Vaccination Planning in Peru using Constraint Programming

Willy Ugarte

Universidad Peruana de Ciencias Aplicadas (UPC), Lima, Peru

Keywords:

CSP, Pandemic, CoViD, Constraint Programming.

Abstract:

Vaccination has been proven to be the most effective method to prevent infectious diseases, specially nowadays

with the global pandemic of CoViD19. Millions of people are not immunized yet in various countries because

of low vaccine availability resulting from inefﬁciencies and/or lack of access to the vaccines. We propose a

constraint programming model, kwnown as Constraint Satisfaction Problem (CSP) as a distribution model for

vaccination to address the unique characteristics and challenges facing vaccine dose assignation. This CSP

model capture the uncertainties of demand for vaccinations such as the age range of the vaccination campaign

and the location of vaccination centers. The objective is to maximize the percentage of fully immunized people

facilitating the access by location and capacity of the vaccination centers while respecting the health ministry

dispositions (e.g., age range, number of doses, etc.). Our research examines how these can be optimized with

a constraint optimization problem in a single objective function. We tested the model using Peru open data on

vaccination planning of their national health ministry. We make many experiments to show the feasibility of

our proposal to increase their immunization coverage.

1 INTRODUCTION

Due to the fact that a pandemic was generated by the

CoViD19, in order to retain its transmission, many

countries had to enter a situation of social isolation.

This isolation brought with it many problems such

as the closure of different public areas and negative

effects on the country’s economy. At present, after

various months, a vaccine was developed that allows

us to ﬁght this virus, thus, vaccination centers need

to organize themselves to be able to attend to unvac-

cinated people, therefore, a method is needed that re-

lates these people, distributed in 5 different districts of

Lima with the respective vaccination centers. Even-

more, Data about the vaccination process, such as

vaccine identiﬁcation and the number of persons who

have been vaccinated, is crucial for vaccine manufac-

ture and distribution in order to reach the target level

of immunization in a country (Carniel et al., 2021).

Constraint Programming (CP) is a paradigm that

allows solving combinatorial problems based on dif-

ferent techniques and that its success is based on its

simple formulation, since it describes the problems in

the form of decision variables where values are as-

signed and restrictions are established to ﬁnd assign-

ments to the variables with the established rules. For

https://orcid.org/0000-0002-7510-618X

example, the solution of an optimization problem be-

gins with a conversion of said problem to an appropri-

ate CSP, which treats the objective function as one of

the constraints, which after that evaluates the value of

said function with all feasible solutions, and choose

the best solution found.

Apart from this, another of the important char-

acteristics of CP is its expressive modeling power

and its great capacity to guarantee the determination

of optimal solutions. That is, expressive modeling

would help to create compact models that adapt to

each different type of problem, while domain reduc-

tion through constraint propagation ensures the de-

termining optimal global solutions, without getting

caught up in local solutions. Besides, there are al-

ready various applications of CP for problems related

to the Pandemic (Manlove et al., 2017; Nasrabadi

et al., 2020; Ugarte, 2020).

From all that has been said previously, in this work

we are going to rescue the main theories in order to

create a CSP model in which it will be focused on vac-

cinations to Peruvian people in various regions. Thus,

accelerating the vaccination process of the entire pop-

ulation, considering the distance of the patients to the

vaccination centers, the age of the patients and the ca-

pacity of patients that a hospital has.

Our contribution are as follows:

Ugarte, W.

Vaccination Planning in Peru using Constraint Programming.

DOI: 10.5220/0010899700003116

In Proceedings of the 14th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2022) - Volume 3, pages 757-764

ISBN: 978-989-758-547-0; ISSN: 2184-433X

757

• We propose that the challenge of assigning vac-

cinations to people be formulated as a Constraint

Satisfaction Problem (CSP). The CSP model de-

scribed here is simple to comprehend, and when

combined with a CP toolkit

, it produces an ideal

outcome utilizing real data from Peruvian people.

