Stationary distribution and bifurcation analysis for a stochastic SIS model with nonlinear incidence and degenerate diffusion

In this paper, a stochastic SIS model with nonlinear incidence and degenerate diffusion is investigated. Firstly, the existence of a uniquely stable stationary distribution of this model is obtained by applying Markov semigroup theory, Fokker–Planck equation and Khasminskiĭ function. In addition, the bifurcation for this two-dimensional stochastic SIS model is discussed. Specifically, phenomenological bifurcation (P-bifurcation) is analyzed by approximately solving the stationary probability density of Fokker–Planck equation for linearizing system of the model. Subsequently, dynamical bifurcation (D-bifurcation) is thoroughly investigated by utilizing the method of Lyapunov exponent. At last, numerical simulations are performed to elaborate the dynamics and the characteristics of distribution for solutions of the model under the variations of different parameters. These findings demonstrate that: (i) appropriate parameters could cause the shape of a stationary probability distribution to shift from monotonic to unimodal; (ii) P-bifurcation caused by the alteration of transmission rate seem to be more obvious than those caused by the change of recovery rate; (iii) P-bifurcation induced by noises also exists even if D-bifurcation would not occur.