  Viruses present an amazing genetic variability. An ensemble of infecting
viruses, also called a viral quasispecies, is a cloud of mutants centered
around a specific genotype. The simplest model of evolution, whose equilibrium
state is described by the quasispecies equation, is the Moran--Kingman model.
For the sharp peak landscape, we perform several exact computations and we
derive several exact formulas. We obtain also an exact formula for the
quasispecies distribution, involving a series and the mean fitness. A very
simple formula for the mean Hamming distance is derived, which is exact and
which do not require a specific asymptotic expansion (like sending the length
of the macromolecules to $\infty$ or the mutation probability to $0$). We try
also to extend these formulas to a general fitness landscape. We obtain an
equation involving the covariance of the fitness and the Hamming class number
in the quasispecies distribution. With the help of these formulas, we discuss
the phenomenon of the error threshold and the notion of quasispecies. We
recover the limiting quasipecies distribution in the long chain regime. We go
beyond the sharp peak landscape and we consider fitness landscapes having
finitely many peaks and a plateau--type landscape. We finally prove rigorously
within this framework the possible occurrence of the survival of the flattest,
a phenomenon which has been previously discovered by Wilke, Wang, Ofria, Lenski
and Adami and which has been investigated in several works.