• We illustrate how the CSP model can be used as

a foundation for imposing additional restrictions,

allowing us to handle versions of the bed assign-

ment problem that may arise naturally in practice

but are difﬁcult to solve with existing algorithms.

• We show that CSPs are a simple paradigm that

does not need training a model or obtaining any

previous data, and that they are an elegant solution

to address this type of problem by explaining the

results.

This paper is organized as follows. Section 2 dis-

cusses related work. Section 3 introduces the rele-

vant concepts and deﬁnes the problem formally, then

presents our approach and our development. This is

evaluated in Section 4 after which we show the results

of our experiments, and ﬁnally we conclude.

2 RELATED WORKS

As various studies reveal, the majority of efforts on

perishable inventory deal with food, particularly fresh

produce (Bakker et al., 2012; Goyal and Giri, 2001;

Li, 2010; Gregor et al., 2018). Although the study is

based on public health, it is comparable to our envi-

ronment in that it only evaluates one party, not the full

supply chain.

In (Duijzer et al., 2018), the authors survey the

literature on vaccine supply chains. The majority of

these research focus on supply chain issues for one of

two types of vaccinations: seasonal vaccines like in-

ﬂuenza and non-seasonal vaccines like pediatric im-

munizations. Only a few studies in the literature

present models for designing a cold supply chain net-

work for inﬂuenza vaccines (Hovav and Tsadikovich,

2015). Some research look into the supply chain con-

tracting issue with inﬂuenza vaccines (Chick et al.,

2008; Dai et al., 2012). In contrast, our work does

not aim to tackle the supply chain of vaccination, but

rather its assignment process for the people.

Other research look at vaccine allocation deci-

sions during pandemics (Westerink-Duijzer et al.,

2017; Westerink-Duijzer et al., 2020). The focus of

our research is only for CoViD vaccines.

OR-tools - https://developers.google.com/

optimization

Other methodologies, such as lean, simulation,

and Markov decision process (MDP) models, have

been employed in recent research on relevant topics in

addition to these mathematical modeling approaches.

In (Mofrad et al., 2014), the authos recently devel-

oped a Markov Decision Process model that calcu-

lates the best time to preserve vials based on the cur-

rent vial inventory, time of day, and remaining clinic

days until the next reﬁll. The authors provide a re-

alistic strategy for minimizing open-vial waste while

giving sufﬁcient vaccines. Based on this ﬁnding,

the authors of (Mofrad et al., 2016) compared opti-

mum and heuristic policies in the presence of random

vial yield and examined numerous operational tech-

niques to maximize coverage while controlling open

vial waste. A simple-to-use decision-making tool was

also created and made available online.

Patient scheduling duties fall into three categories,

according to (Cardoen et al., 2010; Marynissen and

Demeulemeester, 2019): Dynamic patient schedul-

ing, dispersed patient scheduling, and coordinated pa-

tient scheduling. Unlike our strategy, which focuses

on vaccination center assignment during a pandemic,

all of these focus on enhancing hospital resources to

reduce patient waiting times (Vermeulen et al., 2009).

There are numerous options for resolving the as-

signment dilemma. As an example, treating this prob-

lem with mixed integer-programming (Ben Bachouch

et al., 2012; Turhan and Bilgen, 2017). In real-world

scenarios, though, it’s occasionally permissible to ig-

nore some restrictions in order to cover others that are

more important to their model. Hence the importance

of soft constraints.

Contrarily, a few publications have attempted to

solve this problem using different types of optimiza-

tion techniques (e.g., genetic algorithms, local search,

. . . ). In (Demeester et al., 2010), for example, the au-

thors suggest a hybrid tabu search method that assigns

patients to hospital beds automatically. Another ex-

ample of a two-level metaheuristic to handle the op-

erating room scheduling and assignment problem is

described in (Aringhieri et al., 2015). They, on the

other hand, interpret all patient requests as hard limits,

whereas our approach incorporates preferences into

the objective function.

