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\begin{document}

\title{Quantum Mechanical Modeling of Perpetual Immunity to SARS-CoV-2
(COVID-19) Infection}



\author[1]{Muhammad Maqbool}%
\author[2]{Asad Ullah}%
\author[3]{Mujib Ullah}%
\author[4]{Ghafar Ali}%
\author[1]{Tino Unlap}%
\author[5]{Tahirzeb Khan}%
\affil[1]{The University of Alabama at Birmingham}%
\affil[2]{Saint Mary's University Department of Chemistry}%
\affil[3]{Stanford University School of Medicine}%
\affil[4]{Pakistan Institute of Nuclear Science and Technology}%
\affil[5]{Abdul Wali Khan University Mardan}%


\vspace{-1em}



  
  \date{May 25, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
Survivors of SARS-CoV-2 (Covid-19) infections have a unique immune
response that allows them to either delay reinfection or outsmarts the
virus's whims. The emergence of new SARS-CoV-2 variants and immunization
have furthered this immune response by producing high-affinity
anti-SARS-CoV-2 antibodies. There is limited data on the extent to which
this immune response confers protective immunity against recurrent
infections, as well as the duration of this protection. Here we present
the first-ever quantum mechanical model to provide an answer to this
question. The model is well applicable because of the 20 nm -500 nm size
range of coronavirus. We call this model the `Quantum Perturbation
Model'. The model relates the strength of the COVID-19 attack to a wave
function containing information about the system (person infected by the
SARS-CoV-2) and quantized energy states, which shows the chances of
reoccurrence of the disease. By applying the energy corrections provided
by the Quantum Perturbation Theory to the SARS-CoV-2 attack under
reinfections and various pre-existing conditions we have provided
possible interpretations. When pre-existing problems exist at the time
of SARS-COV-2 infection, the model illustrates how the influence of
COVID-19 accumulates up. The model also formulates the deficit in the
intensity of the COVID-19 effect in recurrence or in the presence of
other variables strengthening the body's antibody affinity and immunity
to resist the virus variants.%
\end{abstract}%



\sloppy


\textbf{Quantum Mechanical Modeling of Perpetual Immunity to SARS-CoV-2
(COVID-19) Infection}

Asad Ullah\textsuperscript{1}, Mujib Ullah\textsuperscript{2}, Ghafar
Ali\textsuperscript{3}, Tino Unlap\textsuperscript{4}, Tahirzeb
Khan\textsuperscript{5}, and Muhammad Maqbool*\textsuperscript{6}

Department of Chemistry, Saint Mary's University, Nava Scotia, Canada.

Department of Immunology and Transplantation, School of Medicine,
Stanford University,

California, USA.

Nanomaterials Research Group, Physics Division, PINSTECH, Islamabad
44000, Pakistan.

Biotechnology Program, Department of Clinical \& Diagnostic Sciences,
the University of

Alabama at Birmingham, Birmingham, AL 35294, USA.

Department of Physics, Abdul Wali Khan University, Mardan, Pakistan.

Health Physics Program, Department of Clinical \& Diagnostic Sciences,
the University of

Alabama at Birmingham, Birmingham, AL 35294, USA.

* Corresponding author: Tel, (1)765-212-9260, E-mail: mmaqbool@uab.edu

\textbf{ABSTRACT}

Survivors of SARS-CoV-2 (Covid-19) infections have a unique immune
response that allows them to either delay reinfection or outsmarts the
virus's whims. The emergence of new SARS-CoV-2 variants and immunization
have furthered this immune response by producing high-affinity
anti-SARS-CoV-2 antibodies. There is limited data on the extent to which
this immune response confers protective immunity against recurrent
infections, as well as the duration of this protection. Here we present
the first-ever quantum mechanical model to provide an answer to this
question. The model is well applicable because of the 20 nm -500 nm size
range of coronavirus. We call this model the `Quantum Perturbation
Model'. The model relates the strength of the COVID-19 attack to a wave
function containing information about the system (person infected by the
SARS-CoV-2) and quantized energy states, which shows the chances of
reoccurrence of the disease. By applying the energy corrections provided
by the Quantum Perturbation Theory to the SARS-CoV-2 attack under
reinfections and various pre-existing conditions we have provided
possible interpretations. When pre-existing problems exist at the time
of SARS-COV-2 infection, the model illustrates how the influence of
COVID-19 accumulates up. The model also formulates the deficit in the
intensity of the COVID-19 effect in recurrence or in the presence of
other variables strengthening the body's antibody affinity and immunity
to resist the virus variants.

