The implications of the failure of duality in $\overline{B}B$ scattering are discussed in the framework of a quark model in which all quarks of the target interact simultaneously with all quarks in the projectile through two-body quark-antiquark interactions. It is assumed that the peculiarity of $\overline{B}B$ interactions, which at the particle level manifests itself through annihilation channels, is due to the strong $q\overline{q}$ interaction at the quark level which inhibits the presence of non-active quarks (spectators) in the scattering process. This leads to the replacement of the additivity approach of quark amplitudes by factorization of quark amplitudes. Factorization of quark amplitudes implies, in general, nonfactorization of particle amplitudes and appearance of effects similar to cuts (dips versus peaks in the forward direction) due to the simultaneous exchange of natural and unnatural parity at the quark level. Exchange of exotic quantum numbers appears as a natural consequence of the model, without assuming the existence of exotic particles. It is, however, inhibited by the smallness of charge- and strangeness-exchange cross sections. This model should apply both for large- and small-angle scattering, the difference in these two regions manifesting itself only through different relations between the helicity amplitudes at the quark level. Predictions are made for quasielastic $\overline{B}B$ scattering at small and large momentum transfer. For the reactions where data exist, the agreement between the relations predicted by the model and experiment is satisfactory.

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