  In this work modified Patankar-Runge-Kutta (MPRK) schemes up to order four
are considered and equipped with a dense output formula of appropriate
accuracy. Since these time integrators are conservative and positivity
preserving for any time step size, we impose the same requirements on the
corresponding dense output formula. In particular, we discover that there is an
explicit first order formula. However, to develop a boot-strapping technique we
propose to use implicit formulae which naturally fit into the framework of MPRK
schemes. In particular, if lower order MPRK schemes are used to construct
methods of higher order, the same can be donw with the dense output formulae we
propose in this work. We explicitly construct formulae up to order three and
demonstrate how to generalize this approach as long as the underlying
Runge-Kutta method possesses a dense output formulae of appropriate accuracy.
We also note that even though linear systems have to be solved to compute an
approximation for intermediate points in time using these higher order dense
output formulae, the overall computational effort is reduced compared to using
the scheme with a smaller step size.
