# Validating Optimal COVID-19 Vaccine Distribution Models

Mahzabeen Emu, Dhivya Chandrasekaran, Vijay Mago, Salimur Choudhury

VValidating Optimal COVID- VaccineDistribution Models

Mahzabeen Emu [0000 − − − (cid:63) , DhivyaChandrasekaran [0000 − − − X ] (cid:63) † , Vijay Mago [0000 − − − , andSalimur Choudhury [0000 − − − X ] Department of Computer Science, Lakehead University, Thunder Bay, Ontario,Canada, P7B 5E1 {memu,dchandra,vmago,schoudh1}@lakeheadu.ca

Abstract.

With the approval of vaccines for the coronavirus diseaseby many countries worldwide, most developed nations have begun, anddeveloping nations are gearing up for the vaccination process. This hascreated an urgent need to provide a solution to optimally distribute theavailable vaccines once they are received by the authorities. In this pa-per, we propose a clustering-based solution to select optimal distributioncenters and a Constraint Satisfaction Problem framework to optimallydistribute the vaccines taking into consideration two factors namely pri-ority and distance. We demonstrate the eﬃciency of the proposed modelsusing real-world data obtained from the district of Chennai, India. Themodel provides the decision making authorities with optimal distribu-tion centers across the district and the optimal allocation of individualsacross these distribution centers with the ﬂexibility to accommodate awide range of demographics.

Keywords:

COVID-19 · Constraint satisfaction problem · Vaccine dis-tribution · Operational research · Policy making.

The ongoing pandemic caused by the Severe Acute Respiratory Syndrome Coro-navirus 2 (SARS-CoV-2) called the coronavirus disease (COVID-19), has notonly caused a public health crisis but also has signiﬁcant social, political, andeconomic implications throughout the world [2,13,19]. More than 92.1 millioncases and 1.9 million deaths have been reported worldwide as of January 2021[6]. One of the widely used solutions, in preventing the spread of infectious dis-eases is vaccination. Vaccination is deﬁned by WHO as “

A simple, safe, andeﬀective way of protecting people against harmful diseases before they come intocontact with them. It uses the body’s natural defenses to build resistance to spe-ciﬁc infections and makes the immune system stronger” . Various researchers (cid:63) Equal contribution † Corresponding Author a r X i v : . [ c s . C Y ] F e b i M. Emu et al. and pharmaceutical corporations began research work on identifying potentialvaccines to combat the spread of COVID-19. Of the number of vaccines undertrial, Tozinameran (BNT162b2) by Pﬁzer and BioNTech, and mRNA-1273 byModerna have achieved an eﬃcacy of more than 90% [12,15]. Various nationshave begun the process of approval of these vaccines for mass distribution andhave placed orders to facilitate a continuous ﬂow of supply of vaccines to meetthe demands.In this paper we propose a Constraint Satisfaction Programming (CSP)framework based model to optimize the distribution of vaccine in a given ge-ographical region. We aim to maximize the distribution of vaccine among thegroup of the population with higher priority while minimizing the average dis-tance travelled by any individual to obtain the vaccine. In order to justify theeﬃciency of the proposed model we compare the performance of the followingfour optimization models namely, • Basic Vaccine Distribution Model (B-VDM) • Priority-based Vaccine Distribution Model (P-VDM) • Distance-based Vaccine Distribution Model (D-VDM) • Priority in conjunction with Distance-based Vaccine Distribution Model (PD-VDM).and present how the model PD-VDM provides the most optimal solution forthe distribution of vaccines. We perform a Case Study using the demographicdata obtained from Chennai - a well-renowned city in Southern India. This casestudy highlights how the model can be used by decision making authorities ofa city with an population of 5.7 million. The model aids the decision makingauthorities to choose an optimal number of vaccine distribution centers (DCs),and to optimally assign an individual to a hospital such that the individuals inthe priority groups are vaccinated ﬁrst while minimizing the distance they travelto the vaccine DCs. In section 2 we discuss the various steps and challengesinvolved in the process of vaccine distribution, in section 3 we describe in detailthe procedure followed to build the proposed models. The case-study is discussedin section 4 followed by the discussion of results in section 5. We conclude withfuture research directions in section 6.

