Bifurcation analysis of a two-infection transmission model with explicit vector dynamics.

Dengue fever is a major public health problem and has been extensively modeled. Understanding the role of explicit vector dynamics in vector-borne diseases such as dengue fever is essential for accurately capturing transmission patterns and improving control strategies. In this study, we extend the minimalistic two-infection host-host SIRSIR model by introducing the SIRSIR-UV model, which explicitly incorporates vector population dynamics. Our aim is to investigate how these explicit vector dynamics influence the behavior of the system. In doing so, we extend previous models that assumed implicit vector effects in addition to immunity and disease enhancement factors. Using tools from nonlinear dynamics and bifurcation theory, we derive analytical conditions for transcritical and tangent bifurcations, formalize backward bifurcation using center manifold theory, and compute Hopf and global homoclinic bifurcation curves. We also show that seasonal influences in the vector populations, mimicking the seasonality of mosquitoes, contribute to the occurrence of chaotic behavior in disease transmission, reflecting the current patterns observed in epidemiological data. We thoroughly characterize the dynamics of the SIRSIR-UV model and explore the implications of including explicit vector dynamics. Finally, we discuss our results with the previous SIRSIR model and conclude that the bifurcation structures observed in the SIRSIR-UV model are consistent with those of the minimalistic SIRSIR model. This unexpected result has important implications for the modeling of vector-borne diseases. It suggests that simplifying assumptions, such as the use of implicit vector dynamics, can effectively capture important aspects of disease transmission while reducing the complexity of the mathematical analysis.