  A compact complex manifold $V$ is called Vaisman if it admits an Hermitian
metric which is conformal to a K\"ahler one, and a non-isometric conformal
action by $\mathbb C$. It is called quasi-regular if the $\mathbb C$-action has
closed orbits. In this case the corresponding leaf space is a projective
orbifold, called the quasi-regular quotient of $V$. It is known that the set of
all quasi-regular Vaisman complex structures is dense in the appropriate
deformation space. We count the number of closed elliptic curves on a Vaisman
manifold, proving that their number is either infinite or equal to the sum of
all Betti numbers of a K\"ahler orbifold obtained as a quasi-regular quotient
of $V$. We also give a new proof of a result by Rukimbira showing that the
number of Reeb orbits on a Sasakian manifold $M$ is either infinite or equal to
the sum of all Betti numbers of a K\"ahler orbifold obtained as an
$S^1$-quotient of $M$.
