The IceCube report of a $sim 3.5sigma$ excess of $13pm5$ neutrino events in the direction of the blazar TXS 05056+056 in 2014-2015 and the 2017 detection of a high-energy neutrino, IceCube-170922A, during a gamma-ray flare from the same blazar, have revived the interest in scenarios for neutrino production in blazars. We perform comprehensive analyses on the long-term electromagnetic emission of TXS 05056+056 using optical, X-ray, and gamma-ray data from the All-Sky Automated Survey for Supernovae (ASAS-SN), the Neil Gehrels Swift Observatory (Swift), the Monitor of All-sky X-ray Image (MAXI), and the Fermi Large Area Telescope (Fermi-LAT). We also perform numerical modeling of the spectral energy distributions (SEDs) in four epochs prior to 2017 with contemporaneous gamma-ray and lower energy (optical and/or X-ray) data. We find that the multi-epoch SEDs are consistent with a hybrid leptonic scenario, where the gamma-rays are produced in the blazar zone via external inverse Compton scattering of accelerated electrons, and high-energy neutrinos are produced via the photomeson production process of co-accelerated protons. The multi-epoch SEDs can be satisfactorily explained with the same jet parameters and variable external photon density and electron luminosity. Using the maximal neutrino flux derived for each epoch, we put an upper limit of $sim0.4-2$ on the muon neutrino number in ten years of IceCube observations. Our results are consistent with the IceCube-170922A detection, which can be explained as an upper fluctuation from the average neutrino rate expected from the source, but in strong tension with the 2014-2015 neutrino flare.
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