Recurrent events can occur more than once in the same individual; such events may be of different types, known as multi-type recurrent events. They are very common in longitudinal studies. Often there is a terminating event, after which no further events can occur. The risk of any event including terminating events such as death or cure is typically affected by prior events. We propose a flexible new multi-type recurrent events model that explicitly provides the absolute risk for every event type, terminating and non-terminating, and predicts event-free survival probability over a given time period; provides the risk increment, or decrement, conferred by each non-terminating event to every event type; and yields the change in risk for each event due to subject characteristics including number and type of prior events. The model is parametric, so robust standard errors are easily implemented, mitigating possible model misspecification. We illustrate the model with applications to the Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial (1994-2002) and provide discussion of the results and model features.