# Dynamic Prioritization of COVID-19 Vaccines when Social Distancing is Limited for Essential Workers
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*[J. H. Buckner, G. Chowell, and M. R. Springborn, Dynamic prioritization of COVID-19 vaccines when social distancing is limited for essential workers, PNAS April 20, 2021 118 (16) e2025786118](https://www.pnas.org/content/pnas/118/16/e2025786118.full.pdf)*
- [Supplemental Information](https://www.pnas.org/content/pnas/suppl/2021/04/02/2025786118.DCSupplemental/pnas.2025786118.sapp.pdf)
- existing published studies of vaccination prioritization
- *[Matrajt, J. Eaton, T. Leung, E. R. Brown, Vaccine optimization for COVID-19: Who to vaccinate first? Sci. Adv. 7, eabf1374
(2021).](https://advances.sciencemag.org/content/7/6/eabf1374.abstract)*
- *[K. M. Bubar et al., Model-informed COVID-19 vaccine prioritization strategies by age and serostatus. Science 371, 916–921 (2021).](https://science.sciencemag.org/content/371/6532/916.abstract)*
- cited by
- *[K. M. Bubar, K. Reinholt, S. M. Kissler, Model-informed COVID-19 vaccine prioritization strategies by age and serostatus, Science 26 Feb 2021, pp. 916-921](https://science.sciencemag.org/content/371/6532/916.abstract)*
- *[D. A. Swan, C. Bracis, H. Janes, COVID-19 vaccines that reduce symptoms but do not block infection need higher coverage and faster rollout to achieve population impact, Sci. Rep. vol 11, 5531 (2021)](https://www.nature.com/articles/s41598-021-94719-y)*
- *[E. Wrigley-Field, M. V Kiang, A. R Riley, Geographically-targeted COVID-19 vaccination is more equitable than age-based thresholds alone](https://www.medrxiv.org/content/10.1101/2021.03.25.21254272v1.full.pdf)*

## Abstract

- compartmental model
- 
- demographic group
- ages
- essential worker
- 
- decision period
- policy objective
- minimizing infections
- minimizing years of life lost
- minimizing deaths
- scenarios
- base
- alternative
- weak NPI
- low supply
- ramp up, etc

## Results

- sensitivity of vaccine prioritization


- a broad set of alternative scenarios

- a gradient over four key parameters

- changes to model structures

## Discussion
- Our allocation is **dynamic**, responding to changing epidemiological conditions over a 6-mo period, which is different from other existing published analysis like [_Matrajt et al._](https://doi.org/10.1126/sciadv.abf1374) and [_Bubar et al._](https://doi.org/10.1126/science.abe6959)
- Our results indicate that optimal policies initially target groups with **high risk of infection** and then switch to targeting groups with **high infection fatality**
- We find that working-age adults are a key priority group, particularly **older essential workers**, which may arise from either our allowance for social distancing or dynamic allocation
- Caveats:
- In reality, the risk of infection varies continuously across individuals, even between different “essential” occupations.
- Policy makers should strongly consider occupation-differentiated vaccine allocation strategies
- Further extensions for future work:
- If certain groups experience significant vaccination **side effects**, prioritization might shift away from these groups
- In the longer run, how long lasting immune memory will be is a key unknown
- **Vaccine hesitancy** that is concentrated in a particular community or demographic group could also change the optimal prioritization strategy
- Besides death, YLL, and symptomatic infections, other health-related metrics such as protecting the most vulnerable and **social values** such as returning to school, work, and social life are also important to consider

