  Analytic models for hadron-hadron scattering are characterized by analytical
parametrizations for the forward amplitudes and the use of dispersion relation
techniques to study the total cross section $\sigma_{tot}$ and the $\rho$
parameter. In this paper we investigate four aspects related to the application
of the model to $pp$ and $\bar{p}p$ scattering, from accelerator to cosmic-ray
energies: 1) the effect of different estimations for $\sigma_{tot}$ from
cosmic-ray experiments; 2) the differences between individual and global
(simultaneous) fits to $\sigma_{tot}$ and $\rho$; 3) the role of the
subtraction constant in the dispersion relations; 4) the effect of distinct
asymptotic inputs from different analytic models. This is done by using as a
framework the single Pomeron and the maximal Odderon parametrizations for the
total cross section. Our main conclusions are the following: 1) Despite the
small influence from different cosmic-ray estimations, the results allow us to
extract an upper bound for the soft pomeron intercept: $1 + \epsilon = 1.094$;
2) although global fits present good statistical results, in general, this
procedure constrains the rise of $\sigma_{tot}$; 3) the subtraction constant as
a free parameter affects the fit results at both low and high energies; 4)
independently of the cosmic-ray information used and the subtraction constant,
global fits with the odderon parametrization predict that, above $\sqrt s
\approx 70$ GeV, $\rho_{pp}(s)$ becomes greater than $\rho_{\bar{p}p}(s)$, and
this result is in complete agreement with all the data presently available. In
particular, we infer $\rho_{pp} = 0.134 \pm 0.005$ at $\sqrt s = 200$ GeV and
$0.151 \pm 0.007$ at 500 GeV (BNL RHIC energies).
