Threshold Dynamics, Backward Bifurcation, And Sensitivity Analysis in An Adaptive Behavioral-Immune Feedback Epidemic Model

Abstract

Mathematical modeling of infectious disease spread mechanisms has so far considered behavior adaptation and immunization processes separately from one another, making these models unable to explain some important epidemiological effects, such as waves of epidemics, shift in thresholds and multi-stability. We propose here an innovative approach in form of a new Adaptive Immune-Behavioral (AIB) Feedback Model, which couples the process of behavioral adaptation with the process of acquiring immunity. Our main contribution is the use of adaptive feedback in the sense that the value of transmission rate is adaptive and depends on the actual values of prevalence and immunity-induced protection.  A careful analysis of qualitative properties of the proposed model is conducted, which includes proof of positivity and boundedness of all solutions, calculation of the basic reproductive number, as well as finding disease-free and endemic equilibrium points. Local stability of these equilibria is obtained via analysis of Jacobian matrix, whereas global stability is obtained under certain parametric restrictions. In order to verify the practical relevance of the proposed model, we conduct numerous numerical simulations in a broad range of parameters. It turns out that the inclusion of the behavioral feedback produces highly interesting nonlinear effects such as bistability, hysteresis, and the phenomenon of epidemic resurgence—which do not occur in classical models where behavioral-immunity feedback is ignored. The numerical results show that in the proposed model the feedback adaptation can result in existence of multiple stable equilibria with the same parameters, hence, proving the key importance of the choice of initial conditions in the process of epidemics.