ODE-04

SIR slope I'

Epidemic snapshot.

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ODESIR

Governing equation

I'=\\beta S I-\\gamma I

$S \in \left[0, 1\right]$

where

\beta: β (1/day)
\gamma: γ (1/day)
S: S (—)
I: I (—)
I': I' (1/day)
R_t: Rt (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

I'=\\beta S I-\\gamma I

$S \in \left[0, 1\right]$

$S = 0$ $I' = -0.01$

Along the curve

②

Domain of the variable — blue

S_{\min} = 0,\quad S_{\max} = 1

$S \in \left[0, 1\right]$

$S = 0$ $S = 0$

Along the curve

③

Response curve — green

I'=\\beta S I-\\gamma I

$S \in \left[0, 1\right]$

$S = 0$ $I' = -0.01$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}I'}{\mathrm{d}S}

$S \in \left[0, 1\right]$

$S = 0.006$ $I'^{\prime} = 0.02$

Along the curve

Lecture brief

Historical brief

Exponential decay, the harmonic oscillator, SIR epidemics, Lotka–Volterra and Bessel are the first solvable ODEs of nature and populations. The lab is rate, period and phase portrait in closed form. This sheet (ODE-04 — SIR slope I') is the form associated with SIR. Working symbols: $\beta$ , $\gamma$ , $S$ , $I$ $\rightarrow$ $I'$ , $R_t$ . Epidemic snapshot.

Purpose

Purpose: compute

I'

R_t

from

\beta

\gamma

S

I

in Differential equations via

I'=\\beta S I-\\gamma I

Epidemic snapshot. Use it when a real differential equations question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\beta = 0.400\,\mathrm{1/day}

\gamma = 0.200\,\mathrm{1/day}

S = 0.900\,\mathrm{—}

I = 0.050\,\mathrm{—}

, the governing relation

I'=\\beta S I-\\gamma I

yields

I' = 0.0080\,\mathrm{1/day}

R_t = 1.800\,\mathrm{—}

. Epidemic snapshot. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

β \beta0.4 1/dayγ \gamma0.2 1/dayS S0.9 —I I0.05 —

Outputs

I' I'0.0080 1/day
Rt R_t1.800 —

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Narration of this film

Epidemic snapshot.

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