Fuzzy logic for the management of vaccination during pandemics: a spread-rate-based approach

IAES International Journal of Artificial Intelligence (IJ-AI)
Vol. 13, No. 3, September 2024, pp. 2808~2815
ISSN: 2252-8938, DOI: 10.11591/ijai.v13.i3.pp2808-2815  2808
Journal homepage: http://ijai.iaescore.com
Fuzzy logic for the management of vaccination during
pandemics: a spread-rate-based approach
Abdul Kareem, Varuna Kumara
Department of Electronics and Communication Engineering, Moodlakatte Institute of Technology, Kundapura, India
Article Info ABSTRACT
Article history:
Received Oct 29, 2023
Revised Feb 11, 2024
Accepted Feb 28, 2024
Pandemics, such as coronavirus disease (COVID-19) are known to cause
massive damage to the world's economic growth and their impacts are serious
and influence across every aspect of social structure. The most inevitable
factor in responding to the disaster of pandemics is the right management in
terms of allocating a limited vaccine supply. The focus of this research work
is to utilize a fuzzy logic inference system in the allocation of vaccine doses
to the regional authorities by a central authority. The objective is obtained by
designing a system based on fuzzy logic that considers the spread rate as the
input to infer the vaccination rate of the local population. This system makes
it possible for sufficient doses of vaccines to be allotted to the prioritized
regions where the severity of the spread rate is aconcern, and vaccines arenot
held up in regions wherethe severity of the spread rate is lesser. Thedesigned
system is verified using MATLAB software, which shows that this method
can ensure an effective and efficient allocation of vaccination in the local
regions and aid the fight against the disastrous spread of the disease.
Keywords:
Fuzzy logic
Mamdani inference
Matrix laboratory
Pandemic
Vaccination
This is an open access article under the CC BY-SA license.
Corresponding Author:
Abdul Kareem
Department of Electronics and Communication Engineering, Moodlakatte Institute of Technology
Kundapura, Karnataka, India
Email: afthabakareem@gmail.com
1. INTRODUCTION
Pandemics like coronavirus disease (COVID-19) have severely damaged global economies and
caused widespread morbidity. They have created a global health crisis and the pandemics have put up cultural
and geographical intolerance. The pandemics are far fromover and are predicted to continue their crutches in
the coming days [1]‒[11]. Vaccination is the most effective tool for protecting people against pandemics
[7]‒[15]. Hence, the global efforts to develop vaccines to protect against pandemics have been unrivaled in the
history ofpublic health.As vaccine production scales up and new products are authorized,the allocation criteria
will broaden until supply enables the widespread use of vaccines. The officials are facing an issue in the proper
management in terms of optimal allocation of the vaccines to the different regions in pandemic situations. It
will be a critical challenge to ensure that the allocation of vaccines is managed in a quick, effective, and
unbiased way [14], [15].
The traditional method of distribution of vaccines by a central government or an authority to local
state authorities is to allocate vaccine doses proportional to the population of the states and this method was
recommended by the WHO during the COVID-19 pandemic [3]. A population-based distribution scheme
seems to be expressing equality in terms of moral concern and may be considered to be politically tenable.
However, it considers that equality implies treating different regions identically rather than equitably
responding to their varying needs. Equally populous states can face different levels of spread of the pandemic.
Providing aid merely based on population is unjust and against human reasoning. For example, it would be

