Consider the IVP
and denote
If there is a continuum of equilibria, then
First, remark that
For
In practice, values of
As a consequence, no solution with
Existence of
We could detail more precisely (positive IC
From the invariance dans the boundedness of the total population
Arino, Brauer, PvdD, Watmough & Wu. A final size relation for epidemic models. Mathematical Biosciences and Engineering 4(2):159-175 (2007)
Suppose system can be written in the form
where
IC are
Suppose
If no demopgraphy (epidemic model), then just
Assume no demography, then system should be writeable as
For
Define the row vector
then
Suppose incidence is mass action, i.e.,
Then for
then substitute into
which is a final size relation for the general system when
If incidence is mass action and
In the case of more general incidence functions, the final size relations are inequalities of the form, for
where
Here,
Incidence is mass action so
For final size, since
Suppose
If
Fraction
with
Here,
So
and the final size relation is