\textbf{KEYWORDS:} SARS-COV-2, immunity system response, quantum
mechanical modeling, quantum perturbation theory,

\textbf{INTRODUCTION}

The outbreak of severe acute respiratory syndrome (SARS)-like
coronavirus, SARS-CoV-2, around the globe has caused ripples and spurred
researchers to take alternative strategies to deal with the new
SARS-CoV-2 infection (Baric et al., 2020). With 169 million cases and
3.51 million deaths worldwide, the World Health Organization (WHO), the
United States Centers for Disease Control and Prevention (CDC), and the
European Center for Disease Prevention and Control (ECDC) have declared
COVID-19 one of the deadliest pandemics in the modern human history
(Baric 2020; Lv et al., 2020). SARS-CoV-2 causes COVID-19 when it enters
the body through the nose, mouth, or eyes and finds its way to our cells
through its surface spike protein (Pillalamarri et al. 2021). Its entry
into host cells is a key factor in viral pathogenicity, infectivity, and
immunity (Li et al., 2016). Once inside the cells, SARS-CoV-2 replicates
itself utilizing host cellular machinery, producing virally encoded
proteins that copy its viral genetic material and induce infection (Khan
et al., 2020). When a cell is infected with viruses, it ruptures
(Chatterjee et al., 2021). This causes the cell to die, and the virus
particles can go on to infect more cells (Lv et al., 2020; Singhal et
al., 2020; Shereen et al. 2020). The virus evades our immune system by
changing over time, resulting in genetic mutations in the population of
circulating viral strains that dampen our immune response ( Li et al.,
2021). Coughing or sneezing of a COVID-19 affected person is a leading
cause of the spread of this virus and the disease. The infected person
can spray droplets as far as 6 feet away during a cough or sneeze (Lv et
al., 2020; Shereen et al. 2020). . The latest~data on the COVID-19
indicates that only 17\% of people infected with SARS-CoV-2 show no
symptoms and these individuals are 42\% less likely to transmit the
virus (Byambasuren et al., 2020).

Despite the fact that SARS-CoV-2 has the ability to hijack our immune
system, we must remember that our body's natural defenses are intended
to develop efficient antibodies against a specific invader (Ullah et
al., 2021; Vardhana et al., 2020; Yazdanpanah et al., 2020). Our immune
system remembers how it dealt with pathogens previously (Yazdanpanah et
al., 2020). When an individual is re-infected with the same pathogen,
our immune system recognizes it and protects us (Gasteiger et al.,
2017). In the case of SARS-CoV-2 infection, recovery does provide a
degree of immunity, but there have been cases where the second infection
was fatal (Bi et al., 2020; Parry et al., 2020). To date, there is no
information available that anyone has been infected a third time, though
the possibility may exist as circulation of variants such as B.1.1.7 or
B.1.1.28 has been reported in several countries (Voloch et al., 2021;
Sabino et al., 2021). In the event, if a third re-infection occurs and
the patient recovers, it will aid in the building of long-term immunity,
eventually leading to eternal immunity, based on our existing immunology
understanding.