## Methods
### Model
- A total of eight demographic groups in the set $J$:
1. 0 to 4 years old **(0-4)**
2. 5 to 19 years old **(5-19)**
3. 20 to 39 years old, nonessential workers **(20-39)**
4. 20 to 39 years old, essential workers **(20-39\*)**
5. 40 to 59 years old, nonessential workers **(40-59)**
6. 40 to 59 years old, essential workers **(40-59\*)**
7. 60 to 74 years old **(60-74)**
8. 75 years and older **(75+)**
- For each demographic group, we tracked nine epidemiological states:
1. susceptible **($S$)**
2. protected by a vaccine **($P$)**
3. vaccinated but unprotected **($F$)**
4. exposed **($E$)**
5. presymptomatic **($I_{pre}$)**
6. symptomatic **($I_{sym}$)**
7. asymptomatic **($I_{asym}$)**
8. recovered **($R$)**
9. deceased **($D$)**
- Transitions between epidemiological states:
- Transmission dynamics for each demographic group (indexed by $i$):
$$
\begin{aligned}
\frac{{\rm d}}{{\rm d}t}S_i & = -qs_i\theta \left[\sum_{j\in J}\sum_{m\in M}\tau_mr_{m,i,j}\frac{I_{m,j}}{N_j}\right]S_i - \mu_iv \\
\frac{{\rm d}}{{\rm d}t}F_i & = -qs_i\theta \left[\sum_{j\in J}\sum_{m\in M}\tau_mr_{m,i,j}\frac{I_{m,j}}{N_j}\right]F_i + (1-\epsilon_i)\mu_iv \\
\frac{{\rm d}}{{\rm d}t}E_i & = qs_i\theta \left[\sum_{j\in J}\sum_{m\in M}\tau_mr_{m,i,j}\frac{I_{m,j}}{N_j}\right](S_i+F_i) - \frac{1}{\gamma_{exp}}E_i \\
\frac{{\rm d}}{{\rm d}t}P_i & = \epsilon_i\mu_iv \\
\frac{{\rm d}}{{\rm d}t}I_{pre,i} & = \frac{1}{\gamma_{exp}}E_i - \frac{1}{\gamma_{pre}}I_{pre,i} \\
\frac{{\rm d}}{{\rm d}t}I_{asym,i} & = \frac{\sigma_{asym}}{\gamma_{pre}}I_{pre,i} - \frac{1}{\gamma_{asym}}I_{asym,i} \\
\frac{{\rm d}}{{\rm d}t}I_{sym,i} & = \frac{(1-\sigma_{asym})}{\gamma_{pre}}I_{pre,i} - \frac{1}{\gamma_{sym}}I_{sym,i} \\
\frac{{\rm d}}{{\rm d}t}R_i & = \frac{1}{\gamma_{asym}}I_{asym,i} + \frac{(1-\delta_i)}{\gamma_{sym}}I_{sym,i} \\
\frac{{\rm d}}{{\rm d}t}D_i & = \frac{\delta_i}{\gamma_{sym}}I_{sym,i}
\end{aligned}
$$
- The **transmission rate** was given by the product of:
1. transmission probability **($q$)**
2. age-specific susceptibility **($s_i$)**
3. strength of NPIs **($\theta$)**
4. relative infectiousness of each symptom type **($\tau_m$**, where $m\in M\equiv \{pre, asym, sym\}$**)**
5. the rate of contact **($r_{m,i,j}$)** between infected individuals with symptom type $m$ from group $j$ and susceptible individuals from group $i$
- Individuals in group $i$ are vaccinated at a rate of $\mu_iv(t)$, and a fraction of the those **($\epsilon_i$)** are protected
- Individuals move from $S_i$ to $P_i$ at rate $\epsilon_i\mu_iv$
- Individuals move from $S_i$ to $F_i$ at rate $(1-\epsilon_i)\mu_iv$
- Once infected, individuals move from $E_i$ to $I_{pre,i}$ at rate $\gamma_{exp}^{-1}$
- Let $\sigma_{asym}$ be the infection asymptomatic rate:
- Individuals move from $I_{pre,i}$ to $I_{asym,i}$ at rate $\sigma_{asym}/\gamma_{pre}$
- Individuals move from $I_{pre,i}$ to $I_{sym,i}$ at rate $(1-\sigma_{asym})/\gamma_{pre}$
- Individuals move from $I_{asym,i}$ to $R_i$ at rate $\gamma_{asym}^{-1}$
- Let $\delta_i$ be the age-specific infection fatality rate:
- Individuals move from $I_{sym,i}$ to $D_i$ at rate $\delta_i/\gamma_{sym}$
- Individuals move from $I_{sym,i}$ to $R_i$ at rate $(1-\delta_i)/\gamma_{sym}$

