Diseases cycle as a response to endogenous and exogenous factors. For infectious diseases caused by vector-transmitted pathogens, the exogenous factors are commonly equated to climatic forces and the endogenous factors to the recruitment of new susceptible individuals. Mathematical models that explicitly (parametrically) consider both types of factor are, however, very rare. An approach is presented to model the effects of endogenous and exogenous factors parametrically, using a time series for American cutaneous leishmaniasis (ACL) from Costa Rica. The seasonality of the disease is modelled using a seasonal autoregressive approach. The latter has the advantage of allowing the use of semi-mechanistic frameworks that consider infection clearance, while explicitly introducing the feedbacks produced by the transition between immune classes, as well as climatic forcing. It also uses a relatively small number of degrees of freedom (compared with the numbers involved in semi-parametric approaches), making it useful for relatively short time series and series with abrupt changes. Compared with non-mechanistic models built for prediction purposes, this way of modelling seems to increase the likelihood of the data being explained by a plausible mechanism. The approach used in this study of ACL could be useful in investigating the changes that occur in other diseases that show non-stationary seasonal dynamics, and can be easily adapted to model the dynamics of other infectious diseases that show trends or breakpoints. The present results support the view that humans affected by ACL are mostly incidental hosts, and indicate that, at the population level, there is a delay of about 5 months between human infection with the causative parasites and the onset of clinical symptoms. They encourage the development of surveillance systems, for monitoring the prevalence of infection in the sandflies that act as vectors, and the use of sentinel hosts, so that control measures can be rapidly applied or strengthened before a serious outbreak occurs. The development of more accurate mathematical models of ACL will depend largely on advances in the ecology of the disease and of all the hosts of the causative parasites.