  We use a Monte Carlo radiative transfer model (MCRTM) to simulate the UBVRI
light curves, images and linear polarization of a light echo from supernova
SN$~$1987A in the Large Magellanic Cloud (LMC) using various dust cloud shapes,
sizes, and optical properties. We compare the theoretical simulations to the
observations of AT2019xis, a light echo detected at a large angular distance
(4.05$^{'}$) from SN$~$1987A. We estimate the size and optical thickness of the
dust cloud based on the simulation results and the observations of Optical
Gravitational Lensing Experiment (OGLE-IV) Transient Detection System (OTDS)
I-band light curve. The mass of the dust cloud is calculated using the
estimated size, optical thickness and extinction coefficient. If the dust cloud
is assumed to correspond to a gas-to-dust ratio of 300, the total mass of the
dust cloud is approximately 7.8-9.3 $M_{\odot}$. Based on these theoretical
models, we show that the morphological shapes of the light echoes in the
wavelength range in or shorter than the U-band to be very different from those
in the longer wavelength bands, and the difference carries important
information on the early UV radiation of SN$~$1987A.
