Modeling the Spread of Tuberculosis in Semiclosed Communities

Table 5

Different possible orderings for 𝛽_𝑅0, 𝛽_𝐵, and 𝛽_𝐶. In every case 𝐴>0, Δ₁ is the cubic discriminant of the equation 𝑃(𝑥)=0, Δ is the discriminant of the quadratic equation 𝑃^′(𝑥)=0, where 𝑃(𝑥) is the polynomial (20).

Interval	Coefficients	Equilibria
𝛽𝑅0<𝛽𝐶<𝛽𝐵
𝛽<𝛽𝑅0	𝐵>0, 𝐶>0, 𝐷>0	Disease-free equilibrium
𝛽𝑅0<𝛽<𝛽𝐶	𝐵>0, 𝐶>0, 𝐷<0	Unique endemic equilibrium
𝛽𝐶<𝛽<𝛽𝐵	𝐵>0, 𝐶<0, 𝐷<0	Unique endemic equilibrium
𝛽𝐵<𝛽	𝐵<0, 𝐶<0, 𝐷<0	Unique endemic equilibrium
𝛽𝐶<𝛽𝑅0<𝛽𝐵
𝛽<𝛽𝐶	𝐵>0, 𝐶>0, 𝐷>0	Disease-free equilibrium
𝛽𝐶<𝛽<𝛽𝑅0	𝐵>0, 𝐶<0, 𝐷>0	Two equilibria if 𝑃(𝑥2)≤0 or Δ1<0; none if 𝑃(𝑥2)≥0 or Δ1>0
𝛽𝑅0<𝛽<𝛽𝐵	𝐵>0, 𝐶<0, 𝐷<0	One equilibrium for Δ1<0 or Δ1>0
𝛽𝐵<𝛽	𝐵<0, 𝐶<0, 𝐷<0	Unique endemic equilibrium
𝛽𝐶<𝛽𝐵<𝛽𝑅0
𝛽<𝛽𝐶	𝐵>0, 𝐶>0, 𝐷>0	Disease-free equilibrium
𝛽𝐶<𝛽<𝛽𝐵	𝐵>0, 𝐶<0, 𝐷>0	Two equilibria if 𝑃(𝑥2)≤0 or Δ1<0; none if 𝑃(𝑥2)≥0 or Δ1>0
𝛽𝐵<𝛽<𝛽𝑅0	𝐵<0, 𝐶<0, 𝐷>0	Two equilibria (Δ1<0) or none (Δ1>0)
𝛽𝑅0<𝛽	𝐵<0, 𝐶<0, 𝐷<0	Unique endemic equilibrium
𝛽𝑅0<𝛽𝐵<𝛽𝐶
𝛽<𝛽𝑅0	𝐵>0, 𝐶>0, 𝐷>0	Disease-free equilibrium
𝛽𝑅0<𝛽<𝛽𝐵	𝐵>0, 𝐶>0, 𝐷<0	Unique endemic equilibrium
𝛽𝐵<𝛽<𝛽𝐶	𝐵<0, 𝐶>0, 𝐷<0	One equilibrium (Δ1>0), three equilibria (Δ1<0)
𝛽𝐶<𝛽	𝐵<0, 𝐶<0, 𝐷<0	Unique endemic equilibrium
𝛽𝐵<𝛽𝑅0<𝛽𝐶
𝛽<𝛽𝐵	𝐵>0, 𝐶>0, 𝐷>0	Disease-free equilibrium
𝛽𝐵<𝛽<𝛽𝑅0	𝐵<0, 𝐶>0, 𝐷>0	Two equilibria (Δ1<0) or none (Δ1>0)
𝛽𝑅0<𝛽<𝛽𝐶	𝐵<0, 𝐶>0, 𝐷<0	One equilibrium (Δ1>0), three equilibria (Δ1<0)
𝛽𝐶<𝛽	𝐵<0, 𝐶<0, 𝐷<0	Unique endemic equilibrium
𝛽𝐵<𝛽𝐶<𝛽𝑅0
𝛽<𝛽𝐵	𝐵>0, 𝐶>0, 𝐷>0	Disease-free equilibrium
𝛽𝐵<𝛽<𝛽𝐶	𝐵<0, 𝐶>0, 𝐷>0	Two equilibria (Δ1<0) or none (Δ1>0)
𝛽𝐶<𝛽<𝛽𝑅0	𝐵<0, 𝐶<0, 𝐷>0	Two equilibria (Δ1<0) or none (Δ1>0)
𝛽𝑅0<𝛽	𝐵<0, 𝐶<0, 𝐷<0	Unique endemic equilibrium