Abstract: We consider a long range infection model on finite dimensional lattices with no recovery and L\'evy interaction with exponent $\alpha$. Using monotonicity, one can prove that the growth rate decreases as $\alpha$ increases. We prove existence of four different growth regimes with thresholds depending on the dimension:
a) for $\alpha$ < d instantaneous growth,
b) for d < $\alpha$ < 2d stretched exponential growth,
c) for 2d< $\alpha$ < 2d+1 super linear growth and
d) for $\alpha$>2d+1 linear growth,
where $d$ is the dimension. In one-dimension we characterize the asymptotic distributional limits, which shows existence of a new ``fluctuation'' transition threshold. Finally, we will mention partial results and conjectures in higher dimension and when recovery is added to the model. Prior knowledge about epidemics models is not required for this talk.