Int J Artif Intell ISSN: 2252-8938 
Fuzzy logic for the management of vaccination during pandemics:a spread-rate-based … (Abdul Kareem)
2809
unjust and illogical to allocate antiretrovirals for HIV based on population, rather than on HIV cases [4]. In
short,the schemes based on population have two disadvantages, the states having a severe spread rate mayfacea
shortage ofvaccines,resulting in superspreading and the states having a lesserspread rate may waste vaccines
because of negligence to take vaccines due to pseudo security feeling. Hence, a fair and logical distribution of
vaccines should respond to the pandemic’s spread rate in different states, the spread rate ofthe region has to be
prioritized to avoid widespread or super spread in those states with a severe spread rate [12].
Not much research work is available in the literature on the direction of incorporating the spread rate
in the decision-making ofthe distribution of vaccines.One approach considering the spread rate was proposed
in [4], in which the fair priority model was developed. This model considers the premise that the regions
affected by highly severe spread rates are given top priority, but all regions have to be ultimately allocated
adequate doses of vaccines to break the chain of spread. In this approach, the estimation of vaccine allocation
demands the model integration with data and forecasts based on experience. However, empirical uncertainties
make this approach very difficult or almost impractical to implement [4].
In this research effort, an algorithm is developed, one that considers the spread rate, which is
characterized by two quantities, the number of confirmed cases and the rate at which the confirmed cases vary
for ascertaining the rate of vaccination. Nevertheless, there are no hard and fast rules or precise analytical
models that define the functional relation that associates the inputs to the output, but some experience-based
approximate rules are available. Fuzzy logic is a prominent soft computing tool that embeds structured
experience-based rules into computer algorithms, and it incorporates approximate human reasoning modes in
computer algorithms [16]‒[23]. Hence, we propose an implication system utilizing fuzzy logic that considers
the number of confirmed cases and the rate at which confirmed cases vary for ascertaining the ratio of scarce
vaccine supply to be allocated to various regions, and therefore to determine the vaccination effort ofdifferent
regions. This novelscheme ensures that sufficient doses ofvaccines are allotted to the states on priority where
spread rates are higher, and vaccines are not wasted in states where the spread rates are lower and ensures
effective and efficient distribution of the available vaccine doses to the states and enhance the fight against
pandemics.
The rest of this paper is structured in the following sections. Section 2 describes the method of
developing a fuzzy logic algorithm that considers the severity of the spread rate to ascertain the vaccination
effort. Section 3 presents the results and deliberations that establish the efficacy of the proposed technique.
Finally, the conclusions with relevant discussions and findings are presented in section 4.
2. METHOD
The input quantities, the number of confirmed cases, and the rate at which confirmed cases vary are
scaled into the normal range of [0,1] by appropriate scaling factors. The normalized number of confirmed cases
and the normalized rate at which confirmed cases vary are the inputs to the fuzzy logic system. The output of
the system is the vaccination effort, normalized in the range [0,1].
2.1. Fuzzification
The normalized variables, number of confirmed cases, and the rate at which confirmed cases vary are
applied to the fuzzification block that uses triangular membership functions of the knowledge base which are
given respectively in Figures 1 and 2. The fuzzy linguistic variables of the input “number of confirmed cases”
are very low (VL), low (L), medium (M), high (H), and very high (VH) and those of the input “rate of change
of number of active cases” are negative big (NB), negative medium (NM), negative small (NS), zero (Z),
positive small (PS), positive medium (PM), and positive big (PB). The fuzzy sets of the output “vaccination
rate” are VL, L, M, H, and VH as shown in Figure 3.
Figure 1. Knowledge base: number of confirmed cases

 ISSN: 2252-8938
Int J Artif Intell, Vol. 13, No. 3, September 2024: 2808-2815
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Figure 2. Knowledge base: the rate at which confirmed cases vary
Figure 3. Knowledge base: vaccination rate
2.2. Rule base
The rule base of the proposed method for inferring vaccination based on the spread rate is designed
using human experience-based approximate rules. The spread rate is determined by the number of confirmed
cases and the rate at which confirmed cases vary. The rules are given in Table 1. The rules are designed such
that the vaccination effort is higher ifthe severity ofthe spread rate is higher and the vaccination effort is lower
if the severity ofthe spread rate is lower. The fuzzy surface showing the relationship between inputs and output
relationship is as in Figure 4.
Table 1. Rule base
Rate of changeof active cases Active cases
VL L M H VH
NB VL VL VL L M
NM VL VL L M H
NS VL L M H VH
Z L M H VH VH
PS M H VH VH VH
PM H VH VH VH VH
PB VH VH VH VH VH
Figure 4. Relationship between inputs and output

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2.3. Fuzzy inference
The proposed algorithm uses the Mamdani Interference algorithm. In the Mamdani algorithm, the “if”
part of every rule is a connected statement involving fuzzy sets of input variables using the “and” logic and the
“then” part is a fuzzy set of the output variable [22]‒[26]. For instance, in this system, rules are of the form “if
the number of confirmed cases is VB and the rate of vary of the number of confirmed cases is PH, then the
vaccination rate is VH”.As the input fuzzy sets are connected by the “and” logic,the operator “min” is operated
on the membership values of two inputs to find the truth value of the corresponding rule, which is applied to
the output fuzzy variable “VB”.This procedure gives fuzzy outputs for allthe rules,and they are then combined
by applying the “max” operator on the fuzzy outputs to obtain a final fuzzy output.
2.4. Defuzzification
The final output of the fuzzy implication is defuzzified using the center of gravity scheme, in which
the center of gravity of the fuzzy output is calculated using (1). The center of gravity scheme is preferred in
this technique as it produces highly smooth and precise output [22], [26].
𝑧∗
=
∫ 𝜇𝐶
(𝑧).𝑧 𝑑𝑧
∫ 𝜇𝐶
(𝑧) 𝑑𝑧
(1)
where 𝑧∗
represents the defuzzified value of the output z, ∫ 𝜇𝐶
(𝑧) holds the membership function of the
aggregated fuzzy output.
3. RESULTS AND DISCUSSION
For studying the efficacy of the proposed algorithm, consider the case of allocating vaccines to six
regions: region 1, region 2, region 3, region 4, region 5, and region 6, where the normalized value of the number
of confirmed cases and normalized value of the rate at which confirmed cases vary are as shown in Table 2.
The computation of the proposed algorithm is implemented using the fuzzy logic toolbox of MATLAB, the
results are shown in Figure 5 to Figure 10. The rates of vaccination in various regions inferred by the proposed
algorithm are tabulated in Table 3.
Table 2. Input parameters
Region Normalizedvalue of thenumber ofconfirmedcases Normalizedvalue of rateat whichconfirmedcases vary
Region 1 1 1
Region 2 1 -1
Region 3 0.5 0.5
Region 4 0.5 -0.5
Region 5 0.1 0.5
Region 6 0.1 -0.5
Figure 5. Inference of region 1

