A kernel-modulated SIR model for Covid-19 contagious spread from county to continent

Abstract

The tempo-spatial patterns of Covid-19 infections are a result of nested personal, societal, and political decisions that involve complicated epidemiological dynamics across overlapping spatial scales. High infection "hotspots" interspersed within regions where infections remained sporadic were ubiquitous early in the outbreak, but the spatial signature of the infection evolved to affect most regions equally, albeit with distinct temporal patterns. The sparseness of Covid-19 infections in the United States was analyzed at scales spanning from 10 to 2,600 km (county to continental scale). Spatial evolution of Covid-19 cases in the United States followed multifractal scaling. A rapid increase in the spatial correlation was identified early in the outbreak (March to April). Then, the increase continued at a slower rate and approached the spatial correlation of human population. Instead of adopting agent-based models that require tracking of individuals, a kernel-modulated approach is developed to characterize the dynamic spreading of disease in a multifractal distributed susceptible population. Multiphase Covid-19 epidemics were reasonably reproduced by the proposed kernel-modulated susceptible-infectious-recovered (SIR) model. The work explained the fact that while the reproduction number was reduced due to nonpharmaceutical interventions (e.g., masks, social distancing, etc.), subsequent multiple epidemic waves still occurred; this was due to an increase in susceptible population flow following a relaxation of travel restrictions and corollary stay-at-home orders. This study provides an original interpretation of Covid-19 spread together with a pragmatic approach that can be imminently used to capture the spatial intermittency at all epidemiologically relevant scales while preserving the "disordered" spatial pattern of infectious cases.

Keywords: Fourier analysis; coronavirus disease; multifractals; population agglomeration; susceptible–infectious–recovered approach.

Fig. 1.

(A) The q moments of the Covid-19 cases and their scaling from 10 up to 2,600 km used in the determination of K(q). (B) The moment scaling function K(q); the convex shape indicates multifractality. (C) The fitted spectral slope at various dates. (D) Example multifractal field generated using the UM model using the parameter values estimated from the population. Notice that both K(q) and Fourier spectral slope approach that of the underlying population, indicating increased spatial correlation that trends toward a “saturated” threshold set by the population agglomeration.

Fig. 2.

(A) Schematics of kernel-based model, diffusion-based model, and immobile model. Label 1 represents the spreading mechanism of the diffusion-based model that is driven by the concentration gradient between two adjacent cells. Label 2 represents a more realistic spreading mechanism of the kernel-based model that allows individuals to “travel” beyond one cell with a Gaussian-distributed probability. Label 3 represents the spreading mechanism of the immobile model that each cell is well mixed but isolated from other cells. (B) Example multifractal field and the new field postprocessed by the Fast Fourier Transform–based convolution using the Gaussian-distributed kernel function. (C) TSA check point travel numbers for 2020 and 2019 (https://www.tsa.gov/coronavirus/passenger-throughput) and major US Department of Defense response timeline (https://www.defense.gov/Explore/Spotlight/Coronavirus/DOD-Response-Timeline/).

Fig. 3.

Temporal evolution of (A–C) power spectral density function and (D–F) moment scaling function K(q) for infected cases derived from the kernel-based SIR model, diffusion-based SIR model, and immobile SIR model, respectively. The kernel-based SIR model (C and F) better reproduced the spatial correlation of geographic spread of Covid-19 evolving to that of the underlying population agglomeration compared to the diffusion-based model and accelerates the spatial correlation trending toward the “saturation” threshold set by the population agglomeration.

Fig. 4.

(A) Observed daily Covid-19 cases for the United States and each US state. Note that the time origin is Feb. 15, 2020, and 7 d moving average is taken for each US state. (B) Simulated Covid-19 cases using the kernel-based SIR model, taking into account the multiphase population release. Here, we selected daily confirmed cases reported in the United States, New Jersey, California, and Colorado as representative scenarios showing two peaks occurring before and after the reopening, with a single peak occurring before the reopening, a single peak occurring after the reopening, and multiple peaks, respectively. Note that the model acceptably reproduces the US Covid-19 cases. (C) Values of R_b used for the simulations. (D) Calibrated piecewise linear function F_s.