PMC:7047374 / 3781-17052 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":"Methods\n\nData source\nThe reported cases of SARS-CoV-2, which have been named as COVID-19, were collected for the modelling study from a published literature [3]. As reported by Li et al. [3], the onset date of the first case was on 7 December, 2020, and the seafood market was closed on 1 January, 2020 [11]. The epidemic curve from 7 December, 2019 to 1 January, 2020 was collected for our study, and the simulation time step was 1 day.\n\nSimulation methods and statistical analysis\nBerkeley Madonna 8.3.18 (developed by Robert Macey and George Oster of the University of California at Berkeley. Copyright©1993–2001 Robert I. Macey \u0026 George F. Oster) was employed for the curve fitting. The fourth-order Runge–Kutta method, with tolerance set at 0.001, was used to perform curve fitting. While the curve fitting is in progress, Berkeley Madonna displays the root mean square deviation between the data and best run so far. The coefficient of determination (R2) was employed to assess the goodness-of-fit. SPSS 13.0 (IBM Corp., Armonk, NY, USA) was employed to calculate the R2.\n\nThe Bats-Hosts-Reservoir-People (BHRP) transmission network model\nThe BHRP transmission network model was posted to bioRxiv on 19 January, 2020 [12]. We assumed that the virus transmitted among the bats, and then transmitted to unknown hosts (probably some wild animals). The hosts were hunted and sent to the seafood market which was defined as the reservoir of the virus. People exposed to the market got the risks of the infection (Fig. 1). The BHRP transmission network model was based on the following assumptions or facts: The bats were divided into four compartments: susceptible bats (SB), exposed bats (EB), infected bats (IB), and removed bats (RB). The birth rate and death rate of bats were defined as nB and mB. In this model, we set ɅB = nB × NB as the number of the newborn bats where NB refer to the total number of bats. The incubation period of bat infection was defined as 1/ωB and the infectious period of bat infection was defined as 1/γB. The SB will be infected through sufficient contact with IB, and the transmission rate was defined as βB.\nThe hosts were also divided into four compartments: susceptible hosts (SH), exposed hosts (EH), infected hosts (IH), and removed hosts (RH). The birth rate and death rate of hosts were defined as nH and mH. In this model, we set ɅH = nH × NH where NH refer to the total number of hosts. The incubation period of host infection was defined as 1/ωH and the infectious period of host infection was defined as 1/γH. The SH will be infected through sufficient contact with IB and IH, and the transmission rates were defined as βBH and βH, respectively.\nThe SARS-CoV-2 in reservoir (the seafood market) was denoted as W. We assumed that the retail purchases rate of the hosts in the market was a, and that the prevalence of SARS-CoV-2 in the purchases was IH/NH, therefore, the rate of the SARS-CoV-2 in W imported form the hosts was aWIH/NH where NH was the total number of hosts. We also assumed that symptomatic infected people and asymptomatic infected people could export the virus into W with the rate of μP and μ’P, although this assumption might occur in a low probability. The virus in W will subsequently leave the W compartment at a rate of εW, where 1/ε is the lifetime of the virus.\nThe people were divided into five compartments: susceptible people (SP), exposed people (EP), symptomatic infected people (IP), asymptomatic infected people (AP), and removed people (RP) including recovered and death people. The birth rate and death rate of people were defined as nP and mP. In this model, we set ɅP = nP × NP where NP refer to the total number of people. The incubation period and latent period of human infection was defined as 1/ωP and 1/ω’P. The infectious period of IP and AP was defined as 1/γP and 1/γ’P. The proportion of asymptomatic infection was defined as δP. The SP will be infected through sufficient contact with W and IP, and the transmission rates were defined as βW and βP, respectively. We also assumed that the transmissibility of AP was κ times that of IP, where 0 ≤ κ ≤ 1.