Why do some ideas take off – and others fail to spread? Why are some diseases predictable, and others swamped in uncertainty? And what about the outbreaks that never happen at all? We live in a world that’s more connected than ever before. But even as we see our lives being shaped by the spread of ideas, trends – and even diseases – we sometimes struggle to grasp how it actually works. Outbreaks seem to be driven by randomness and hidden laws, and in order to understand them, we need to start thinking like mathematicians. Here, epidemiologist Adam Kucharski reveals how new mathematical approaches are transforming what we know about contagion – from the revolutionary initiatives that helped tackle gun violence in Chicago to the truth behind the spread of fake news.