Cross immunity protection and antibody dependent enhancement: A distributed delay dynamic model

Dengue fever is endemic in tropical and sub-tropical countries, and some of the important features of Dengue fever spread continues to pose challenges for mathematical modelling. Here, we propose a system of integro-differential equations (IDE) to study the disease transmission dynamics that involves multiserotypes and cross immunity. Our main objective is to incorporate and analyze the effect of a general time delay term describing acquired cross immunity protection and the effect of antibody dependent enhancement (ADE), both characteristics of Dengue fever. We perform qualitative analysis of the model and obtain results to show the stability of the epidemiologically important steady solutions that is completely determined by the basic reproduction number and the invasion reproduction number. We establish the global dynamics, by constructing suitable Lyapunov functions. We also conduct some numerical experiments to illustrate bifurcation structures, indicating the occurrence of periodic oscillations for specific range of values of a key parameter representing the ADE.