EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 2, 2020, 369-389
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Developing Optimal Policies to Fight Pandemics and
COVID-19 Combat in the United States

Eyüp Çetin1,∗, Serap Kiremitci2, Baris Kiremitci1

1 New York Business Global, Silver Spring, Maryland, USA
2 School of Transportation and Logistics, Istanbul University, Turkey

Abstract. The world has faced many outbreaks and pandemics with hundreds of millions of
deaths throughout its history. Those epidemics are global health concerns and as well as serious
economic issues. There is certain need to allocate scarce sources efficiently to fight such epidemics
in both personal and global dimensions. Here we develop and propose two optimization models
to maximize the total protection from any epidemic, pandemic or bioterror attacks; the first one
is personal protection model or protocol and the other one is mass protection model that is in-
spired by the well-known weapon-target assignment problem of operations research. These efforts
are optimal allocation of scarce medical and economic resources to save millions of lives- gift of
life. We implement our general mathematical programming models with real-world data to fight
the coronavirus pandemic for a person and the United States. Our personal protection protocol
provides 99.99% protection from COVID-19 for an American through personal strategies when the
mass protection model supplies 96.961% protection on average from coronavirus pandemic for the
United States through countrywide policies. The mass protection model which recommends gen-
eral policy frame for health care policy makers could be applied for any epidemic at any level from
county to city, to state and country as well as in global scale. The mathematical relation between
the personal protection protocol and the mass protection model also highlights the fact of unus pro
omnibus, omnes pro uno (one for all, all for one) for fighting epidemics- particularly the moving
enemy, novel coronavirus which is double invisible due to its viral nature and the availability of the
high level of asymptomatic cases. Recognize the enemy, as protecting yourself means protecting
people, love life, follow the rules and stay at home. That is the greatest ever social impact.

2020 Mathematics Subject Classifications: 49Mxx, 90Cxx, 97M40 , 65Cxx

Key Words and Phrases: COVID-19, novel coronavirus, SARS-CoV-2, 2019-nCoV, pandemic,
epidemic, outbreak, health policy, public health, weapon-target assignment

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i2.3700

Email addresses: eycetin@gmail.com (E. Çetin),
serapy@istanbul.edu.tr (S. Kiremitci), bariskiremitci@gmail.com (B. Kiremitci)

http://www.ejpam.com 369 c© 2020 EJPAM All rights reserved.



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 370

1. Introduction

The world has faced an important number of outbreaks and pandemics with death
tolls of hundreds of millions throughout its history such as The Great Plague of London in
1665, the Spanish Flu in 1918, 1957-1958 Pandemic (H2N2 virus), 1968 Pandemic (H3N2
virus) and 2009 H1N1 Pandemic [7]. More recently, there have been other epidemics
that are serious threats to global public health including Severe Acute Respiratory Syn-
drome (SARS-CoV) which was recognized in February 2003 and Middle East Respiratory
Syndrome (MERS-CoV) which was recognized in September 2012. Both SARS-CoV and
MERS-CoV are caused by a large family of viruses called coronaviruses (CoV)[36]. A novel
coronavirus (2019-nCoV or SARS-CoV-2) was first publicized from Wuhan, China, on 31
December 2019 as a new strain[39]. This new viral disease outbreak that struck mainly
China and spreads across countries was re-named as Coronavirus Disease (COVID-19) on
11 February 2020.

The Virus Has Been Going Viral

World Health Organization (WHO) declared COVID-19 is a Public Health Emergency
of International Concern on 30 January 2020. As number of infected countries grows, the
WHO defined the COVID-19 outbreak as a pandemic on 11 March 2020. It is the first
pandemic caused by a coronavirus and the first pandemic which is also controllable[37].
WHO Risk Assessment is currently “Very High” at Global Level as of 18 April 2020 03:00
GMT+3, there are 2,160,207 globally confirmed cases with a death toll of 146,088, and
the virus is now in 213 countries[35]. A combination of false and misleading information,
fake news and true information about the coronavirus has been circulating especially on
social media which causes an infodemic that has already become a pandemic much earlier
than the novel coronavirus itself. Based on experts’ opinions, The Economist reports a
broad guess that 25-70% of the population of any infected country may catch the disease
and that the virus could kill 60,000 Americans in a worse-scenario year when the death
toll could be in the millions across the world[11]. In a more recently official guess, the
White House forecasts the novel coronavirus death toll may be 100,000 to 240,000 despite
the measures taken in the US[19] which is a new epicenter for the novel coronavirus.

Weapons to Fight Pandemics and COVID-19

Fighting tools for any pandemic may be classified into two categories in general which
are also valid for the COVID-19 outbreak: pharmaceutical that are both treatment and
protection instruments and nonpharmaceutical tools.

