Assume that $S(0)>=0, I_w(0)>=0, I(0)>=0$ and $F(t)=\mu+\beta_1\frac{I_W(t)}{N}+\beta_2 \frac{I(t)}{N}$(where $\mu, \beta_1,\beta_2$ and N are constants). If we have that:
$\frac{d}{dt}\{S(t)exp(\int_0^t F(s)ds)\}>=0$, can(by integrating) we conclude that $S(t)>=0$ or $S(t)>0$? We also have followings:
$\frac{dS(t)}{dt}=\lambda_1-\mu S(t)-\beta_1 \frac{S(t)I_W(t)}{N}-\beta_2\frac{S(t)I(t)}{N},\\ \frac{dI_W(t)}{dt}=\lambda_2+\beta_1 \frac{S(t)I_W(t)}{N}+\beta_2\frac{S(t)I(t)}{N}-(\mu+\alpha_1+\alpha_2)I_W(t),\\ \frac{dI(t)}{dt}=\lambda_3+\alpha_1I_W(t)-(\mu+\alpha_3)I(t). $
Thanks for any help.
The first equation of the system can be written $\frac{d S(t)}{d t} + F(t)S(t)= \lambda_1$. Multiplying by $\exp(\int_0^t F(s) d s)$ one gets, assuming $\lambda_1 > 0$, \begin{equation} \frac{d}{d t}\left\{S(t)e^{\int_0^t F(s) d s}\right\} = \lambda_1 e^{\int_0^t F(s) d s}> 0 \end{equation} Hence $\varphi(t) = S(t)e^{\int_0^t F(s) d s}$ increases strictly with time and $\varphi(0) \ge 0$. It follows that $\varphi(t)>0$ for $t>0$, hence also $S(t)> 0$.