Epidemiological models with polynomial equation for endemic equilibria

Question

The polynomial equation is:
$c_1x^4+c_2x^3+c_3x^2+c_4x+c_5=0$
with and using the change sign rule at $c_ 2, c_ 3, c_ 4, c_5$ and $c_1$ is always positive i got the possible positive roots of the polynomial equation as follows:
$1$ when $R_0 >1$
$1,3$ when $R_0 > 1$
$0,2$ when $R_0 < 1$
$0,2,4$ when $R_0 < 1$
($R_0$ is basic reproduction number) so the purpose Epidemiological models with polynomial equation for endemic equilibria is