Reaction-diffusion equations (RDEs) are often derived as continuum limits of lattice-based discrete models. Recently, a discrete model which allows the rates of movement, proliferation and death to depend upon whether the agents are isolated has been proposed, and this approach gives various RDEs where the diffusion term is convex and can become negative (Johnston et al., 2017), i.e. forward-backward-forward diffusion. Numerical simulations suggest these RDEs support both smooth and shock-fronted travelling waves. In this talk, I will formalise these preliminary numerical observations by analysing the smooth and shock-fronted travelling waves.