A nonlinear relapse model with disaggregated contact rates: Analysis of a forward-backward bifurcation.

Throughout the progress of epidemic scenarios, individuals in different health classes are expected to have different average daily contact behavior. This contact heterogeneity has been studied in recent adaptive models and allows us to capture the inherent differences across health statuses better. Diseases with reinfection bring out more complex scenarios and offer an important application to consider contact disaggregation. Therefore, we developed a nonlinear differential equation model to explore the dynamics of relapse phenomena and contact differences across health statuses. Our incidence rate function is formulated, taking inspiration from recent adaptive algorithms. It incorporates contact behavior for individuals in each health class. We use constant contact rates at each health status for our analytical results and prove conditions for different forward-backward bifurcation scenarios. The relationship between the different contact rates heavily influences these conditions. Numerical examples highlight the effect of temporarily recovered individuals and initial conditions on infected population persistence.