  Eigen's quasi-species model describes viruses as ensembles of different
mutants of a high fitness "master" genotype. Mutants are assumed to have lower
fitness than the master type, yet they coexist with it forming the
quasi-species. When the mutation rate is sufficiently high, the master type no
longer survives and gets replaced by a wide range of mutant types, thus
destroying the quasi-species. It is the so-called "error catastrophe". But
natural selection acts on phenotypes, not genotypes, and huge amounts of
genotypes yield the same phenotype. An important consequence of this is the
appearance of beneficial mutations which increase the fitness of mutants. A
model has been recently proposed to describe quasi-species in the presence of
beneficial mutations. This model lacks the error catastrophe of Eigen's model
and predicts a steady state in which the viral population grows exponentially.
Extinction can only occur if the infectivity of the quasi-species is so low
that this exponential is negative. In this work I investigate the transient of
this model when infection is started from a small amount of low fitness
virions. I prove that, beyond an initial regime where viral population
decreases (and can go extinct), the growth of the population is
super-exponential. Hence this population quickly becomes so huge that selection
due to lack of host cells to be infected begins to act before the steady state
is reached. This result suggests that viral infection may widespread before the
virus has developed its optimal form.
