For Questions 1-3, use the SEVIRHD model described in the Model Definition below. For Question 4, you will be using a different model (defined in the question).
0. Model Definition
The compartments for this SEVIRHD model are Susceptible, Exposed, Vaccinated, Infected, Hospitalized, Recovered, and Dead.
The base model is defined as follows:
$$\begin{align}
\frac{dS}{dt} & = -\beta\cdot\frac{S\cdot I}{N} - (\nu\cdot S) \
\frac{dE}{dt} & = \beta\cdot\frac{S\cdot I}{N} + \epsilon\cdot\frac{V\cdot I}{N} -\sigma\cdot E \
\frac{dV}{dt} & = \nu\cdot S - \epsilon\cdot\frac{V\cdot I}{N} \
\frac{dI}{dt} & = \sigma\cdot E - (\gamma + \delta + \alpha)\cdot I \
\frac{dR}{dt} &= \delta\cdot I + \rho\cdot H \
\frac{dH}{dt} &= \gamma\cdot I - (\rho + \mu)\cdot H \
\frac{dD}{dt} &= \alpha\cdot I + \mu\cdot H
\end{align}$$
Where:
$N = S + E + V+ I +R + H = 100000$ people is the total population excluding deceased individuals
$\beta = 0.60 \text{πππ€ ππππππ‘ππππ πππ ππππππ‘ππ ππππ ππ/ πππ¦}$ is the transmission rate,
$\nu = 0.005 / ππy$ is the vaccination rate,
$\sigma = 0.10 / πππ¦$ is the rate exposed and unvaccinated individuals become infected,
$\epsilon = 0.05 / πππ¦$ is the rate exposed and vaccinated individuals become infected,
$\delta = 0.05 / πππ¦$ is the rate infected individuals recover,
$\gamma = 0.02 / πππ¦$ is the rate infected individuals become hospitalized,
$\alpha = 0.01 / πππ¦$ is the death rate for infected individuals who are not hospitalized,
$\rho = 0.10 / πππ¦$ is the rate hospitalized individuals recover, and
$\mu = 0.02 / πππ¦$ is the death rate for hospitalized individuals
For initial conditions, assume $π(0) = 99900$ people, $πΌ(0) = 100$ people, and all other states begin with $0$ people.
Sensitivity Analysis: For this question we do a sensitivity analysis to understand how model outcomes are impacted by sources of uncertainty (which are not under the control of the decisionmaker) β see Figure 2 describing a sensitivity analysis example.
Figure 2. Sensitivity analysis example plotted against two uncertainty axes.
For this question, use the model defined in Q0 and assume there is uncertainty in the following parameters:
β’ $\gamma = 0.02 /πππ¦$, the rate at which infected individuals become hospitalized
β’ $\epsilon = 0.05/πππ¦$, the rate exposed and vaccinated individuals become infected
Vary each of these parameters from Β½ of their given value in Q0, to $1.5$ times their given value. In other words, let $\gamma$ vary from $0.01/πππ¦$ to $0.03/πππ¦$ and let $\epsilon$ vary from $0.025/πππ¦$ to $0.075/πππ¦$. Explore the sensitivity of the model output to these two parameters.
Decisionmaker Question: For this question we construct a decisionmaker question looking at multiple intervention options (which are under control of the decisionmaker) with no sources of uncertainty in the parameters. See Figure 3 describing a decision-maker example.
