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

Xiaolong Geng, Gabriel G. Katul, Firas Gerges, Elie Bou-Zeid, Hani Nassif, Michel C. Boufadel

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16 Scopus citations


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.

Original languageEnglish (US)
Article numbere2023321118
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number21
StatePublished - May 25 2021

All Science Journal Classification (ASJC) codes

  • General


  • Coronavirus disease
  • Fourier analysis
  • Multifractals
  • Population agglomeration
  • Susceptible–infectious–recovered approach


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