In multiple case study reports, researchers have discovered that
pre-existing chronic conditions such as heart disease, cancer, or
digestive diseases are reinforcing factors for SARS-CoV-2 and that a
more severe viral attack is expected in such patients. In simple words,
a person with a pre-existing condition like heart disease can be
affected more by the COVID-19 (Sanyaolu et al., 2020; Zheng et al.,
2020; Mao et al., 2020). Based on the reported data of 76993 patients,
Emami \emph{et al.} reported that cardiovascular diseases, hypertension,
smoking, diabetes mellitus, chronic obstructive pulmonary disease
(COPD), malignancy, and chronic kidney disease were among the most
prevalent underlying diseases among hospitalized COVID-19 patients,
respectively (Emami et al., 2020).

The SARS-CoV-2 infection can damage the heart in many ways. For example,
the virus may directly attack and inflame the heart muscle, affecting
blood flow and the supply of oxygen to the body part. COVID-19 seems to
promote the development of cardiovascular disorders, such as myocardial
injury, arrhythmias, acute coronary syndrome (ACS), and venous
thromboembolism (Shi et al., 2020; Dai et al., 2020; Shi et al., 2020).
Children with COVID-19 have also been reported to develop
hyperinflammatory shock with features akin to~Kawasaki disease,
including cardiac dysfunction and coronary vessel abnormalities (Dai et
al., 2020). Similarly, cancer patients are also at the forefront of risk
from COVID-19 and have a disproportionately high death rate (Williamson
et al., 2020; Sud et al., 2020; Tillett et al., 2021; Azam et al.,
2020). Clearly, some of these factors relate to immune suppression and
chemotherapy and hence we can say that COVID-19 could have a more severe
effect on those who have pre-existing serious disorders.

Through case studies of several COVID-19 patients, researchers
discovered that re-infection and recurrence of COVID-19 is possible in
patients who survived the first COVID-19 attack. According to a recent
study, those who had COVID 19 had an 84 \% lower chance of reinfection
and a 93 \% reduced risk of symptomatic SARS-CoV-2 infection (Hall et
al., 2021). Clinicians and researchers presented a case study on a range
of patients and investigated SARS-COV-2 infection as well as the impact
of COVID-19 on patients with various conditions. They reported eleven
COVID-19 patients having experienced a second clinically- and
virologically confirmed acute COVID-19 episode (Bonifacio et al., 2020;
Gousseff et al., 2020). Several cases of recurrence have also been
reported in other parts of the world (Jiang et al., 2020).

The purpose of this study is to develop a proper model based on the
Quantum Perturbation Theory that provides information and a pathway to
recurrence of COVID-19 under various conditions affecting the immunity
and affinity of the body.

\textbf{BACKGROUND AND SIGNIFICANCE}

Keeping the severity of the COVID-19 attack in the presence of a
pre-existing major disease and recurrence of the attack, it is important
to model the effect of SARS-CoV-2 infection and its progress.
Researchers have reported various aspects of COVID-19 attack, the
severity of the developed infection, and the outcomes. However, a search
is still going on to find a proper model explaining how the strength of
COVID-19 infection is affected by pre-existing diseases and how the
strength and chances of recurrence vary under various conditions and
factors.

In this paper, we developed the first Quantum Mechanical Model based on
Quantum Perturbation Theory. We call it the ``Quantum Perturbation
Model'' to model COVID-19 infections and the immune response of the
affected body. The model describes how the effect of COVID-19 adds up
when pre-existing major diseases exist at the time of coronavirus
attack. The model also formulizes the weakness in the strength of the
COVID-19 effect in recurrence or in the presence of other factors
boosting the affinity and immunity of the body to resist. This quantum
perturbation model is uniquely significant because the model is well
suited to formulize COVID-19 because of the 20 nm -500 nm size range of
coronavirus, which is exactly matching with the quantum mechanical
limits of explaining the phenomenon and outcomes that cannot be
explained and solved by the traditional approaches. Many phenomena and
effects like single-electron tunneling in a biological system, alpha
decay, electrons tunneling in physical and biological systems, and
behavior of neurons have been successfully modeled and explained when
all other approaches have failed. We hypothesize that a microscopic
cellular event must be viewed as a quantum entity (system) in the
context that it is complex and confined in space and time. These
cellular events rely on a limited (unlimited) number of molecules that
are densely packed and can occur within a relatively short period,
leaving them accessible to the possibility that quantum mechanical
processes can play an important role in biological systems.

