Journal: Scientific Reports

Article Title: A high-resolution flux-matrix model describes the spread of diseases in a spatial network and the effect of mitigation strategies

doi: 10.1038/s41598-022-19931-w

Figure Lengend Snippet: Comparison of the (fitted) model predictions with reported values using ( a ) the infectivity matrix derived from SafeGraph fluxes (same model as Fig. ), ( b ) an infectivity matrix with uniform transmission rates, and ( c ) the infectivity matrix derived from SafeGraph fluxes truncated to geographical distances \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_c < {400} \; \text{km}$$\end{document} d c < 400 km . Local epidemic sizes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_a$$\end{document} T a are shown. One symbol is associated to one given community and one given day.

Article Snippet: Figure 5 Comparison of the (fitted) model predictions with reported values using ( a ) the infectivity matrix derived from SafeGraph fluxes (same model as Fig. ), ( b ) an infectivity matrix with uniform transmission rates, and ( c ) the infectivity matrix derived from SafeGraph fluxes truncated to geographical distances \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_c < {400} \; \text{km}$$\end{document} d c < 400 km .

Techniques: Comparison, Infection, Derivative Assay, Transmission Assay

Journal: Scientific Reports

Article Title: A high-resolution flux-matrix model describes the spread of diseases in a spatial network and the effect of mitigation strategies

doi: 10.1038/s41598-022-19931-w

Figure Lengend Snippet: Limiting long-distance travels without local lockdown. ( a ) Only transmission rates for communities separated by a distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{ij} < d_c$$\end{document} d ij < d c are retained. ( b ) Daily new infections for increasing values of the cutoff distance. ( c ) Spatial visualization of the daily new infections using a cutoff \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_c = {200}\;{\text {km}}$$\end{document} d c = 200 km . See also Movie .

Article Snippet: Figure 5 Comparison of the (fitted) model predictions with reported values using ( a ) the infectivity matrix derived from SafeGraph fluxes (same model as Fig. ), ( b ) an infectivity matrix with uniform transmission rates, and ( c ) the infectivity matrix derived from SafeGraph fluxes truncated to geographical distances \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_c < {400} \; \text{km}$$\end{document} d c < 400 km .

Techniques: Transmission Assay

Journal: Scientific Reports

Article Title: A high-resolution flux-matrix model describes the spread of diseases in a spatial network and the effect of mitigation strategies

doi: 10.1038/s41598-022-19931-w

Figure Lengend Snippet: Random-matrix properties and mitigation strategies. ( a ) Level spacing distribution (unfolded , – ) of infectivity matrices between 2020-03-01 and 2021-02-15 (pooled by 7-day windows). The solid lines denote the Wigner-Dyson and the Poisson statistics. The level distribution is shown in Fig. . ( b ) The epidemic size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega /\Omega _0$$\end{document} Ω / Ω 0 versus the relative number of cuts \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_c/N/(N-1)$$\end{document} N c / N / ( N - 1 ) for the 3 mitigation methods: distance cutoff, bandwidth, and edge-betweenness centrality. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _0$$\end{document} Ω 0 is the epidemic size for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_c=0$$\end{document} N c = 0 . Level-spacing entropy S and epidemic size as a function of ( c ) the cutoff distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_c$$\end{document} d c in the range 170-12,000 km, and ( d ) the half-bandwidth \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1{\le }B<1023$$\end{document} 1 ≤ B < 1023 .

Article Snippet: Figure 5 Comparison of the (fitted) model predictions with reported values using ( a ) the infectivity matrix derived from SafeGraph fluxes (same model as Fig. ), ( b ) an infectivity matrix with uniform transmission rates, and ( c ) the infectivity matrix derived from SafeGraph fluxes truncated to geographical distances \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_c < {400} \; \text{km}$$\end{document} d c < 400 km .

Techniques: Infection

Journal: Scientific Reports

Article Title: A high-resolution flux-matrix model describes the spread of diseases in a spatial network and the effect of mitigation strategies

doi: 10.1038/s41598-022-19931-w

Figure Lengend Snippet: Model for the spreading of SARS-CoV-2 in a network of communities in the USA. ( a ) One community i interacting with three other communities j , k and l , with the transmission rates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{ij}$$\end{document} β ij , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{ik}$$\end{document} β ik and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{il}$$\end{document} β il respectively. ( b ) Infectivity matrix. ( c ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2^{10}$$\end{document} N = 2 10 communities in the USA. Each community aggregates a number of Census Block Groups (CBGs).

Article Snippet: Figure 5 Comparison of the (fitted) model predictions with reported values using ( a ) the infectivity matrix derived from SafeGraph fluxes (same model as Fig. ), ( b ) an infectivity matrix with uniform transmission rates, and ( c ) the infectivity matrix derived from SafeGraph fluxes truncated to geographical distances \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_c < {400} \; \text{km}$$\end{document} d c < 400 km .

Techniques: Transmission Assay, Infection, Blocking Assay

Journal: Scientific Reports

Article Title: A high-resolution flux-matrix model describes the spread of diseases in a spatial network and the effect of mitigation strategies

doi: 10.1038/s41598-022-19931-w

Figure Lengend Snippet: Existence of a wave with nearest-neighbors-only interactions. ( a ) Equation is solved on a square lattice of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^n \times 2^m$$\end{document} 2 n × 2 m sites. ( b ) Traveling front of infected individuals moving along the x direction (left to right). We took \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = \beta = \gamma = {0.1}$$\end{document} α = β = γ = 0.1 , so that the rescaled recovery rate is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\gamma } = \gamma / (a \beta ) = {0.2}$$\end{document} γ ~ = γ / ( a β ) = 0.2 . ( c ) The dynamics simulated for different values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} β . The position m ( t ) ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(t, m(t))=(1 + S_\infty )/2$$\end{document} S ( t , m ( t ) ) = ( 1 + S ∞ ) / 2 ) of the wave is asymptotically linear in time, hence a constant velocity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{v} = \sqrt{1-\tilde{\gamma }}$$\end{document} v ~ = 1 - γ ~ . ( d ) The simulated wave profiles (symbols) are in agreement with the profiles predicted by Eq. (solid lines). The corresponding S,I,R profiles are shown in Fig. . A simulation with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\gamma } = {0.2}$$\end{document} γ ~ = 0.2 is shown in Movie .

Article Snippet: Figure 5 Comparison of the (fitted) model predictions with reported values using ( a ) the infectivity matrix derived from SafeGraph fluxes (same model as Fig. ), ( b ) an infectivity matrix with uniform transmission rates, and ( c ) the infectivity matrix derived from SafeGraph fluxes truncated to geographical distances \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_c < {400} \; \text{km}$$\end{document} d c < 400 km .

Techniques: Infection