5 Surprising Mathematical Models That Shape Our World

In everyday life, many seemingly ordinary phenomena are governed by mathematical models. From traffic scheduling to environmental protection, from biological evolution to economic decisions, these models act as universal keys that reveal the rules of a complex world.

1. Lotka‑Volterra Model: Predator‑Prey Dynamics

The Lotka‑Volterra (predator‑prey) model describes the interaction between predators and their prey in an ecosystem.

Mathematical expression

x – prey population (e.g., rabbits)

y – predator population (e.g., foxes)

α, β, γ, δ – parameters representing birth rate, predation rate, death rate, and conversion efficiency

Life insights

The model is not limited to ecology; it can also describe market competition, where each company can act as both predator and prey.

2. PageRank Algorithm: How Google Ranks Pages

Google’s search engine relies on the PageRank algorithm, a mathematical model that evaluates the importance of web pages by analyzing their link structure.

Mathematical expression

The basic PageRank formula is:

PR(i) = (1‑d) / N + d * Σ_{j∈M(i)} PR(j) / L(j)

PR(i) – importance score of page i

d – damping factor (usually 0.85)

M(i) – set of pages linking to i

L(j) – number of outbound links from page j

Life insights

PageRank reshaped information distribution on the internet, enabling users to find high‑quality content more quickly.

3. SIR Model: Tracing Epidemic Spread

The SIR model, a cornerstone of epidemiology, divides a population into Susceptible (S), Infectious (I), and Recovered (R) groups to predict disease dynamics.

Mathematical expression

\frac{dS}{dt} = -βSI,
\frac{dI}{dt} = βSI - γI,
\frac{dR}{dt} = γI

S, I, R – numbers of susceptible, infectious, and recovered individuals

β – transmission rate

γ – recovery rate

Application example

During the COVID‑19 pandemic, governments used SIR‑type models to assess the impact of lockdown policies on disease transmission.

4. Nash Equilibrium: Wisdom of Game Theory

Nash equilibrium, popularized by the film "A Beautiful Mind," defines a set of strategies where no player can improve their payoff by unilaterally changing their own strategy.

Mathematical expression

A strategy profile σ* is a Nash equilibrium if, for every player i ,

u_i(σ*_i, σ*_{-i}) \ge u_i(σ_i, σ*_{-i}) \quad \forall σ_i

σ_i – strategy of player i

σ_{-i} – strategies of all other players

u_i – payoff function of player i

Life insights

The concept helps explain optimal decision‑making in competitive environments such as business promotions and international negotiations.

5. Random Walk Model: Quantifying Randomness

A random walk describes a path consisting of successive random steps, useful for modeling stock price fluctuations, animal foraging paths, and more.

Mathematical expression

X_{n+1} = X_n + ξ_{n+1}

X_n – position after n steps

ξ_{n+1} – random increment (e.g., +1 or –1)

Application example

Brownian motion in financial markets is based on random walk theory, aiding the analysis of asset price volatility.

These seemingly simple models have profoundly transformed our world, from ecology and economics to the internet and epidemic control. Your next innovative idea might also become a world‑changing model through the power of mathematics.