In order to eﬀectively distribute vaccines, it is necessary to understand the sup-ply chain of vaccines. The supply chain of vaccines is divided into four majorcomponents namely the product, production, allocation, and ﬁnally the distri-bution [7]. The ﬁrst concern of the decision making authorities is to decide onwhich vaccine to choose for distribution in their country, province, or region.For example, for countries in tropical regions, the storage temperature of thevaccine is an important factor. Similarly, while developed countries are able toaﬀord vaccines at a higher price, most developing countries prefer the vaccinewhich has an aﬀordable price. In the present scenario, three of the prominentvaccines in play are the BNT162b2, mRNA-1273, and AZD1222 [10]. They have alidating Optimal COVID- Vaccine Distribution Models iii storage temperatures of -70℃, -20℃, and 0℃ and cost USD 20, USD 50, andUSD 4 respectively . Based on the storage it is safe to assume that while coun-tries in temperate regions like the United States, Canada, and Russia wouldhave the option of purchasing any one of these vaccines while tropical coun-tries like India, Bangladesh, Pakistan would prefer AZD1222. The cost of thesaid vaccines also has a signiﬁcant impact on the decision-making process ofdeveloping nations which mostly cater to a greater number of people. Once theproduct is chosen, the production of vaccines has to be scheduled according tothe demand. Factors that are taken into consideration at this stage include theproduction time, capacity for manufactures, supply-demand analysis, and so on[3]. Depending on the stage of the epidemic and the severity, the demand for thevaccine and the optimal timeline for the supply of vaccines may vary. Allocationand distribution stages go hand in hand in the vaccine supply chain. Depend-ing on the distribution strategy, the allocation of vaccines at any level is tunedto achieve the best result. Allocation at a global level may depend on priorityestablished through contracts and agreements among pharmaceuticals and gov-ernments. However, once a country receives the vaccines, further allocating thevaccines to provinces, states, or subgroups of the population is a critical decisionthat in turn has an impact on the distribution strategy. The distribution stageof the supply chain addresses the challenges of establishing an eﬀective routingprocedure, infrastructure of the vaccine distribution centers, inventory control,workforce, etc.Operation Research (OR) involves the development and use of various sta-tistical and mathematical problem-solving strategies to aid in the process ofdecision making. Various OR models are proposed over time to optimize thedistribution of vaccines from the distribution centers. Ramirez-Nafarrate et al.[17] proposes a genetic algorithm to optimize the distribution of vaccines byminimizing the travel and waiting time. Huang et al. [9] formulated a vaccinedeployment model for the inﬂuenza virus that ensures geographical priority basedequity in Texas. However, their mathematical model might have generalizationissues when applied to smaller or larger than state-level regions. Lee et al. [11]developed the RealOpt © a tool to aid in identifying optimal location for vac-cine distribution centers, resource allocation and so on. Researchers have alsoprovided models to accommodate speciﬁc locations, for example Aaby et al. [1]proposes a simulation model to optimize the allocation of vaccine distributioncenters at Montgomery county, Maryland. This model aims to minimize the vac-cination time and increase the number of vaccinations. While the above modelsconsider the distribution centers to be stationary, Halper et al. [8] and Rachani-otis et al. [16] consider the vaccine distribution centers to be mobile and addressthis as a routing problem. While the later model proposes that various mobileunits serve diﬀerent nodes in a network, the latter considers that a single mobileunits serves various areas with a goal to minimize the spread of infection. Someof the OR models used in epidemics’ control include non-linear optimization, Quadratic Programming (QP), Integer Linear Programming (ILP) and MixedInteger Linear Programming (MILP) [5]. ILP, MILP, and QP models are notsuitable for many practical use cases due to its time expensive nature and infea-sibility issues prevailing with irrational model designs. The mathematical designprocess and selection of pre-deﬁned numerical bounds can lead to several techni-cal glitches in the models. Despite the apparent similarities with ILP and MILP,CSP can eliminate all the aforementioned drawbacks and ensure sub-optimalsolution in non-deterministic polynomial time by applying boolean satisﬁabilityproblem formulation. In the ﬁeld of computer science, CSP is considered as apowerful mechanism to address research challenges regarding scheduling, logis-tics, resource allocation, and temporal reasoning [4]. Hence in this article weemploy CSP to propose four models to optimize the distribution minimizing thetraveling distance and maximizing the vaccination of high priority population.

