We propose and analyze a mathematical model of a vector-borne disease that includes vector feeding preference for carrier hosts and intrinsic incubation in hosts. Analysis of the model reveals the following novel results. We show theoretically and numerically that vector feeding preference for carrier hosts plays an important role for the existence of both the endemic equilibria and backward bifurcation when the basic reproduction number $R_0$ is less than one. Moreover, by increasing the vector feeding preference value, backward bifurcation is eliminated and endemic equilibria for hosts and vectors are diminished.
Therefore, the vector protects itself and this benefits the host. As an example of these phenomena, we present a case of Andean Cutaneous Leishmaniasis (ACL) in Peru. We use parameter values from previous studies, primarily from Peru to introduce bifurcation diagrams and compute global sensitivity of $R_0$ in order to quantify and understand the effects of the important parameters of our model. Global sensitivity analysis via partial rank correlation coefficient (PRCC) shows that $R_0$ is highly sensitive to both sandflies feeding preference and mortality rate of sandflies.