A mathematical model for coupling within-host and between-host dynamics in an environmentally-driven infectious disease

This work presents a new model for the linking of within- and between-host dynamics. We use this as a conceptual model for the dynamics of Toxoplasma gondii, in which the parasite's life cycle includes interactions with the environment. We postulate the infection process to depend on the size of the infective inoculum that susceptible hosts may acquire by interacting with a contaminated environment. Because the dynamical processes associated with the within- and between-host occur on different time scales, the model behaviors can be analyzed by using a singular perturbation argument, which allows us to decouple the full model by separating the fast- and slow-systems. We define new reproductive numbers for the within-host and between host dynamics for both the isolated systems and the coupled system. Particularly, the reproduction number for the between-host (slow) system dependent on the parameters associated with the within-host (fast) system in a very natural way. We show that these reproduction numbers determine the stability of the infection-free and the endemic equilibrium points. Our model may present a so-called backward bifurcation.