  In the present work we analyze the problem of adaptation and evolution of RNA
virus populations, by defining the basic stochastic model as a multivariate
branching process in close relation with the branching process advanced by
Demetrius, Schuster and Sigmund ("Polynucleotide evolution and branching
processes", Bull. Math. Biol. 46 (1985) 239-262), in their study of
polynucleotide evolution. We show that in the absence of beneficial forces the
model is exactly solvable. As a result it is possible to prove several key
results directly related to known typical properties of these systems like (i)
proof, in the context of the theory of branching processes, of the lethal
mutagenesis criterion proposed by Bull, Sanju\'an and Wilke ("Theory of lethal
mutagenesis for viruses", J. Virology 18 (2007) 2930-2939); (ii) a new proposal
for the notion of relaxation time with a quantitative prescription for its
evaluation and (iii) the quantitative description of the evolution of the
expected values in four distinct regimes: transient, "stationary" equilibrium,
extinction threshold and lethal mutagenesis. Moreover, new insights on the
dynamics of evolving virus populations can be foreseen.
