Traditional models of disease evolution are based upon the deterministic competition between strains that confer complete cross-immunity, and predict the selection of strains with higher basic reproductive ratios (R(0)). In contrast, evolution in a stochastic setting is determined by a complex mixture of influences. Here, to isolate the impact of stochasticity, we constrain all competing strains to have an equal basic reproductive ratio - thereby eliminating deterministic selection. The resulting stochastic models predict an evolutionary unstable strategy, which separates a region favouring the evolution of rapid-transmission (acute) strains from one favouring persistent (chronic) strains. We find this to be a generic phenomenon with strain evolution consistently driven towards extremes of epidemiological behaviour. Even in the absence of an equal R(0) constraint, such stochastic selective pressures operate in addition to standard deterministic selection and will therefore influence the evolutionary behaviour of disease in all scenarios.