Nonlinear Incidence of Waterborne Diseases

Many waterborne diseases have two significant transmission pathways. These diseases can be contracted through interpersonal contact as well as through environmental water sources. Additionally, the dose response curve, the relationship between dose of pathogen in the water source and probability of infection, is often nonlinear. In this work we explore how these two factors affect disease dynamics using a dynamical systems approach. We derive a geometric condition for the existence/nonexistence of multiple equilibria. Using this condition, we examine several classes of dose response curves commonly used in epidemiology, such as the Beta-Poisson curve and the Hill Type 2 curve. Of the dose response curves, the Log-Normal and Hill Type 2 curves were found to be capable of multiple equilibria. Additionally, we numerically explored the model dynamics for a variety of parameter ranges and found periodic orbits consistent with seasonal disease patterns. By incorporating nonlinear incidence into a two transmission pathway model for realistic parameter ranges, we observe a wider variety of system behaviors than is possible for a single transmission linear dose response model. This suggests that for diseases with a significant secondary transmission pathway, incorporating nonlinear incidence may yield a more accurate picture of disease dynamics.