Analytical solution of l-i SEIR model–Comparison of l-i SEIR model with conventional SEIR model in simulation of epidemic curves

by Xiaoping Liu

The Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model has been commonly used to analyze the spread of infectious diseases. This 4-compartment (S, E, I and R) model uses an approximation of temporal homogeneity of individuals in these compartments to calculate the transfer rates of the individuals from compartment E to I to R. Although this SEIR model has been generally adopted, the calculation errors caused by temporal homogeneity approximation have not been quantitatively examined. In this study, a 4-compartment l-i SEIR model considering temporal heterogeneity was developed from a previous epidemic model (Liu X., Results Phys. 2021; 20:103712), and a closed-form solution of the l-i SEIR model was derived. Here, l represents the latent period and i represents the infectious period. Comparing l-i SEIR model with the conventional SEIR model, we are able to examine how individuals move through each corresponding compartment in the two SEIR models to find what information may be missed by the conventional SEIR model and what calculation errors may be introduced by using the temporal homogeneity approximation. Simulations showed that l-i SEIR model could generate propagated curves of infectious cases under the condition of l>i. Similar propagated epidemic curves were reported in literature, but the conventional SEIR model could not generate propagated curves under the same conditions. The theoretical analysis showed that the conventional SEIR model overestimates or underestimates the rate at which individuals move from compartment E to I to R in the rising or falling phase of the number of infectious individuals, respectively. Increasing the rate of change in the number of infectious individuals leads to larger calculation errors in the conventional SEIR model. Simulations from the two SEIR models with assumed parameters or with reported daily COVID-19 cases in the United States and in New York further confirmed the conclusions of the theoretical analysis.