American Institute of Mathematical Sciences

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April  2018, 15(2): 337-359. doi: 10.3934/mbe.2018015

The potential impact of a prophylactic vaccine for Ebola in Sierra Leone

Erin N. Bodine , , Connor Cook and  Mikayla Shorten

Rhodes College, Department of Mathematics & Computer Science, 2000 N. Parkway, Memphis, TN 38112, USA

* Corresponding author: Erin N. Bodine.

Received  July 12, 2016 Accepted  April 05, 2017 Published  June 2017

The 2014 outbreak of Ebola virus disease (EVD) in West Africa was multinational and of an unprecedented scale primarily affecting the countries of Guinea, Liberia, and Sierra Leone. One of the qualities that makes EVD of high public concern is its potential for extremely high mortality rates (up to 90%). A prophylactic vaccine for ebolavirus (rVSV-ZEBOV) has been developed, and clinical trials show near-perfect efficacy. We have developed an ordinary differential equations model that simulates an EVD epidemic and takes into account (1) transmission through contact with infectious EVD individuals and deceased EVD bodies, (2) the heterogeneity of the risk of becoming infected with EVD, and (3) the increased survival rate of infected EVD patients due to greater access to trained healthcare providers. Using fitted parameter values that closely simulate the dynamics of the 2014 outbreak in Sierra Leone, we utilize our model to predict the potential impact of a prophylactic vaccine for the ebolavirus using various vaccination strategies including ring vaccination. Our results show that an rVSV-ZEBOV vaccination coverage as low as 40% in the general population and 95% in healthcare workers will prevent another catastrophic outbreak like the 2014 outbreak from occurring.

Keywords: Ebolavirus, prophylactic vaccine, vaccine coverage, ring vaccination, ordinary differential equations, epidemiology, population dynamics, Sierra Leone.
Mathematics Subject Classification: Primary: 92B05, 92D30; Secondary: 92-08.
Citation: Erin N. Bodine, Connor Cook, Mikayla Shorten. The potential impact of a prophylactic vaccine for Ebola in Sierra Leone. Mathematical Biosciences & Engineering, 2018, 15 (2) : 337-359. doi: 10.3934/mbe.2018015
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References:
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[5]

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[6]

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[7]

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[8]

