The two-step exponential decay reaction network: analysis of the solutions and relation to epidemiological SIR models with logistic and Gompertz type infection contact patterns

Abstract

We mathematically analyze the solutions to the dynamical system induced by the two-step exponential (growth-)decay (2SED) reaction network involving three species and two rate parameters. We study the influence of the rate parameters on the shape of the solutions. We compare the latter to those of the classic Kermack–McKendrick epidemiological SIR model. We then discuss the similarities and differences between the 2SED and the SIR models from the perspective of chemical reaction network theory (CRNT), as well as from epidemiological modelling view-point. The CRNT approach suggests that the classical SIR model, based on the logistic reaction mechanism, describes well epidemic events related to diseases spreading via a ‘one-to-one’ contact pattern between individuals. On the other side, the 2SED model can be used to simulate epidemic data coming from non-communicable diseases. Our comparative analysis naturally suggests the formulation of a SIR-type model, which is situated between the classic SIR model and the 2SED model, such that the logistic ‘one-to-one’ contact mechanism is replaced by a catalytic (Gompertzian) one. The proposed G-SIR model can be considered as an intermediate step between the SIR and the 2SED models. We compare the shapes of the solutions to the three discussed models and formulate a hypothesis that relates the characteristics of the solution shapes to the model reaction mechanism, resp. to the contact patterns of the particular disease.