In this section, we initially determine the optimal number and location of vaccineDCs using K-medoids clustering algorithm. Provided with the various locationsof possible vaccine DCs, the algorithm determines the optimal number of clustersinto which the region can be divided into, in order to eﬀectively distribute thevaccines across the chosen region. On selecting the number of clusters and thecluster heads, we further propose four diﬀerent vaccine distribution simulationmodels to optimize vaccine distribution based on two factors namely distanceand priority. The clustering algorithm and the simulation models are explainedin the subsections below.

Our proposed Algorithm 1 imitates the core logic of K-medoids clustering tech-nique [14]. Firstly, the algorithm determines the optimal number of vaccine DCsto be selected from a set of hospitals ˜ H based on silhouette score analysis [18],as mentioned in line number . We determine the silhouette score for to ˜ h ˜ H − number of potential vaccine DCs, where ˜ H = { ˜ h , ˜ h , ˜ h , ..., ˜ h ˜ H } . Then, we se-lect the optimal number of vaccine DCs as η that retains the highest silhouettescore. As per the line number , we randomly select η hospitals as vaccine DCsinto H . Later, we initiate the clustering process. Each hospital is assigned toits closest vaccine DC to form η clusters, according to line numbers − . Thecluster informations indicating which hospital is associated to which vaccine DCare recorded in C . Next, the algorithm reassigns the vaccine DCs H to the oneswith the minimum total distance to all other hospitals under the same cluster,executed in line numbers − . We let the algorithm repeat the entire cluster-ing process until vaccine DC assignments do not change. Hence, the terminationcriteria depends on the stability of the clustering process. To summarize, weemploy this algorithm with inputs of a set of hospitals/potential vaccine DCs, ˜ H and a D dist matrix deﬁning the distances of one hospital to every otherhospital. The output of the algorithm is the set of optimally selected vaccineDCs H based on the distance metric. The primary idea is to choose vaccine DCs alidating Optimal COVID- Vaccine Distribution Models v

Algorithm 1:

K-medoids algorithm to choose vaccine DCs from a set of hospitals

Input: ˜ H : A set of hospitals, dist : Squared matrix representing the distances of onehospital to every other hospital Result: H : A set of COVID- vaccine DCs η ← Determine the number of optimal vaccine distribution centers using silhouette score H ←

