We create and analyze a hierarchy of reduced order models for the spread of a Wolbachia bacteria infection in mosquitoes. Mosquitoes that are infected with some strains of the Wolbachia bacteria are much less effective at transmitting zoonotic diseases, including Zika, chikungunya, dengue fever, and other mosquito-borne diseases. The infection will persist in a wild mosquito population only if the fraction of infected mosquitoes exceeds a minimum threshold. Mathematical models can be used to understand the complex maternal transmission of Wolbachia infection and to guide efforts for keeping the infection above the threshold. This threshold can be characterized as a backward bifurcation for a system of nine ordinary differential equations modeling the Wolbachia infection in a heterosexual mosquito population. Although this system captures the detailed transmission dynamics, they are difficult to analyze. We describe the mathematical approach used in each model reduction to create a system of seven, four, or two ordinary differential equations that capture the important properties of the original system. We evaluate the quality of the approximation at each step by comparing the associated important dimensionless numbers, analyzing the critical threshold condition for each reduced model, and using phase plane analysis to demonstrate that all the reduced models accurately reproduce the dynamics of the full system.