We have the following stochastic SEIR model $dS=\Lambda - \beta SI - \mu S - \sigma SI dB(t)$ $dE=\beta SI - (\lambda +\mu) E+\sigma SI dB(t)$ $dI=\lambda E-(\gamma +\alpha +\mu) I$ $dR=\gamma I-\mu R$
The initial conditions $S_0,E_0,I_0,R_0$ are all positives
By using itô's formula about $lnE(t)$, i found
$dlnE(t)=\left(\frac{\beta SI}{E}-(\lambda +\mu)-\frac{1}{2}\frac{\sigma^2\beta S^2I^2}{E^2}\right)dt+\frac{\sigma SI}{E}dB(t)$
For the persistence of the disease I saw in some paper that As $R_0>1$ , thus when $t$ tends to infinite, it has that $E(t)$ and $I(t)$ in average are $E*$ and $I*$, so that
$\frac{\beta SI}{E}=\frac{\beta SI*}{E*}$
Where $I*$ and $E*$ are the endemic equilibrium point and $R_0$,is the reproduction number of the deterministic SEIR system.
I did not understand exactly this process, and is there another way to study the persistence