To the best of our knowledge, none of these sys-

tems deal with simultaneous geolocation for multiple

vaccination centers; rather, they employed it to locate

people within a single health center or medical facil-

ity in order to optimize resources and reduce waiting

times.

ICAART 2022 - 14th International Conference on Agents and Artiﬁcial Intelligence

758

(a) Map of regions of Peru (b) Corresponding graph (c) Colored map of regions of Peru

Figure 1: Map of regions of Peru.

3 CP FOR VACCINATION

PLANNING

Now, we introduce the main notions for deﬁning the

problem formally, for the sake of presenting our mod-

eling in CP and its development.

3.1 Preliminary Concepts

Many countries’ health systems are currently failing,

particularly their capacities, due to a lack of coop-

eration and other issues. This problem is exacer-

bated for third-world countries, such as Peru, because

their health systems were under-prepared and under-

funded prior to the pandemic.

3.1.1 Problem Statement

Given the current scenario, it is critical to prepare a

vaccination campaign with the goal of vaccinating the

majority of the population in the event of a COVID-

19 pandemic. Assigning people to vaccination centers

is a difﬁcult task that necessitates meeting a number

of criteria. Like the number of dosages available at

each site, the age of the individual, the location of the

people, and the location of the centers are all factors

to consider. With the aforementioned criteria, it is re-

quired that as many people as possible be assigned to

vaccination centers as soon as possible. This problem

can be simpliﬁed to two principles:

1. People should be assigned to the nearest center.

2. Older people should be assigned earlier than the

rest.

3.1.2 Constraint Programming

Deﬁnition 1 (Constraint Satisfaction Problem

(CSP) (Rossi et al., 2006)). A Constraint Satisfaction

Problem (CSP) P = (X , D, C ) is deﬁned as:

• a ﬁnite set of variables X = {x

, x

, . . . , x

• a set of domains D, mapping every variable x

∈

X to a set of values D(x

• a ﬁnite set of constraints C .

The goal is to ﬁnd a variable-to-value mapping

that maps each variable x

to a value in its domain

D(x

) while meeting all of the C constraints. A solu-

tion to the CSP is the name given to this mapping.

Example 1 (Graph Coloring). Graph Coloring is

the process of coloring the vertices (v

) of a graph

G = (V, E) so that no two neighboring vertices are

the same color. The following CSP (see Deﬁnition 1)

can be used to model this problem:

• X = {c

, c

, . . . , c

}, where c

is the color variable

for each node v

in G.

• D = {1, 2, . . . , m}, where each number represents

a label for a color (e.g., 1 is blue, 2 is red, . . . ).

• C = {c

6= c

, ∀(i, j) ∈ E}, where each inequality

ensures having no two adjacent vertices are of the

same color.

Fig. 1a depicts a map of Peru divided into regions.

This map can be represented by a graph, as shown

in Fig. 1b. The colored map matching to the Graph

Coloring of 25 locations is shown in Fig. 1c.

Vaccination Planning in Peru using Constraint Programming

759

Deﬁnition 2 (Constraint Optimization Problem

(COP) (Chen et al., 2020; Rossi et al., 2006)). A

constraint optimization problem (COP) is deﬁned as

= P ∪ { f }, where P is a constraint solving problem

(CSP) (see Deﬁnition 1) and f is the objective func-

tion to be optimized (either maximized or minimized).

Apart from identifying mappings from variables

to values, the task is to optimize (either maximize

or minimize) these mappings in order to discover the

best mapping based on an objective function f .

Example 2. Example 1 is being continued. Consider

the graph G = (V, E) and the COP (see Deﬁnition 2)

for Graph Coloring. P = (X = {c

, c

, . . . , c

}, D =

{1, 2, . . . , m}, C = {c

6= c

, ∀(i, j) ∈ E}) and f =

#o f colors are deﬁned as P

= P ∪ { f }.

Figure 1c shows the Graph Coloring of 25 regions

with only 4 colors. This result is optimal since, it can-

not be done with fewer colors.

