COMPLEX STRUCTURE IN THE ENDEMIC EQUILIBRIUM SET OF AN SIS EPIDEMIC PATCH MODEL WITH THE MASS-ACTION INFECTION MECHANISM

We investigate the structure of the endemic equilibrium (EE) set in a multipatch susceptible-infectious-susceptible (SIS) epidemic model with mass-action transmission mechanism. For the corresponding single-patch model, the basic reproduction number R-0 completely determines the disease dynamics: a unique stable EE exists if and only if R-0 > 1. In contrast, the multipatch setting exhibits far more complex behavior. First, we show that as the dispersal rate of susceptible individuals d(S) varies, the EE set consists of a finite union of disjoint curves. Under mild conditions, when R-0 < 1, the closure of each curve forms a loop in the d(S) x || I || (1)-plane, where || I ||(1) denotes the total infected population at equilibrium. When R-0 > 1, the EE set contains two distinct types of curves: bounded and unbounded. Moreover, the structure of the EE set as d(S) tends to zero provides explicit spatial patterns of the EEs, offering insights into how restricting susceptible movement influences disease dynamics. Second, using R-0 as the bifurcation parameter, we establish that the EE set forms a simple unbounded curve in the R-0 x || I || (1)-plane, which may exhibit an Sshape and undergo either a backward or forward transcritical bifurcation at R-0 = 1. These results refine and extend previous findings by revealing novel and intricate structures within the EE set. They underscore the interplay between population movement, total population size, and spatial heterogeneity, providing new insights into the long-term dynamics of infectious diseases.