Fig 1.
Graphs of incidence rate function.
(a) c ≥ 0; (b) .
Fig 2.
(a) ; (b)
; (c)
; (d)
.
Fig 3.
The bifurcation curves in (β, b) for system (5) when a ≠ 0.
Fig 4.
Graph of h(I) with different signs of Δ1 when b1 < 0.
When I2 = Hm, I2 = HM or I2 = HM = Hm, Hopf bifurcation occurs. BT bifurcation of codimension 2 occurs when I* = Hm or I* = HM and BT bifurcation of codimension 3 occurs when I* = Hm = HM.
Fig 5.
Graphs of Bifurcation curve in parameters plane (β, b) and the phase trajectory for system (5).
(a) Curve q(β, b) = 0. The green curve is supercritical Hopf bifurcation; The red curve is subcritical Hopf bifurcation. σ1 becomes 0 at the DH point. (b) Two limit cycles bifurcation from the weak focus E2.
Fig 6.
The bifurcation diagram of BT of codimension 3.
(a) The parameter space and the trace of the bifurcation diagram on the S(ϵ1 ≤ 0); (b) The sign of the BT is positive if the coefficient of the term XY in the norm form is positive, otherwise it is negative [24].
Fig 7.
The bifurcation diagram in plane (β, b).
There are two types of Bogdanov-Takens bifurcation, BT+ and BT−. The green curve represents supercritical Hopf bifurcation, the red curve represents subcritical Hopf bifurcation. The blue dash (solid) curve represents saddle-node bifurcation (neutral saddle curve).
Fig 8.
The phase diagram of system (5).
The blue curve represents unstable manifold, green curve represents stable manifold. (a) b = 0.1, β = 0.339; (b) b = 0.1, β = 0.340; (c) b = 0.1, β = 0.3415; (d) b = 0.18, β = 0.367; (e) b = 0.18, β = 0.3737; (f) b = 0.21, β = 0.3815; (g) b = 0.21, β = 0.4; (h) b = 0.1587, β = 0.3683. In the (b), there is an unstable limit cycle marked black curve near the epidemic equilibrium E2. In the (e) and (f), there is a stable limit cycle marked red curve. In the (h), we find that there are two limit cycle, the small one is unstable, the another one is stable.
Fig 9.
Bifurcation diagram in (β, I) with different b.
The red dash(solid) represents unstable epidemic equilibrium(limit cycle). The blue curve represents stable epidemic equilibrium or limit cycle. (a) b = 0.145; (b) b = 0.14.
Fig 10.
Bifurcation digram near the Bogdanov-Takens.
Fig 11.
Phase portraits for parameters in different regions of Fig 10.
Fig 12.
The curve q(β, b) = 0 with different values of a.
The blue curve, yellow curve, red curve and green curve are drawn according to a = 0, a = 0.5, a = 1 and a = 2 respectively.
Fig 13.
Bifurcation in the plane (b, I) with different μ1.
β = 0.39, a = 0. (a) ; (b)
; (c)
.
Fig 14.
The bifurcation diagram of system (3) in parameters plane (β, b).
A = 3, d = 0.3, β = 0.5, c = 0.185, a = 0.2, μ1 = 3.1728627, μ0 = 1.5,
.
Fig 15.
The bifurcation diagram of system (3) in parameters plane (β, b).
A = 3, d = 0.3, β = 0.5, c = 0.1, a = 0.2, μ1 = 3.1728627, μ0 = 1.5, BT+(0.337066, 0.072821), BT−(0.382572, 0.172627).
Fig 16.
Phase portraits of system (3) in the plane (S, I) with different d.
A = 3, β = 0.375, α = 0.5, μ0 = 0.5, μ1 = 0.7, a = 0.7, b = 0.01, c = −1.65. (a) d = 1.49; (b) d = 1.483783; (c) d = 1.4. In case (a) (b) and (c), E1 is always a saddle and E0 is a stable node. E2 is a stable node in case (b) and (c), while it is a unstable node in case (a). There is an unstable limit cycle near E2 in case (b).