Population and epidemic models

Level UniversityDifficulty ★★★★★Application⌖ Open in the map

What is it?

Computational biology with ODEs: logistic growth, predator–prey cycles, and SIR epidemics whose basic reproduction number $R_0 = \beta/\gamma$ decides whether an outbreak grows. These models informed policy during COVID-19.

Formulas

$\dot I = (\beta S - \gamma)\,I$
infections grow while $\beta S/\gamma > 1$

The mathematics behind it

Exponential functions★★★★★fundamental
Early epidemic growth and unconstrained populations are exponential, $N(t) = N_0 e^{rt}$ .
Ordinary differential equations★★★★★fundamental
Logistic growth, predator–prey and SIR epidemics are ODE models.
Systems of ODEs★★★★★fundamental
SIR/SEIR epidemics and Lotka–Volterra predator–prey models are nonlinear ODE systems.
Separable equations★★★★★frequent
The logistic equation of population and adoption curves is separable; its solution is a shifted sigmoid.
Bifurcations★★★★★frequent
Discrete population models show period doubling and chaos as the growth rate increases (May, 1976).

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.