I have an epidemic model with 9 compartments, where all epidemic parameters are non-negative. \begin{equation} \label{model} \begin{split} &\frac{dS}{dt} = \beta N -\delta S-\phi \frac {SI}{N} \qquad\qquad [I=Is+Ia]\\ &\frac{ dE}{dt} = \phi \frac {SI}{N}-(\delta +\psi+\rho+\mu)E\\ &\frac{dQ}{dt} = \mu E-(\alpha+\sigma+r+\delta ) Q\\ &\frac{ dIs}{dt} = \psi E +\alpha Q-(\kappa+h+\delta)Is\\ &\frac{dIa}{dt} = \rho E+\sigma Q-(m+l+\delta)Ia \\ &\frac{dH}{dt} = \kappa Is+l Ia+\nu Hm-(\epsilon +n+\delta) H\\ &\frac{dHm}{dt} = m Ia+h Is-(q+\delta+\eta+\nu)Hm \\ &\frac{ dRc}{dt} = n H+p Ia+q Hm +rQ-\delta Rc\\ &\frac{dD}{dt} = \epsilon H +\eta Hm \end{split} \end{equation}
with the conditions,
\begin{equation}
\begin{split}
&S(t)+E(t)+Q(t)+I_s(t)\\
&+I_a(t)+H(t)+H_m(t)+R_c(t)+D(t)=N\\
&S(0)+E(0)+Q(0)+I_s(0)+I_a(0)\\
&+H(0)+H_m(0)+R_c(0)+D(0)=N_0
\end{split}
\end{equation}
I have to prove a therem : The solution of each of the compartmental values $S, E, I_s, I_a, Q, Hm, H, R, D$ of system uniquely exists in $\scriptsize R_+^{9}$ and is non negative for given initial conditions at any time $t$.
Problem are Establishing Lipschitz Continuity, Lipschitz constant as some ODE have non-linear terms.