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Table 3. Vaccination effort
Region Normalizedvalue of vaccination rate
Region 1 0.92
Region 2 0.5
Region 3 0.905
Region 4 0.375
Region 5 0.648
Region 6 0.212
In region 1, where normalized values of number of confirmed cases and the rate at which confirmed
cases vary are respectively 1 and 1 (both are higher), the vaccination rate is 0.92 (higher). In region 2, where
normalized values ofthe number ofconfirmed cases and the rate at which confirmed cases vary are respectively
1 (higher) and -1 (the severity is falling at a higher rate), the vaccination rate is 0.5 (medium). In region 3,
where normalized values of the number of confirmed cases and the rate at which confirmed cases vary are
respectively 0.5 (medium) and 0.5 (severity is rising at the medium rate), the vaccination rate is 0.905 (higher).
In region 4, where normalized values of the number of confirmed cases and the rate at which confirmed cases
vary are respectively 0.5 (medium) and -0.5 (the severity is falling at the medium rate), the vaccination rate is

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0.375 (lower). In region 5, where normalized values of number of confirmed cases and the rate at which
confirmed cases vary respectively 0.1 (lower) and 0.5 (the severity is rising at the medium rate), the vaccination
rate is 0.648 (medium). In region 6, where normalized values of the number of confirmed cases and the rate at
which confirmed cases vary are respectively 0.1 (lower) and -0.5 (the severity is falling at the medium rate),
the vaccination rate is 0.212 (lower). The computation results prove that the proposed fuzzy algorithm
considers the spread rate for ascertaining the vaccination rate allocated to different regions. This inference
system makes sure that adequate doses of vaccines are allotted to the prioritized regions where the severity of
spread rates is complex, and vaccines are not held up in regions where the spread rate is not that severe. The
proposed allocation system enables proper allocation of the existing vaccine supply and hence an effectual
vaccination, and aid in containing the disastrous spread of pandemics.
The proposed allocation scheme has the limitation that the population of the region is not at all a factor
in deciding the vaccination rate,and it fails to consider the natural concern that all regions should be ultimately
allocated adequate vaccine doses to break the chain of transmission.In near future work,we propose to develop
a fuzzy algorithm based on both the spread rate and population to infer the vaccination rate. The objective is
to ensure that adequate doses ofvaccines are allocated to the prioritized regions where the severity ofthe spread
rate is higher and vaccines are not held up in regions where the severity is lower, simultaneously, all regions
are ultimately allocated adequate vaccine doses to break the chain of transmission.
4. CONCLUSION
In this paper, a fuzzy logic algorithm for the right management of vaccination by conjecturing the
allocation of the constrained vaccine doses available from a central authority to regional authorities. The
proposed algorithmis based on a fuzzy inference systemthat considers the severity of the spread ofthe disease
to compute the vaccine doses to be distributed to various regions of a central authority. This scheme makes
sure that adequate doses of vaccines are allocated to the prioritized regions in which the severity of the
transmission of the disease is higherand vaccines are not held up in regions in which the severity is lesser.The
proposed algorithmis evaluated using the fuzzy logic toolboxof MATLAB.The results imply that the proposed
algorithm ensures the appropriate distribution of the available vaccine supply and hence an effectual
vaccination of all the regions and boosts the fight against the disastrous transmission of the pandemic disease.
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BIOGRAPHIES OF AUTHORS
Abdul Kareem holds a Doctor of Philosophy fromSt Peter’s Institute of Higher
Education and Research, Chennai, India. He also received his M.Tech. from Visvesvaraya
Technological University, Belagavi, India in 2008. He is currently the Principal and a
Professor of Electronics and Communication Engineering at Moodlakatte Institute of
Technology, Kundapura, India. His research interests are in artificial intelligence, machine
learning, control systems, and microelectronics. He is a senior member of IEEE. He can be
contacted at email: afthabakareem@gmail.com.
Varuna Kumara is a Research Scholar in the Department of Electronics
Engineering at JAIN Deemed to be University, Bengaluru, India. He also received his
M.Tech. from Visvesvaraya Technological University, Belagavi, India in 2012. He is
currently an Assistant Professor of Electronics and Communication Engineering at
Moodlakatte Instituteof Technology, Kundapura, India. His research interests arein artificial
intelligence, signal processing, and control systems. He can be contacted at email:
vkumarg.24@gmail.com.

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