ODE-04

SIR slope I'

Epidemic snapshot.

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ODESIR

Governing equation

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

where

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

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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