  The aim of this paper is to study the stochastic SIR equation with general
incidence functional responses and in which both natural death rates and the
incidence rate are perturbed by white noises. We derive a sufficient and almost
necessary condition for the extinction and permanence for an epidemic system
with multi noises \begin{equation*} \begin{cases}
dS(t)=\big[a_1-b_1S(t)-I(t)f(S(t),I(t))\big]dt + \sigma_1 S(t) dB_1(t)
-I(t)g(S(t),I(t))dB_3(t),\\ dI(t)=\big[-b_2I(t) + I(t)f(S(t),I(t))\big]dt +
\sigma_2I(t) dB_2(t) + I(t)g(S(t),I(t))dB_3(t). \end{cases} \end{equation*}
Moreover, the rate of all convergences of the solution are also established. A
number of numerical examples are given to illustrate our results
