Helmut Meister, Eugen Grycko
Fair Distribution of Vaccines Among Population Groups
Lehrgebiet Stochastik Forschungsbericht
Fakultät für
Mathematik und
Informatik
Fair Distribution of Vaccines Among Population Groups
Helmut Meister, Eugen Grycko
Department of Mathematics and Computer Science FernUniversit¨at
Universit¨atsstraße 1 D-58084 Hagen, Germany email: meisterhelmut1@t-online.de email: eugen.grycko@fernuni-hagen.de
June 23, 2021
Abstract
Vaccination of a population of persons under limited resources is al- ways a matter of controversial discussions. The common proceeding to overcome these medical and ethical problems as agreed in Germany in 2021 is a ranking of risk groups according to their lethal rates during the spread of the pandemics. The question arises whether this approach is re- ally an optimal response to the pandemic development. One may argue, it is better to immunize those groups with the most intensive social activity first, because they are the most dangerous group to other people. For an analysis of the situation we develop a simulation model for the spread of an infectious decease within a population divided into groups with differ- ent social activity and different lethal rates to examine the lethality rate of the whole population depending on the prioritization of vaccination. The result of the simulation runs suggests that the most vulnerable groups should be vaccinated first, but does not identify a complete ranking of vaccination over all groups. Therefore in a second approach we model the situation as a comparative game with the focus on a fair distribution of a limited amount of vaccines. Game Theory provides the most suit- able concepts to solve conflict situations with multiple actors competing against each other about vaccine application rates. We make use of such a concept for data generation. It seems that this approach prefers groups with high social activity but is also sensible to the lethal rates of other groups.
Keywords: Pandemic spread, Prioritized Vaccination, Simulation Model, Comparative Game, Equilibrium
1 Modeling the Situation
1.1 Elements of the Model
Simulation models can contribute to analyze different prioritization strategies for the vaccination of a population divided into groups with different vulnera- bility. In Germany this strategy was aligned with the vulnerability of groups and therefore puts the groups of higher age in front. The question whether this is an acceptable approach will be answered through the generation of data by a simulation model taking the lethal rates within the groups and the social contacts into account.
1. The setup of the model can be described by a population of agents who can take various states described by numbers:
• 0 = not infected
• 1 = infected
• 2 = symptomatic
• 3 = cured, immunized
• 4 = dead
2. The population divides into groups of agents according to their vulnera- bility:
• Group 1: lethal rate close to zero
• Group 2: low lethal rate
• Group 3: medium lethal rate
• Group 4: high lethal rate
3. The agents are linked to some other agents according to their social ac- tivity rate. The number of links is generated by a matrix representing the social connectivity of the groups.
4. The frequency of realization of contacts is adjusted by a model parameter.
As a first guess we use the following specification for the model input:
1. Lethality rates within the given population groups (% of infected indi- viduals in the respective group): The Infection Fatality Rate (IFR) in different age groups is taken from a graphics published by the German Robert-Koch-Institute (also presented in the television show ”‘Quarks”’
of WDR) and is aggregated to our situation. For the IFR in our setting we obtain the following values:
Age IFR
Group 1 0.005%
Group 2 0.61%
Group 3 9.5%
Group 2 19.6%
2. The input data concerning contact frequencies are only hypothetical val- ues. The authors did not find any sources of real data. The initial matrix of contact frequency within the given population groups (as lines) has the following inputs:
3 0 5 1
0 2 5 5
1 1 10 2
1 1 5 3
(1)
It deserves to be mentioned that the contact matrix needs not be sym- metric and represents only the most frequent contacts of group members.
We assume that contacts follow certain degrees of kinship and common activities. We do not know any data sources on such relations and rely therefore on plausibility. For small sizes of groups there may arise a con- sistency problem, but this will not occur in our context.
With these input data a social network is created. The size of the 4 groups is correlated with the actual German population pyramid (see Figure 1). However, we consider the groups as generic representation of population groups which may suffer from the decease with different severity. We make use of the NetLogo Agent Based Framework ([4]) for the implementation of the model.
Figure 1: German population pyramid
Source: https://www.populationpyramid.net/germany/
The different colors in Figure 2 of the actors have the following meaning:
Grey actors: group 1 Red actors: group 2 Orange actors: group 3 Brown actors: group 4
The network looks as in Figure 2 and consists of 100 actors:
1.2 Prioritized Vaccination
The vaccination algorithm uses all different sequences of ranking within the 4 groups. This is done calculating all permutations of 4 members. For each such permuted sequence the available vaccines are distributed alongside the prioritization of the groups. We are aware that prioritization is in practice more complicated. Nevertheless, we do not want to go in too much detail, because
Figure 2: Network of actors
even the rough model gives some insight in different vaccination rankings. The algorithm works as follows.
