Weak measurements are a subset of measurement processes in quantum mechanics wherein the system, which is being measured, interacts very weakly with the measuring apparatus. Measurement values of observables undergoing a weak interaction and their amplification are concepts that have sharpened our understanding of interaction processes in quantum mechanics. Recent experiments show that naturally occurring processes such as resonance fluorescence from excited states of an atom can exhibit weak value amplification effect. In this paper we theoretically analyze the process of elastic resonance fluorescence from a V -type three-level atomic system, using the well-known Weiskopff-Wigner (WW) theory of spontaneous emission. Within this theory we show that a weak interaction regime can be identified and for suitable choices of initial and final excited states the mean scattering time between these states show an amplification effect during interaction with the vacuum bath modes of the electromagnetic field. We thus show that a system-bath interaction can show weak value amplification. Using our theory we reproduce the published experimental results carried out in such a system. More importantly, our theory can calculate scattering time scales in elastic resonance scattering between multiple excited states of a single atom or between common excited state configurations of interacting multiatom systems.