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We study the phase portraits of a susceptible-infective-removed-susceptible epidemic model with a nonlinear incidence rate, where we consider the power of the population of the susceptible involved in the nonlinear incidence rate is 2. Previous results mainly considered the case when the power is 1. We reduce the model system of three first order equations into a system of two equations involving the fractions of infective and removed populations. We find out the ranges of the four parameters involved in the IR system under which the equilibria are positive. By carrying out the qualitative analysis, we show that the disease free equilibrium can be a saddle-node, saddle point or stable node, the endemic equilibrium with a smaller number of infected individuals may be a stable focus or node, and the endemic equilibrium with large infectives may be a saddle. This is different from other models with the power 1, where the endemic equilibrium with a smaller number of infected individuals may be a saddle while the endemic equilibrium with a larger number of infected individuals may be a stable, unstable node or focus.
How to Cite this Article:
Thi Doan Doan, Changrong Zhu, Kunquan Lan, Phase plane analysis of the susceptible-infected-removed-susceptible (SIRS) epidemic models with nonlinear incidence rates, Journal of Nonlinear Functional Anlaysis 2016 (2016), Article ID 30.