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While cosmological and astrophysical probes suggest that dark matter would make up for 85% of the total matter content of the Universe, the determination of its nature remains one of the greatest challenges of fundamental physics. Assuming the $Λ$CDM cosmological model, Weakly Interacting Massive Particles would annihilate into Standard Model particles, yielding $γ$-rays, which could be detected by ground-based telescopes. Dwarf spheroidal galaxies represent promising targets for such indirect searches as they are assumed to be highly dark matter dominated with the absence of astrophysical sources nearby. Previous studies have led to upper limits on the annihilation cross-section assuming single exclusive annihilation channels. In this work, we consider a more realistic situation and take into account the complete annihilation pattern within a given particle physics model. This allows us to study the impact on the derived upper limits on the dark matter annihilation cross-section from a full annihilation pattern compared to the case of a single annihilation channel. We use mockdata for the Cherenkov Telescope Array simulating the observations of the promising dwarf spheroidal galaxy Sculptor. We show the impact of considering the full annihilation pattern within a simple framework where the Standard Model of particle physics is extended by a singlet scalar. Such a model shows new features in the shape of the predicted upper limit which reaches a value of $\langle σv \rangle = 3.8\times10^{-24}~\rm{cm}^{-3}\rm{s}^{-1}$ for a dark matter mass of 1 TeV at 95% confidence level. We suggest to consider the complete particle physics information in order to derive more realistic limits.