  This paper considers a general stochastic SIR epidemic model driven by a
multidimensional Levy jump process with heavy tailed increments and possible
correlation between noise components. In this framework, we derive new
sufficient conditions for disease extinction and persistence in the mean. Our
method differs from previous approaches by the use of Kunita's inequality
instead of the Burkholder-Davis-Gundy inequality for continuous processes, and
allows for the treatment of infinite Levy measures by the definition of new
threshold values. An SIR model driven by a tempered stable process is presented
as an example of application with the ability to model sudden disease outbreak,
illustrated by numerical simulations. The results show that persistence and
extinction are dependent not only on the variance of the processes increments,
but also on the shapes of their distributions.