Figure 3. Decision-maker examples in 1D and 2D, mapped on intervention axes
Consider the following intervention options:
Baseline (change nothing from Q0 model configuration)
Option 1
Limited mask intervention: assume 15% decrease in transmission rates
Limited vaccination campaign: increase vaccination rate to 1% of susceptible population being vaccinated daily
Option 2
Stringent mask intervention: assume 50% decrease in transmission rates
Limited vaccination campaign: increase vaccination rate to 1% of susceptible population being vaccinated daily
Option 3
Limited mask intervention: assume 15% decrease in transmission rates
Stronger vaccination campaign: increase vaccination rate to 1.5% of susceptible population being vaccinated daily
Option 4
Stringent mask intervention: assume 50% decrease in transmission rates
Stronger vaccination campaign: increase vaccination rate to 1.5% of susceptible population being vaccinated daily
These options are summarized in the following parameter table:
Q2 Parameter Table
Masking
Vaccination
Option 1
Limited masking
Limited vaccination
$\beta = 0.51$ new infections per infected person/day
$\nu = 0.01/\text{day}$
Option 2
More stringent masking
Limited vaccination
$\beta = 0.30$ new infections per infected person/day
$\nu = 0.01/\text{day}$
Option 3
Limited masking
Higher vaccination rate
$\beta = 0.51$ new infections per infected person/day
$\nu = 0.015/\text{day}$
Option 4
More stringent masking
Higher vaccination rate
$\beta = 0.30$ new infections per infected person/day
$\nu = 0.015/\text{day}$
What are the expected model outputs (cumulative infections, cumulative hospitalizations, total deaths) after 200 days, for each of these 4 intervention options, and which option shows the greatest improvement over the baseline?
Value of Information: For this question, we build upon Q2, and explore how choice of intervention (which is under control of the decisionmaker) is impacted by sources of uncertainty (which are not under control of the decisionmaker)? See Figure 4 describing a value of information example.
Figure 4. Value of information example mapped to uncertainty and intervention axes
In this question we keep the same decisionmaker intervention options from Q2, and we additionally consider uncertainty in the following two parameters:
$\gamma = 0.02/\text{day}$, the rate at which infected individuals become hospitalized
$\epsilon = 0.05/\text{day}$, the rate exposed and vaccinated individuals become infected
Vary each of these parameters from Β½ of their given value in Q0, to 1.5 times their given value. In other words, let $\gamma$ vary from $0.01/\text{day}$ to $0.03/\text{day}$ and let $\epsilon$ vary from $0.025/\text{day}$ to $0.075/\text{day}$.
With this parameter uncertainty, how do model outputs (cumulative infections, cumulative hospitalizations, total deaths) after 50 days, for each of the 4 decisionmaker intervention options, compare with Q2 (which did not have parameter uncertainty)? How is your choice of the best intervention option affected by the uncertainty in $\gamma$ and $\epsilon$?
Horizon Scanning: For this question, we consider a new model, and there are no dependencies on Q0-Q3. Assume there is an emerging pathogen with unknown transmissibility and unknown severity. We are interested in conducting a horizon scan to get a sense of how lethal the pathogen could be.
Where:
$N = S + E + I + R = 80000$ people is the total population,
$\beta = 0.50$ new infections per infected person/day is the transmission rate,
$\sigma = 0.15$/day is the rate exposed individuals become infected, and
$\mu = 0.075$/day is the rate infected individuals die.
Use the following initial conditions:
$S(0) = 79900$ people
$E(0) = 0$ people
$I(0) = 100$ people
$D(0) = 0$ people
Given the huge amount of uncertainty around the emerging pathogen, conduct a horizon scan by simulating model outcomes when the lower and upper bounds of transmission rate $\beta$ are 0.2/day and 0.8/day, and simultaneously, the lower and upper bounds of death rate $\mu$ are 0.05/day and 0.10/day. Compare total deaths and total infections at the end of the simulation as the outcomes of interest.
Scenario 3: Scenario Templates
For Questions 1-3, use the SEVIRHD model described in the Model Definition below. For Question 4, you will be using a different model (defined in the question).
