PMC:7547104 / 28682-32906 JSONTXT

{"project":"LitCovid-PD-CLO","denotations":[{"id":"T90","span":{"begin":375,"end":380},"obj":"http://purl.obolibrary.org/obo/CLO_0006985"},{"id":"T91","span":{"begin":375,"end":380},"obj":"http://purl.obolibrary.org/obo/CLO_0006987"},{"id":"T92","span":{"begin":408,"end":413},"obj":"http://purl.obolibrary.org/obo/CLO_0007052"},{"id":"T93","span":{"begin":419,"end":422},"obj":"http://purl.obolibrary.org/obo/CLO_0051142"},{"id":"T94","span":{"begin":864,"end":869},"obj":"http://purl.obolibrary.org/obo/CLO_0006985"},{"id":"T95","span":{"begin":864,"end":869},"obj":"http://purl.obolibrary.org/obo/CLO_0006987"},{"id":"T96","span":{"begin":897,"end":902},"obj":"http://purl.obolibrary.org/obo/CLO_0007052"},{"id":"T97","span":{"begin":908,"end":911},"obj":"http://purl.obolibrary.org/obo/CLO_0051142"},{"id":"T98","span":{"begin":1931,"end":1933},"obj":"http://purl.obolibrary.org/obo/CLO_0053733"},{"id":"T99","span":{"begin":2943,"end":2948},"obj":"http://purl.obolibrary.org/obo/CLO_0007052"},{"id":"T100","span":{"begin":3979,"end":3980},"obj":"http://purl.obolibrary.org/obo/CLO_0001020"},{"id":"T101","span":{"begin":4074,"end":4076},"obj":"http://purl.obolibrary.org/obo/CLO_0053733"},{"id":"T102","span":{"begin":4115,"end":4116},"obj":"http://purl.obolibrary.org/obo/CLO_0001020"}],"text":"The network model of Italy we adopt in this study is, for i = 1, …, 20,7 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot S_i = - \\mathop {\\sum}\\limits_{j = 1}^M {\\mathop {\\sum}\\limits_{k = 1}^M {\\rho _j} } \\beta \\phi _{ij}\\left( t \\right)S_i\\frac{{\\phi _{kj}\\left( t \\right)I_k}}{{N_j^p}},$$\\end{document}S˙i=−∑j=1M∑k=1MρjβϕijtSiϕkjtIkNjp,8 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot I_i = \\mathop {\\sum}\\limits_{j = 1}^M {\\mathop {\\sum}\\limits_{k = 1}^M {\\rho _j} } \\beta \\phi _{ij}\\left( t \\right)S_i\\frac{{\\phi _{kj}\\left( t \\right)I_k}}{{N_j^p}} - \\alpha _iI_i - \\psi _iI_i - \\gamma I_i,$$\\end{document}I˙i=∑j=1M∑k=1MρjβϕijtSiϕkjtIkNjp−αiIi−ψiIi−γIi,9 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot Q_i = \\alpha _iI_i - \\kappa _i^HQ_i - \\eta _i^QQ_i + \\kappa _i^QH_i,$$\\end{document}Q˙i=αiIi−κiHQi−ηiQQi+κiQHi,10 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot H_i = \\kappa _i^HQ_i + \\psi _iI_i - \\eta _i^HH_i - \\kappa _i^QH_i - \\zeta \\left( {H_i{\\mathrm{/}}T_i^H} \\right)H_i,$$\\end{document}H˙i=κiHQi+ψiIi−ηiHHi−κiQHi−ζHi/TiHHi,11 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot D_i = \\zeta \\left( {H_i{\\mathrm{/}}T_i^H} \\right)H_i,$$\\end{document}D˙i=ζHi/TiHHi,12 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot R_i = \\gamma I_i + \\eta _i^QQ_i + \\eta _i^HH_i$$\\end{document}R˙i=γIi+ηiQQi+ηiHHi13 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$N_i^p = \\mathop {\\sum}\\limits_{k = 1}^M {\\phi _{ki}} \\left( t \\right)\\left( {S_k + I_k + R_k} \\right)$$\\end{document}Nip=∑k=1MϕkitSk+Ik+Rkwhere in addition to the parameters and states described above, we included the fluxes ϕij(t) between regions; \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\phi _{ij}\\left( t \\right):{\\Bbb R} \\to \\left[ {0,1} \\right]$$\\end{document}ϕijt:R→0,1 denoting the ratio of people from region i interacting with those in region j at time t, such that \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\mathop {\\sum}\\nolimits_j {\\phi _{ij}} \\left( t \\right) = 1.$$\\end{document}∑jϕijt=1. Note that, as a result of the identification procedure illustrated in Supplementary Notes, in Eqs. (10) and (11) the mortality rate ζ is expressed as a function of the saturation of the regional health systems whose expression is given in Supplementary Notes."}