3.2 CP Model

To model our problem as a CSP, we must ﬁrst deﬁne a

triplet (X , D, C ) that a solver will process and solve.

3.2.1 Variables

Let C a set of vaccination centers, V

the set of avail-

able vaccine doses in center i and P be the set of peo-

ple. The set of variables is deﬁned as:

i jk

∈ {0, 1} where (i, j, k) ∈ C ×V

× P(1)

∈ {0, 1, 2} where k ∈ P (2)

age

∈ [0..100] where k ∈ P (3)

If at the center i, the vaccination dose j is taken

by the person k, then x

i jk

= 1. nd

is the number of

doses received by the person k. age

is the age of the

person k. The goal is to develop a set of variables that

satisfy all of our requirements in order to correlate

each vaccine dose of a center with a matching person.

3.2.2 Constraints

Constraints are crucial in a CSP model because they

determine whether or not a variable assignment is

possible and whether or not the problem can be ad-

dressed.

The following are the essential principles (i.e.,

hard limitations) that vaccination centers must follow

in order to solve this problem:

• There Must Be at Most a Single Person As-

signed to Each Vaccine Dose. This can be mod-

eled as:

∀i ∈ C, ∀ j ∈ V

∑

k∈P

i jk

≤ 1 (4)

• There Must Be at Most a Single Vaccine Dose

Assigned to Every Person. This can be modeled

as:

∀k ∈ P,

∑

i∈C

∑

j∈V

i jk

≤ 1 (5)

• If the Person is Already Vaccinated, He/She

Must not Be Assigned to Any Center. This can

be modeled as:

∀i ∈ C, ∀ j ∈ V

, ∀k ∈ P, x

i jk

⇐⇒ (nd

< 2) (6)

• If the Person Is Not in the Age Range for Vac-

cination, He/She Must not Be Assigned to Any

Center. This can be modeled as:

∀i ∈ C, ∀ j ∈ V

, ∀k ∈ P, x

i jk

⇐⇒ (age

>= min

age

)

(7)

Therefore, the CSP P

vacc

= (X , D, C ) for this

problem is deﬁned as:

• X =











i jk

| ∀(i, j, k) ∈ C ×V

× P}

{nd

| ∀k ∈ P}

{age

| ∀k ∈ P}











• D =







i jk

∈ {0, 1} ∀(i, j, k) ∈ C ×V

× P}

∈ {0, 1, 2} ∀k ∈ P

age

∈ {0, .., 100} ∀k ∈ P







• C =











{

∑

k∈P

i jk

≤ 1 | ∀(i, j) ∈ C ×V

}

{

∑

i∈C

∑

j∈V

i jk

≤ 1 | ∀k ∈ P}

i jk

⇐⇒ (nd

< 2) | ∀(i, j, k) ∈ C ×V

× P}

i jk

⇐⇒ (age

>= min

age

) | ∀(i, j, k) ∈ C ×V

× P}











3.2.3 Preferences

Preferences, also known as soft constraints (Bistarelli

et al., 1995; Cooper et al., 2010), are highly desired;

in other words, a solution to this problem should aim

to fulﬁll them as much as feasible, but they are not re-

quired to ﬁnd a solution. These are some of the rules:

• Ideally, every person should have access to a vac-

cination center if necessary. This can be modeled

as:

max

sol

∑

i∈C

∑

j∈V

∑

k∈P

i jk

(8)

• If possible, everyone should be vaccinated at the

closest vaccination center. This can be modeled

as:

min

sol

∑

i∈C

∑

j∈V

∑

k∈P



i jk

× distance(i, k)



(9)

where distance(i, j) is the distance from center i

from patient k.

