Successful invasion and persistence were more often observed for the constant function than for the other functions (figures 3 and 4). Even at low population sizes, successful invasion was almost certain for infectious periods of 30 days and longer, and was even observed for an infectious period of 7 days in sufficiently large populations. By contrast, for the other functions, successful invasion was never observed below infectious periods of 7 days, and even with an infectious period of 30 days invasion was rare for the sigmoid function. For the linear function, invasion probabilities were lower than for the power function, whereas persistence probabilities for these two functions were similar. The sigmoid function resulted in the lowest invasion or persistence success, where persistence was rarely observed for an infectious period of 30 days. Even when the infectious period was 60 days, persistence was observed only when population size was sufficiently large (e.g. 50% for a population size of 5000). Figure 3.Invasion probabilities for the different contact–density functions, for a range of infectious periods 1/γ and initial population sizes N0 (transmission rate p = 50). Simulations were conducted for all values indicated by tick marks on the axes, and results are interpolated between these values for illustration. Figure 4.Persistence probabilities for the different contact–density functions, for a range of infectious periods 1/γ and initial population sizes N0 (transmission rate p = 50). Persistence probabilities were calculated using only simulation runs in which there was successful invasion. Simulations were conducted for all values indicated by tick marks on the axes, and results are interpolated between these values for illustration.