recovered and are immune, or have died. People move one way only, from S to I to R. The total never changes.

Two rates drive the traffic. Each infectious person makes contacts that would transmit at some rate, usually written beta, and only the fraction of those contacts that land on a susceptible person produce a new case. Each infectious person also leaves the I box at a rate gamma, so the average time spent infectious is one over gamma. That is the whole model: three boxes, two rates, and the assumption that everyone mixes with everyone else evenly.

One ratio and what it does not tell you

Divide beta by gamma and the result is the basic reproduction number, R0: the number of new cases one case would produce in a population where everybody is susceptible. If it is above one, the first cases more than replace themselves and the infection takes off. If it is below one, they do not, and the count of infectious people can only shrink.

The number is easy to overread. A review in a public health journal lists the common mistakes plainly: R0 is not a biological constant for a pathogen, not a rate over time, and not a measure of how severe a disease is. It is a count of cases per case. Because it contains the contact rate, it belongs to a population and a setting as much as to a germ, and the review finds limited evidence that a value carries over outside the region where it was calculated.

The same review gives a sobering example. The often quoted range of 12 to 18 for measles rests on data gathered in the United States between 1912 and 1928 and in England and Wales between 1944 and 1979, and more recent estimates span a much wider range across settings. A single R0 printed next to the name of a disease hides all of that. The presets on this page are therefore named only by their number.

Nor does R0 say how fast anything happens. Two outbreaks with the same ratio can differ tenfold in speed, because one has both rates ten times larger. Speed comes from the early growth rate, beta minus gamma when nearly everyone is susceptible, and the doubling time is the natural logarithm of two divided by that.

Why the curve turns

As the outbreak runs, the S box empties. Each case now wastes more of its contacts on people who are already infectious or immune. The count of infectious people grows only while each case still replaces itself, which means while the susceptible share stays above one over R0. The moment it falls to exactly that level, the curve is at its peak. Nothing dramatic happens that day. Replacement simply drops below one for one.

The same level, read as a share immune, is the herd immunity threshold of the model: 1 - 1/R0. If that share is immune before the infection arrives, a handful of cases cannot grow into an outbreak. The Centers for Disease Control and Prevention training text puts the idea in words: with a high enough resistant share, the few who are susceptible are protected because the pathogen is unlikely to find them. It adds a warning the model cannot see. In practice outbreaks have occurred in populations with immunization levels of 85% to 90%, partly because the susceptible few are often clustered together.

Here is a worked example that can be reproduced with the tool above. Choose the preset R0 2.5: a contact rate of 0.5 a day, five infectious days, 1,000 people and three infectious at the start. The threshold is 60%. The smooth curve peaks near day 21 with about 235 people infectious at once, and the count of infections doubles every 2.3 days at first.

Now look at the end. The peak came when 60% were no longer susceptible, yet the outbreak ends with 89.3% ever infected. The difference is overshoot. At the peak there are still 235 infectious people, and they go on infecting others while their numbers decline. The final share solves a small equation that Kermack and McKendrick’s model implies: the share ever infected, z, satisfies z = 1 - exp(-R0 z) when almost everyone starts susceptible. For 2.5 that gives 0.893. For 1.5 it gives 0.583.

The same training text describes the pattern from the field side. A propagated outbreak, passed from person to person, usually wanes after a few generations, either because the number of susceptible persons falls below a critical level or because control measures take effect. The model contains only the first of those reasons unless a second is added by hand.

The tool allows one such addition: a window of days with fewer contacts. Run it and the peak drops. End the window early and a second rise can follow, because the susceptible share was left above the threshold. A study of exactly this problem in the same model found that the best time-limited measures are fragile, since small errors in timing produce large increases in the peak. That is a statement about the equations. What any real measure achieves is a question for data, not for this page.

Chance in small outbreaks

The smooth curve treats people as a fluid. Real populations come in whole persons, and the first few cases are a gamble. Draw the same model as a sequence of random events, each infection and each recovery arriving after a random wait, and every run is different. The method is the one chemists use for reactions among small numbers of molecules: compute the total rate of all possible events, draw an exponential waiting time, then pick which event happened in proportion to its rate.

Early on, such a run behaves like a family tree that may die out. With exponential infectious periods, the line founded by one case goes extinct with probability one over R0, so even at R0 1.5 two introductions in three fizzle. Britton’s survey reports 10,000 simulated outbreaks of that kind in a community of 1,000: the fraction that stayed minor was 0.678, against a theoretical two thirds.

With several first cases, all their lines must fail, and the chances multiply. In the worked example three first cases at R0 2.5 give a die-out chance of 6.5%, so in forty random runs about two or three should fizzle and the rest should take off. The tool counts them. In a large population the runs that take off hug the smooth curve; in a small one they scatter around it, and the scatter is information.

Both kinds of curve here are computed, not looked up. The smooth one comes from the standard fourth-order Runge-Kutta rule with a fixed step. The random runs start from a seed fed to a published xorshift generator, so the same seed replays the same outbreak, and a different seed tells a different story from identical rates.

What three boxes leave out

Everything rests on even mixing, and nobody mixes evenly. Households, schools, workplaces and distance shape who meets whom. People change their behavior when an outbreak is in the news, which changes beta while the model holds it constant. Infectious periods are not exponential. Immunity can fade, and births refill the S box, which is why the page offers those options and why the curve can then settle at a steady level and not at zero.

Then there is the question of data. The review of R0 notes that the number is nearly always estimated after the fact, from models with their own assumptions, and that counting cases during an epidemic is extremely difficult. A curve drawn here from chosen rates is exact about the model and silent about the world.

So use the three boxes for what they do well. They show why growth must end, why the end leaves some people untouched, why a threshold exists at all, and why chance decides the fate of a few first cases. A century on, that is still a great deal to get from two rates and a division.