Randomly select η hospitals from ˜ H C ← ∅ while there is no change in H do foreach ˜ h a ∈ ˜ H do foreach h b ∈ H do h b ← Find the closest h b to ˜ h a using dist matrix C ← C ∪ (˜ h a , h b ) foreach h b ∈ H do temp ← ∅ foreach ˜ h a ∈ ˜ H do if (˜ h a , h b ) ∈ C then temp ← temp ∪ ˜ h a ∪ h b q ← argmin ˆ ha ∈ temp (cid:80) h ∗ a ∈ temp dist (ˆ h a , h ∗ a ) Swap h b with ˜ h q in C Update h b by ˜ h q in H optimally in a sparse manner to facilitate reachability for people living in anypart of the considered region. In this subsection, we proceed by explaining the system model for vaccine distri-bution. We denote H = { h , h , h , ..., h H } to be the set of COVID − vaccinedistribution centers selected by Algorithm , where H ⊆ ˜ H and h i is the i th vac-cine DC in H . Moreover, we deﬁne U = { u , u , u , ..., u U } as the set of availablestaﬀ, where every u j ∈ U i . We denote U i as the available set of staﬀ in vaccineDC h i and U i ⊆ U . Subsequently, we can infer that U = ∪ |H| i =1 U i . We furtherassume that E = { e , e , e , ..., e E } represents the set of people required to bevaccinated, and the k th person is e k . In order to specify the priority of people forvaccination purpose, we use the set P = { p , p , p , ..., p P } . Hence, p k deﬁnes thepriority level of a person e k ∈ E , and |E| = |P| . It is noteworthy that, the higherthe priority level, the faster the vaccination service deployment is desired. Thedistance between a distribution center h i ∈ H and a speciﬁc person e k ∈ E isrepresented using D i,k . In this research, we consider the solution binary decisionvariable as x i,j,k ∈ { , } . The value of the decision variable x i,j,k is , in case adistribution center h i ∈ H allocates a staﬀ u j ∈ U i to vaccinate a person e k ∈ E ,otherwise remains . In this paper, we formulate the opted vaccine distribution research enigma as aCSP model. The CSP framework includes a set of aforementioned decision vari-ables that should be assigned values in such a way that a set of hard constraintsare satisﬁed. Hard constraints are essential to be satisﬁed for any model to reacha feasible solution. Suppose, our proposed model iterates over T = { t , t , ..., t T } i M. Emu et al. times to complete vaccination, where each time instance t n ∈ T refers to pertime frame for vaccine deployment decision making. Finally, N represents thetotal amount of available vaccine throughout the entire time periods T . All ofour proposed CSP based models are subject to the following hard constraintsthat have been translated into integer inequalities, for any time frame t n ∈ T : C (cid:88) e k ∈E x i,j,k ≤ , ∀ h i ∈H , ∀ u j ∈U i (1) C (cid:88) h i ∈H (cid:88) u j ∈U i x i,j,k ≤ , ∀ e k ∈E (2) C x i,j,k ∈ { , } , ∀ h i ∈H , ∀ u j ∈U i , ∀ e k ∈E (3)The constraint C veriﬁes that every staﬀ from any distribution center can vac-cinate at most one person at a time. Thus, for every staﬀ u j ∈ U i of distributioncenter h i ∈ H , either there is a unique person e k ∈ E assigned for vaccination,or the staﬀ remains unassigned. Then, constraint C ensures that every person e k ∈ E is allocated at most one vaccine through a single staﬀ u j ∈ U i from aunique distribution center h i ∈ H . Finally, C is a binary constraint representingthe value of decision variable to be , in case a staﬀ u j ∈ U i of distribution cen-ter h i ∈ H is assigned to vaccinate a person e k ∈ E , otherwise , as mentionedpreviously. C (cid:88) t n ∈ T (cid:88) h i ∈H (cid:88) u j ∈U i (cid:88) e k ∈E x i,j,k ≤ N (4)The constraint C conﬁrms that the total vaccine distribution should not bemore than the available vaccine by any means throughout the entire periods T considered for vaccine deployment. Now, let us assume Ω be the set of all thefeasible solutions that satisfy all hard constraints. Ω = { x i,j,k | C , C , C , C } (5)Apart from hard constraints, our proposed CSP formulation incorporatesa set of soft constraints as well. Whilst hard constraints are modeled as in-equalities, soft constraints are outlined through expressions intended to be even-tually minimized or maximized. Soft constraints are not mandatory for ﬁnd-ing a solution, rather highly desirable to improvise the quality of the solutionsbased on the application domain. The soft constraint C strives to maximizethe number of overall vaccinated people. The focus of another soft constraint C remains to vaccinate the people with higher priority levels beforehand. Inother words, this constraint leads to maximize the summation of priorities ofall vaccinated people. Subsequently, the soft constraint C refers that every peo-ple should be vaccinated by staﬀ from the nearest vaccine distribution center. C Ω (cid:88) h i ∈H (cid:88) u j ∈U i (cid:88) e k ∈E x i,j,k (6) C Ω (cid:88) h i ∈H (cid:88) u j ∈U i (cid:88) e k ∈E x i,j,k × p k (7) C Ω (cid:88) h i ∈H (cid:88) u j ∈U i (cid:88) e k ∈E x i,j,k × D i,k (8) alidating Optimal COVID- Vaccine Distribution Models vii

