Can SIRS occur without Sepsis from Infection?
According to the classification of sepsis, it is SIRS with a suspected or known source of infection. By definition therefore, sepsis without SIRS is not sepsis but SIRS can exist without sepsis. In all fairness though, SIRS can exist if you run for a while (elevated heart rate and respiratory rate by themselves qualify for a diagnosis of SIRS). 

Sepsis is however not the same as a bacteraemia which may be caused even by brushing your teeth (although the bacteraemia in this case is transient).