Constrained Optimization in COVID-19 Vaccine Allocation [Part II]

In order to estimate the number of susceptible individuals in each county i, we need to consider the population size N_i and the total current COVID-19 cases C_i. Considering these factors, the estimate is given by :

N_i − C_i   

The N_i of the 14 Massachusetts counties ranges from 11,801 of Nantucket County to 1,618,160 of Middlesex County as of 2021. C_i is determined by data from the Massachusetts Department of Public Health (Table 1). It is important to take N_i − C_i into account instead of just N_i because recovered cases often obtain some immunity and vaccination will not notably affect those currently infected.

Table 1: Total COVID-19 cases per county  Table 2: Population density and size by county

Next, contact rate is important as even in small populations, more individuals may be infected due to greater population density. As can be seen by Table 2, although Middlesex County had the greatest N_i, Suffolk County had the greatest 𝛿_i (population density in each county i), proving the importance of contact rate and population density consideration. The equation below equates contact rate in terms of population density. (exp refers to an exponential function of e)

1−exp(−𝛿_i / max{𝛿_i })

Finally, we need to consider the age ratio as each age has a different probability of transmission and death susceptibility. When dividing the age group into two, <65 years and >65 years, >65 are more vulnerable while <65 have greater contact rates due to more active social relationships. According to a study performed by Xin Chen et al., allocating around 92% of the vaccines to >65 and around 8% to <65 minimized the deaths. Finding the optimal balance of distribution is necessary for an accurate constrained optimization result. Setting E_i as the >65 percentage in each county i, we can define the age ratio as:

E_i(100-E_i)

In Massachusetts counties, the ranking below shows the E_i for each county (Figure 4).

           Figure 2: Ranking of E_i per county

All three factors are crucial to determining the counties and age groups to allocate 23,000 doses effectively. For instance, if authorities allocate the vaccines in order of susceptible individuals, they may fail to take into account the age ratio in each county or may fail to account for the contact rates. The same goes for any other instances where one factor is considered a priority. 

1. Possible Solution and Limitations

Table 4: Top 7 ranking for each considered factor by county

Because our priority is to minimize the death rates, not transmission, the age ratio must be the most impactful factor in deciding the county to allocate the vaccines first. According to table 4, out of the top 7 ranked counties of age ratio, Essex county is the only one that appears in the top 7 of the other factors. Despite prioritizing the age ratio, it is unreasonable to consider Barnstable county, the county with the greatest age ratio, as the first county to allocate vaccines. This is because Barnstable has very low contact rates and a low number of susceptible individuals, showing that the elderly are unlikely to be infected. On the other hand, Essex stands middle ground in terms of age ratio but ranks in the top 4 for the other two factors. This shows that Essex is a county that is highly populated, has a high transmission rate, and has a fairly high number of vulnerable individuals >65. Therefore, it is most reasonable to allocate the vaccines first in Essex.

Essex has around 644,570 susceptible individuals and an age ratio of 17%. This means that around 109, 577 individuals(644,570 * 0.17) are >65. Considering Essex as our county of vaccine allocation, the 70,000 doses of available vaccines should be implemented in a 92:8 ratio, according to the study of Xin Chen et al. mentioned above. This results in 64,400 doses for >65 and 5,600 doses for <65 in Essex county being the optimal vaccine allocation strategy under a limited vaccine supply of 70,000 doses.

Although this optimization in respect to the 3 factors reflects a real-life situation, there are limitations as not all factors were considered. For instance, when considering the vulnerability to COVID-19, it is necessary to consider underlying health issues and jobs(front line health workers and social care workers) of the individuals. Moreover, vaccine effectiveness was considered as constant while in real life, different vaccines have different effectiveness. In addition, the age group was only divided into two groups. Despite these limitations, this optimization can be used to aid the general distribution of COVID-19 vaccines, be used as a basis for future pandemics, and may help discover unanticipated patterns in vaccine distribution.