By leveraging diﬀerent combinations of soft constraints, we propose four diﬀerentvariations of vaccine distribution models: a) Basic - Vaccine Distribution Model(B-VDM), b) Priority based - Vaccine Distribution Model (P-VDM), c) Distancebased - Vaccine Distribution Model (D-VDM), and d) Priority in conjunctionwith Distance based - Vaccine Distribution Model (PD-VDM).B-VDM is a rudimentary vaccine distribution model that solely concentrateson the soft constraint C to maximize the overall vaccine distribution, irrespec-tive of any other factors. A gain co-eﬃcient α has been introduced to the ultimateobjective function of the model in Eq. 9. C ⇐⇒ max Ω (cid:88) h i ∈H (cid:88) u j ∈U i (cid:88) e k ∈E α × x i,j,k (9)P-VDM ensures maximum vaccine distribution among the higher priority groupsof people, by reducing soft constraints C and C into one objective function inEq. 10. We denote β as the gain factor associated to soft constraint C . C ∧ C ⇐⇒ max Ω (cid:88) h i ∈H (cid:88) u j ∈U i (cid:88) e k ∈E x i,j,k × ( α + β × p k ) (10)Contrarily, D-VDM encourages vaccination of the people located closer to dis-tribution centers. This model can be speciﬁcally useful for rural regions, wheredistribution centers and people are sparsely located, including higher travellingexpenses. For this model, we incorporate soft constraints C and C , by multi-plying gain coeﬃcients α and γ , respectively. C ∧ C ⇐⇒ max Ω (cid:88) h i ∈H (cid:88) u j ∈U i (cid:88) e k ∈E x i,j,k × ( α − γ × D i,k ) (11)Finally, the PD-VDM merges all the soft constraints simultaneously. Ourproposed PD-VDM considers maximization of vaccine distribution in prioritygroups and minimization of distance factored in transportation expenditure,collaboratively. Furthermore, α , β , and γ have been introduced as gain factorsto equilibrate the combination of soft constraints and then presented as a multi-objective function in the Eq. 12. Hence, this model can optimize priority anddistance concerns conjointly based on the adapted values of gain factors. C ∧ C ∧ C ⇐⇒ max Ω (cid:88) h i ∈H (cid:88) u j ∈U i (cid:88) e k ∈E x i,j,k × ( α + β × p k − γ × D i,k ) (12)The gain parameters of all the proposed models can be tuned to balance or inclinetowards more focused convergent vaccine distribution solutions. For instance, α , β , and γ are individually responsible for maximum distribution, maximization ofpriorities, and minimization of distance focused vaccine distribution solutions,respectively. Moreover, the policymakers can exploit these models and adjust iii M. Emu et al. gain parameters according to the region speciﬁcs and domain knowledge of vac-cine distribution centers and population density. The regulation of these gainfactors can aid to analyze and ﬁgure out the applicability of our various pro-posed models relying on diﬀerent contextual targets, environment settings, anddemand-supply gaps. We demonstrate the proposed models using real-world data obtained from one ofthe popular cities in the southern part of India - Chennai. As listed by Rachan-iotis et al. [16] most of the articles in literature make various assumptions todemonstrate the performance of their models, Similarly, we make a few reason-able assumptions to accommodate the lack of crucial data required to implementthe model. Various input parameters of the models and their method of estima-tion or assumptions made to reach the decisions are described below. • The entire city is divided into 15 zones for administrative purposes and weassume the distribution of vaccines is carried out based on these 15 zones aswell. • To determine the vaccine distribution centers, we assume that the vaccineswill be distributed from hospitals or primary health centers. While there areapproximately 800 hospitals in Chennai, based on ‘on-the-ground’ knowl-edge, we select 45 hospitals (3 hospitals per zone) to enable us to determinethe distance between the hospitals. The selected hospitals are classiﬁed asprivate (PVT) and public funded (PUB) based on their administration. Atleast one publicly funded hospital or primary health center is chosen perzone. • A 45 x 45 distance matrix is constructed with each row and column repre-senting the hospitals. The cells are populated with the geographic distanceobtained from Google Maps . Using these distance measures, the optimalvaccine DCs are chosen by implementing k-medoids clustering algorithm de-scribed in Section 3.1. • Total population to be vaccinated ( E ): As per the census records col-lected in 2011, Chennai has a population of 4.6 million distributed across anarea of 175 km with a population density of 26,553 persons/km . Based onthe growth in the overall population of India, we estimate the current popu-lation of Chennai to be 5,128,728 with a population density of 29,307/km . • Set of Vacccine Distribution centers ( H ): The optimal vaccine DCs arechosen from using the clustering phase of the model using silhouette widthsuch that the chosen DCs are evenly spread across the entire district. Vaccine Distribution Models ix • Staﬀ for vaccination per DC ( U ): Based on the capacity of the hospitalin terms of facilities, workforce, etc. the hospitals are classiﬁed as ‘SMALL’,‘MED’, and ‘LARGE’. We assume that small, medium, and large hospitalsallocate 5, 20, and 40 health-care workers for vaccination purposes respec-tively. • Priority levels ( P ): While the priority levels can be decided by the au-thorities based on wide variety of parameters, for our simulation we assumethat the priority depends on the age of the individual such that the elderlypeople are vaccinated earlier. The distribution of population across variousage groups is calculated based on the age-wise distribution calculated duringthe census 2011. Table 1 shows the distribution of the population across sixpriority groups.