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12].">Figure 1.  Histogram of mortality rates for the 19 outbreaks of EVD that have had more than 5 reported cases (excluding the 2014 West Africa Outbreak); outbreaks of Zaire ebolavirus are shown in solid red while all other outbreaks are shown in cross-hatched blue. Data taken from [12].
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Figure 2.  Timeline showing the implementation of different control strategies over the course of the outbreak. The date for time $t=0$ is 30 days prior to the initial date that the CDC began recording outbreak data.
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Figure 3.  Flow diagram of the model given in System (1). Shaded compartments indicate high risk of exposure (red), low risk of exposure (orange), and ability to transmit EVD (yellow). Green arrows indicate movement into the population, blue arrows indicate background death, and dashed line arrows indicate vaccination.
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9] of cumulative infections (A) and deaths (B) over the five distinct intervention periods given in Figure 2.">Figure 4.  Model (solid curve) fit to CDC data (points) [9] of cumulative infections (A) and deaths (B) over the five distinct intervention periods given in Figure 2.
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Figure 5.  $\mathcal{R}_A(t)$ for response periods $[0,95]$ (red), $[95,202]$ (blue), $[202,248]$ (orange), $[248,460]$ (green), and $[460,647]$ (purple).
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Figure 6.  Matrix plots displaying the sensitivity of the parameters $\alpha$, $x$, $\nu$, and $\rho_L$ to the cumulative number of infected individuals (CI) during a ring vaccination scenario with $\epsilon=1$, $\rho_H={\raise0.5ex\hbox{$\scriptstyle 0.95$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 21$}}$, and $\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.99$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 7$}}$.
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Table 1.  Parameters and values used in the model.
Parameter Units Value Description Source
Fitted Parameters $\beta_1^H$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from infectious individuals to high risk susceptible individuals
$\beta_1^L$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from infectious individuals to low risk susceptible individuals
$\beta_1^W$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from infectious individuals to susceptible healthcare workers
$\beta_2^H$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from deceased individuals to high risk susceptible individuals
$\beta_2^L$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from deceased individuals to low risk susceptible individuals
$\beta_2^W$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from deceased individuals to susceptible healthcare workers
$\beta_{3}$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from recovering, still infectious individuals to low risk individuals
$\alpha$ -- See Table 3 proportion of symptomatic individuals who go to hospitals
$\nu$ -- See Table 3 scaling constant where $1-\nu$ represents effectiveness of healthcare workers in reducing EVD death rate
Vax $\epsilon$ -- $1$ proportion of individuals in whom the vaccine is effective [31]
$\rho_L$ -- See Section 5 proportion of low susceptible population ($S_L$) who get vaccinated per year
$\rho_H$ -- See Section 5 proportion of high susceptible population ($S_H$) who get vaccinated per year
$\hat{\rho}$ -- See Section 5 proportion of healthcare workers ($S_W$) who get vaccinated per year
$\psi$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See section 3 migration rate of healthcare workers into the population
$\mu$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ ${\raise0.5ex\hbox{$\scriptstyle 11.03$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle (1,000)(365)$}}$ natural death rate per day per 1,000 individuals [16]
$\Omega$ ${\raise0.5ex\hbox{$\scriptstyle people$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ ${\raise0.5ex\hbox{$\scriptstyle 37.4$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle (1,000)(365)$}}$ natural birth rate per day per 1,000 individuals [16]
