Structure of epidemic models: toward further applications in economics

Abstract

In this paper, we review the structure of various epidemic models in mathematical epidemiology for the future applications in economics. The heterogeneity of population and the generalization of nonlinear terms play important roles in making more elaborate and realistic models. The basic, effective, control and type reproduction numbers have been used to estimate the intensity of epidemic, to evaluate the effectiveness of interventions and to design appropriate interventions. The advanced epidemic models includes the age structure, seasonality, spatial diffusion, mutation and reinfection, and the theory of reproduction numbers has been generalized to them. In particular, the existence of sustained periodic solutions has attracted much interest because they can explain the recurrent waves of epidemic. Although the theory of epidemic models has been developed in decades and the development has been accelerated through COVID-19, it is still difficult to completely answer the uncertainty problem of epidemic models. We would have to mind that there is no single model that can solve all questions and build a scientific attitude to comprehensively understand the results obtained by various researchers from different backgrounds.

Keywords: Behavior change; Epidemic model; Intervention; Reproduction number.

Fig. 1

Transfer diagram of SIR, SEIR, SIS and SIRS models

Fig. 2

Conceptual diagram of the two-group SIR model

Fig. 3

Time variation of the infective population I(t) (black) and the effective reproduction number ${\mathcal{R}}_t$ (red) for the SIR model (1) without vital dynamics

Fig. 4

Time variation of the infective population I(t) (black) and the effective reproduction number ${\mathcal{R}}_t$ (red) for the SIR model (3) with vital dynamics

Fig. 5

Transfer diagram of SIQR and SVIR models

Fig. 6

g(I) for considering a the saturation effect and b the psychological effect

Fig. 7

Truncated exponential distribution given by (10)

Fig. 8

Time variation of the infective population in the SIR model (3) with the force of infection (9), the truncated exponential distribution (10) and parameters (11)

Fig. 9

Parameter region where the periodic solution exists or not

Fig. 10

Daily number of newly reported cases of COVID-19 in Japan from 14 January, 2020 to 31 July, 2021 (WHO, 2021)