Modeling parasite clearance and transmission delays in American Cutaneous Leishmaniasis transmission dynamics

American Cutaneous Leishmaniasis (ACL) is a multi-host vector-borne disease with complex transmission dynamics involving sand fly vectors, reservoirs, and incidental hosts. In this study, we develop and analyze two delay differential equation (DDE) models to explore the role of biological time delays in ACL transmission dynamics. The first model incorporates a delay in the parasite clearance term of the incidental host, reflecting the minimum parasite clearance time between infection and recovery. The second model introduces a delay that represents delays in skin lesion development following exposure to infective sand fly bites. For both models, we derived the basic reproduction number $R_0$ and performed an elasticity analysis which revealed that vector-reservoir transmission and vector mortality are key drivers of outbreak potential. For both models we studied the stability of the disease-free and endemic equilibria. Using linearization, we identified critical delay thresholds that generate Hopf bifurcations. We show that time delays can destabilize the endemic equilibrium and induce periodic oscillations. Despite having the same $R_0$ , the two models exhibit distinct qualitative behaviors due to differences in the delay structure. Simulations support the analytical findings, illustrating how increasing delays can lead to oscillations in transmission. These findings highlight the importance of incorporating biologically inspired delays when modeling ACL transmission.