For diseases that infect humans or livestock, transmission dynamics are at least partially dependent on human activity and therefore human behaviour. However, the impact of human behaviour on disease transmission is relatively understudied, especially in the context of heterogeneous contact structures such as described by a social network. Here, we use a strategic game, coupled with a simple disease model, to investigate how strategic agent choices impact the spread of disease over a contact network. Using beliefs that are based on disease status and that build up over time, agents choose actions that stochastically determine disease spread on the network. An agent's disease status is therefore a function of both his own and his neighbours actions. The effect of disease on agents is modelled by a heterogeneous payoff structure. We find that the combination of network shape and distribution of payoffs has a non-trivial impact on disease prevalence, even if the mean payoff remains the same. An important scenario occurs when a small percentage (called noncooperators) have little incentive to avoid disease. For diseases that are easily acquired when taking a risk, then even when good behavior can lead to disease eradication, a small increase in the percentage of noncooperators (less than 5%) can yield a large (up to 25%) increase in prevalence.