NovDemo-3: Scenario Templates

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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:

• $N = S + E + V+ I +R + H = 100000$ people is the total population excluding deceased individuals
• $\beta = 0.60 \text{𝑛𝑒𝑤 𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛𝑠 𝑝𝑒𝑟 𝑖𝑛𝑓𝑒𝑐𝑡𝑒𝑑 𝑝𝑒𝑟𝑠𝑜𝑛/ 𝑑𝑎𝑦}$ is the transmission rate,
• $\nu = 0.005 / 𝑑𝑎y$ is the vaccination rate,
• $\sigma = 0.10 / 𝑑𝑎𝑦$ is the rate exposed and unvaccinated individuals become infected,
• $\epsilon = 0.05 / 𝑑𝑎𝑦$ is the rate exposed and vaccinated individuals become infected,
• $\delta = 0.05 / 𝑑𝑎𝑦$ is the rate infected individuals recover,
• $\gamma = 0.02 / 𝑑𝑎𝑦$ is the rate infected individuals become hospitalized,
• $\alpha = 0.01 / 𝑑𝑎𝑦$ is the death rate for infected individuals who are not hospitalized,
• $\rho = 0.10 / 𝑑𝑎𝑦$ is the rate hospitalized individuals recover, and
• $\mu = 0.02 / 𝑑𝑎𝑦$ is the death rate for hospitalized individuals

For initial conditions, assume $𝑆(0) = 99900$ people, $𝐼(0) = 100$ people, and all other states begin with $0$ people.

1. 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) – 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:
• $\gamma = 0.02 /𝑑𝑎𝑦$, the rate at which infected individuals become hospitalized
• $\epsilon = 0.05/𝑑𝑎𝑦$, the rate exposed and vaccinated individuals become infected
Vary each of these parameters from ½ of their given value in Q0, to $1.5$ times their given value. In other words, let $\gamma$ vary from $0.01/𝑑𝑎𝑦$ to $0.03/𝑑𝑎𝑦$ and let $\epsilon$ vary from $0.025/𝑑𝑎𝑦$ to $0.075/𝑑𝑎𝑦$. Explore the sensitivity of the model output to these two parameters.
3. 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?

3. 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 ½ 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 $\epsilon$?

4. 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.

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:

• $S(0) = 79900$ 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.