# Coast to Coast Seminar Series: "Oscillations in a Patchy Environment Disease Model"

*Dr. Lin Wang*

### Abstract

For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R0 is identified. It is shown that the disease free equilibrium is globally asymptotically stable if R0 < 1. For R0 > 1, local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided.

For a two-patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that 1) travel can reduce oscillations in both patches; 2) travel may enhance oscillations in both patches; 3) travel can also switch oscillations from one patch to another.