Age group % of distribution

Table 1: Distribution of the population of Chennai across 6 age groups • Time per vaccination ( t ): Based on the data provided during the recenttrial dry run carried out in India, the government estimates to carry out 25vaccinations in 2 hours span, we approximate that the time taken for thevaccination of one person to be 5 minutes. • Time per vaccination ( N ): Keeping in mind the deﬁcit in the supply ofvaccines in the early stages of vaccination we assume that only vaccine dosesfor 50% of the total population is available currently. However, the numbercan be increased based on increase in production These described parameters can be tuned by the decision making authoritiesto accommodate the distribution at the area under consideration. For all theexperimental settings, we consider the gain factors α , β , and γ as × | (cid:15) | , × |P| × | (cid:15) | , and respectively. Yet, as mentioned earlier, these gain factors can beexplored and set according to the solutions desired by policy makers and decisionmaking authorities. We demonstrate the proposed models in two scenarios with randomly generateddata and two scenarios with real world data from Chennai. We discuss in detailthe inferences from each scenario in this section. In each scenario we comparethe four models and highlight how PD-VDM optimizes the two parameters takeninto consideration - priority and distance.

For the two random scenarios we consider that there are 12 vaccine DCs in totaland based on the silhouette width measures as shown in Fig. 1a we select three DCs optimally distributed across the area. We assume the total population size( E ) to be 200 and the total number of vaccines available ( N ) to be 85. Of thethree chosen DCs we assume that each has a capacity ( U ) of 15, 30, 45. Themodel considers ﬁve diﬀerent diﬀerent priority levels ( P ) with 1 being the leastand 5 being the most. The population is distributed among these priority levelsas shown in Table 2. To demonstrate the impact of the population distribution Priority group

Raw Count

43 35 50 45 27

Table 2: Distribution of population across priority groups in random simulation.parameter, we run the simulation model under two diﬀerent distributions namely, • Random-case -1 (RC-1): Uniform random distribution • Random-case -2 (RC-2): Poisson like distribution where the population isdense at some regions and spares at others. (a) Random simulation (b) Case Study

Fig. 1:

Distribution of vaccine across various priority groups

The distribution of vaccines at any time instance t n by all four models, forboth random cases is depicted in Fig. 2. The percentage of individuals vaccinatedin each priority group in random case 1 and random case 2 are depicted in Fig.3aand Fig. 3b respectively. The B-VDM vaccinates 41.86%, 37.14%, 42%, 46.67%,and 44.44% of individuals across priorities 1 to 5 respectively, in both RC1 andRC2. We can see that 55.56% of the individuals from the highest priority groupare left out. The D-VDM optimizes only the distance parameter and vaccinates37.21%, 51.43%, 38%, 51.11% and 33.33% of individuals across priorities 1 to 5respectively in RC1 and almost the same results in RC2. Again we can notice,that a greater percentage of the individuals in the highest priority group are notvaccinated. While both P-VDM and PD-VDM attempt to vaccinate 100% of thehigh priority individuals in both RC1 and RC2, on studying the average distancetravelled by each individual of the population as depicted in Fig. 4a, we canclearly identify that the PD-VDM reduces the distance parameter by more than40% in both RC1 and RC2 . We can also see that the ‘distribution of population’parameter does not impact the performance of the models and PD-VDM is the alidating Optimal COVID- Vaccine Distribution Models xi(a) RC1 B-VDM (b) RC1 P-VDM (c) RC1 D-VDM (d) RC1 PD-VDM(e) RC2 B-VDM (f) RC2 P-VDM (g) RC2 D-VDM (h) RC2 PD-VDM

Fig. 2:

Snapshot of the simulation of the models in the random case studies(a) RC1 (b) RC2(c) CS1 (d) CS2

Fig. 3:

Distribution of vaccine across various priority groupsii M. Emu et al.

