# A brief history of mathematical modeling

Mostly for me, I am writing a little history of mathematical models. A timeline really.

1202, Fibonacci tried to model rabbits with the eponymous sequence.

1780, Malthus tries to model human populations as a deterministic birth death process, not valid as time goes to infinity of course, but he has

$\dot{N}=\alpha N$

where $\alpha=b-d$ the difference between birth and death rates.

1820, Verhulst makes this more realistic by adding a carrying capacity that depends on resources. Thus he uses

$\dot{N}=\alpha(1-\frac{N}{K})N$

so that $N>K$ makes for $\dot{N}<0$ and a decline in population, whereas below the carrying capacity, the population grows.

1925, Lotka writes his famous differential equations to describe predator-prey dynamical systems (1926 Volterra independently publishes… allegedly). Here I write them in terms of rabbits $R$ and foxes $F$

$\dot{R} = \alpha_R R - \beta RF$

$\dot{F} = \beta\omega RF - \delta_F F$

1936ish, Kolmogorov generalizes these equations

1927, Kermack and McKendrick write the famous SIR model, describing an epidemic process in susceptible, infected and recovered individuals. Typically this model is a fractional model using $S+I+R=N$ and rates are written in terms of fractions

$\dot{S}=-{\frac {\beta IS}{N}}$

$\dot{I}={\frac {\beta IS}{N}}-\gamma I$

$\dot{R}=\gamma I$

von Foerster

Zipf’s law