Another unique and significant outcome of this model is that any
COVID-19 patient who shows a different path of variation in the Covid-19
effect (other than predicted and prescribed by this model) has either
been suffering from another underlying disease which is not been
diagnosed yet or has another affinity factor which is yet to be
detected.

\textbf{FUNDAMENTAL MECHANISM OF PERPETUAL IMMUNITY TO SARS-COV-2
INFECTION}

Our immune system is very versatile, capable of fighting a wide range of
infections and possessing a memory that allows for a robust response to
previously encountered pathogens. When an antigen attacks our body, the
immune system responds to it through cell-mediated immunity and antibody
production. Because SARS-CoV-2 is a novel virus for humans, initially it
causes an innate immune response that activates an elaborate signaling
cascade using proteins called cytokines (Boechat et al., 2021; Vetter et
al., 2020). Within a week or two of the cytokines' storm, the adaptive
system becomes active that creates natural antibodies (IgM, IgD, and
IgA). These antibodies interact with the virus and eventually kill or
disable it (Sette et al., 2021). The innate immune response plays a
vital role in B cell development, proliferation, and isotype switching
(Viau et al., 2005). When B. lymphocytes bind to SARS-CoV-2, they divide
and mature into identical cells. Millions of high-affinity SARS-CoV-2
specific antibodies are released into the bloodstream and lymphatic
system by these mature memory B cells. As these SARS-CoV-2 driven
antibodies circulate, they bind to SARS-CoV-2 virus particles and
neutralize them by changing their chemical composition, preventing them
from entering other cells. Concurrent interaction with new virus strains
or booster vaccinations increases the affinity of these antibodies even
further. We believe that this process of SARS-CoV-2-driven B cell
activation is repeated several times in order to sustain the efficiency
of the anti-SARS-CoV-2 antibodies, which could lead to what is known as
''perpetual immunity.'' Various pre-existing diseases have been linked
to an increased chance of dying from COVID 19, albeit not all
comorbidities carry the same risk (Callender et al., 2020) . This
mechanism of perpetual immunity development is shown in Figure 1.

\textbf{Hosted file}

\verb`image1.emf`  available at \url{https://authorea.com/users/737238/articles/712351-quantum-mechanical-modeling-of-perpetual-immunity-to-sars-cov-2-covid-19-infection}

\textbf{Figure 1.} SARS-CoV-2 and perpetual immunity: A significant
number of high-affinity antibodies are created, which outwit the virus.

\textbf{QUANTUM PERTURBATION MODEL OF COVID-19 ATTACK AND IMMUNITY
RESPONSE}

Based on the published data regarding COVID-19 cases, a person who has
recovered from a COVID-19 may be re-infected by the virus several times.
However, the body's immunity system comes into play and develops
resistance to the virus by producing high-affinity antibodies. When we
have concurrent infections, our immune system adapts to the virus and
produces a diverse repertoire of immunoglobulins (Ig) that have a higher
affinity to take down the virus as described in Figure 1 (Gazumyan et
al., 2012; Kumar et al., 2014). Furthermore, the strength of the body's
immunity grows with additional factors such as vaccination and the
number of times the virus infects. One of the most effective methods for
predicting the strength and impact of SARS-CoV-2 attacks is to create a
model and explain it in terms of some known parameters.

We have developed a model based on the Quantum Perturbation Theory. We
call this model the `Quantum Perturbation Model'. The model relates the
strength of the COVID-19 attack to a wave function that contains
information about the system (a person infected with SARS-CoV-2) and
quantized energy levels that indicate the likelihood of illness
recurrence.