$\kappa$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 5.5$}}$ rate at which individuals become symptomatic and infectious [45]
${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle \gamma$}}$ $days$ $9$ infectious period of individuals who are not in hospitals [50]
${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle \hat{\gamma}$}}$ $days$ $7$ infectious period of individuals who are in hospitals
$\delta$ -- $0.73$ proportion of individuals who die from Ebola [12]
$\phi$ $people$ $0.00039N(0)$ number of healthcare workers in the population prior to outbreak [16]
$\lambda$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 5$}}$ burial rate of deceased individuals who were not in hospitals
$\hat{\lambda}$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}$ burial rate of deceased individuals who were in hospitals [50]
$\eta$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 150$}}$ rate at which recovering and still infectious individuals become non-infectious [18,15]
$\sigma_H$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ $\kappa$ rate at which individuals move from low risk to high risk susceptible populations
$\sigma_L$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ $1/({\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle \gamma$}}+{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle \lambda$}})$ rate at which individuals move from high risk to low risk susceptible populations
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Parameter Units Value Description Source
Fitted Parameters $\beta_1^H$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from infectious individuals to high risk susceptible individuals
$\beta_1^L$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from infectious individuals to low risk susceptible individuals
$\beta_1^W$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from infectious individuals to susceptible healthcare workers
$\beta_2^H$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from deceased individuals to high risk susceptible individuals
$\beta_2^L$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from deceased individuals to low risk susceptible individuals
$\beta_2^W$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from deceased individuals to susceptible healthcare workers
$\beta_{3}$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See Table 3 transmission rate from recovering, still infectious individuals to low risk individuals
$\alpha$ -- See Table 3 proportion of symptomatic individuals who go to hospitals
$\nu$ -- See Table 3 scaling constant where $1-\nu$ represents effectiveness of healthcare workers in reducing EVD death rate
Vax $\epsilon$ -- $1$ proportion of individuals in whom the vaccine is effective [31]
$\rho_L$ -- See Section 5 proportion of low susceptible population ($S_L$) who get vaccinated per year
$\rho_H$ -- See Section 5 proportion of high susceptible population ($S_H$) who get vaccinated per year
$\hat{\rho}$ -- See Section 5 proportion of healthcare workers ($S_W$) who get vaccinated per year
$\psi$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ See section 3 migration rate of healthcare workers into the population
$\mu$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ ${\raise0.5ex\hbox{$\scriptstyle 11.03$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle (1,000)(365)$}}$ natural death rate per day per 1,000 individuals [16]
$\Omega$ ${\raise0.5ex\hbox{$\scriptstyle people$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ ${\raise0.5ex\hbox{$\scriptstyle 37.4$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle (1,000)(365)$}}$ natural birth rate per day per 1,000 individuals [16]
$\kappa$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 5.5$}}$ rate at which individuals become symptomatic and infectious [45]
${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle \gamma$}}$ $days$ $9$ infectious period of individuals who are not in hospitals [50]
${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle \hat{\gamma}$}}$ $days$ $7$ infectious period of individuals who are in hospitals
$\delta$ -- $0.73$ proportion of individuals who die from Ebola [12]
$\phi$ $people$ $0.00039N(0)$ number of healthcare workers in the population prior to outbreak [16]
$\lambda$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 5$}}$ burial rate of deceased individuals who were not in hospitals
