Modeling Secondary Infections with Temporary Immunity and Disease Enhancement Factor: Mechanisms for Complex Dynamics in Simple Epidemiological Models

Modeling insights for epidemiological scenarios characterized by chaotic dynamics have been largely unex-plored. A rigorous analysis of such systems are essential for a real predictive power and a more accurate disease control decision making. Motivated by dengue fever epidemiology, we study a basic SIR-SIR type model for the host population, capturing differences between primary and secondary infections. This model is the minimalistic version to previously suggested multi-strain models for dengue fever in which deterministic chaos was found in wider parameter regions. Without strain structure of pathogens, we consider temporary immunity after a primary infection and disease enhancement in a subsequent infection to identify to which extent these biological mechanisms can generate complex behavior in simple epidemiological models.Stability analysis of the system is performed using the classical linearization theory, and the qualitative behavior of the model is investigated with a detailed bifurcation analysis. Rich dynamical structures are identified, including the Bogdanov-Takens, cusp and Bautin bifurcations which has never been described in dengue fever epidemiology. Besides the conventional transcritical bifurcation, a backward bifurcation occurs for higher disease enhancement in secondary infections, exhibiting bi-stability when biological temporary immunity period is assumed. The backward bifurcation is formalized using the center manifold theory. While the Hopf and the global homoclinic bifurcation curves were computed numerically, analytical expressions for the transcritical and tangent bifurcations are obtained. The combination of temporary immunity and disease enhancement play a significant role in the complexity of the system dynamics, with chaotic behavior observed after including seasonal forcing.