I need to establish an inequality containing $19$ terms in $7$ variables.
My problem arises in the context of proving that the HIV-only Quasi Disease-Free Equilibrium of a HIV-TB co-epidemic model is locally asymptotically stable. I already know that the expression is negative (by biological meaning) - I just need to show it.
I need to show that: $$a^2b-a^2c-ae^2-a^2e-a^2f-abd+abe+acd-ace+ade+adf+bcf-aef+bdf+ad^2g-a^2dg+bd^2g+bcdg-adeg<0 $$
given that:
$1. \ \ 0<a,b,c,d,e,f,g<\infty $
$2. \ \ a>c+d$
$3. \ \ b<c+e$
I am 99% done with the process - verifying this inequality is the last step. I just don't know how to show it, or how to use software to do it?
Thanks for any help.
Your expression can be rewritten $$ a(b-c-e)(a-d-c) + (a+b)(f+dg)(c+d-a) + a(c+e+f+dg)(b-c-e) $$ and all the signs follow from the givens.