\nFig. 1 Flowchart of the Bats-Hosts-Reservoir-People transmission network model\nThe parameters of the BHRP model were shown in Table 1.\nTable 1 Definition of those parameters in the Bats-Hosts-Reservoir-People (BHRP) model\nParameter Description\nnB The birth rate parameter of bats\nnH The birth rate parameter of hosts\nnP The birth rate parameter of people\nmB The death rate of bats\nmH The death rate of hosts\nmP The death rate of people\n1/ωB The incubation period of bats\n1/ωH The incubation period of hosts\n1/ωP The incubation period of people\n1/ω’P The latent period of people\n1/γB The infectious period of bats\n1/γH The infectious period of hosts\n1/γP The infectious period of symptomatic infection of people\n1/γ’P The infectious period of asymptomatic infection of people\nβB The transmission rate from IB to SB\nβBH The transmission rate from IB to SH\nβH The transmission rate from IH to SH\nβP The transmission rate from IP to SP\nβW The transmission rate from W to SP\na The retail purchases rate of the hosts in the market\nμP The shedding coefficients from IP to W\nμ’P The shedding coefficients from AP to W\n1/ε The lifetime of the virus in W\nδP The proportion of asymptomatic infection rate of people\nκ The multiple of the transmissibility of AP to that of IP.\n\nThe simplified reservoir-people transmission network model\nWe assumed that the SARS-CoV-2 might be imported to the seafood market in a short time. Therefore, we added the further assumptions as follows: The transmission network of Bats-Host was ignored.\nBased on our previous studies on simulating importation [13, 14], we set the initial value of W as following impulse function: \\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}$$ Importation= impulse\\left(n,{t}_0,{t}_i\\right) $$\\end{document}Importation=impulsent0ti\nIn the function, n, t0 and ti refer to imported volume of the SARS-CoV-2 to the market, start time of the simulation, and the interval of the importation.\nTherefore, the BHRP model was simplified as RP model and 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}$$ \\left\\{\\kern0.5em \\begin{array}{c}\\frac{d{S}_P}{dt}={\\varLambda}_P-{m}_P{S}_P-{\\beta}_P{S}_P\\left({I}_P+\\upkappa {A}_P\\right)-{\\beta}_W{S}_PW\\kern11em \\\\ {}\\frac{d{E}_P}{dt}={\\beta}_P{S}_P\\left({I}_P+\\upkappa {A}_P\\right)+{\\beta}_W{S}_PW-\\left(1-{\\delta}_P\\right){\\upomega}_P{E}_P-{\\delta}_P{\\upomega}_P^{\\prime }{E}_P-{m}_P{E}_P\\kern0.5em \\\\ {}\\frac{d{I}_P}{dt}=\\left(1-{\\delta}_P\\right){\\upomega}_P{E}_P-\\left({\\gamma}_P+{m}_P\\right){I}_P\\kern16.5em \\\\ {}\\frac{d{A}_P}{dt}={\\delta}_P{\\upomega}_P^{\\prime }{E}_P-\\left({\\gamma}_P^{\\prime }+{m}_P\\right){A}_P\\kern18.75em \\\\ {}\\frac{d{R}_P}{dt}={\\gamma}_P{I}_P+{\\gamma}_P^{\\prime }{A}_P-{m}_P{R}_P\\kern20em \\\\ {}\\frac{dW}{dt}={\\mu}_P{I}_P+{\\mu}_P^{\\prime }{A}_P-\\varepsilon W\\kern20.5em \\end{array}\\right. $$\\end{document}dSPdt=ΛP−mPSP−βPSPIP+κAP−βWSPWdEPdt=βPSPIP+κAP+βWSPW−1−δPωPEP−δPωP′EP−mPEPdIPdt=1−δPωPEP−γP+mPIPdAPdt=δPωP′EP−γP′+mPAPdRPdt=γPIP+γP′AP−mPRPdWdt=μPIP+μP′AP−εW\nDuring the outbreak period, the natural birth rate and death rate in the population was in a relative low level. However, people would commonly travel into and out from Wuhan City mainly due to the Chinese New Year holiday. Therefore, nP and mP refer to the rate of people traveling into Wuhan City and traveling out from Wuhan City, respectively.\nIn the model, people and viruses have different dimensions. Based on our previous research [15], we therefore used the following sets to perform the normalization: \\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}$$ {s}_P=\\frac{S_P}{N_P},{e}_P=\\frac{E_P}{N_P},{i}_P=\\frac{I_P}{N_P}, {a}_P=\\frac{A_P}{N_P},{r}_P=\\frac{R_P}{N_P},w=\\frac{\\varepsilon W}{\\mu_P{N}_P},\\kern0.5em {\\mu}_P^{\\prime }=c{\\mu}_P,\\kern0.5em {b}_P={\\beta}_P{N}_P,\\mathrm{and}\\ {b}_W=\\frac{\\mu_P{\\beta}_W{N}_P}{\\varepsilon .} $$\\end{document}sP=SPNP,eP=EPNP,iP=IPNP,aP=APNP,rP=RPNP,w=εWμPNP,μP′=cμP,bP=βPNP,andbW=μPβWNPε.