function [pInf] = p_inf_fn_of_true_beta(beta_f,k, betas,q,r)
% 
% Written by:
% -- 
% John L. Weatherwax                2007-07-01
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

num1 = ( beta_f^2 - betas.^2 ) * q; 
den1 = 2 * betas .* ( beta_f + k ) .* ( betas + beta_f + k ); 
pInf = ( k^2 * r + q )/( 2 * (beta_f + k) ) + num1 ./ den1;