{"project":"LitCovid-PD-CHEBI","denotations":[{"id":"T21","span":{"begin":430,"end":434},"obj":"Chemical"},{"id":"T22","span":{"begin":919,"end":923},"obj":"Chemical"},{"id":"T23","span":{"begin":1004,"end":1009},"obj":"Chemical"},{"id":"T24","span":{"begin":1032,"end":1037},"obj":"Chemical"},{"id":"T25","span":{"begin":1385,"end":1390},"obj":"Chemical"},{"id":"T26","span":{"begin":2569,"end":2574},"obj":"Chemical"}],"attributes":[{"id":"A21","pred":"chebi_id","subj":"T21","obj":"http://purl.obolibrary.org/obo/CHEBI_10545"},{"id":"A22","pred":"chebi_id","subj":"T22","obj":"http://purl.obolibrary.org/obo/CHEBI_10545"},{"id":"A23","pred":"chebi_id","subj":"T23","obj":"http://purl.obolibrary.org/obo/CHEBI_30216"},{"id":"A24","pred":"chebi_id","subj":"T24","obj":"http://purl.obolibrary.org/obo/CHEBI_30212"},{"id":"A25","pred":"chebi_id","subj":"T25","obj":"http://purl.obolibrary.org/obo/CHEBI_30216"},{"id":"A26","pred":"chebi_id","subj":"T26","obj":"http://purl.obolibrary.org/obo/CHEBI_30212"}],"text":"The network model of Italy we adopt in this study is, for i = 1, …, 20,7 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot S_i = - \\mathop {\\sum}\\limits_{j = 1}^M {\\mathop {\\sum}\\limits_{k = 1}^M {\\rho _j} } \\beta \\phi _{ij}\\left( t \\right)S_i\\frac{{\\phi _{kj}\\left( t \\right)I_k}}{{N_j^p}},$$\\end{document}S˙i=−∑j=1M∑k=1MρjβϕijtSiϕkjtIkNjp,8 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot I_i = \\mathop {\\sum}\\limits_{j = 1}^M {\\mathop {\\sum}\\limits_{k = 1}^M {\\rho _j} } \\beta \\phi _{ij}\\left( t \\right)S_i\\frac{{\\phi _{kj}\\left( t \\right)I_k}}{{N_j^p}} - \\alpha _iI_i - \\psi _iI_i - \\gamma I_i,$$\\end{document}I˙i=∑j=1M∑k=1MρjβϕijtSiϕkjtIkNjp−αiIi−ψiIi−γIi,9 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot Q_i = \\alpha _iI_i - \\kappa _i^HQ_i - \\eta _i^QQ_i + \\kappa _i^QH_i,$$\\end{document}Q˙i=αiIi−κiHQi−ηiQQi+κiQHi,10 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot H_i = \\kappa _i^HQ_i + \\psi _iI_i - \\eta _i^HH_i - \\kappa _i^QH_i - \\zeta \\left( {H_i{\\mathrm{/}}T_i^H} \\right)H_i,$$\\end{document}H˙i=κiHQi+ψiIi−ηiHHi−κiQHi−ζHi/TiHHi,11 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot D_i = \\zeta \\left( {H_i{\\mathrm{/}}T_i^H} \\right)H_i,$$\\end{document}D˙i=ζHi/TiHHi,12 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot R_i = \\gamma I_i + \\eta _i^QQ_i + \\eta _i^HH_i$$\\end{document}R˙i=γIi+ηiQQi+ηiHHi13 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$N_i^p = \\mathop {\\sum}\\limits_{k = 1}^M {\\phi _{ki}} \\left( t \\right)\\left( {S_k + I_k + R_k} \\right)$$\\end{document}Nip=∑k=1MϕkitSk+Ik+Rkwhere in addition to the parameters and states described above, we included the fluxes ϕij(t) between regions; \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\phi _{ij}\\left( t \\right):{\\Bbb R} \\to \\left[ {0,1} \\right]$$\\end{document}ϕijt:R→0,1 denoting the ratio of people from region i interacting with those in region j at time t, such that \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\mathop {\\sum}\\nolimits_j {\\phi _{ij}} \\left( t \\right) = 1.