\nIn the normalization, parameter c refers to the relative shedding coefficient of AP compared to IP. The normalized RP model is changed 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}$$ \\left\\{\\begin{array}{c}\\frac{d{s}_P}{dt}={n}_P-{m}_P{s}_P-{b}_P{s}_P\\left({i}_P+\\upkappa {a}_P\\right)-{b}_W{s}_Pw\\\\ {}\\frac{d{e}_P}{dt}={b}_P{s}_P\\left({i}_P+\\upkappa {a}_P\\right)+{b}_W{s}_Pw-\\left(1-{\\delta}_P\\right){\\upomega}_P{e}_P-{\\delta}_P{\\upomega}_P^{\\prime }{e}_P-{m}_P{e}_P\\\\ {}\\frac{d{i}_P}{dt}=\\left(1-{\\delta}_P\\right){\\upomega}_P{e}_P-\\left({\\gamma}_P+{m}_P\\right){i}_P\\\\ {}\\frac{d{a}_P}{dt}={\\delta}_P{\\upomega}_P^{\\prime }{e}_P-\\left({\\gamma}_P^{\\prime }+{m}_P\\right){a}_P\\kern26.5em \\\\ {}\\frac{d{r}_P}{dt}={\\gamma}_P{i}_P+{\\gamma}_P^{\\prime }{a}_P-{m}_P{r}_P\\\\ {}\\frac{dw}{dt}=\\varepsilon \\left({i}_P+c{a}_P-w\\right)\\kern28.2em \\end{array}\\right. $$\\end{document}dsPdt=nP−mPsP−bPsPiP+κaP−bWsPwdePdt=bPsPiP+κaP+bWsPw−1−δPωPeP−δPωP′eP−mPePdiPdt=1−δPωPeP−γP+mPiPdaPdt=δPωP′eP−γP′+mPaPdrPdt=γPiP+γP′aP−mPrPdwdt=εiP+caP−w\n\nThe transmissibility of the SARS-CoV-2 based on the RP model\nIn this study, we used the R0 to assess the transmissibility of the SARS-CoV-2. Commonly, R0 was defined as the expected number of secondary infections that result from introducing a single infected individual into an otherwise susceptible population [13, 16, 17]. If R0 \u003e 1, the outbreak will occur. If R0 \u003c 1, the outbreak will toward an end. In this study, R0 was deduced from the RP model by the next generation matrix approach [18].\n\nParameter estimation\nThe parameters were estimated based on the following facts and assumptions: The mean incubation period was 5.2 days (95% confidence interval [CI]: 4.1–7.0) [3]. We set the same value (5.2 days) of the incubation period and the latent period in this study. Thus, ωP = ω’P = 0.1923.\nThere is a mean 5-day delay from symptom onset to detection/hospitalization of a case (the cases detected in Thailand and Japan were hospitalized from 3 to 7 days after onset, respectively) [19–21]. The duration from illness onset to first medical visit for the 45 patients with illness onset before January 1 was estimated to have a mean of 5.8 days (95% CI: 4.3–7.5) [3]. In our model, we set the infectious period of the cases as 5.8 days. Therefore, γP = 0.1724.\nSince there was no data on the proportion of asymptomatic infection of the virus, we simulated the baseline value of proportion of 0.5 (δP = 0.5).\nSince there was no evidence about the transmissibility of asymptomatic infection, we assumed that the transmissibility of asymptomatic infection was 0.5 times that of symptomatic infection (κ = 0.5), which was the similar value as influenza [22]. We assumed that the relative shedding rate of AP compared to IP was 0.5. Thus, c = 0.5.\nSince 14 January, 2020, Wuhan City has strengthened the body temperature detection of passengers leaving Wuhan at airports, railway stations, long-distance bus stations and passenger terminals. As of January 17, a total of nearly 0.3 million people had been tested for body temperature [23]. In Wuhan, there are about 2.87 million mobile population [24]. We assumed that there was 0.1 million people moving out to Wuhan City per day since January 10, 2020, and we believe that this number would increase (mainly due to the winter vacation and the Chinese New Year holiday) until 24 January, 2020. This means that the 2.87 million would move out from Wuhan City in about 14 days. Therefore, we set the moving volume of 0.2 million per day in our model. Since the population of Wuhan was about 11 million at the end of 2018 [25], the rate of people traveling out from Wuhan City would be 0.018 (0.2/11) per day. However, we assumed that the normal population mobility before January 1 was 0.1 times as that after January 10. Therefore, we set the rate of people moving into and moving out from Wuhan City as 0.0018 per day (nP = mP = 0.0018).\nThe parameters bP and bW were estimated by fitting the model with the collected data.\nAt the beginning of the simulation, we assumed that the prevalence of the virus in the market was 1/100000.\nSince the SARS-CoV-2 is an RNA virus, we assumed that it could be died in the environment in a short time, but it could be stay for a longer time (10 days) in the unknown hosts in the market. 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