Stability of compartmental disease transmission models via Lyapunov functions and basic reproduction number

Stability of the disease-free equilibrium is the crucial part of investigation of a compartmental disease transmission model. One of the most effective tools in study of stability are Lyapunov functions which in case of epidemiological models are strictly connected with the basic reproduction number. We make a short review of the theory of Lyapunov functions and an extension of the direct Lyapunov’s method as well as present some new results concerning equilibria on the boundary of the domain. We also describe the construction of the basic reproduction number using the concept of a next generation matrix which was introduced by P. van den Driessche and J. Watmough.