Dynamics of a model of Toxoplasmosis disease in cat and human with varying size populations

A mathematical model with varying human population size and vertical transmission for the transmission of Toxoplasmosis disease in human and cat populations is proposed. By the basic reproductive number, the stabilities of equilibria are analyzed. If the basic reproduction number is less than one, then the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is larger than one, then the endemic equilibrium point is globally asymptotically stable. Our results indicate that the introduction of varying human population size does not modify the conclusions from a model with human constant population size. Additionally, the introduction of the vertical transmission in human lowers the level of infected individuals, but does not affect the extinction of the disease.