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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 $\dot{x}=f(x).$ The vector function $f$ is said to be essentially nonnegative if for all $i=1,\ldots,n, f_i(X)\geq 0,$ where $X\in\mathbb{R}^n\geq 0$ such that $X_i=0,$ where $X_i$ denotes the $i$-th element of $X.$

So for my system: \begin{align} \dot{x}&= \lambda − dx − \beta vx \\ \dot{y}&= \beta vx − ay − pyz \\ \dot{v}&= ky − uv − qvw \\ \dot{w}&= gvw − hw \\ \dot{z}&= cyz − bz \end{align} I have an issue with proving that, the r.h.s. of $\dot{y}$ for $y=0$ is $\beta vx,$ is non-negative for all $v.$

Gonçalo
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Anas TABET
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  • For some basic information about writing mathematics at this site see, e.g., here, here, here and here. – Another User Jul 08 '22 at 18:24
  • You need that $\dot X_i\ge 0$ whenever $X_i=0$. This is easily checked as true for this system. $βvx$ has all factors non-negative. – Lutz Lehmann Jul 08 '22 at 18:51
  • @LutzLehmann for beta and x which are both non-negative its true, but how is it the case for v since we don't really whether its positive or not for now. if you wouldn't mind clarifying this, or maybe its something I didn't understand in the theorem. – Anas TABET Jul 08 '22 at 19:06
  • So you have to check that $f_i\geq 0$ whenever $X_i=0$. If $\beta, \lambda, k \geq 0$, then that's completely obvious, unless there is something I do not understand. For $\dot{y}$ you mention for example, well $\beta v y\geq 0$ since the three factors are each greater or equal to zero. For $\dot{x}$, you just need $\lambda\geq 0$. – Gateau au fromage Jul 08 '22 at 20:41
  • You only consider points in the positive orthant and its boundary, that is, $X\ge 0$, and test that the direction of the vector field keeps the system there in forward time. For inner points this is trivial, so it remains to test the boundary planes. – Lutz Lehmann Jul 09 '22 at 03:45

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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.

Sgg8
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Gateau au fromage
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  • The parameters lambda, beta and k are indeed all positive. I have simply not read properly the theorem since it requires to check fi≥0 where X∈Rn≥0 such that Xi=0. So my issue with the v is cleared with the "X∈Rn≥0" – Anas TABET Jul 08 '22 at 21:46
  • Can you "approve" my answer then, please? – Gateau au fromage Jul 09 '22 at 12:33
  • @Gateauaufromage Can you let me know what is "that paper" in your answer? The link is broken and it sounds like an interesting reference. thanks. – Snoop Dogg May 09 '24 at 06:26
  • @Snoop Dogg Sure, it is https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.868.7534&rep=rep1&type=pdf – Gateau au fromage May 10 '24 at 10:14
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    @Snoop Dogg In case it still does not work: Bernstein, Dennis S., and Santosh P. Bhat. "Nonnegativity, reducibility, and semistability of mass action kinetics." In Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No. 99CH36304), vol. 3, pp. 2206-2211. IEEE, 1999. – Gateau au fromage May 10 '24 at 10:16