1.1 Algorithm: Given an amount of vaccine a with unit in % of the whole population, the distribution algorithm follows the prioritization permutation (π(1), . . . , π(4)) of the 4 groups and starts with group π(1). If this group can completely be vaccinated, the remaining vaccines are assigned to the next group π(2) and so on, until the complete vaccine reserve is distributed.
After a simulation run the status of actors can be determined as from Figure 3.
Figure 3: Status of actors
The statistics of several simulation runs depending on the vaccination rate
and the prioritization of certain groups is shown in diagram 4.
Figure 4: Lethality rates
1.3 Results
• The analysis shows that prioritized handling of groups 3 and 4 has a significant effect on the lethality rate in the course of infection diffusion.
It must be noticed that the social activity of the groups is kept constant during the simulation runs. This means, there is no influence of social contact reduction during the spread of infection.
• There is, of course, an effect of randomization in the simulation runs. The number of repetitions was adjusted to 20 to make the runs not too time consuming. Not only the spread of the infection is a stochastic process but also the actor 0, who is infected first, is chosen by random. This may explain, why some values exceed the expected values sometimes signifi- cantly.
• No big difference in lethality can be realized between the prioritized vac- cination of group 3 and 4. This fact shows that there is no benefit of a strict prioritization as long as groups 3 and 4 are vaccinated first. A weak tendency can be observed that group 2 in the first position may be disadvantageous.
2 A Comparative Game of Vaccine Distribution
From a game theoretic view one would argue that a prioritized handling of certain groups could be unfair. Particularly, other groups with higher social activity might object that a deviation from the strategy of prioritizing the most vulnerable groups would be beneficial for the whole population and could re- sult in a lower lethality rate. The group might therefore argue that it should
be vaccinated first to stop the infection diffusion. This argumentation can be considered as a typical game theoretic problem and suggests an investigation of Nash-type solutions ([3]).
2.1 Game Theoretic Setting
In this sense, we introduce a population dynamic model based on a risk analysis for the whole population with some groups being partially vaccinated while maintaining contacts to other people. The risk matrix V is built from the matrixCof group-specific contact frequencies as already mentioned in the first section. The contacts are calibrated by the strength of the population groups and the lethality rates, i.e. the coefficientsvij of the risk matrix are defined by
vij :=pi·pj·lj·cij, (2)
where (p1, . . . , p4) is the distribution of group strengths, (l1, . . . , l4) is the vector of lethality probabilities for the groups andcij is the corresponding coefficient of the contact frequency matrix. Pure strategies in this model are distribution vectors (q1, . . . , q4) of vaccine quantities, assigned to the 4 population groups and are calculated by the algorithm 1.1.
In this way, the main entries of the model are given by the vaccine reservea and the prioritization permutations (π(1), . . . , π(4)). The distribution procedure ends up in at most 4! pure strategiesq(π):= (q(π)(1), . . . , q(π)(4))1. For practical reasons, we will assume that the quantitiesq1(π), . . . , q4(π) denote the portion of vaccination in each group (0≤qj(π)≤1). After this step of calculating all pure strategies, we can form the risk matrixV∗ with coefficients
vij∗ :=q(πi)V q(πj)T
, (3)
whereπ1, . . . , πnis the system of all relevant permutations andq:= (1, . . . ,1)− q. The interpretation of this setting can be formulated as follows: we evaluate the contacts between not-vaccinated actors when they get together according to the contact frequencies and lethal rates given by the matrixV.
The intention of this approach is the formulation of optimality conditions for a fair vaccine distribution along with the given risk assessment by the matrix V∗. A first proposal would be simply to minimize the objective function
u(x) :=xV∗xT (4)
within the unit simplex ∆n ⊂R. Then, by definition of the pure strategies in this game, a simple calculation would show thatq(x) := Pn
i=1xiq(πi) gives a distribution of vaccines among the groups. Unfortunately, in a game theoretic logic this will not always be a distribution, on which all groups can agree. The following example illustrates this fact.
1Some of them may result in the same vaccine distribution.
2.1 Example: For simplicity, we consider a 2×2-matrix of the type V∗:=
0 a
b 0
(5) with a, b < 0. The solution of the minimization problem of the expression xV∗xT is given byx:= (12,12) independent of the concretion ofaandb. In case a < b, the second group will argue that for the given distribution strategy xa deviation toy:= (0,1) would end up inxV∗yT =a2 <a+b2 =xV∗xT, and would therefore reduce the risk. Hence, the given strategyxcannot be considered as stable from game theoretic principles. Stability of the distribution strategy will be ensured by the requirement
xV∗xT ≤xV∗yT for all y∈∆2. (6) This requirement will be satisfied byx:= (a+ba ,a+bb ) and takes the higher risk of group 2 into account.
This example serves as motivation for a Nash-type equilibrium concept in distribution games with a risk assessment matrix.
