Visceral leishmaniasis (VL), a vector-borne disease caused by protozoan flagellates of the genus Leishmania, is transmitted by sand flies. After malaria, VL is the second-largest parasitic killer, responsible for an estimated 500,000 infections and 51,000 deaths annually worldwide. Mathematical models proposed for VL have included the impact of dogs versus wild canids in disease dissemination and models developed to assist in control approaches. However, quantitative conditions that are required to control or eradicate VL transmission are not provided and there are no mathematical methods proposed to quantitatively calculate optimal control strategies for VL transmission. The research objective of this work was to model VL disease transmission system (specifically Zoonotic VL), perform bifurcation analysis to discuss control conditions, and calculate optimal control strategies. Three time-dependent control strategies involving dog populations, sand fly population, and humans are mainly discussed. Another strategy sometimes used in attempts to control zoonotic VL transmission, dog culling, is also evaluated in this paper.