Threshold quantities and Lyapunov functions for ordinary differential equations epidemic models with mass action and standard incidence functions

This paper presents a novel algebraic method for the construction of Lyapunov functions to study global stability of the disease-free equilibrium points of deterministic epidemic ordinary differential equation models with mass action and standard incidence functions. The method is named as Jacobian-Determinant method. In our method, a direct algebraic procedure that also relies only on determinant of the Jacobian matrix of the infected subsystem is developed to determine a threshold quantity, R & PRIME;0 akin to the basic reproduction number, R0 of such class of models. The developed technique is applied on a wide variety of models to construct Lyapunov functions to study the global stability of the infection-free critical points. Further, implementation of our method reveals that the threshold quantity is the same as (or the square) of the basic reproduction numbers as obtained using the next-generation matrix method. It is further observed that even for models that do not use the standard or mass action incidence, the threshold quantity is still related to the basic reproduction numbers as obtained with the next-generation matrix method.

Automated summary

This paper presents a novel algebraic method, named as Jacobian-Determinant method, for constructing Lyapunov functions to study the global stability of disease-free equilibrium points in deterministic epidemic ordinary differential equation models. The method relies on a direct algebraic procedure and determines a threshold quantity, R'0, which is analogous to the basic reproduction number, R0. The method is applied to various models and reveals that the threshold quantity is related to the basic reproduction numbers obtained using the next-generation matrix method, even for models that do not use standard or mass action incidence.