Multi-serotype nested immuno-epidemiological model for dengue hemorrhagic fever involving backward bifurcation and serotype invasion

Reinfection with the same dengue serotype is generally benign, as individuals develop protective immunity. On the other hand, in the case of reinfection with a different serotype, pre-existing antibodies can increase the risk of developing Dengue Hemorrhagic Fever (DHF), by inducing Antibody-Dependent Enhancement (ADE). To model this dynamic, we introduce a multi-scale immuno-epidemiological system. The immunological part is described by a system of ODEs representing the interaction between two antibodies (from previous and current infection) and the virus. The epidemiological part is represented by an infection-age structured SIRS system (for both the primary and secondary infections) and a recovery-age structured equation (for the first infection). A detailed mathematical analysis of the equilibrium points of the multi-scale reinfection model, including disease-free, mono-endemic and bi-endemic states, is performed. We establish necessary and sufficient conditions for the existence of backward bifurcations and derive an expression for the invasion reproduction number, which shows that the second serotype can invade the population after a mono-endemic first serotype. We also investigate the dependence of the basic and invasion reproduction numbers on the immunological parameters of the first and second infections. This gives us a better understanding of the relationship between DHF and ADE during secondary infection.