$$\\end{document}∑jϕijt=1. Note that, as a result of the identification procedure illustrated in Supplementary Notes, in Eqs. (10) and (11) the mortality rate ζ is expressed as a function of the saturation of the regional health systems whose expression is given in Supplementary Notes."}

{"project":"LitCovid-sentences","denotations":[{"id":"T134","span":{"begin":0,"end":3964},"obj":"Sentence"},{"id":"T135","span":{"begin":3965,"end":4224},"obj":"Sentence"}],"namespaces":[{"prefix":"_base","uri":"http://pubannotation.org/ontology/tao.owl#"}],"text":"The network model of Italy we adopt in this study is, for i = 1, …, 20,7 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot S_i = - \\mathop {\\sum}\\limits_{j = 1}^M {\\mathop {\\sum}\\limits_{k = 1}^M {\\rho _j} } \\beta \\phi _{ij}\\left( t \\right)S_i\\frac{{\\phi _{kj}\\left( t \\right)I_k}}{{N_j^p}},$$\\end{document}S˙i=−∑j=1M∑k=1MρjβϕijtSiϕkjtIkNjp,8 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot I_i = \\mathop {\\sum}\\limits_{j = 1}^M {\\mathop {\\sum}\\limits_{k = 1}^M {\\rho _j} } \\beta \\phi _{ij}\\left( t \\right)S_i\\frac{{\\phi _{kj}\\left( t \\right)I_k}}{{N_j^p}} - \\alpha _iI_i - \\psi _iI_i - \\gamma I_i,$$\\end{document}I˙i=∑j=1M∑k=1MρjβϕijtSiϕkjtIkNjp−αiIi−ψiIi−γIi,9 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot Q_i = \\alpha _iI_i - \\kappa _i^HQ_i - \\eta _i^QQ_i + \\kappa _i^QH_i,$$\\end{document}Q˙i=αiIi−κiHQi−ηiQQi+κiQHi,10 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot H_i = \\kappa _i^HQ_i + \\psi _iI_i - \\eta _i^HH_i - \\kappa _i^QH_i - \\zeta \\left( {H_i{\\mathrm{/}}T_i^H} \\right)H_i,$$\\end{document}H˙i=κiHQi+ψiIi−ηiHHi−κiQHi−ζHi/TiHHi,11 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot D_i = \\zeta \\left( {H_i{\\mathrm{/}}T_i^H} \\right)H_i,$$\\end{document}D˙i=ζHi/TiHHi,12 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot R_i = \\gamma I_i + \\eta _i^QQ_i + \\eta _i^HH_i$$\\end{document}R˙i=γIi+ηiQQi+ηiHHi13 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$N_i^p = \\mathop {\\sum}\\limits_{k = 1}^M {\\phi _{ki}} \\left( t \\right)\\left( {S_k + I_k + R_k} \\right)$$\\end{document}Nip=∑k=1MϕkitSk+Ik+Rkwhere in addition to the parameters and states described above, we included the fluxes ϕij(t) between regions; \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\phi _{ij}\\left( t \\right):{\\Bbb R} \\to \\left[ {0,1} \\right]$$\\end{document}ϕijt:R→0,1 denoting the ratio of people from region i interacting with those in region j at time t, such that \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\mathop {\\sum}\\nolimits_j {\\phi _{ij}} \\left( t \\right) = 1.$$\\end{document}∑jϕijt=1. Note that, as a result of the identification procedure illustrated in Supplementary Notes, in Eqs. (10) and (11) the mortality rate ζ is expressed as a function of the saturation of the regional health systems whose expression is given in Supplementary Notes."}

