The Failure of 𝑅₀

Figure 5

The effects of backward bifurcations. Solid curves indicate stable equilibria, while dashed curves indicate unstable equilibria. (a) A backward bifurcation at 𝑅₀=1 may result in persistence of the disease when 𝑅₀<1. There is a point 𝑅_𝑎<1 such that the endemic equilibrium exists for 𝑅_𝑎<𝑅₀<1 and a third, unstable, equilibrium also exists. Hence, the disease-free equilibrium is only globally stable if 0<𝑅₀<𝑅_𝑎. (b) Backward bifurcations at other points may also affect the outcome. Although the disease persists for all 𝑅₀>1 and is eradicated when 𝑅₀<1 (due to the transcritical bifurcation at 𝑅₀=1), there is a region 𝑅_𝑚<𝑅₀<𝑅_𝑛, where three equilibria coexist. In this region, the outcome depends on the initial conditions.

(a)

(b)