We study an expansion method of the general multidimensional master equation, based on a system-size expansion of the time evolution equations of the moments. The method turns out to be more accurate than the traditional van Kampen expansion for the first and second moments, with an error that scales with system-size similarly to an alternative expansion, also applied to the equations of the moments, called Gaussian approximation, with the advantage that it has less systematic errors. Besides, we analyze a procedure to find the solution of the expansion method and we show different cases where the solution greatly simplifies. This includes the analytical solution of the average value and fluctuations of the number of infected nodes of the SIS epidemic model in complex networks, under the degree-based approximation.
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