PMC:7047374 / 17054-23868 JSONTXT

{"target":"https://pubannotation.org/docs/sourcedb/PMC/sourceid/7047374","sourcedb":"PMC","sourceid":"7047374","source_url":"https://www.ncbi.nlm.nih.gov/pmc/7047374","text":"Results\nIn this study, we assumed that the incubation period (1/ωP) was the same as latent period (1/ω’P) of human infection, thus ωP = ω’P. Based on the equations of RP model, we can get the disease free equilibrium point as: \\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}$$ \\left(\\frac{\\varLambda_P}{m_P},0,0,0,0,0\\right) $$\\end{document}ΛPmP00000\\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}$$ F=\\left[\\begin{array}{cccc}0\u0026 {\\beta}_P\\frac{\\varLambda_P}{m_P}\u0026 {\\beta}_P\\kappa \\frac{\\varLambda_P}{m_P}\u0026 {\\beta}_W\\frac{\\varLambda_P}{m_P}\\\\ {}0\u0026 0\u0026 0\u0026 0\\\\ {}0\u0026 0\u0026 0\u0026 0\\\\ {}0\u0026 0\u0026 0\u0026 0\\end{array}\\right],{V}^{-1}=\\left[\\begin{array}{cccc}\\frac{1}{\\omega_P+{m}_P}\u0026 0\u0026 0\u0026 0\\\\ {}A\u0026 \\frac{1}{\\gamma_P+{m}_P}\u0026 0\u0026 0\\\\ {}B\u0026 0\u0026 \\frac{1}{\\gamma_P^{\\hbox{'}}+{m}_P}\u0026 0\\\\ {}B\u0026 E\u0026 G\u0026 \\frac{1}{\\varepsilon}\\end{array}\\right] $$\\end{document}F=0βPΛPmPβPκΛPmPβWΛPmP000000000000,V−1=1ωP+mP000A1γP+mP00B01γP'+mP0BEG1ε\nIn the matrix: \\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}$$ A=\\frac{\\left(1-{\\delta}_P\\right){\\upomega}_P}{\\left({\\upomega}_P+{m}_P\\right)\\left({\\gamma}_P+{m}_P\\right)} $$\\end{document}A=1−δPωPωP+mPγP+mP\\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}$$ B=\\frac{\\delta_P{\\upomega}_P}{\\left({\\upomega}_P+{m}_P\\right)\\left({\\gamma}_p^{\\prime }+{m}_P\\right)} $$\\end{document}B=δPωPωP+mPγp′+mP\\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}$$ D=\\frac{\\left(1-{\\delta}_P\\right){\\mu \\upomega}_P}{\\left({\\upomega}_P+{m}_P\\right)\\left({\\gamma}_P+{m}_P\\right)\\varepsilon }+\\frac{\\mu^{\\prime }{\\delta}_P{\\upomega}_P}{\\left({\\upomega}_P+{m}_P\\right)\\left({\\gamma}_p^{\\prime }+{m}_P\\right)\\varepsilon } $$\\end{document}D=1−δPμωPωP+mPγP+mPε+μ′δPωPωP+mPγp′+mPε\\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}$$ E=\\frac{\\mu }{\\left({\\gamma}_P+{m}_P\\right)\\varepsilon } $$\\end{document}E=μγP+mPε\\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}$$ G=\\frac{\\mu^{\\prime }}{\\left({\\gamma}_p^{\\prime }+{m}_P\\right)\\varepsilon } $$\\end{document}G=μ′γp′+mPε\nBy the next generation matrix approach, we can get the next generation matrix and R0 for the RP model: \\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}$$ F{V}^{-1}=\\left[\\begin{array}{cccc}{\\beta}_p\\frac{\\varLambda_P}{m_P}A+{\\beta}_P\\kappa \\frac{\\varLambda_P}{m_P}+{\\beta}_W\\frac{\\varLambda_P}{m_P}D\u0026 \\ast \u0026 \\ast \u0026 \\ast \\\\ {}0\u0026 0\u0026 0\u0026 0\\\\ {}0\u0026 0\u0026 0\u0026 0\\\\ {}0\u0026 0\u0026 0\u0026 0\\end{array}\\right] $$\\end{document}FV−1=βpΛPmPA+βPκΛPmP+βWΛPmPD∗∗∗000000000000\\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}$$ {R}_0=\\rho \\left(F{V}^{-1}\\right)={\\beta}_P\\frac{\\varLambda_P}{m_P}\\frac{\\left(1-{\\delta}_P\\right){\\omega}_P}{\\left({\\omega}_P+{m}_P\\right)\\left({\\gamma}_P+{m}_P\\right)}+{\\beta}_P\\kappa \\frac{\\varLambda_P}{m_P}\\frac{\\delta_P{\\omega}_P}{\\left({\\omega}_P+{m}_P\\right)\\left({\\gamma}_P^{\\hbox{'}}+{m}_P\\right)}+{\\beta}_W\\frac{\\varLambda_P}{m_P}\\frac{\\left(1-{\\delta}_P\\right)\\mu {\\omega}_P}{\\left({\\omega}_P+{m}_P\\right)\\left({\\gamma}_P+{m}_P\\right)\\varepsilon }+\\beta W\\frac{\\varLambda_P}{m_P}\\frac{\\mu^{\\hbox{'}}{\\delta}_P{\\omega}_P}{\\left({\\omega}_P+{m}_P\\right)\\left({\\gamma}_P^{\\hbox{'}}+{m}_P\\right)\\varepsilon } $$\\end{document}R0=ρFV−1=βPΛPmP1−δPωPωP+mPγP+mP+βPκΛPmPδPωPωP+mPγP'+mP+βWΛPmP1−δPμωPωP+mPγP+mPε+βWΛPmPμ'δPωPωP+mPγP'+mPε\nThe R0 of the normalized RP model is shown as follows: \\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}$$ {R}_0={b}_p\\frac{n_P}{m_p}\\frac{\\left(1-{\\delta}_P\\right){\\omega}_P}{\\left[\\left(1-\\delta p\\right){\\omega}_P+{\\delta}_P{\\omega}_P^{\\hbox{'}}+{m}_P\\right]\\left({\\gamma}_P+{m}_P\\right)}+\\kappa {b}_P\\frac{n_P}{m_P}\\frac{\\delta_P{\\omega}_P^{\\hbox{'}}}{\\left[\\left(1-{\\delta}_P\\right){\\omega}_P+{\\delta}_P{\\omega}_P^{\\hbox{'}}+{m}_P\\right]\\left({\\gamma}_P^{\\hbox{'}}+{m}_P\\right)}+{b}_W\\frac{n_P}{m_P}\\frac{\\left(1-{\\delta}_p\\right){\\omega}_p}{\\left[\\left(1-{\\delta}_p\\right){\\omega}_P+{\\delta}_P{\\omega}_P^{\\hbox{'}}+{m}_p\\right]\\left({\\gamma}_P+{m}_P\\right)}+{b}_W\\frac{n_P}{m_P}\\frac{c{\\delta}_P{\\omega}_P^{\\hbox{'}}}{\\left[\\left(1-{\\delta}_p\\right){\\omega}_P+{\\delta}_P{\\omega}_P^{\\hbox{'}}+{m}_p\\right]\\left({\\gamma}_P^{\\hbox{'}}+{m}_P\\right)} $$\\end{document}R0=bpnPmp1−δPωP1−δpωP+δPωP'+mPγP+mP+κbPnPmPδPωP'1−δPωP+δPωP'+mPγP'+mP+bWnPmP1−δpωp1−δpωP+δPωP'+mpγP+mP+bWnPmPcδPωP'1−δpωP+δPωP'+mpγP'+mP\nOur modelling results showed that the normalized RP model fitted well to the reported SARS-CoV-2 cases data (R2 = 0.512, P \u003c 0.001) (Fig. 2). The value of R0 was estimated of 2.30 from reservoir to person, and from person to person and 3.58 from person to person which means that the expected number of secondary infections that result from introducing a single infected individual into an otherwise susceptible population was 3.58.\nFig. 2 Curve fitting results of the RP 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