  We propose and study a shifted SICA epidemic model, extending the one of
Silva and Torres (2017) to the stochastic setting driven by both Brownian
motion processes and jump L\'evy noise. L\'evy noise perturbations are usually
ignored by existing works of mathematical modelling in epidemiology, but its
incorporation into the SICA epidemic model is worth to consider because of the
presence of strong fluctuations in HIV/AIDS dynamics, often leading to the
emergence of a number of discontinuities in the processes under investigation.
Our work is organised as follows: (i) we begin by presenting our model, by
clearly justifying its used form, namely the component related to the L\'evy
noise; (ii) we prove existence and uniqueness of a global positive solution by
constructing a suitable stopping time; (iii) under some assumptions, we show
extinction of HIV/AIDS; (iv) we obtain sufficient conditions assuring
persistence of HIV/AIDS; (v) we illustrate our mathematical results through
numerical simulations.
