Transmission of Japanese Encephalitis in a 3-population model

The model considered for the spread of Japanese Encephalitis (JE) in a human population of varying size from a reservoir population (pigs, cattle, equines, birds, etc.) through a vector population (particular species of mosquitos) is of SIRS (susceptible-infective-recovered-susceptible) type for the human and reservoir populations and SIS (susceptible-infective-susceptible) type for the vector population. We have considered the logistic differential equation with density-dependent birth rate for the vector population whereas the reservoir population is of constant size. We assume that the human population is regulated by the disease. We also assume that there is a constant recruitment rate of susceptibles into the human population. We perform an equilibrium and stability analysis to find a threshold condition. If the threshold is exceeded, then there is a unique equilibrium with disease present which is locally stable to small perturbations and global stability depends on death rates and the ratio of the equilibrium population sizes of the infected vector and total human populations.