The $eta$ mesic nucleus is considered to be one of the interesting exotic many body systems and has been studied since 1980s theoretically and experimentally. Recently, the formation of the $eta$ mesic nucleus in the fusion reactions of the light nuclei such as $d + d rightarrow (eta + alpha) rightarrow X$ has been proposed and the experiments have been performed by WASA-at-COSY. We develop a theoretical model to evaluate the formation rate of the $eta$ mesic nucleus in the fusion reactions and show the calculated results. We find that the $eta$ bound states could be observed in the reactions in cases with the strong attractive and small absorptive $eta$-nucleus interactions. We compare our results with existing data of the $d + d rightarrow eta + alpha$ and the $d + d rightarrow {^3 rm He} + N + pi$ reactions. We find that the analyses by our theoretical model with the existing data can provide new information on the $eta$-nucleus interaction.
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