The pharmaceutical methods include vaccines and antiviral drugs. WHO says that,
to date, there is no vaccine and no specific antiviral medicine to prevent or treat COVID-
19[40], and that globally more than 20 vaccines are in development, and several therapeu-
tics are in clinical trials[8]. It is well known that the availability of approved medications
will take months when vaccines optimistically take a year.



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 371

Apart from vaccines and drugs, nonpharmaceutical interventions may cover basic per-
sonal protection tools such as frequently washing hands, hand hygiene[38], using facemasks
and N95 masks and social mitigation strategies including social distancing, life style educa-
tion like avoiding close contact and cover coughs and sneezes, and school closure as well as
more strict measures like travel restriction, isolation and quarantine[3] and stay-at-home
order.

Quarantine as it is now employed is a public health tool and a collective action for the
common good. Modern quarantines are more likely to involve a few people exposed to
contagion in a small area, such as at a hospital, on an airplane or at a public gathering,
and only unusually are applied to entire cities or communities[18]. Although it is rare
and new to science and modern applications to quarantine a whole city or country, which
seems to be a ’new normal’ in fighting epidemics. Some whole countries are on lockdown
to contain COVID-19. The lockdown goes viral as the novel coronavirus does. One third
of the world is currently on lockdown as if the planet earth has been closed. Also, 2020
Tokyo Olympics has been postponed to 2021 due to the coronavirus pandemic. Besides,
a major number of countries have currently shutdown their borders, and implemented
airlines cancelling and cancelling sailings with some infected countries. As a remarkable
instance, the US cancels all travel to and from continental Europe as of 13 March 2020 for
the next 30 days except US citizens since the new epicenter of the coronavirus pandemics
has become Europe. The UK and Turkey are exempted.

Almost all countries and governments as well as major international actors including
United Nations (UN), The World Bank, International Monetary Fund (IMF), International
Chambers of Commerce, World Economic Forum, and FIFA have been working to fight
the novel coronavirus in collaboration with WHO. The WHO massively supports coun-
tries in all dimensions of knowledge, mentoring, financial, medical supplies. Also, almost
all countries and international agents declare financial supports for fighting COVID-19.
As instances, the World Bank reports up to $12 Billion immediate support for country
response[31] when the IMF makes available $50 Billion[15]. United Nations Foundation,
the Swiss Philanthropy Foundation and WHO launches Solidarity Response Fund to sup-
port the work of the WHO and partners to help countries respond to the COVID-19
pandemic[41].

The COVID-19 pandemic has also been damaging economy, stock markets, finance
sector, businesses and as well as supply chain processes of many products- shaking the
world as if chains of heavy global economic and health earthquakes. The IMF states the
global economy is projected to contract sharply by –3% in 2020, much worse than during
the 2008–09 financial crisis. The U.S. economy is projected to shrink this year by 5.9%
and the euro area by 7.5% as a result of COVID-19[16].

As the coronavirus infodemic spreads, WHO calls on industry and countries to increase
manufacturing of personal protective equipment such as N95 respirators, facemasks, gowns
and gloves by 40% to meet rising global demand due to the COVID-19 and associated panic
buying. There would be a global shortage to meet the demand even from frontline health
responders.

All pharmaceutical and nonpharmaceutical fighting measures for all epidemics and as



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 372

well as COVID-19 pandemic are already scarce resources and carrying costs, and thus
naturally have limitations. In a nutshell, fighting costs and constraints are always matter
for any epidemic, outbreak, pandemic or bioterror attack.

The WHO also strictly encourages countries to innovate and improve to find new
ways to prevent infections, save lives and minimize impact in addition to basic efforts like
detection, prevention and reduction[42].

It is an undoubtedly inevitable must to put efforts to maximize personal and overall
protection from a pandemic, and hence save lives under some functional and resource
constraints. That is optimal allocation of scarce resources to save millions of lives- gift of
life.

In this work, we propose two mathematical programming models for fighting any pan-
demic at micro and macro levels, respectively; the first one is personal protection model or
protocol and the other one is mass protection model for a country. We also apply these the-
oretical models to COVID-19 pandemic with real-world data for a person and the United
States of America. In other words, we put the general policy framework for maximum
protection of a person and the United States of America from COVID-19 pandemic on the
stage.

The literature in mathematical models in epidemiology is relatively scarce in compari-
son with the huge literature in epidemiology and a major part of the existing quantitative
epidemiological studies is generally about the understanding and predicting the behavior
of viruses and epidemics. The works involving mathematical modeling on the protection
and fighting against epidemics and pandemics are much more limited as they shape a small
subset in the literature. To mention a few, as instances, Yurt[43] derives a country-based
protection model from pandemic A(H1N1) with a real data implementation for Turkey.
Peng et al.[26] discuss the prevention of epidemics by public vaccination and individual
protection. Also, as a part of protection matter, Myers et al.[23] develop a mathematical
model for assessing the effectiveness of protective devices such as respiratory protective
devices in reducing risk of infection by inhalable droplets.