Every measurable quantity is represented by an operator say \emph{Ồ} and
the state of a system are represented by a wave function \selectlanguage{greek}\emph{ψ} \selectlanguage{english}in
Quantum Mechanics. When the operator operates on the wave function, it
gives us the same wave function multiplied by certain numbers. The
numbers are called \emph{Eigen Values} and provide the measurable value
of the parameter; one is interested to calculate. For example, the total
energy operator is represented by Hamiltonian H. This operator operates
on wave function \selectlanguage{greek}ψ \selectlanguage{english}as given below (Griffith, 1994; Liboff, 2002).

\selectlanguage{greek}\textbf{HΨ = EΨ} \selectlanguage{english}(1)

where E gives us the energy of the system.

When a quantum system perturbs the energy of the system also changes
slightly. The change in energy depends upon the nature and strength of
perturbation.

The rules and equations of Quantum Mechanics are applied to a system
with size in nanometer (nm) or smaller. The size of coronavirus is from
20 nm up to as big as 500 nm (Williamson et al., 2020; Sud et al.,
2020). This size range is very appropriate to be discovered and
investigated by the laws and equations of Quantum Mechanics. Therefore,
we have applied Quantum Perturbation Theory to the attack and effect of
SARS-CoV-2 on humans.

Let the energy operator be given by Hamiltonian H of the Quantum
Mechanical system is compatible to the full strength of the attack by
coronavirus on a person then the energy eigenvalue E will give the
effect of COVID-19 on the health of the person. Let us further assume
that the first attack in full strength by coronavirus on a person is
represented by an un-perturbed system then mathematically it can be
expressed as (Griffith, 1994; Liboff, 2002; Reed, 2007),

\selectlanguage{greek}\textbf{H\textsubscript{0} Ψ\textsubscript{n}\textsuperscript{(0)} =
E\textsubscript{n}\textsuperscript{(0)}Ψ\textsubscript{n}\textsuperscript{(0)}}\selectlanguage{english}
(2)

Where H\textsubscript{0} is the Hamiltonian (total strength of the
coronavirus) and \selectlanguage{greek}Ψ\selectlanguage{english}\textsubscript{n}\textsuperscript{(0)} is the
unperturbed wave function of the system in its nth state. The energy of
the unperturbed system is also called the zeroth-order correction to
energy. Equation (2) gives the unperturbed zeroth-order correction in
energy further simplified in equation (3).

\selectlanguage{greek}\textbf{E\textsubscript{n}\textsuperscript{(0)}= \textless{}
Ψ\textsubscript{n}\textsuperscript{(0)}}\selectlanguage{english}
|\textbf{H\textsubscript{0}|Ψn\textsuperscript{(0)}\textgreater{}}\selectlanguage{english} (3)

Then small perturbation is applied to find the approximate solution for
the same system under a slight perturbation.

\selectlanguage{greek}\textbf{H Ψ\textsubscript{n} = E\textsubscript{n} Ψ\textsubscript{n}}\selectlanguage{english}(4)

To see the effect of perturbation, we expand the Hamiltonian into a
modified form,

\selectlanguage{greek}\textbf{H = H 0 + λH p}\selectlanguage{english} (5)

Where \selectlanguage{greek}λ \selectlanguage{english}is a dimensionless parameter meant to keep track of the degree
of ``smallness''. H\textsubscript{p} is the perturbed Hamiltonian. When
\selectlanguage{greek}λ \selectlanguage{english}- \selectlanguage{english}0, H\textsubscript{0} - H.