Priority group 1 2 3 4 5 6Case-study scenario 1

546 598 2199 307 160 90

Case-study scenario 2

Table 3: Distribution of population across six priority groups for Case studyscenariosmost eﬃcient across both Uniform and Poisson distribution. Although there isa slight increase in the average distance travelled in the second distribution thePD-VDM still achieves the most optimal results. (a) RC1 and RC2 (b) CS1 and CS2

Fig. 4:

Distribution of vaccine across various priority groups

In both the case-study scenarios we consider that the vaccination process con-tinues for ﬁve hours each day and as mentioned in Section 4 each vaccinationtakes ( t n ) 5 mins. Hence there are 60 vaccinations carried out by each healthcare worker for any given day. We compile and present the results of the vac-cination process at the end of any given day. Although in the initial stages ofthe vaccination it is likely that individuals from one priority group will only bevaccinated, as time progresses people across various priority groups will need tobe considered. Hence the population considered by the model for each day istaken as a stratiﬁed sample from the age-wise distribution of the overall popula-tion of Chennai provided in Table 1. Initially we identify the optimal number ofDCs needed for eﬀectively serving the district of Chennai based on the distancebetween the hospitals considered. On analyzing the silhouette width of various‘number of clusters’ as depicted in Fig. 1b we present two diﬀerent scenariosnamely, • Case study scenario -1 (CS1): Three optimal DCs • Case study scenario -2 (CS2): Twelve optimal DCs

Case study scenario -1:

For this scenario, we consider 3 optimal vaccine DCs( H ) with capacities ( U ) of 5, 20, 40 since the chosen DCs fall under ‘SMALL’,‘MED’ and ‘LARGE’ respectively. Based on these factors the total populationto be vaccinated ( E ) in any given data is estimated to be 3,900. As mentioned inSection 4 we assume, that the total vaccines ( N ) available to be half the totalpopulation which in this case sums upto 1,950. The entire population falls undersix priority groups ( P ) based on their age. The actual number of people in eachpriority group is provided in Table 3. alidating Optimal COVID- Vaccine Distribution Models xiii

Case study scenario -2:

For this scenario, we consider 12 optimal vaccineDCs ( H ) and among these 12 chosen DCs there are 3, 2 and 7 DCs with astaﬀ capacity ( U ) of 5 (‘SMALL’), 20 (‘MED’), and 40 (‘LARGE’) respectively.Hence the total population ( E ) that can be vaccinated at any given 5 hour day is20100. Similar to scenario 1 we assume that the total vaccines ( N ) available tobe half the total population which amounts to 10,050 and that the population isdistributed across six diﬀerent priority groups ( P ) based on their age as shownin Table 3.The vaccination percentage across the priority groups for both case studyscenarios are provided in Fig. 3c and Fig. 3d respectively. Unlike the randomscenarios, it is interesting to note that in both CS1 and CS2, the B-VDM thoughnot optimized to satisfy a speciﬁc hard constraint, it produces results that arealmost identical to P-VDM vaccinating 100% of the highest priority group in-dividuals. This can be attributed to the similarity in the eﬀect of α and β gainparamters in both these models in CS1 and CS2. Though the PD-VDM modelvaccinates less than 90% of the individuals in the three highest priority groupswe can clearly see that it signiﬁcantly reduces the ‘average distance travelled’ byan individual in the population by more than 70%, which makes it more eﬃcientthan all the other models. While D-VDM achieves the least ‘average distancetravelled’ value it sacriﬁces vaccinating almost 50% of the high priority groups.Thus, we demonstrate how PD-VDM eﬃciently distributes the available vac-cines by modifying various parameters like the distribution of population, totalpopulation, total number of available vaccines, the number of vaccine DCs andthe capacity of each DCs. The model provides ﬂexibility for the decision-makingauthorities of any given demographic to optimize the distribution of vaccine inthe desired region. In this paper, we propose an optimization model (PD-VDM) based on ConstraintSatisfaction Programming framework to ﬁnd the most eﬀective way to distributevaccines in a given demographic region, in terms of distance and a priority (age,exposure, vulnerability, etc). We compare the eﬃciency of the model with threeother models which take into consideration either one or none of the two op-timization constraints. While this model can be adapted across a wide varietyof scenarios as demonstrated in our case studies, it is essential to understandthat due to resource constraints we have demonstrated only two of the manyavailable constraints. Expanding the scope of the model to allow optimizing awide variety of parameters can be carried out in future research, along with anattempt to replace some of the assumptions made in our model with real worlddata.

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