Fig 1.
Qualitative illustration of forward bifurcation by plotting I* versus R0.
The red dotted line represent unstable equilibria while the black solid line represent stable equilibria.
Fig 2.
Qualitative illustration of backward bifurcation at R0 = 1.
The critical value of R0, namely R0 = Rc = 0.53. The red dotted line represent unstable equilibria while the black solid line represent stable equilibria.
Fig 3.
Schematic diagram of the main processes involved in TB infection according to model Eq (1).
Table 1.
Description of variables and parameters of model (1).
Fig 4.
Illustration of backward bifurcation when there is no recurrent TB (i.e. θ = 0).
Parameters are defined in Table (1) except p = 0.09 > pc = 0.0658, k = 0.0002, q = 0.05, c = 60, β ∈ {0.4, 0.7} and βc = 0.5099 corresponding to Rc = 0.8852. The blue solid line represent the stable equilibria while the red dotted line represent unstable equilibria.
Fig 5.
A plot of the critical value of Rc as a function of the level of exogenous reinfection (p).
NBB and BB respectively denote no backward bifurcation and backward bifurcation regions. In the region denoted by NBB, the level of exogenous reinfection is too low to induce backward bifurcation while in region denoted by BB the level of exogenous reinfection is sufficient to cause multiple equilibria. Parameters used remain as shown in Table (1).
Fig 6.
Infected population, I* at equilibrium plotted as a function of the parameter β.
Illustrates backward bifurcation when recurrent TB parameters are included. Parameters used include; p = 0.09 > pc = 0.0639., σ = 0.05, θ = 0.3, μ = 0.0167, μd = 0.1, r = 2, q = 0.05, Λ = 100 c = 60, k = 0.0002 and β ∈ {0.3, 0.8}. Note that the blue solid line denote stable equilibria while dotted red line denote unstable equilibria. To aid visualisation, bifurcation structure is plotted with semi-log axes.
Fig 7.
Infected population, I* at equilibrium plotted as a function of the parameter β.
Shows forward bifurcation when recurrent TB parameters are included. Parameters used are same as in Fig 6 except p = 0.06 < pc = 0.0639. To aid visualisation, bifurcation structure is plotted with semi-log axes.
Fig 8.
Shows forward bifurcation with hysteresis where the multiple equilibria are strictly to the right of R0 = 1.
Parameters: μ = 0.0167, μd = 0.1, k = 0.0002, θ = 0.5, r = 2, q = 0.05, σ = 0.2, c = 60, Λ = 100, β ∈ (0.2, 0.8) and p = 0.057 < pc = 0.0639. Note that the blue solid line denote stable equilibria while dotted red line denote unstable equilibria. To aid visualisation, the bifurcation structure is plotted with semi-log axes.
Fig 9.
Shows forward bifurcation with hysteresis where there are multiple equilibria to the left and to the right of R0 = 1.
Parameters used are similar to Fig 8 except p = 0.058 < pc = 0.0639. For a clear view, the bifurcation structure is plotted with semi-log axes.
Table 2.
Generalization of the model equilibria of Eq (1).
Fig 10.
Shows forward bifurcation in a scenario where reinfection path A is omitted (i.e no reinfection of exposed individuals, p = 0).
Parameters used include: μ = 0.0167, μd = 0.1, k = 0.0002, r = 2, q = 0.05, σ = 0.2, c = 60, β ∈ (0.5, 0.65) and θ = 4 < θc = 5. Semi-logarithmic scales are used to aid visualisation.
Fig 11.
Shows backward bifurcation structure when reinfection path A is excluded.
Parameters used are the same as in Fig 10 except θ = 6 > θc = 5. The solid blue line represent stable equilibrium while the dotted red line represent unstable equilibrium. Semi-logarithmic scales are used to aid visualisation.
Fig 12.
Shows backward bifurcation structure when reinfection path A is excluded.
Parameters used are the same as in Fig 10 except θ = 7.5 > θc = 5. The solid blue line represent stable equilibrium while the dotted red line represent unstable equilibrium. Semi-logarithmic scales are used to aid visualisation.
Fig 13.
Shows the effect of recurrent TB on backward bifurcation due to incorporation of reinfection pathways B and C.
Parameter values: μ = 0.0167, μd = 0.1, k = 0.0002, r = 2, q = 0.05, σ = 0.05, p = 0.09, c = 60, Λ = 100, β ∈ [0.45, 0.7]. With no recurrent TB (θ = 0), p = 0.09 > pc = 0.0658 while with recurrent TB (θ = 0.3). Parameters used are the same but pc is altered, i.e p = 0.09 > pc = 0.0647. The blue solid line represent stable equilibria while the dotted red line represent unstable equilibria. Semi-log scales are used for a clear view.
Fig 14.
Contour plots of force of infection at equilibrium as a function of θ and p.
Increasing recurrent TB increases TB prevalence.