A switching model for the impact of toxins on the spread of infectious diseases.

To study the effects of an environmental toxin, such as fine particles in hazy weather, on the spread of infectious diseases, we derive a toxin-dependent dynamic model that incorporates the birth rate with the toxin-dependent switching mode, the mortality rate, and infection rate with the toxin-dependent saturation effect. We analyze the model by showing the positive invariance, existence and stability of equilibria, and bifurcations. Numerical simulation is adopted to verify the mathematical results and exhibit transcritical and Hopf bifurcations. Our theoretical results show that there exists a threshold value of the environmental toxin: if the environmental toxin concentration is lower than the threshold, the system has a disease-free equilibrium and an interior equilibrium; if the environmental toxin concentration is higher than the threshold, the system has the extinction equilibrium. For the case where the disease-induced death is ignored, we show the global stability results. Numerical simulations clearly show that the environmental toxin facilitates the spread of infectious diseases. This study provides a theoretical basis for uncovering the impact of toxins on the spread of infectious diseases and for guiding the decision making by disease control agencies and governments.