2. Developing the Models

We develop a personal protection model or protocol that maximizes total protection
probability of a person subject to required number of tools and budget. We also pro-
pose a mass protection model that maximizes total weighted protection probability in a
population and that can be applied to any county, city, state, country, continent and the
world.

2.1. Formulating Personal Protection Model

We propose maximum personal protection protocols for combating epidemics, pan-
demics or bioterror attacks. The recipe that is a binary nonlinear programming model
can be originated in the following way.



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 373

As a personal point of view, a person can adopt protection and treatment tools from
the sets of pharmaceutical and nonpharmaceutical instruments such as vaccine, antivi-
ral medicines, use of hand hygiene products, 50% social distancing and voluntary self-
quarantine during a reasonable planning horizon.

The notations are defined as in;
w: the number of weapons; pharmaceuticals and nonpharmaceutical tools,
i= 1, 2, . . . ,v, . . . ,f , . . . ,w: weapon index; treatment and protection tools index,
v: the number of vaccine types,
f : the number of pharmaceutical tool types,
f − v: the number of tools other than vaccines; eg, antiviral medicine types,
w−f : the number of nonpharmaceutical tool types; eg, use of surgical mask,
B: total budget,
T : total number of required agents,
Tph: total number of required pharmaceutical agents,
Tnph: total number of required nonpharmaceutical agents,
pi: estimated probability of protection of the person by use of a single tool i per agent

(per dose or unit)
A personal protection protocol can be defined by a w-dimensional binary vector x that

satisfies

xi =


1 if a tool i is adopted,

0 otherwise.

From the definition of xi, we have

(1− pi)
xi =

{
1 if xi=0,

1− pi if xi=1.

Hence, (1− pi)
xi represents the probability that a protection protocol x will not protect

the person if tool i adopted, that is, fail probability.
In general, under the assumption of statistically independence, the probability that a

protection protocol x protects the person is

1−
w∏
i=1

(1− pi)xi

It follows that the personal protection protocol that maximizes the probability of
protection from epidemic disease solves the objective function

max 1−
w∏
i=1

(1− pi)xi

Subject to following constraints. Total assignment must not exceed the total budget;

w∑
i=1

cixi ≤ B, ∀ i = 1, . . . , w.



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 374

There are required numbers of pharmaceutical, nonpharmaceutical and total agents, re-
spectively;

f∑
i=1

xi ≥ Tph, ∀i = 1, 2, . . . , f.

w∑
i=f+1

xi ≥ Tnph, ∀i = f + 1, . . . , w.

w∑
i=1

xi ≥ T , ∀i = 1, 2, . . . , w.

Some agents particularly pharmaceuticals may not be used at the same time; as an in-
stance, all antivirals cannot be adopted simultaneously, we impose this using either-or
constraints;

xixj = 0, for appropriate pair of i and j, and i 6= j, i, j ∈ {1, . . . , w}
All decision variables are binary variables,

xi ∈ {0, 1}, ∀i = 1, 2, . . . , w.

2.2. Building Mass Protection Model

We propose a general mathematical model that maximizes mass weighted protection
probability of people from an epidemic, pandemic or bioterror attack under some certain
scarce resources, functional constraints and budget. The model could work at the levels
from county to city, to state and across country as well as in the global scale. Our mass
fighting model is inspired by the weapon-target assignment (WTA) problem of operations
research. Further details and related literature about the WTA are available in Çetin and
Tolun Esen[9] and Kline, Ahner and Hill[17].

The model which is a nonlinear integer programming model for fighting pandemics,
epidemics or bioterror attacks to maximize total weighted protection probability in a
population stemming from well-known weapon-target assignment problem in the following
way.

We segment whole population into medical protection targets of sub-populations, and
partitions of target segments are persons in our model language as well. Those target
segments may be chronic patients, pregnant women, health care workers, infants and other
specific groups, and the remaining people. Our weapons are treatment and protection
instruments; pharmaceuticals such as vaccines and antiviral drugs and nonpharmaceutical
tools including use of surgical masks, N95 respirators, hand hygiene products, airlines
cancelling, voluntary self-quarantine and lockdowns. In this work, a priori tools are
designated as the tools that have been previously decided for adopting. As instance,
quarantine regions, border shutdown, airline cancelling, and cancelling sailings may be
previously decided for implementation. Their decisions are independent of the model,
but affects the model components. In the model, we assume the contributions of the
people characterized by a priori decisions to related target segments are only compactly



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 375

represented by their segment protection probabilities. Also, an agent means an appropriate
one of a person, a unit or quantity of dose. The model outputs can be implemented during
a planning horizon.

