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

• Published in: Journal of computational science Vol. 37; p. 101027
• Main Authors: Ehrhardt, Matthias, Gašper, Ján, Kilianová, Soňa
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2019
• Subjects: Basic reproduction number; Discrete model; Effective reproduction number; Finite difference scheme; Measles; Ordinary-integral differential equation; SIR model; Vaccination strategy; Waning immunity; Vaccination strategy; Finite difference scheme; Basic reproduction number; Waning immunity; Discrete model; SIR model; Measles; Ordinary-integral differential equation; Effective reproduction number
• ISSN: 1877-7503; 1877-7511
• DOI: 10.1016/j.jocs.2019.101027

•A new SIR model describing vaccination as well as waning immunity.•This model is derived on a purely discrete level and hence corresponds to a novel finite difference scheme.•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 immunity.•Efficient numerical scheme to solve this reduced model, based on finite differences.•The specialty here is that we first model on a discrete level and then pass to the continuous level. 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 R0 and the effective reproduction number R(t) of the reduced model and show the behavior of solutions for examples with R0>1 and R0<1.