ICAART 2022 - 14th International Conference on Agents and Artiﬁcial Intelligence

760

• Mostly, elder people should be prioritized when

there are not enough vaccines. This can be mod-

eled as:

max

sol

∑

i∈C

∑

j∈V

∑

k∈P



i jk

× age



(10)

Preferences are not treated as a (in)equality, but

as a maximization (resp. a minimization), because

we prefer to satisfy them as much (resp. as least) as

possible. These three preferences can be combined

into a single goal function that can be optimized as

follows:

f =

∑

i∈C

∑

j∈V

∑

k∈P



i jk



1 −

distance(i, k)

max

dist

age

max

age



(11)

The objective function (11) requests that the over-

all patient attention be maximized. It’s worth not-

ing that if x

i jk

= 0, the person isn’t counted in the

total. The distance distance(i, k) (resp. the age

age

) must be normalized by dividing it by the maxi-

mum value max

dist

(resp. max

age

), yielding

distance(i,k)

max

dist

(resp.

age

max

age

). Nonetheless, because the distance

should be minimized and f is maximized, the dis-

tance is negative. Therefore, the COP is deﬁned as:

vacc

= {P

vacc

∪ max( f )}.

4 EXPERIMENTS

In this section, experiments will be carried out to

show the feasibility of our approach, starting from the

experimental protocol and the results, as well as its

discussion.

4.1 Experimental Protocol

All of the tests were carried out on a personal

computer with a Linux operating system, an i5-8600

CPU core processor running at 3.10 GHz, and 16GB

of RAM. The implementation was carried out in

OR-tools

. All source codes and data sets are pub-

licly available at https://colab.research.google.com/

drive/13ndIi8BsgoY9gQrwpq YcO s-uTTwvVh?

usp=sharing.

Data. In this case, two main sources of information

were needed:

• Vaccination Centers: The informa-

tion on vaccination centers in Peru, and

their geographical location were found at

https://www.datosabiertos.gob.pe/dataset/

centros-de-vacunacion then exactly plotted with

Figure 2: Example of people Distribution.

geojson with the Polygonal District Map of Peru

at https://github.com/juaneladio/peru-geojson/

blob/master/peru distrital simple.geojson.

For instance, for districts of Cieneguilla, Ate,

Chaclacayo, Lurigancho and La Molina, the 8

principal health centers are listed in Table 1.

• People: For deﬁning the quantity of peo-

ple per district, information was used

from the 2017 census of National Insti-

tute of Statistics and Information (INEI)

at https://www.inei.gob.pe/estadisticas/

indice-tematico/poblacion-y-vivienda/. Fi-

nally, for the vaccination capacity at the district

level, we used statistics from the National Min-

istry of Health at https://www.minsa.gob.pe/

reunis/data/vacunas-covid19.asp.

The National Authority of Personal Data Pro-

tection protects people’s geolocation data un-

der Peru’s National Law on Personal and Health

Data (see http://bvs.minsa.gob.pe/local/MINSA/

5118.pdf).

Lacking real information about the location of in-

fected people, it was required to generate loca-

tions following their pupulation distribution (see

Fig. 2).

4.2 Results

Now, we report and discuss the numerical results on

synthetic problems when comparing vaccination as-

signments of various combinations of districts.

Figures 3a and 3b show the location of vaccina-

tion centers for some districts in Lima, Peru, the lo-

cation of people as colored points (red points are as-

signed according to their distance to the vaccination

center and the age range allowed by the ministry,

green points are people already vaccinated and blue

points are people that are infected, and thus must wait

in quarantine before vaccination).

Now, two test scenarios will be described. For the

ﬁrst one, only the districts of Cieneguilla, Ate, Cha-

clacayo, Lurigancho and La Molina will be consid-

ered, while in the second one, only the districts of

Surco, San Isidro, San Borja, Surquillo y Miraﬂores

will be considered.

Vaccination Planning in Peru using Constraint Programming

761

Table 1: Lisf of 8 health centers.