We define the notations as follows;
w: the number of weapons; pharmaceuticals and nonpharmaceutical tools,
t: the number of target segments,
i= 1, 2, . . . ,v, . . . ,f , . . . , a, . . . ,w: weapon index; treatment and protection tools index,
j= 1, 2, . . . ,r, . . . ,t: target segment index,
v: the number of vaccine types,
f : the number of pharmaceutical tool types,
f − v: the number of tools other than vaccines; eg, antiviral medicine types,
w−f : the number of nonpharmaceutical tool types,
w−a: the number of a priori tool types; already adopted tools like quarantine regions,
a−f : the number of non a priori nonpharmaceutical tool types,
r: the number of target segments for featured segments like risk groups for diseases,
P : total population,
Pj : population of target segment j,
Pi: population characterized by a priori tool i, eg, population of quarantined county,
Uj : priority weight for target j; relative importance,
pij : estimated probability of protection of target segment j by use of a single tool

i per agent (per dose, person or unit); ie, estimated protected amount of
partitioned target j by assigning tool i per agent,

xij : the number of assigned agents (person, unit, quantity of dose,) i to target segment
j,

cij : unit variable cost of assigning tool i to target segment j,
Bs: budget allocated by the government,
Bp: budget allocated by the non-government resources; people or private sector,
B: total budget,
Wi: available amount of tool i; upper bound for tool i,
Wi: total number of required agents of tool i; lower bound for tool i,
Tj : maximum allowable total number of agents for target segment j,
Tj : total number of required agents for target segment j,
Tj : the number of required agents from selected tool to target segment j,

ρ: the ratio of all cost that non-government resources should meet,

The objective function is the maximization of weighted estimated protection prob-
ability of the total population from the pandemic under the constraints, which can be
expressed as;

max
t∑

j=1

Uj

(
1−

w∏
i=1

(1− pij)xij

)
There are naturally some constraints. There is a total budget for all assignment operations.
A certain ratio ρ of the total cost must be met by non-government resources for any



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 376

appropriate tool h, and where the total budget is B =
∑t

j=1

∑w
i=1 cijxij ; for all i =

1, . . . , w, and j = 1, 2, . . . , t.

ρ
t∑

j=1

w∑
i=1

cijxij ≤ Bp, h ∈ i = {1, . . . , w} ,

t∑
j=1

(
w∑
i=1

cijxij − ρ
h∑

i=1

cijxij

)
≤ Bs, h ∈ i = {1, . . . , w}

The sum of government and non-government budgets cannot exceed the total budget;

Bp +Bs ≤ B,

There are upper bounds for the available agents;

t∑
j=1

xij ≤Wi, ∀i = 1, 2, . . . , w.

There may be some required amounts of selected tools that must be assigned;

t∑
j=1

xij ≥Wi, for selected i = 1, 2, . . . , a.

The number of assigned agents must be within the minimum required and maximum
allowed levels and any target segment may adopt multiple agents;

Tj ≤
w∑
i=1

xij ≤ Tj , ∀j = 1, 2, . . . , t.

For any selected pair of tool i and target segment j, there must be some required number
of agents to exposure to the related target segment; eg, there may be a certain amount of
vaccine dose that health care workers must get;

xij ≥ Tj , for any appropriate pair of i and j,

The decision variables must not exceed the populations of target segments;

xij ≤ Pj ,∀j = 1, 2, . . . , t.

The decision variables for a priori tools (a priori decision variables) must equal to the
related population shares (target segment distribution) of population of a priori tools Pi

where the target segment distribution is
{
P1/P, P2/P, . . . , PT /P

}
;

xij ≤
Pj

P
Pi, ∀i = a+ 1, . . . , w, ∀j = 1, 2, . . . , w.



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 377

xij ≥
Pj

P
Pi, ∀i = a+ 1, . . . , w, ∀j = 1, 2, . . . , t.

All decision variables must be nonnegative and integer;

xij ≥ 0 and integer, ∀i = 1, 2, . . . , w, ∀j = 1, 2, . . . , t.

As the WTA problem is an NP-complete problem[9], from which the nonlinear integer
programming model stemming is also in the class of NP-complete.

3. Applications of the Models for COVID-19 Combat in the United
States

The proposed mathematical models are implemented for COVID-19 combat of a person
and the United States as real-world illustrative examples, respectively. The data that
inputs the models is collected in the following way.

3.1. Data Collection

The data that the models fed are taken from open reliable sources and published works.
There is a wide uncertainty about the novel coronavirus, its behavior and response, and
its interaction with protection and fighting tools, and hence there is lack of published data
for the novel coronavirus.

As we specify the planning horizon is 6 weeks for the model implementation, we set
the units of the agents fired from the tools for as follows. The unit for vaccine is a single
dose, the unit of antivirals is the required drug dose regimen (ie, for 10 capsules, required
75 mg twice daily 5 days). The unit for surgical mask is a box of 50 Pcs, and for N95
respirator is is 2 boxes of 3M7185 respirator particulate N95 10/Box (20 items). A unit
of hand hygiene product is 1000 ml hand sanitizer (70% alcohol hand sanitizer fluid-1
liter bottle). The units for the remaining nonpharmaceutical and a priori agents are per
person.

