Fig 1.
Heat-map of the cumulative numbers of infected cases showing the spatial spread of the Omicron variant of COVID-19 during the initial phase of the outbreak in Shanghai city during 1–17 March 2022.
The source of the basemap shapefile was from the open access platform: National Platform for Common Geospatial Information Services (https://www.tianditu.gov.cn/; note that this link works well in P.R. China, but may be blocked in some other countries).
Table 1.
Definitions and values of parameters and variables in model (3) and model (4).
Fig 2.
Schematic diagram illustrating the recruitment process of susceptible individuals.
Within the epicentre, the classic SIR compartment model remains applicable for describing the transmission of the pathogen. The crucial aspect lies in incorporating the recruitment process of the susceptible population into the classic SIR compartment model, recognizing the spatio-temporal heterogeneity of the susceptibles. We term the susceptible individuals in the epicentre as the effective susceptible population, denoted by Se, and those in RoEC are referred to as the reserved susceptible population, denoted by Sr. Considering the continuous spatial spread of infections, the epicentre is expanded to encompass more susceptibles. As a result, the susceptible population in class Sr will transition into the newly added epicentre and become Se. Letting f(t) represent the transition rate from Sr(t) to Se(t), we establish the following generalized modelling framework in terms of the transition process of susceptibles.
Fig 3.
(a) Time series of the numbers of daily reported cases of the Omicron variant in the 16 districts of Shanghai city from 1 March to 2 July 2022. (b) Frequency distribution histogram of the time-interval from infection to report and the fitting results of the probability distribution. (c) Processed data of the numbers of daily new infections.
Fig 4.
Model calibrations and solutions.
(a) Training result of the UDE model (red dashed curve) and the best fitting result of the mechanistic model (blue solid curve), here the blue shaded area is the 95% confidence interval of the fitting result of the mechanistic model. (b) Estimated effective reproduction number from the two models. (c) Estimated recruitment rate of the susceptible population f(t). (d)-(g) Solutions of the UDE model and the mechanistic model by fixing the parameters and initial conditions as the estimated values listed in Table 1.
Fig 5.
Solutions of the mechanistic model (model (4)) with three different values of k and the homogeneous model (model (1)) by varying the transmission rate β (corresponding to different variants of COVID-19).
Here, we plotted the solution of S(t) of model (1) in the first panel. The corresponding effective reproduction numbers are shown in the last panel.
Table 2.
Prediction of key epidemic indices for different scenarios using model (4).
Fig 6.
Comparison of four epidemic indices predicted by models (4) and (1), with the basic reproduction number varied between 1.4 and 10.
The horizontal bars represent the reference values of the attack rate for the wild type with R0 ∈ (1.4, 3.9) [41], Delta variant with R0 ∈ (2.43, 5.11) [42], and Omicron variant strains with R0 ∈ (7.4, 9) [43], respectively. The reference value for Omicron is derived from survey data in [44]. The values for the wild type strain and the Delta variant were calculated by analyzing the rates of epidemics in Belarus under weak intervention measures. The data from 1 March to 1 October 2020, are used for the wild type strain, and data from 1 October 2020, to 20 December 2021, are used for the Delta variant.