Owing to their rapid reproductive rate and the severe penalties for reduced fitness, diseases are under immense evolutionary pressure. Understanding the evolutionary response of diseases in new situations has clear public-health consequences, given the changes in social and movement patterns over recent decades and the increased use of antibiotics. This paper investigates how a disease may adapt in response to the routes of transmission available between infected and susceptible individuals. The potential transmission routes are defined by a computer-generated contact network, which we describe as either local (highly clustered networks where connected individuals are likely to share common contacts) or global (unclustered networks with a high proportion of long-range connections). Evolution towards stable strategies operates through the gradual random mutation of disease traits (transmission rate and infectious period) whenever new infections occur. In contrast to mean-field models, the use of contact networks greatly constrains the evolutionary dynamics. In the local networks, high transmission rates are selected for, as there is intense competition for susceptible hosts between disease progeny. By contrast, global networks select for moderate transmission rates because direct competition between progeny is minimal and a premium is placed upon persistence. All networks show a very slow but steady rise in the infectious period.