$\hat{\lambda}$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}$ burial rate of deceased individuals who were in hospitals [50]
$\eta$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 150$}}$ rate at which recovering and still infectious individuals become non-infectious [18,15]
$\sigma_H$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ $\kappa$ rate at which individuals move from low risk to high risk susceptible populations
$\sigma_L$ ${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle days$}}$ $1/({\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle \gamma$}}+{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle \lambda$}})$ rate at which individuals move from high risk to low risk susceptible populations
Table 3.  Values of fitted parameters for each response period. The range of the uniform distribution for each parameter for each response period is shown above the fitted parameter value.
$\mathbf{t\!\in\![0,95]}$ $\mathbf{t\!\in\![95,202]}$ $\mathbf{t\!\in\![202,248]}$ $\mathbf{t\!\in\![248,460]}$ $\mathbf{t\!\in\![460,647]}$
$\mathbf{\beta^H_1}$ $[0.45,0.80]$ $[0.30,0.80]$ $[0.30,0.80]$ $[0.20,0.80]$ $[0.20,0.80]$
$0.651$ $0.441$ $0.336$ $0.586$ $0.640$
$\mathbf{\beta^H_2}$ $[0.45,0.80]$ $[0.30,0.80]$ $[0.30,0.80]$ $[0.20,0.80]$ $[0.20,0.80]$
$0.471$ $0.437$ $0.671$ $0.572$ $0.376$
$\mathbf{\beta^L_1}$ $[0.10,0.45]$ $[0.10,0.30]$ $[0.10,0.30]$ $[0.01,0.20]$ $[0.01,0.20]$
$0.163$ $0.263$ $0.212$ $0.198$ $0.182$
$\mathbf{\beta^L_2}$ $[0.10,0.45]$ $[0.10,0.30]$ $[0.10,0.30]$ $[0.01,0.20]$ $[0.01,0.20]$
$0.294$ $0.139$ $0.204$ $0.143$ $0.146$
$\mathbf{\beta^W_1}$ $[0.10,0.80]$ $[0.10,0.80]$ $[0.10,0.80]$ $[0.01,0.80]$ $[0.01,0.80]$
$0.436$ $0.319$ $0.296$ $0.356$ $0.532$
$\mathbf{\beta^W_2}$ $[0.10,0.80]$ $[0.10,0.80]$ $[0.10,0.80]$ $[0.01,0.80]$ $[0.01,0.80]$
$0.720$ $0.753$ $0.325$ $0.348$ $0.322$
$\mathbf{\beta_3}$ $[1 \times 10^{-7},\,0.001]$ $[1 \times 10^{-7},\,0.001]$ $[1 \times 10^{-7},\,0.001]$ $[1 \times 10^{-7},\,0.001]$ $[1 \times 10^{-7},\,0.001]$
$0.000765$ $0.0000861$ $0.000961$ $0.000596$ $0.000996$
$\mathbf{\alpha}$ $[0.25,0.75]$ $[0.25,0.75]$ $[0.25,0.75]$ $[0.25,0.75]$ $[0.25,0.75]$
$0.407$ $0.572$ $0.744$ $0.659$ $0.723$
$\mathbf{\nu}$ $[0.0,1.0]$ $[0.0,1.0]$ $[0.0,1.0]$ $[0.0,1.0]$ $[0.0,1.0]$
$0.691$ $0.019$ $0.022$ $0.483$ $0.156$
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$\mathbf{t\!\in\![0,95]}$ $\mathbf{t\!\in\![95,202]}$ $\mathbf{t\!\in\![202,248]}$ $\mathbf{t\!\in\![248,460]}$ $\mathbf{t\!\in\![460,647]}$
$\mathbf{\beta^H_1}$ $[0.45,0.80]$ $[0.30,0.80]$ $[0.30,0.80]$ $[0.20,0.80]$ $[0.20,0.80]$
$0.651$ $0.441$ $0.336$ $0.586$ $0.640$
$\mathbf{\beta^H_2}$ $[0.45,0.80]$ $[0.30,0.80]$ $[0.30,0.80]$ $[0.20,0.80]$ $[0.20,0.80]$
$0.471$ $0.437$ $0.671$ $0.572$ $0.376$
$\mathbf{\beta^L_1}$ $[0.10,0.45]$ $[0.10,0.30]$ $[0.10,0.30]$ $[0.01,0.20]$ $[0.01,0.20]$
$0.163$ $0.263$ $0.212$ $0.198$ $0.182$
$\mathbf{\beta^L_2}$ $[0.10,0.45]$ $[0.10,0.30]$ $[0.10,0.30]$ $[0.01,0.20]$ $[0.01,0.20]$
$0.294$ $0.139$ $0.204$ $0.143$ $0.146$
$\mathbf{\beta^W_1}$ $[0.10,0.80]$ $[0.10,0.80]$ $[0.10,0.80]$ $[0.01,0.80]$ $[0.01,0.80]$
$0.436$ $0.319$ $0.296$ $0.356$ $0.532$
$\mathbf{\beta^W_2}$ $[0.10,0.80]$ $[0.10,0.80]$ $[0.10,0.80]$ $[0.01,0.80]$ $[0.01,0.80]$
$0.720$ $0.753$ $0.325$ $0.348$ $0.322$
$\mathbf{\beta_3}$ $[1 \times 10^{-7},\,0.001]$ $[1 \times 10^{-7},\,0.001]$ $[1 \times 10^{-7},\,0.001]$ $[1 \times 10^{-7},\,0.001]$ $[1 \times 10^{-7},\,0.001]$
$0.000765$ $0.0000861$ $0.000961$ $0.000596$ $0.000996$
$\mathbf{\alpha}$ $[0.25,0.75]$ $[0.25,0.75]$ $[0.25,0.75]$ $[0.25,0.75]$ $[0.25,0.75]$
$0.407$ $0.572$ $0.744$ $0.659$ $0.723$
$\mathbf{\nu}$ $[0.0,1.0]$ $[0.0,1.0]$ $[0.0,1.0]$ $[0.0,1.0]$ $[0.0,1.0]$
$0.691$ $0.019$ $0.022$ $0.483$ $0.156$
Table 2.  Weight equation parameter values for each response period.
$\mathit{\boldsymbol{t}}\mathbf{\in[0,95]}$ $\mathit{\boldsymbol{t}}\mathbf{\in[95,202]}$ $\mathit{\boldsymbol{t}}\mathbf{\in[202,248]}$ $\mathit{\boldsymbol{t}}\mathbf{\in[248,460]}$ $\mathit{\boldsymbol{t}}\mathbf{\in[460,647]}$
$\mathit{\boldsymbol{a}}$ 0.015 0.030 0.060 0.013 0.013
$\mathit{\boldsymbol{b}}$ 40 20 10 50 45
