Optimal vaccine allocation: modelling population interaction and administrative constraints.

The COVID-19 pandemic and associated vaccine shortages in many places have caused immense suffering. This paper proposes a theoretical framework for finding the optimal vaccine allocation among interconnected populations. In particular, the analysis focuses on modelling how interaction and administrative constraints affect the optimal vaccine allocation. The proposed framework suggests that both factors affect the optimal vaccine allocation differently, especially when the shortage of vaccines is severe. Four hypotheses are put forward, which should be verified empirically in order to falsify the plausibility of the proposed theoretical framework.


Introduction
The number of deaths worldwide caused by COVID-19 has exceeded 6 million (Ritchie et al., 2020). Scientists developed vaccines several months after the first cases of COVID-19 were confirmed (AJMC Staff, 2021), and many countries have faced vaccine shortages (Feinmann, 2021). Given this reality, in order to limit human suffering, it is imperative to understand how to allocate vaccines optimally. In this paper, I look at how the interaction between populations and constraints on the number of vaccines administered daily affect the optimal vaccine allocation. Using a simple pandemic model and an optimisation algorithm, I show that both factors affect the optimal vaccine allocation differently, especially when the number of vaccines available is small.

Literature Review
Major pandemics have had a significant influence on the history of humankind (Piret and Boivin, 2021). To understand them, scholars have developed mathematical models that allow us to describe and predict the flow of a pandemic with a set of equations. SIR model, developed in the 1920s (Kermack and McKendrick, 1927), is one of the simplest ways of modelling pandemics. Its basic version divides the population into three compartments (Susceptible, Infected, Recovered) but scholars have extended the model to account for quarantines, lockdowns, vaccinations, and deaths (Fosu et al., 2020).
One of the reasons why scholars are interested in modelling pandemics is the need to understand how to fight them effectively. In particular, one of the questions discussed in the literature is how to allocate vaccines optimally between different populations, given the constraints policymakers face. This problem can be formulated as a constrained optimisation problem and has attracted the attention of economists (Kitagawa and Wang, 2021). Dujizer et al. (2017) showed the importance of the herd effect in determining the optimal allocation. This paper attempts to extend the discussion of the optimal vaccine allocation problem in two ways. Firstly, I look at how administrative constraints (the number of people that can be vaccinated daily) affect the optimum. Secondly, I investigate how different patterns of interaction between populations (allowing for infections in population A to affect population B) affect the optimum.

The model
I use an extended version of the SIR model to understand how administrative constraints and population interconnectedness affect the optimal allocation. I assume that in each population, there are four different compartments: Susceptible, Infected, Recovered, and Dead. At any point in time, each person from every society is in one of them. Once someone is in the Recovered compartment, they can no longer move back to being infected (they are immune). Lastly, I assume it is possible to identify susceptible individuals, and vaccination (which is fully effective and takes no time) transfers people from the Susceptible to the Recovered compartment.
The number of susceptible, infected, recovered, and dead people in population i are denoted by Si, Ii, Ri, and Di. Recovery and death rates are denoted by γi and αi, respectively. The number of people vaccinated in each period in population i is denoted by Vi (this is a fixed number for every population). Lastly, βi and ωij capture the transmission of the disease. The former coefficient describes the increase in infections resulting from contact inside population i, whereas the latter describes the increase in infections in population i resulting from the contact with population j (note that ωij is not necessarily equal to ωji). Ω denotes the matrix containing all possible ωij. Figure  1 summarises the model.

Figure 1: A diagram of a SIR model for population i with four compartments, vaccination, and interaction between populations. ωi* denotes the ith row of matrix Ω.
The pandemic flow for k societies can be described as a system of differential equations (Ni denotes the size of population i):

Minimisation algorithm
The problem of optimal vaccine allocation can be described as a constrained minimisation problem: Where D i T denotes the number of deaths in society i at the end of the pandemic, Wi denotes the number of vaccines allocated to society i, and P denotes the number of vaccines available.
Instead of solving this problem explicitly, I designed a simple greedy minimization algorithm. It builds on the presumption that marginal gains from adding a small number of vaccines to every population are equal for all populations at the optimum. The algorithm works as follows: ; while the decrease in total deaths > ε: 3: compute marginal gain (decrease in total deaths from marginally increasing vaccine stockpile) for each society; 4: transfer 10 vaccines from the society with the lowest marginal gain to the one with the highest; return Wi for all i This algorithm ensures arriving at the locally optimal solution, but not necessarily the globally optimal one. To address this problem, for each of the cases described below, I ran the algorithm 100 times. Afterwards, from all allocations suggested by the algorithm, I chose the one with the smallest total number of deaths.

The four scenarios
The two phenomena I would like to analyse using the model are the presence of administrative constraints and interaction between populations. To understand them better, I look at the optimal vaccine allocation between three populations under four scenarios: the base scenario (with no interaction and no administrative constraint, corresponding to the case discussed by Duijzer et al. (2017)), the administrative constraint scenario (only 1% of the population can be vaccinated daily), the first interaction scenario (all populations interact with each other), and the second interaction scenario (Population 2 is affected by Population 1 and Population 3 is affected by Population 3, but not vice-versa). Table 1 summarises the parameter values used for each scenario.

Results
I computed the optimal vaccine allocation for each of the four scenarios with different vaccine stockpiles. Compared to the base scenario, the optimal allocation is more equitable under the administrative constraint scenario. That may be because a more equitable distribution allows vaccination of more people in the early days of the pandemic. Hence, under administrative constraints, maximising the number of people vaccinated in the early days of the pandemic may be the optimal strategy.
When we include interaction between all societies (the network scenario), the largest population receives more vaccines. The larger the population size, the more a population can potentially affect populations interacting with it (since the impact of interaction is proportional to the total number of people infected). Hence, when focusing on larger populations, policymakers can slow the spread of the pandemic by limiting the effects of interaction. Secondly, under the chain interaction scenario, the population earlier in the chain receives more vaccines. This should not be a surprise since the earlier a population is in the chain, the more influence it effectively has on other groups, given the interaction structure. That would suggest policymakers should prioritise populations earlier in the chain, other things being equal.
Lastly, the optimal allocation is similar under all four scenarios when the number of vaccines available is high. Moreover, under all four scenarios, we can see the importance of the herd effect.
In every case, populations receive vaccines until a certain threshold, after which the marginal gains from adding more vaccines diminish.
The results suggest four main hypotheses about the optimal vaccine allocation: (1) when faced with administrative constraints, policymakers should aim to maximise the total number of people vaccinated in the early days of the pandemic; (2) when several populations are equally interconnected, policymakers should prioritise the largest population; (3) when there is a chain of populations, policymakers should give more weight to those earlier in the chain; (4) given the herd effect, policymakers should not vaccinate whole populations when faced with a vaccine shortage.

Conclusion
In this paper, I examined how the interaction between societies and constraints on the number of vaccines administered daily might affect the optimal vaccine allocation. Both of these factors influence the optimum differently. The presented approach and analysis can be easily extended by allowing other parameters (death rates, recovery rates) to vary. Lastly, the hypotheses presented in this paper should be verified empirically.