2.2 Definition: Let be given a n×n-matrixV, for which the entries assign a certain risk to the interaction ofn types of actors. We will address this matrix as Comparative Risk Matrix. An Equilibrium of the game based on this payoff structure will be any vector x ∈ ∆n (i.e. a randomized strategy), for which
xV∗xT ≤xV∗yT for all y∈∆n (7) holds. 2
Characterization and existence of equilibrium in comparative games are de- rived in the research paper [2]. Particularly, the formulation of the solution concept by a linear complementarity problem (Theorem 2.5 in [2]) gives rise to an algorithmic approach. There exist various algorithms, which end up in a solution of the corresponding linear complementarity problem to the optimality condition (7). The pioneering research was published by Lemke and Howson [1]. Their basic algorithm is known under name Lemke-Howson-Algorithm in the respective literature.
2.2 The Vaccination Game with Comparative Risk As- sessment
We will use an own implementation of this algorithm within the NetLogo frame- work (see [4]) to identify solutions in our case of the vaccination problem. We will only summarize a few results of this analysis.
2We make use of the approach as taken in the research paper [2] with the straight forward modification of replacing the utility concept by a risk concept.
vaccination 10%
contact matrix lethality rates distribution of vaccines result 1.1
3 2 5 3
0 50 10 5
5 2 50 2
4 4 10 10
(0.005, 0.61, 9.5, 19.6) (0, 0.15, 0, 0)
result 1.2
3 2 5 3
0 50 10 5
5 2 50 2
4 4 10 10
(1, 1, 10, 20) (0, 0.1, 0.4, 0)
result 1.3
3 2 5 3
0 30 10 5
5 2 30 10
4 4 10 10
(0.005, 0.61, 9.5, 19.6) (0, 0.05, 0.76, 0)
vaccination 25%
contact matrix lethality rates distribution of vaccines result 2.1
3 2 5 3
0 30 10 10
5 2 30 10
4 4 10 10
(0.005, 0.61, 9.5, 19.6) (0, 0.27, 0.86, 0)
result 2.2
3 2 5 3
0 30 10 10
5 2 30 10
4 4 10 10
(1, 1, 10, 20) (0, 0.29, 0.71, 0)
vaccination 50%
contact matrix lethality rates distribution of vaccines result 3.1
3 2 5 3
0 50 10 5
5 2 50 2
4 4 10 10
(0.005, 0.61, 9.5, 19.6) (0, 0.63, 0.88, 0.12)
result 3.2
3 2 5 3
0 30 10 10
5 2 30 10
4 4 10 10
(0.005, 0.61, 9.5, 19.6) (0, 0.61, 0.99, 0.15)
vaccination 70%
contact matrix lethality rates distribution of vaccines result 4.1
3 2 5 3
0 30 10 10
5 2 30 10
4 4 10 10
(0.005, 0.61, 9.5, 19.6) (0, 0.86, 1, 0.7)
2.3 Analysis of Data
• For low availability of vaccines (10%) and low lethal rates in group 4 the model recommends to immunize only part of group 2 (result 1.1), while slightly higher lethal rates in all groups requires also the immunization of a part of group 3 (result 1.2). Relatively more intensive social contacts of group 3 with group 4 implicate more attention to group 3 (result 1.3).
The model reacts rather sensibly to changes in social activities.
• Increased availability of vaccines (25%) shows the dependency from lethal rates of all groups (results 2.1 and 2.2). More attention is required to vaccinate group 2 and 3.
• Even under lower internal social activity in group 2 and 3 the dependency from contacts with group 4 is significant (results 3.1 and 3.2).
• Only in the case of high availability of vaccines (70%) a significant portion of the vaccines must be spent to group 4 (result 4.1) while group 1 is still ignored.
This analysis shows that a strict prioritization of population groups with the only focus on vulnerability is not necessarily a reasonable strategy. Social
activity has besides lethal rates a significant influence on distribution strategies for all groups. These strategies can be considered as acceptable in the sense of fairness.
3 Discussion
Clearly, political, ethical and strategic arguments dominate the discussion how the spread of a pandemics can be stopped. Decision makers have to take all these issues into account. Of course, our simulation model is a simplified approach to reality, but gives at least some insight, how a fair prioritization of groups in the vaccination ranking could be achieved. The results of the first section confirm the German course of action roughly. The game theoretic approach is more sophisticated, and the results are not an overall optimization of the vaccination process. Nevertheless, the model highlights the problem of social competition on limited resources and offers a stable solution for the conflict.
References
[1] Lemke, C. E., Howson J. T., Equilibrium points of bimatrix games, SIAM Journal of Applied Mathematics, 1964
[2] Meister, H., 2020,Games with Comparative Utility Functions, Research Pa- per, https://ub-deposit.fernuni-hagen.de/.
[3] Nash, John Forbes, 1950, Non-cooperative games, Dissertation, Princeton University.
[4] Wilensky, U., 1999, NetLogo, http://ccl.northwestern.edu/netlogo/, Center for Connected Learning and Computer-Based Modeling, Northwestern Uni- versity, Evanston, IL.