{"project":"LitCovid-PubTator","denotations":[{"id":"90","span":{"begin":3536,"end":3542},"obj":"Species"},{"id":"91","span":{"begin":4082,"end":4091},"obj":"Disease"}],"attributes":[{"id":"A90","pred":"tao:has_database_id","subj":"90","obj":"Tax:9606"},{"id":"A91","pred":"tao:has_database_id","subj":"91","obj":"MESH:D003643"}],"namespaces":[{"prefix":"Tax","uri":"https://www.ncbi.nlm.nih.gov/taxonomy/"},{"prefix":"MESH","uri":"https://id.nlm.nih.gov/mesh/"},{"prefix":"Gene","uri":"https://www.ncbi.nlm.nih.gov/gene/"},{"prefix":"CVCL","uri":"https://web.expasy.org/cellosaurus/CVCL_"}],"text":"The network model of Italy we adopt in this study is, for i = 1, …, 20,7 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot S_i = - \\mathop {\\sum}\\limits_{j = 1}^M {\\mathop {\\sum}\\limits_{k = 1}^M {\\rho _j} } \\beta \\phi _{ij}\\left( t \\right)S_i\\frac{{\\phi _{kj}\\left( t \\right)I_k}}{{N_j^p}},$$\\end{document}S˙i=−∑j=1M∑k=1MρjβϕijtSiϕkjtIkNjp,8 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot I_i = \\mathop {\\sum}\\limits_{j = 1}^M {\\mathop {\\sum}\\limits_{k = 1}^M {\\rho _j} } \\beta \\phi _{ij}\\left( t \\right)S_i\\frac{{\\phi _{kj}\\left( t \\right)I_k}}{{N_j^p}} - \\alpha _iI_i - \\psi _iI_i - \\gamma I_i,$$\\end{document}I˙i=∑j=1M∑k=1MρjβϕijtSiϕkjtIkNjp−αiIi−ψiIi−γIi,9 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot Q_i = \\alpha _iI_i - \\kappa _i^HQ_i - \\eta _i^QQ_i + \\kappa _i^QH_i,$$\\end{document}Q˙i=αiIi−κiHQi−ηiQQi+κiQHi,10 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot H_i = \\kappa _i^HQ_i + \\psi _iI_i - \\eta _i^HH_i - \\kappa _i^QH_i - \\zeta \\left( {H_i{\\mathrm{/}}T_i^H} \\right)H_i,$$\\end{document}H˙i=κiHQi+ψiIi−ηiHHi−κiQHi−ζHi/TiHHi,11 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot D_i = \\zeta \\left( {H_i{\\mathrm{/}}T_i^H} \\right)H_i,$$\\end{document}D˙i=ζHi/TiHHi,12 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\dot R_i = \\gamma I_i + \\eta _i^QQ_i + \\eta _i^HH_i$$\\end{document}R˙i=γIi+ηiQQi+ηiHHi13 \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$N_i^p = \\mathop {\\sum}\\limits_{k = 1}^M {\\phi _{ki}} \\left( t \\right)\\left( {S_k + I_k + R_k} \\right)$$\\end{document}Nip=∑k=1MϕkitSk+Ik+Rkwhere in addition to the parameters and states described above, we included the fluxes ϕij(t) between regions; \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\phi _{ij}\\left( t \\right):{\\Bbb R} \\to \\left[ {0,1} \\right]$$\\end{document}ϕijt:R→0,1 denoting the ratio of people from region i interacting with those in region j at time t, such that \\documentclass[12pt]{minimal} \t\t\t\t\\usepackage{amsmath} \t\t\t\t\\usepackage{wasysym} \t\t\t\t\\usepackage{amsfonts} \t\t\t\t\\usepackage{amssymb} \t\t\t\t\\usepackage{amsbsy} \t\t\t\t\\usepackage{mathrsfs} \t\t\t\t\\usepackage{upgreek} \t\t\t\t\\setlength{\\oddsidemargin}{-69pt} \t\t\t\t\\begin{document}$$\\mathop {\\sum}\\nolimits_j {\\phi _{ij}} \\left( t \\right) = 1.$$\\end{document}∑jϕijt=1. Note that, as a result of the identification procedure illustrated in Supplementary Notes, in Eqs. (10) and (11) the mortality rate ζ is expressed as a function of the saturation of the regional health systems whose expression is given in Supplementary Notes."}