The population of the United States for 2020 is estimated as 331,002,647[27]. The
age distribution of population is also considered[32] and re-arranged with respect to the
population of risk factors to obtain target segment distribution. The numbers of risk
factors are obtained as follows. By 2020, it is projected that 157 million Americans will
have chronic diseases, with 81 million having multiple conditions[24]. This means that 47%
of the US population have at least one chronic illness, which is in compatible with other
published estimations. We take the number of health care workers which is another risk
factor is 16,866,020 from the 2018 statistics[33]. We consider the number of pregnancies is
6,155,000 from 2010 statistics[28] (published in 2019) which seems stable over years. The
number of infants is taken as 3,791,712 which is registered number of births in 2018, down
2% from 2017[6].

Regarding a priori tools, we arbitrarily specify the population of a hypothetical quar-
antined city #1 in US which is 4,150,000. We also assume the number of people affecting
from border shutdown and canceling sailings are 1,250,000 and 2,100,000, respectively.



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 378

The US cancels all travel from/to continental Europe as of 13 March 2020 for the next 30
days except UK, Turkey and US citizens. We only take this concrete mass airline canceling
in the model implementation. The number of passengers from US to Europe was 7,687,718
and 2,360,804 from UK to US in June 2019- a supposition equal to from US to UK in
size[34]. Thus, we take the number of people affected by Europe airline canceling, the dif-
ference, is 5,326,914 as an estimation with the presumption that a traveler from US would
not fly to Europe and be protected from the novel coronavirus. We distribute the numbers
to target segments using target segment distribution that is acquired by normalizing the
segment populations to the US population in 2020.

At present, there are no approved vaccines and antiviral drugs for the novel coron-
avirus. Thus, there is no information about their protection probabilities. We substitute
-as estimators- the protection probabilities of vaccines and antivirals against pandemic
A(H1N1), which are used in Yurt[43] based on published scientific data. Immune re-
sponse at 21 days, seroprotection ratio for single dose is 80% as it is 92-96% for pregnant
women[43]. We take protection probability of vaccine for pregnant women is 0.94, and
0.80 for the other target segments, except infants (zero probability) since they cannot get
vaccine[43]. Similarly, we substitute two antivirals for novel coronavirus by Oseltamivir
and Relenza (Zanamivir) and efficient for A(H1N1) as estimators on the protection prob-
ability base. The use of Tamiflu and Oseltamivir protects with the ratio of 70%-80%[21].
As estimators, we adopt the protection probabilities of Antiviral #1 and Antiviral #2 as
0.79 and assume Antiviral #2 must not be used for infants, so its protection probability
is 0.00 for infants.

The use of face mask and hand hygiene reduces influenza-like illness within 35%-51%
for 4-6 weeks[1]. Since there is no published data for the novel coronavirus on the matter,
in this study, we assume the protection probability of the use of surgical mask is 0.51
with the implicit assumption a person uses surgical mask and hand hygiene products
together. The use of N95 respirators may protect with 95% when the use of hand hygiene
products protects with 89.2% and school closure provides a protection between 30%-50%,
and lifestyle education such as avoiding close contact and cover coughs and sneezes does
77% (except infants)[43]. We take the protection probabilities of school closure is from 0.15
to 0.40 alongside with the target segments. 50% social distancing provide 40.9% protection
compared to no intervention[30]. As there is no published data for the novel coronavirus
in the regards following, we also assume the protection probabilities of voluntary self-
quarantine and compulsory quarantine are 0.95 and that of border shutdown, airlines
canceling and canceling sailings are 0.90 equally for all target segment.

Regarding costs, as there are not vaccines and antivirals at present, we substitute
available vaccine and medicine costs as estimators for COVID-19. For the vaccines, we
consider Pediatric Influenza Vaccine Price List published by the CDC for infants, the
median price of the pediatric influenza vaccine state prices (not private sector price) is
$13.55 per single dose for infants when the median price is $13.25 per single dose for
all remaining target segments from Adult Influenza Vaccine Price List[4]. We assume
two antivirals for COVID-19; as cost estimators, we take Oseltamivir prices for Antiviral
#1, the price of 75 mg Oseltamivir oral capsule $39.50 (for 10 capsules, required 75 mg



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 379

twice daily 5 days) for adults whereas the price of 6 mg/mL Oseltamivir oral powder
reconstitution (60 mililiters) is $71.30 for babies[25]. We substitute Relenza (zanamivir)
price as a substitute price for Antiviral #2 which is $71.10 for 5 mg Relenza inhalation
powder a quantity of 20 (5x4 each) for all segments except infants[29].