Id District Name Latitude Longitude

La Molina La Molina Center -12.090 - 77.017

Cieneguilla Loza Deportiva Municipal -12.092 - 77.070

Cieneguilla Tambo Viejo -12.106 - 77.046

Lurigancho Mall Santa Anita -12.101 - 77.035

Chaclacayo Pachacutec Stadium -12.106 - 77.055

Ate Wholesale Market -12.128 - 77.017

Ate Elitte -12.103 - 77.029

Ate District Health Center -12.115 - 77.033

Scenario 1 (See Figure 3a):

For this case, only the districts of Cieneguilla, Ate,

Chaclacayo, Lurigancho and La Molina will be con-

sidered. Ten execution tests were performed. An av-

erage execution time of 12.2 seconds was achieved,

managing to assign 1,500 people.

Scenario 2 (See Figure 3b):

For this case, only the districts of Surco, San Isidro,

San Borja, Surquillo y Miraﬂores will be considered.

Ten execution tests were performed.

An average execution time of 36.1 seconds was

achieved, managing to take 10,000 people. Also in

Figure 3b, we can see that most of the centers have as-

signments from people nearby. Contrarily, some cen-

ters take assignments further, even if this might seem

counter-intuitive, it may indicate that any of those

center has a limited capacity. This happens because

the objective function (see Equation (11)) tries to min-

imize distance and simultaneously maximize the at-

tention.

Time Analysis

The typical execution times for allocating vaccination

locations for a certain number of centers at a speciﬁc

percentage of capacity with an age range are shown

in Table 2.

For instance, in Scenario 1, it takes 0.79 (resp.

1.71) seconds for 20% (resp. 100%) of centers to as-

sign the people older than 60 years. For instance, in

Scenario 2, it takes 2.41 (resp. 4.32) seconds for 20%

(resp. 100%) of centers to assign the people older than

40 years.

Table 2: Times for assignments (in seconds).

% of Centers

20% 40% 60% 80% 100%

Scenario 1

min

age

= 60 0.79 0.99 1.29 1.36 1.71

min

age

= 50 1.17 1.29 1.49 2.16 2.83

min

age

= 40 1.45 1.49 1.69 2.34 3.11

min

age

= 30 1.57 1.67 1.90 2.67 3.33

min

age

= 18 1.78 1.90 2.14 2.76 3.78

Scenario 2

min

age

= 60 1.67 1.93 2.33 2.46 2.81

min

age

= 50 2.25 2.36 2.47 3.26 3.93

min

age

= 40 2.41 2.45 2.65 3.43 4.32

min

age

= 30 2.48 2.64 2.99 3.72 4.45

min

age

= 18 2.67 3.88 3.21 3.86 4.87

5 CONCLUSIONS

Throughout the development of this work, we have re-

alized that it is possible to solve current problems us-

ing the CP paradigm, with useful information in Peru,

that is easy to access. In addition, on the subject of

heuristics, we have been able to observe that, through

a good modeling of our exercise, it can be simpliﬁed

to a few steps, which gives us to understand the im-

portance of using Constraint Programming, which al-

lows us to, both in the execution time and in the cod-

ing part, it provides us with great ease. This includes

the use of objective functions, as we have seen in the

part of restrictions, which we have been able to use as

minimum objectives.

ICAART 2022 - 14th International Conference on Agents and Artiﬁcial Intelligence

762

(a) Scenario 1

(b) Scenario 2

Figure 3: Comparison of vaccination assignments for Scenario 1 (a) and Scenario 2 (b).

Vaccination Planning in Peru using Constraint Programming

763

On downfall, if this kind of problem is tackled

with machine learning, is that it needs a lot of data to

train, in Peru and in general, many times there is not

enough (available) data. In contrast, CP works to ﬁnd

solutions from the data set that must respect given re-

strictions, optimizing an objective function with sim-

ple heuristics, without training or a lot of data.

As a future work, we would like to scale our ap-

proach, with more instances, or evenmore with new

constraints that appear on the ﬂy, for instance now

some countries have a booster shots policies, that re-

quire to rethink some of the constraints, such as Dy-

namic CSPs (Verfaillie and Schiex, 1994)

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