Table 1.

Stability analysis of boundary equilibria of system (2)

	Equilibrium and coordinate	Feasibility condition	Jacobian matrix and eigenvalues	Stability status
(i)	$E_0(0,0,0,0)$	always	$\left(\begin{array}{cccc}-\mu & 0& 0& 0\\ 0& -(\mu +a)& 0& 0\\ 0& a& -(\gamma +\mu )& 0\\ 0& 0& \gamma & -\mu \end{array}\right)$	Asymptotically
			${\lambda }_1=-\mu ,{\lambda }_2=-(\mu +a),{\lambda }_3=-(\gamma +\mu ),{\lambda }_4=-\mu$	Stable
(ii)	$E_1(N,0,0,0)$	$R_0<1$	$\left(\begin{array}{cccc}-\mu & 0& -\beta & 0\\ 0& -(\mu +a)& \beta & 0\\ 0& a& -(\gamma +\mu )& 0\\ 0& 0& \gamma & -\mu \end{array}\right)$	Asymptotically
			${\lambda }_1=-\mu ,{\lambda }_2=-(\mu +a),{\lambda }_3=-(\gamma +\mu ),{\lambda }_4=-\mu$	Stable
(iii)	$E_{\ast }(S^{\ast },E^{\ast },I^{\ast },R^{\ast })$		$\left(\begin{array}{cccc}-\mu -\frac{\beta I^{\ast }}{N}& 0& -\frac{\beta S^{\ast }}{N}& 0\\ \\ \frac{\beta I^{\ast }}{N}& -(\mu +a)& \frac{\beta S^{\ast }}{N}& 0\\ \\ 0& a& -(\gamma +\mu )& 0\\ \\ 0& 0& \gamma & -\mu \end{array}\right)$
		$m_1>0$		Asymptotically
		$m_2-\frac{m_3}{m_1}>0$	Characteristic equation: ${\lambda }^4+{\mathit{m}}_{\mathbf{1}}{\lambda }^3+{\mathit{m}}_{\mathbf{2}}{\lambda }^2+{\mathit{m}}_{\mathbf{3}}\lambda +{\mathit{m}}_{\mathbf{4}}=0$	Stable
		$m_3-\frac{m_1{m}_4}{m_2-\frac{m_3}{m_1}}>0$		Routh–Hurwitz
		$m_4>0$		Criterion