How can one rule about change predict cooling coffee, populations and epidemics?

One simple rule about change can predict all of these because a rule that says how fast something changes, together with where it starts, fixes everything that follows. A differential equation is exactly such a rule: an equation that relates an unknown function to its derivatives. In applications, the functions represent physical quantities, the derivatives represent their rates of change, and the equation defines the relationship between the two.

Take a cooling cup of coffee. Newton's law of cooling states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment. Scalding coffee in a cold room loses heat quickly, lukewarm coffee loses it slowly, and so the temperature difference decays in an exponential way over time. Isaac Newton published his work on cooling anonymously in 1701.

Population growth follows the same shape of rule. In exponential growth, the quantity grows at a rate directly proportional to its present size: when it is three times as big, it grows three times as fast. In 1838 Pierre François Verhulst, having read Thomas Malthus on unconstrained exponential growth, published the logistic model of self-limiting growth, in which the rate of reproduction is proportional to both the existing population and the amount of available resources.

Epidemics work too. Models split a population into compartments, most commonly Susceptible, Infectious and Recovered, and write a rule for how people flow between them. A new pathogen with no immunity in the population will generally spread exponentially in the early stages, while the supply of susceptible people is plentiful. Differential equations came into existence with the invention of calculus by Newton and Leibniz, and they play a prominent role in engineering, physics, economics and biology.