0. Model Definition
The compartments for this SEVIRHD model are Susceptible, Exposed, Vaccinated, Infected, Hospitalized, Recovered, and Dead. The base model is defined as follows:
$$\begin{align} \frac{dS}{dt} & = -\beta\cdot\frac{S\cdot I}{N} - (\nu\cdot S) \ \frac{dE}{dt} & = \beta\cdot\frac{S\cdot I}{N} + \epsilon\cdot\frac{V\cdot I}{N} -\sigma\cdot E \ \frac{dV}{dt} & = \nu\cdot S - \epsilon\cdot\frac{V\cdot I}{N} \ \frac{dI}{dt} & = \sigma\cdot E - (\gamma + \delta + \alpha)\cdot I \ \frac{dR}{dt} &= \delta\cdot I + \rho\cdot H \ \frac{dH}{dt} &= \gamma\cdot I - (\rho + \mu)\cdot H \ \frac{dD}{dt} &= \alpha\cdot I + \mu\cdot H \end{align}$$
Where:
For initial conditions, assume $π(0) = 99900$ people, $πΌ(0) = 100$ people, and all other states begin with $0$ people.
Figure 2. Sensitivity analysis example plotted against two uncertainty axes.
For this question, use the model defined in Q0 and assume there is uncertainty in the following parameters: β’ $\gamma = 0.02 /πππ¦$, the rate at which infected individuals become hospitalized β’ $\epsilon = 0.05/πππ¦$, the rate exposed and vaccinated individuals become infected Vary each of these parameters from Β½ of their given value in Q0, to $1.5$ times their given value. In other words, let $\gamma$ vary from $0.01/πππ¦$ to $0.03/πππ¦$ and let $\epsilon$ vary from $0.025/πππ¦$ to $0.075/πππ¦$. Explore the sensitivity of the model output to these two parameters.
Figure 3. Decision-maker examples in 1D and 2D, mapped on intervention axes
Consider the following intervention options:
These options are summarized in the following parameter table:
- Limited masking
- Limited vaccination
$\beta = 0.51$ new infections per infected person/day$\nu = 0.01/\text{day}$
- More stringent masking
- Limited vaccination
$\beta = 0.30$ new infections per infected person/day$\nu = 0.01/\text{day}$
- Limited masking
- Higher vaccination rate
$\beta = 0.51$ new infections per infected person/day$\nu = 0.015/\text{day}$
- More stringent masking
- Higher vaccination rate
$\beta = 0.30$ new infections per infected person/day$\nu = 0.015/\text{day}$
What are the expected model outputs (cumulative infections, cumulative hospitalizations, total deaths) after 200 days, for each of these 4 intervention options, and which option shows the greatest improvement over the baseline?
Figure 4. Value of information example mapped to uncertainty and intervention axes
In this question we keep the same decisionmaker intervention options from Q2, and we additionally consider uncertainty in the following two parameters:
Vary each of these parameters from Β½ of their given value in Q0, to 1.5 times their given value. In other words, let $\gamma$ vary from $0.01/\text{day}$ to $0.03/\text{day}$ and let $\epsilon$ vary from $0.025/\text{day}$ to $0.075/\text{day}$.
With this parameter uncertainty, how do model outputs (cumulative infections, cumulative hospitalizations, total deaths) after 50 days, for each of the 4 decisionmaker intervention options, compare with Q2 (which did not have parameter uncertainty)? How is your choice of the best intervention option affected by the uncertainty in $\gamma$ and $\epsilon$?
Use the following SEID model:
$$ \frac{dS}{dt} = -\beta \cdot \frac{S \cdot I}{N} $$
$$ \frac{dE}{dt} = \beta \cdot \frac{S \cdot I}{N} - \sigma \cdot E $$
$$ \frac{dI}{dt} = \mu \cdot I $$
Where:
$N = S + E + I + R = 80000$ people is the total population,
$\beta = 0.50$ new infections per infected person/day is the transmission rate,
$\sigma = 0.15$/day is the rate exposed individuals become infected, and
$\mu = 0.075$/day is the rate infected individuals die.
Use the following initial conditions:
Given the huge amount of uncertainty around the emerging pathogen, conduct a horizon scan by simulating model outcomes when the lower and upper bounds of transmission rate $\beta$ are 0.2/day and 0.8/day, and simultaneously, the lower and upper bounds of death rate $\mu$ are 0.05/day and 0.10/day. Compare total deaths and total infections at the end of the simulation as the outcomes of interest.