Complex dynamics in an SIS epidemic model induced by nonlinear incidence

Ruixia Yuan; Zhidong Teng; Jinhui Li

Firstly, fix y k = 1/2 and solve for a = −2 + 6 (2−2Λ−σ) . Also, fix x k = 1, based on a = −2 + 6 (2−2Λ−σ) , we can get d = . Secondly, substitute these expressions into η and through complicated computation, we can obtain η L 1 = 81(1+2Λ+σ) 2 64(1−2Λ+σ) 6 (32Λ 5 − 80Λ 4 (−1 +σ) + 8Λ 3 (−39 + 2σ(−13 + 5σ)) +4Λ 2 (−103 +σ(99 + 2(24 − 5σ)σ)) +σ(140 −σ(139 +σ(−21 +(−11 +σ)σ))) + 2Λ(−80 +σ(242 +σ(−81 +σ(−38 + 5σ))))), withσ = 1 − σ > 0, which is the first Liapunov number of the equilibrium (0, 0) of (8). Then, we fixσ = 4/5 and solve equation η = 0, then we get only one suitable value 0.5859 for Λ. That is to say, if (σ, Λ, d, a, c) = (1/5, 5859, 0.2, 8, 4.741), then L 1 = 0. Furthermore, it can be seen that E k = E 3 under this group of parameters. In the following, we further compute the second Liapunov number of the equilibrium (0, 0) of system (8) by the successor function method. It is convenient to introduce polar coordinates (r, θ) and rewrite system (8) in polar coordinates by x = r cos θ, y = r sin θ. It is clear that in a small neighborhood of the origin, the successor function D(c 0 ) of system (8) can be expressed by