In this article, we extend MPRK methods, named MPRKO methods, using Oliver's approach (Oliver, 1975) to improve the accuracy of these schemes in the field of nonautonomous systems. The approach does not demand Ae = c in the Butcher tableau (A, b, c), where e = (1, ... , 1)(T). Following the general analysis of MPRK schemes described in Kopecz and Meister (2018), positivity and mass conservation fundamental properties are proven and even conditions concerning the Patankar weights are given to get second order accuracy of the MPRKO methods. Finally, we consider different linear models and a non-linear epidemiological SEIR problem to confirm the theoretical results and to give reliable statements about the accuracy of the novel class of MPRKO methods. (C) 2020 Elsevier B.V. All rights reserved.