There are several specific ways to calculate the reproduction number $R_0$ for the deterministic model I extracted it properly, but in the stochastic model I found a difference in how to extract it. For example in this article, By taking the infected class as: $$ dI=\beta \tau S(t) I(t) +\beta \tau k I(t) V(t)-(\sigma+\mu+\epsilon) I(t) +\beta_3IdW_3 $$ Using Itô's lemma for twice differentiable function $$f(t, I(t)) =ln(t, I(t))$$ After simple calculations we get: $$ d(t, I(t)) =[\beta\tau S(t) +\beta \tau k V(t)-\frac{1}{2} \beta^2_3-(\sigma+\mu+\epsilon)] dt+\beta_3dW_3(t) $$ Now, he choice that: $$ F=\beta\tau S(t) +\beta \tau k V(t)-\frac{1}{2} \beta^2_3 $$ And $$ V=-(\sigma+\mu+\epsilon) $$ F : the rate of appearance of new infected in the compartment I.
V : the transfer rate of individuals entering and leaving the compartment I.
But in other models I noticed that the term $\frac{1}{2}\beta^2_3$ appears in $V$ not $F$ as: $$ F=\beta\tau S(t) +\beta \tau k V(t) $$ And $$ V=-(\sigma+\mu+\epsilon)-\frac{1}{2}\beta^2_3 $$ Here I do not understand which of the two options is correct. Can someone direct me to which one?