Fig 1.
The schematic of the proposed model.
It must be emphasized that the natural birth and death are not illustrated here and all the quantities and parameters of the model are listed in Tables 1 and 2.
Table 1.
List of the quantities of the model.
Table 2.
List of the parameters of the model.
Fig 2.
The simulation of the theorem 0.6 (forward bifurcation).
The plots of S* and I*, which are the coordinates of the fixed points, as functions of R0 where the black and red curves respectively indicate the stable and unstable fixed points, and when the parameters are b = 1, A = 1, d = 1, α = 1, n = 1, and n′ = 1 and when β changes. The quantities and the parameters are defined in the Tables 1 and 2; R0 is defined in the Eq (9).
Fig 3.
The closed curve C.
Fig 4.
The simulation of the theorem 0.7 (backward bifurcation).
The plots of S* and I*, which are the coordinates of the fixed points, as functions of R0 where the black and red curves respectively indicate the stable and unstable fixed points, and when the parameters are b = .01, A = 1, d = 1, α = 1, n = 1, and n′ = 1, and when β changes. The quantities and the parameters are defined in the Tables 1 and 2; R0 is defined in the Eq (9).
Fig 5.
The simulation of the theorem 0.8 (Hopf bifurcation).
The curves of the eigenvalues in the complex plane where the blue and yellow curves indicate different eigenvalues and when the parameters are A = 10, d = .1, α = 1.9, n = 1, and b = 1, and when β changes in the interval (.04, .09) or equally R0 ∈ (1, 2.25). The quantities and the parameters are defined in the Tables 1 and 2; R0 is defined in the Eq (9).
Fig 6.
The simulation of the system (5).
The stream plot S − I (A, A′) and the plots of S, I, H, R (B, B′) and the fluxes (C, C′) as functions of t when the parameters are b = 1, A = 10, d = .1, α = 1.9, n = 1, n′ = 1 and when β = .0408; the initial conditions are X0 = (60, 1, 0, 0) For panels (A, B, C) and X0 = (100, .1, 0, 0) (A′, B′, C′). The quantities and the parameters are defined in the Tables 1 and 2; the fluxes are defined in the Eq (15).
Fig 7.
The simulation of the system (5).
The stream plot S − I (A, A′) and the plots of S, I, H, R (B, B′) and the fluxes (C, C′) as functions of t when the parameters are b = 1, A = 10, d = .1, α = 1.9, n = 1, n′ = 1 and when β = .055; the initial conditions are X0 = (60, 1, 0, 0) for panels (A, B, C) and X0 = (100, .1, 0, 0) for panels (A′, B′, C′). The quantities and the parameters are defined in the Tables 1 and 2; the fluxes are defined in the Eq (15).