Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model

For an age-structured SIR epidemic model, which is described by a system of partial differential equations, the global asymptotic stability of an endemic equilibrium in the situation where the basic reproduction number R0R0 is greater than unity has been an open problem for decades. In the present paper, we construct a multigroup epidemic model regarded as a generalization of the model, and study the global asymptotic stability of each of its equilibria. By discretizing the multigroup model with respect to the age variable under some parameter assumptions, we first rewrite the PDE system into an ODE system, and then, applying the classical method of Lyapunov functions and a recently developed graph-theoretic approach with an original idea of maximum value functions, we prove that the global asymptotic stability of each equilibrium of the discretized system is completely determined by R0R0, namely, the disease-free equilibrium is globally asymptotically stable if R0≤1R0≤1, while an endemic equilibrium exists uniquely and is globally asymptotically stable if R0>1R0>1. A numerical example illustrates that a numerical solution of R0R0 for the discretized ODE system becomes closer to that for the original PDE system as the step size of the discretization decreases.