I am working on this epidemic model of this SIR type:
Any suggestion of a Lyapunov function candidate that can be used to prove the global stability of the disease-free equilibrium ($\frac{\Pi}{k_1+\mu}, 0, 0, 0, \frac{k_1 \Pi}{\mu(k_1 + \mu)}$)? The function usually should be a function of the reproduction number $R_c.$ Then, when $R_c < 1$ we should have the global stability. $R_c$ can be derived from the model equations. We find $$R_c = \frac{\alpha \beta_0 \mu }{b(k_1+\mu)(\alpha+ k_2 + \mu)(\gamma + k_3 + \mu)} \left( b + \delta f \gamma \right)$$