We consider the unit of surgical mask agent is a box of 50 disposable surgical face mask
to be used for 6 weeks. The unit variable cost of 50 disposable surgical mask is $22.19[2].
We take the unit of N95 respirator agent is 2 boxes of 3M7185 respirator particulate N95
10/Box (20 items) to be used for 6 weeks. The unit variable cost of that product was
$189.96 at Amazon on 15 March 2020. We also consider 1000 ml hand sanitizer (70%
alcohol hand sanitizer fluid- 1 liter bottle) as the unit of our hand hygiene product agent
for use during 6 weeks. The unit variable cost was $42.15 at eBay on 15 March 2020 (the
products and links are currently not available). We assume lifestyle education to public
and adaptation is $1.5 per person. The cost of school closures is estimated in a US study
in 2009 as a pandemic mitigation strategy in terms of GDPs. For 6 weeks closure, low cost
estimate measures 0.1%, base estimate calculates 0.4% and high cost estimate quantifies
0.5% of GDP, respectively[20].

We also assume that school closure dose not lead to a reduction in key HCWs. As we
deal with persons in target segments and as agent units for the related tools, we can easily
estimate this cost using GDP per capita. The GDP per capita forecast of the US for 2020
is $ 67,426.8[12]. We consider the base estimate scenario, which is 0.4% of the GDP (per
capita), then we estimate the school closure cost for 6 weeks per person is $269.7072. As
school closure estimations include absenteeism, it would be a good measure to estimate the
followings. We also assume that the cost of 50% social distancing per person for 6 weeks
is 0.4% of the GDP (per capita) which is $269.7072. We assume the costs of voluntary
self-quarantine and compulsory quarantine per person for 6 weeks is 0.5% of the GDP
(per capita), ie. $337.134. We suppose that people are able to work from home while they
are in voluntary or compulsory quarantine. We assume the costs of the tools of border
shutdown, airlines canceling and canceling sailings again per person are in terms of GDP
per capita; we arbitrarily suppose those costs equally per person for 6 weeks as 0.01% of
the GDP (per capita) which is $67.4268.

We assume the budget for the personal protection model is $500 and $ 600 billion for
the mass protection model which is sum of $500 billion that the government must meet
and $100 billion which the nongovernment resources must meet. We specify the available
amounts of agents fired from the tools (in their related units) as shown on the spreadsheets
that the models implemented.

3.2. Achieving the Personal Protection Protocol

We also implement the model with real-world data for a person living in USA in order
to obtain optimal protection strategies from COVID-19. The person is an adult aged
between 18 and 59, and can work from home. Although the model can handle any finite
number of tools, we consider a vaccine and two antiviral medicines as pharmaceutical tools
and using surgical mask, N95 respirator, hand hygiene products, lifestyle adaptation, 50%



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 380

social distancing and self-quarantine as nonpharmaceutical interventions for the adoption
of the person for 6 weeks. We employ either-or constraints for the model so that the
person cannot adopt vaccine and any antiviral or both antivirals at the same time. The
data acquisition for this one-person model is also presented in Data Collection section.
The model is easily optimized by MS Excel Solver as a powerful spreadsheet tool within a
few seconds, and the optimal solution which may also result in other optimal combination
of decision variables- alterna optima is given in Figure 1.

Figure 1: Optimal personal protection strategies

According to our personal protection model, the person should adopt vaccine, surgical
mask, hand hygiene, lifestyle education and self-quarantine strategies for 6 weeks within
his/her $500 budget. S/he should stay home and work from home for 6 weeks after get-
ting vaccine and use hand hygiene products and surgical mask-when needed although the
Centers for Disease Control and Prevention (CDC) does not currently suggest for Amer-
icans, but it seems a policy shift is forthcoming- with careful lifestyle like avoiding close
contact and cover coughs and sneezes. This optimal protection strategy provides 99.991%
protection of the person with a cost of $416.224. That is, the protection probability of the
person from the coronavirus is 0.99991. As it is seen in Figure 1, the costliest strategy is
self-quarantine with 81%.

There are no approved vaccine and antivirals for COVID-19 at present. Thus, based
on the current practice, we easily re-run the model without pharmaceutical tools by just
relaxing the number of pharmaceutical required tools constraint and giving the zero pro-
tection probabilities to vaccine and antivirals on the same spreadsheet, we get an optimal
solution again under new conditions covering only some nonpharmaceutical interventions.
In this case, the optimal protection rate reduces to 99.910% with the surgical mask, hand



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 381

hygiene, lifestyle education and self-quarantine strategies. The protection gap is 0.08%
which is the opportunity cost of vaccine or antivirals.

3.3. Mass Protection Policy for Fighting COVID-19 in the United States

We run the mass protection model with real data of USA to protect the whole coun-
try from COVID-19. Even though the model can handle thousands of target segments
and therapeutic/nontherapeutic and pharmaceutical/non pharmaceutical tools; any finite
numbers of tools and targets (wxt), we consider 14 tools and 9 target segments of USA
population. The planning time horizon is 6 weeks.

