Numerical representations of global epidemic threshold for nonlinear infection-age SIR models

In this paper, the global dynamical behavior of non-linear infection-age SIR models is investigated visually by numerical processes. Firstly, for an age-independent SIR model with diseased mortality and isolation rates, it has been shown that the linearly implicit Euler method reproduces the local dynamical behavior. For the study of the numerical global stability, numerical removed individuals are introduced and then the global behavior is represented by numerical processes for any stepsize. Generally, the numerical global behavior of the linearly implicit Euler method applied to non-linear infection-age SIR model is investigated. Using an infinite-dimensional Leslie operator, the disease-free equilibrium point is globally asymptotically stable for numerical processes when the numerical basic reproduction number of Rh<1. The numerical endemic disease equilibriums are globally asymptotically stable when the numerical threshold Rh>1. It is much more important that the global behavior of the analytical solutions is visually displayed by the numerical processes for the sufficiently small stepsize h. Finally, numerical applications to some the influenza models including HIV models and SIR models with protections are shown to illustrate our analysis.