Proof for positivity of solutions of an ODE system I have an ODE system  that takes a mathematical model describing the dynamics between HCV and the immune system. My question is about the proof that the solution of the ODE system is positive if the initial conditions are all positive. I tried this proof:
Consider the DE
x˙=f(x).
The vector function f is said to be essentially nonnegative if for all i=1,…,n, fi(X)≥0, where X∈Rn≥0 such that Xi=0, where Xi denotes the i-th element of X.
So for my system:
ẋ = λ − dx − βvx
ẏ = βvx − ay − pyz
v̇ = ky − uv − qvw
ẇ = gvw − hw
ż = cyz − bz
I have an issue with proving that, the r.h.s. of y˙ for y=0 is βvx, is non-negative for all v.
 A: I found the theorem written in details in Section 3 of that paper. According to Theorem in that paper, if we have a dynamical system of the form
$$
\dot{\bf{X}}={\bf{f(X)}},\;\;{\bf{X}}\in \mathbb{R}^n.
$$
then we need to check that $f_i\geq 0$ if $X_i=0$ for all $i=1,2...,n$.
In your case, we have
$$
{\bf{X}}=\left(
\begin{array}{c}
x\\y\\v\\w\\z
\end{array}
\right),
$$
and ${\bf{f(X)}}$ given by the RHS of the system given in the question.
Then we compute:
$$
f_1(0,y,v,w)=\lambda,\;\;f_2(x,0,v,w,z)=\beta v x,\;\;f_3(x,y,0,w,z)=k y,\\f_4(x,y,v,0,z)=0,\;\;f_5(x,y,v,w,0)=0.
$$
If the constants $\lambda, \beta,$ and $k$ are assumed to be greater or equal to zero (nonnegative), then it is easy to check that all the expressions above are nonnegative if the variables in the expressions are themselves are nonnegative.
Thus ${\bf{f(X)}}$ is essentially nonnegative and according to Theorem 3.1 of that paper, we have that the solution components of the ODE system are greater or equal to zero if the initial conditions are all greater or equal to zero.