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$\mathit{\boldsymbol{t}}\mathbf{\in[0,95]}$ $\mathit{\boldsymbol{t}}\mathbf{\in[95,202]}$ $\mathit{\boldsymbol{t}}\mathbf{\in[202,248]}$ $\mathit{\boldsymbol{t}}\mathbf{\in[248,460]}$ $\mathit{\boldsymbol{t}}\mathbf{\in[460,647]}$
$\mathit{\boldsymbol{a}}$ 0.015 0.030 0.060 0.013 0.013
$\mathit{\boldsymbol{b}}$ 40 20 10 50 45
Table 4.  $\bar{\mathcal{R}}_A(t)$ for each response period.
Response period $\bar{\mathcal{R}}_A(t)$
$t\in[0,95]$ 3.31
$t\in[95,202]$ 1.88
$t\in[202,248]$ 0.90
$t\in[248,460]$ 0.27
$t\in[460,647]$ 0.16
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Response period $\bar{\mathcal{R}}_A(t)$
$t\in[0,95]$ 3.31
$t\in[95,202]$ 1.88
$t\in[202,248]$ 0.90
$t\in[248,460]$ 0.27
$t\in[460,647]$ 0.16
Table 5.  Impact of prophylactic vaccine distributed during the outbreak on cumulative infections and deaths
Cumulative Infections Cumulative Deaths
at tf Prevented at tf Prevented
Baseline$^*$ 14436 0 4416 0
$\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.9$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ Start vax at $t=95$ 8863 5573 2738 1678
$\rho_L={\raise0.5ex\hbox{$\scriptstyle 0.3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $t=202$ 12583 1853 3837 579
$\rho_H={\raise0.5ex\hbox{$\scriptstyle 0.3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $t=248$ 13225 1211 4050 366
$t=460$ 14325 111 4392 24
$\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.9$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ Start vax at $t=95$ 5138 9298 1656 2760
$\rho_L={\raise0.5ex\hbox{$\scriptstyle 0.9$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $t=202$ 11075 3361 3343 1073
$\rho_H={\raise0.5ex\hbox{$\scriptstyle 0.9$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $t=248$ 12168 2268 3708 708
$t=460$ 14173 263 4357 59
$\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.99$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 7$}}$ Start vax at $t=95$ 13999 437 4283 133
$\rho_L={\raise0.5ex\hbox{$\scriptstyle 0.01$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $t=202$ 14285 151 4369 47
$\rho_H={\raise0.5ex\hbox{$\scriptstyle 0.95$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 21$}}$ $t=248$ 14351 85 4391 25
$t=460$ 14429 7 4415 1
$\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.99$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 7$}}$ Start vax at $t=95$ 8821 5615 2726 1690
$\rho_L={\raise0.5ex\hbox{$\scriptstyle 0.3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $t=202$ 12553 1883 3827 589
$\rho_H={\raise0.5ex\hbox{$\scriptstyle 0.95$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 21$}}$ $t=248$ 13211 1225 4046 370
$t=460$ 14323 113 4391 25
$^*$Cumulative number of infections or deaths at $t=647$, as predicted by System (1) with parameters from Tables 1 & 3, when no prophylactic vaccine is administered.
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Cumulative Infections Cumulative Deaths
at tf Prevented at tf Prevented
Baseline$^*$ 14436 0 4416 0
$\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.9$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ Start vax at $t=95$ 8863 5573 2738 1678
$\rho_L={\raise0.5ex\hbox{$\scriptstyle 0.3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $t=202$ 12583 1853 3837 579
$\rho_H={\raise0.5ex\hbox{$\scriptstyle 0.3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $t=248$ 13225 1211 4050 366
$t=460$ 14325 111 4392 24
$\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.9$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ Start vax at $t=95$ 5138 9298 1656 2760
$\rho_L={\raise0.5ex\hbox{$\scriptstyle 0.9$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $t=202$ 11075 3361 3343 1073
$\rho_H={\raise0.5ex\hbox{$\scriptstyle 0.9$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $t=248$ 12168 2268 3708 708
$t=460$ 14173 263 4357 59
$\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.99$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 7$}}$ Start vax at $t=95$ 13999 437 4283 133
$\rho_L={\raise0.5ex\hbox{$\scriptstyle 0.01$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $t=202$ 14285 151 4369 47
$\rho_H={\raise0.5ex\hbox{$\scriptstyle 0.95$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 21$}}$ $t=248$ 14351 85 4391 25
$t=460$ 14429 7 4415 1
$\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.99$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 7$}}$ Start vax at $t=95$ 8821 5615 2726 1690
