Relative prevalence-based dispersal in an epidemic patch model

In this paper, we propose a two-patch SIRS model with a nonlinear incidence rate: [Formula: see text] and nonconstant dispersal rates, where the dispersal rates of susceptible and recovered individuals depend on the relative disease prevalence in two patches. In an isolated environment, the model admits Bogdanov–Takens bifurcation of codimension 3 (cusp case) and Hopf bifurcation of codimension up to 2 as the parameters vary, and exhibits rich dynamics such as multiple coexistent steady states and periodic orbits, homoclinic orbits and multitype bistability. The long-term dynamics can be classified in terms of the infection rates [Formula: see text] (due to single contact) and [Formula: see text] (due to double exposures). In a connected environment, we establish a threshold [Formula: see text] between disease extinction and uniform persistence under certain conditions. We numerically explore the effect of population dispersal on disease spread when [Formula: see text] and patch 1 has a lower infection rate, our results indicate: (i) [Formula: see text] can be nonmonotonic in dispersal rates and [Formula: see text] ([Formula: see text] is the basic reproduction number of patch i) may fail; (ii) the constant dispersal of susceptible individuals (or infective individuals) between two patches (or from patch 2 to patch 1) will increase (or reduce) the overall disease prevalence; (iii) the relative prevalence-based dispersal may reduce the overall disease prevalence. When [Formula: see text] and the disease outbreaks periodically in each isolated patch, we find that: (a) small unidirectional and constant dispersal can lead to complex periodic patterns like relaxation oscillations or mixed-mode oscillations, whereas large ones can make the disease go extinct in one patch and persist in the form of a positive steady state or a periodic solution in the other patch; (b) relative prevalence-based and unidirectional dispersal can make periodic outbreak earlier.