### Contact Rates and Social Distancing
- The contact matrices $r_m$ used in this model are built from the contact matrices in [_K. Prem, A. R. Cook, M. Jit, Projecting social contact matrices in 152 countries using contact surveys and demographic data. PLoS Comput. Biol. 13, e1005697 (2017)_](https://doi.org/10.1371/journal.pcbi.1005697)
- We constructed contact matrices for each of four locations, $x\in \{home, work, school, other\}$.
- The total contact rate for an asymptomatic individual is given by the weighted sum of these location-specific matrices
$$
r_m = \sum_x \alpha_{m,x} r_x
$$
- The weights $\alpha_{m,x}$, which is the product of **social distancing** policies, depend on symptom status $m$ and location $x$, as specified in this table:

### Initial Conditions
- Because $\{I_{pre}(0), I_{asym}(0), I_{sym}(0), S(0)\}$ are uncertain by the time the initial vaccine doses are deployed, we test a range of values from 1-20 symptomatic cases per 1000
- The portion of each group infected at time $t = 0$ in the Base parameter set:
- The number of **recovered** individuals in the population was set to 8% of each group, which was informed by IHME projections
- We set the number of **deceased** and **vaccinated** individuals in each group to 0

### Vaccine Prioritization Optimization
- We numerically solved for vaccine allocation strategies that minimize the total burden associated with three different health metrics:
1. death
$$
min\left\{\int_0^T\sum_{i\in J}\frac{\delta_i I_{sym,i}(t)}{\gamma_{sym}}dt\right\}
$$
2. YLL
$$
min\left\{\int_0^T\sum_{i\in J}\frac{e_i \delta_i I_{sym,i}(t)}{\gamma_{sym}}dt\right\}
$$
3. symptomatic infections
$$
min\left\{\int_0^T\sum_{i\in J}\frac{I_{sym,i}(t)}{\gamma_{sym}}dt\right\}
$$
where $e_i$ is the years remaining of life expectancy for group $i$, and $T$ is a 180-day time horizon

- We identified the optimal solution using a two-step algorithm:
1. Use a **genetic algorithm** to identify an approximate solution
:::info
1. Sample $N_t$ candidate solutions $\{x_{n,t}\}$ from a Dirichlet distribution with parameter $\alpha_t$
2. Each candidate solution is evaluated with the objective function
3. The best $K_t$ candidates $\{x_{n,t}^{best}\}$ are solved and $\alpha_t$ is updated to $\alpha_{t+1}$ which is the mean of $\{x_{n,t}^{best}\}$ times the entropy parameter $\eta_t$
4. Repeat steps 1 to 3 for $t=0,1,2,...,(T-1)$ and the best candidate solution sampled at any iteration is returned
:::
2. Then use **simulated annealing** to refine the solution from the last part
:::info
1. Sample a candidate solution $x_0$ and initialize a counter $i$
2. Generate a new sample from the distribution $x_t\sim {\cal N}(x_0, \sigma I)$
3. If $obj(x_t) < obj(x_0)$, replace $x_0$ with $x_t$, update $i = i + 1$ and repeat from step 2
4. If $obj(x_t) > obj(x_0)$, sample $\mu \sim (0,1)$. If $\mu > e^{-\frac{x_t-x_0}{T(i)}}$, then replace $x_0$ with $x_t$, update $i = i + 1$ and repeat from step 2. Otherwise save $x_0$ and repeat from step 2.
- $T(i)=T_0/i$ is the temperature function
5. Stop when $i > max\ iter$
:::