$\rho_L={\raise0.5ex\hbox{$\scriptstyle 0.3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $t=202$ 12553 1883 3827 589
$\rho_H={\raise0.5ex\hbox{$\scriptstyle 0.95$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 21$}}$ $t=248$ 13211 1225 4046 370
$t=460$ 14323 113 4391 25
$^*$Cumulative number of infections or deaths at $t=647$, as predicted by System (1) with parameters from Tables 1 & 3, when no prophylactic vaccine is administered.
Table 6.  Effect of vaccination prior to an initial outbreak (1 infected). Vaccination coverage of healthcare workers is 0% when $x=0$ and 95% otherwise.
Vaccination strategy for $t>0$
$\rho_L = 0$ $\rho_L = {\raise0.5ex\hbox{$\scriptstyle 0.3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $\rho_L = {\raise0.5ex\hbox{$\scriptstyle 0.01$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $\rho_L = {\raise0.5ex\hbox{$\scriptstyle 0.3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$
$\rho_H = 0$ $\rho_H = {\raise0.5ex\hbox{$\scriptstyle 0.3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $\rho_H = {\raise0.5ex\hbox{$\scriptstyle 0.95$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 21$}}$ $\rho_H = {\raise0.5ex\hbox{$\scriptstyle 0.95$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 21$}}$
$\hat{\rho} = 0$ $\hat{\rho} = {\raise0.5ex\hbox{$\scriptstyle 0.9$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.99$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 7$}}$ $\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.99$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 7$}}$
CI $\mathbf{t^*}$ CI $\mathbf{t^*}$ CI $\mathbf{t^*}$ CI $\mathbf{t^*}$
0 4156370 495 3743965 516 650846 492 2643460 492
0.10 3064971 648 2506362 694 1853742 644 1845334 644
0.20 1866043 974 1026659 1138 1040091 987 1030788 988
0.29 764174 1990 37 103 188898 2267 188898 2267
0.30 646329 2270 33 0 82717 2546 82717 2546
0.31 529673 2657 29 0 19911 2502 19911 2502
0.32 414210 3213 25 0 3866 2091 3866 2091
0.33 300060 4090 23 0 954 1570 954 1570
0.34 171811 5697 20 0 325 1023 325 1023
0.35 13195 7300 18 0 147 0 147 0
0.36 431 0 16 0 81 0 81 0
0.40 26 0 20 0 23 0 23 0
0.50 7 0 7 0 7 0 7 0
0.60 4 0 4 0 4 0 4 0
0.70 2 0 2 0 2 0 2 0
0.80 2 0 2 0 2 0 2 0
0.90 1 0 1 0 1 0 1 0
$t^*$ is the day at which the peak number of cases occurs; CI represents cumulative infections calculated once $I(t)+\hat{I}(t) < 1$
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Vaccination strategy for $t>0$
$\rho_L = 0$ $\rho_L = {\raise0.5ex\hbox{$\scriptstyle 0.3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $\rho_L = {\raise0.5ex\hbox{$\scriptstyle 0.01$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $\rho_L = {\raise0.5ex\hbox{$\scriptstyle 0.3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$
$\rho_H = 0$ $\rho_H = {\raise0.5ex\hbox{$\scriptstyle 0.3$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $\rho_H = {\raise0.5ex\hbox{$\scriptstyle 0.95$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 21$}}$ $\rho_H = {\raise0.5ex\hbox{$\scriptstyle 0.95$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 21$}}$
$\hat{\rho} = 0$ $\hat{\rho} = {\raise0.5ex\hbox{$\scriptstyle 0.9$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 365$}}$ $\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.99$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 7$}}$ $\hat{\rho}={\raise0.5ex\hbox{$\scriptstyle 0.99$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 7$}}$
CI $\mathbf{t^*}$ CI $\mathbf{t^*}$ CI $\mathbf{t^*}$ CI $\mathbf{t^*}$
0 4156370 495 3743965 516 650846 492 2643460 492
0.10 3064971 648 2506362 694 1853742 644 1845334 644
0.20 1866043 974 1026659 1138 1040091 987 1030788 988
0.29 764174 1990 37 103 188898 2267 188898 2267
0.30 646329 2270 33 0 82717 2546 82717 2546
0.31 529673 2657 29 0 19911 2502 19911 2502
0.32 414210 3213 25 0 3866 2091 3866 2091
0.33 300060 4090 23 0 954 1570 954 1570
0.34 171811 5697 20 0 325 1023 325 1023
0.35 13195 7300 18 0 147 0 147 0
0.36 431 0 16 0 81 0 81 0
0.40 26 0 20 0 23 0 23 0
0.50 7 0 7 0 7 0 7 0
0.60 4 0 4 0 4 0 4 0
0.70 2 0 2 0 2 0 2 0
0.80 2 0 2 0 2 0 2 0
0.90 1 0 1 0 1 0 1 0
$t^*$ is the day at which the peak number of cases occurs; CI represents cumulative infections calculated once $I(t)+\hat{I}(t) < 1$
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