SIR-based mathematical modeling of infectious diseases with vaccination and waning immunity

In this paper we will derive an SIR model describing vaccination as well as waning immunity and propose a finite difference scheme for its solution together with some qualitative results. For the modeling of the waning immunity we assume a statistical distribution for the level of antibodies depending on the time lapsed since individual's full recovery or vaccination. We arrive at a system of two ODEs and two PDEs that we reduce to a model of just two ODEs and a few algebraic equations. Next, we propose and implement an efficient numerical scheme to solve this reduced model, based on finite differences. To illustrate our findings we provide graphical results and discuss some qualitative properties of the solutions. Additionally, we derive formulas for the basic reproduction number R-0 and the effective reproduction number R(t) of the reduced model and show the behavior of solutions for examples with R-0 > 1 and R-0 < 1. (C) 2019 Elsevier B.V. All rights reserved.