The pharmaceutical tools are a vaccine and two antiviral medicines when nonpharma-
ceutical instruments are use of surgical masks, N95 respirators and hand hygiene products,
lifestyle adaptation, school closures, 50% social distancing, voluntary self-quarantine and
a set of a priori tools. The CDC does not currently recommend the use of surgical masks
for healthy citizens in the US; however, the agency may recommend due to that about
25% of infected people may be asymptomatic[22]. A priori decisions are as follows: We
assume that there is already a decision that a city with a population of 4,150,000 must be
on lockdown. The US has made a real decision to ban the travel from/to Europe (except
UK and Turkey) effecting 5,326,914 people for a month. We also take border shutdown
and cancelling sailings influencing, by assumption, 1,250,000 and 2,100,000 people, respec-
tively. The real US data acquisition is also discussed for the mass protection model in
Data Collection section.

We set the target segments of the US population as chronic patients, pregnant women,
health care workers, infants (0-6 month), and people aged 6 month-14 years, 15-24 years,
25-54 years, 55-64 years and 65+ years. Possible risk factors for COVID-19 may in-
clude, but are not limited to, older age, and underlying chronic medical conditions such
as lung disease, cancer, heart failure, cerebrovascular disease, renal disease, liver dis-
ease, diabetes, immunocompromising conditions, and pregnancy[5]. Also, coronaviruses
and enteroviruses/ rhinoviruses affect chronically ill, older patients than healthy, young
adults[13]. Besides, healthcare workers (HCWs) taking care of MERS-CoV patients are
at an increased risk of acquiring the infection[14], and those dealing with novel coron-
avirus, too. Thus, in this study, we consider chronic patients, pregnant women, health
care workers and older people (65+) as risk factors that is why we set them as risk factors
target segments. The estimated protection probabilities (success probabilities) over target
segments are tabulated in Figure 2

Unit variable costs of tools or agents with respect to their units along the target
segments are also presented (in USD) in Figure 3. We assume that people are volunteer to
adopt self-quarantine can work from home. The huge unit costs in the HCWs and Infants
columns are for avoiding assignments to related cells.

The mass protection model is fed by real US data on MS Excel which is a robust spread-
sheet instrument. The availability of tools, required agents to exposure target segments,
priority weights and budgets are given in Figure 4, 5 and 6. We give relative importance
weights to target segments between 1 and 10, the higher weight means more preemptive



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 382

Figure 2: Estimated protection probabilities of tools over target segments.

Figure 3: Unit variable costs of tools over target segments.

target since the objective function is in maximization form. As the most important risk
factors are HCWs, chronic patients and 65+, we give them 10, and 8 to pregnant women.
Also, we designate that HCWs must get vaccine at least 12,000,000 doses (one dose each),
in the model context, we impose the constraint xij ≥ Tj , x13 ≥ 12, 000, 000. There

are also other regulations. As examples, HCWs must not adopt voluntary self-quarantine
strategy, and infants must not get vaccine or cannot wear masks, or they cannot get An-
tiviral #2 when they are able to get Antiviral #1. In order to impose those limitations,
we just give a huge amount of cost to their related costs to guarantee not to assign agents.
This procedure is called the big-M-method which is common in optimization. We assume
that the government budget is $500 Billion and nongovernment budget is $100 billion
which sums to $ 600 billion as total budget of assignment. We also specify that 40% of
the cost of assigned agents of Antiviral #1, Antiviral #2, surgical mask, N95 respirators,
hand hygiene products, 50% social distancing and self-quarantine would be met by people
and private sector (ρ = 0.40).



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 383

The model is solved by MS Excel Solver. The CPU time for the solution of the model
with wxt = 14x9 = 126 integer decision variables is 97 seconds on a computer. The
solution is very practical even for such an NP-complete model with 126 decision variables.
Since the model is NP-complete, we consider the solution as near-optimal assignments.
The near-optimal protection for the US as public health policy against novel coronavirus
is given in Figure 4. It shapes the general framework of protecting people from COVID-19.
The key point is that all target segments should adopt and be exposed appropriately the
assigned number of agents to themselves for 6 weeks. The re-run of the model may also
generate alterna optima solutions.

According to the mass protection model, 99.431% of HCWs, 94.707% of chronic pa-
tients, 99.741% of pregnant women, 99.223% of aged 65+ and, etc. are protected from
COVID-19 if the near-optimal policies are adopted. The average protection ratio or av-
erage protection probability of all target segments, given in Figure 4, is 96.961% which is
very high.

Figure 4: Assignment of tools to target segments and protection ratios of target segments.

