Abstract: In this talk we will discuss some recent progress on the contact process on random graphs. The contact process is a model for the spread of a virus on a network, where people can recover and be reinfected (sounds familiar?). Every infected node can infect healthy neighbours with a certain rate, and the recover at a certain rate. It has been studied by the probability community for several decades, first mainly on special infinite graphs, where the main question is if there is a positive chance of survival, later on random finite graphs, where one can identify a phase transition: on a finite graph the process will almost surely die out, but depending on the infection rate, this may take a short time (order log(N), where N is the number of nodes) or a very long time (order exp(N)). Recently, in joint work with Henk Don, we have developed a method to prove an explicit lower bound on the infection rate to get this exponential extinction time. The method is quite robust to the kind of network we are considering, but it does not always get sharp results. We are also able to give explicit lower bounds on the expected extinction time.