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 an scenarios.