Epidemic Curves

Each sick person infects others at one rate and recovers at another. The ratio of those two rates decides whether an outbreak fizzles out or explodes.

susceptible 99%infected 1.0%recovered 0%

t = 0.0 days

How it works

Nonlinear systems of ODEs

In one sentence

In the SIR model, susceptible people are infected at rate $\beta SI$ and infected people recover at rate $\gamma I$ , so cases grow while $\beta S/\gamma > 1$ and peak when $S$ falls to $\gamma/\beta$ .

General formula

\begin{gathered} {\htmlData{sym=0}{S}}' = -{\htmlData{sym=3}{\beta}} {\htmlData{sym=0}{S}} {\htmlData{sym=1}{I}}, \qquad {\htmlData{sym=1}{I}}' = {\htmlData{sym=3}{\beta}} {\htmlData{sym=0}{S}} {\htmlData{sym=1}{I}} - {\htmlData{sym=4}{\gamma}} {\htmlData{sym=1}{I}} \\ {\htmlData{sym=2}{R}}' = {\htmlData{sym=4}{\gamma}} {\htmlData{sym=1}{I}} \end{gathered}

$\beta SI$ counts new infections from meetings between sick and healthy people. $\gamma I$ is recovery, a fixed share of the sick each day. $R$ just collects the recovered.

What the symbols mean

$S$
Susceptible
People who can still catch the disease.
$I$
Infected
People who are sick right now and can pass it on.
$R$
Recovered
People who are past the illness and can't catch it again.
$\beta$
Transmission rate
How likely a contact between a sick and a healthy person spreads the disease. New cases scale with S·I.
$\gamma$
Recovery rate
The fraction of sick people who recover each day. 1/γ is how many days an infection lasts.

$\beta$ is the transmission rate and $\gamma$ the recovery rate, with $S$ , $I$ , $R$ as fractions of the population. $R_0 = \beta/\gamma$ .
Infections grow only while $\beta S > \gamma$ , so the peak arrives once enough people are immune.
Lowering $\beta$ through distancing or vaccines flattens and delays the peak.

Learn the math

Systems of Differential EquationsDiff Eq · Lesson 30, Systems of Differential Equations Modeling with SystemsDiff Eq · Lesson 36, Systems of Differential Equations