The top four weighted protection values are of HCWs, aged 65+, chronic patients and
pregnant women which are all risk factors. The costliest segment is chronic patients with
a share of 45.06% of total cost when the least costly is infants (0-6 month) with 0.56%.
The weighted protection and costs of all target segments are presented in Figure 5.

The costs of tools are also obtained that the costly tool is voluntary self-quarantine
with a share of 34.479% of total cost as the pharmaceutical tools are associated with
2.469%. we also observe that all available agents are assigned to target segments. The
near-optimal weighted protection of the country, objective function value of the model, is
62.335 unit. The total cost of near-optimal assignment is $ 293,332,400,284 which is less
than half of allocated total budget, and only 14.66% of $ 2 trillion that has been recently
released by the US government and the US Senate to combat and protect from the novel
coronavirus pandemic and economic devastation[10]. The details are shown in Figure 6.



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 384

Figure 5: Weighted protection and costs of target segments.

Figure 6: Total weighted protection, costs of tools, costs to government and non-government resources and
total cost.

Mind the Gap: Opportunity Cost of Vaccines and Antiviral Medicines

There are no approved vaccines and antiviral medicines against novel coronavirus at
present. Thus, for the present-day situation, we also run the mass protection model with-
out pharmaceutical tools, only with the nonpharmaceutical tools by just easily giving zero
protection probabilities to the vaccine and antivirals or adding new constraints impos-
ing them to zeros on the spreadsheet, which is indeed another advantage of spreadsheet
modeling.

We get the total weighted protection of 60.845 unit based on the near-optimal policy.
The average of protection probabilities of all target segments reduces from 96.961% to
94.198% as the total cost decreases by about $7,25 billion. The protection ratio gap
between the models with and without vaccine and antivirals is 2.763% which corresponds
the failure of protecting 8,292,109 people as much as population of the State of Virginia or



E. Çetin, S. Kiremitci, B. Kiremitci / Eur. J. Pure Appl. Math, 13 (2) (2020), 369-389 385

New York City – opportunity cost of vaccines and antiviral drugs. If the available number
of nonpharmaceutical agents would decrease or be in shortage in the nonpharmaceutical
case, the number of expected unprotected people would increase dramatically.

4. Concluding Remarks

Some properties of the personal protection protocol and the mass protection model are
discussed and several recommendations on fighting COVID-19 are drawn using analogy
and metaphor as concluding comments.

Unus pro omnibus, omnes pro uno: Relation Between the Models

It could be asserted that “protecting yourself means protecting people” is an extremely
important principle in epidemiology to combat the outbreaks. Our two proposed models
for fighting all epidemics, pandemics or bioterror attacks in this work are in favor of the
fact unus pro omnibus, omnes pro uno – one for all, all for one. From a standpoint, the
mass protection model with integer decision variables is a mathematical generalization of
the personal protection model with binary decision variables, and conversely, a special
case of the mass protection model is the personal protection model. If you join the mass
protection policy, you will have personal protection strategy in a somewhat and if you
adopt personal protection protocol, you will contribute to the mass protection policy to
fight epidemics, which leads us to unus pro omnibus, omnes pro uno - one for all, all for one.

Triple Corona of Protection and Fighting

We remark that both personal and mass protection models have some crowns of surviv-
ing epidemic diseases. In other words, in our approach, there are hallmarks of protection
and hence fighting against invisible enemy which can be asserted as “triple corona”.

The first crown may be protection probability which is a brilliant parameter for our
procedures. The second corona would be cost which is also vitally important for fighting
pandemics even when it is also thought pandemics are global health concerns as well as
economic issues. The third corona, the coronation of the models, is optimal policy. The
models yield us maximum protection policies in micro and macro planning dimensions.
The first two crowns bloom the third corona to shine the proposed models and the “triple
corona” on the top.

Our personal protection model can treat a finite number of tools (w) when the mass
protection model can handle any finite numbers of tools and target segments (wxt) and
can be applied to any region at the levels from county to city, to state and across coun-
try as well as in the global scale. Once the models are built on spreadsheet, their data
and parameters may be easily updated and re-run due to the benefits and elasticity of
spreadsheet modeling, which makes the implementation of the models user-friendly. The
developed optimization models that are quite efficient may be incorporated into dynamic
frames as future research.



REFERENCES 386

Love in the Time of Corona

As the moving enemy, novel coronavirus, is double invisible due to its viral nature and
the availability of the high level of symptom-free cases of infected people which is guessed
1 in 4 in the US and 2 in 4 in Ireland, the seriousness and difficulty of fighting coronavirus
pandemic should be internalized altogether.

Recognize the enemy, love yourself, love the personal protection protocol and the mass
protection synergy to give lovely gifts of life to yourself first, your family and the people
that you love, to society, to people and your country, and to the rest of the world. Love
life, follow the measure rules and stay at home! That is the greatest ever social impact.
Global health policy makers and governments would join the circle of love in the time of
corona!

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