Incorporating equation (4) into equation (3), we get,

\selectlanguage{greek}\textbf{(H 0 + λH p) Ψ\textsubscript{n} =
E\textsubscript{n}Ψ\textsubscript{n}}\selectlanguage{english} (6)

We can easily the perturbed wave function in terms of the unperturbed
wave function and the strength of perturbation \selectlanguage{greek}λ. \selectlanguage{english}And the energy of the
perturbed system in terms of the energy of the unperturbed system and \selectlanguage{greek}λ.
\selectlanguage{english}Therefore,

\selectlanguage{greek}\textbf{Ψ\textsubscript{n} = Ψ\textsubscript{n}\textsuperscript{(0)} + λ
Ψ\textsubscript{n}\textsuperscript{(1)} +
λ\textsuperscript{2}Ψ\textsubscript{n}\textsuperscript{(2)} +
λ\textsuperscript{3}Ψ\textsubscript{n}\textsuperscript{(3)}}\selectlanguage{english} (7)

And \selectlanguage{greek}\textbf{E\textsubscript{n} =
E\textsubscript{n}\textsuperscript{(0)}+ λ
E\textsubscript{n}\textsuperscript{(1)} +
λ\textsuperscript{2}E\textsubscript{n}\textsuperscript{(2)} +
λ\textsuperscript{3}E\textsubscript{n}\textsuperscript{(3)}}\selectlanguage{english} (8)

Putting equation (7) and equation (8) in equation (4) and collecting
terms with the same exponent of \selectlanguage{greek}λ, \selectlanguage{english}we can get variation or correction to
the energy and wave function of the system.

\selectlanguage{greek}\textbf{H(Ψ\textsubscript{n} = Ψ\textsubscript{n}\textsuperscript{(0)} +
λ Ψ\textsubscript{n}\textsuperscript{(1)} +
λ\textsuperscript{2}Ψ\textsubscript{n}\textsuperscript{(2)} +
λ\textsuperscript{3}Ψ\textsubscript{n}\textsuperscript{(3)} \ldots{}) =}\selectlanguage{english}

(\selectlanguage{greek}\textbf{E\textsubscript{n}\textsuperscript{(0)} + λ
E\textsubscript{n}\textsuperscript{(1)} +
λ\textsuperscript{2}E\textsubscript{n}\textsuperscript{(2)} +
λ\textsuperscript{3}E\textsubscript{n}\textsuperscript{(3)} \ldots{})
Ψ\textsubscript{n}}\selectlanguage{english}(9)

The first order and higher order variations or corrections to the wave
function and to the energy are obtained, by solving equation (9) and
analyzing the terms (Griffith, 1994; Liboff, 2002; Reed, 2007).

\textbf{First Order correction to wave function:}

\selectlanguage{greek}\textbf{Ψ\textsubscript{n}\textsuperscript{(1)} =}\selectlanguage{english} [?]
\{\selectlanguage{greek}\textbf{\textless{} Ψ\textsubscript{n}\textsuperscript{(0)}|H
p|Ψ\textsubscript{j}\textsuperscript{(0)} \textgreater{} / (
E\textsubscript{j}\textsuperscript{(0)} --
E\textsubscript{n}\textsuperscript{(0)} ) \}
Ψ\textsubscript{n}\textsuperscript{(0)}}\selectlanguage{english} (10)

\textbf{First Order energy correction:}

\selectlanguage{greek}\textbf{E\textsubscript{n}\textsuperscript{(1)} = \textless{}
Ψ\textsubscript{n}\textsuperscript{(0)}|H
p|Ψ\textsubscript{n}\textsuperscript{(0)} \textgreater{} / \textless{}
Ψ\textsubscript{n}\textsuperscript{(0)}|Ψ\textsubscript{n}\textsuperscript{(0)}\textgreater{}}\selectlanguage{english}
(11)

\textbf{Second Order energy correction:}

\textbf{E\textsubscript{n}\textsuperscript{(2)} =}
[?]\textbf{\{|\textless{} Ψ\textsubscript{n}\textsuperscript{(0)}|H
p|Ψ\textsubscript{j}\textsuperscript{(0)}\textgreater{}|\textsuperscript{2}
/ ( E\textsubscript{j}\textsuperscript{(0)} --
E\textsubscript{n}\textsuperscript{(0)} )\}} (12)

Similarly, we can obtain 3\textsuperscript{rd} order,
4\textsuperscript{th} order and high order corrections to the energy of
the system.

It must be noted that the perturbation energies are very small as
compared to the original total energy of the system. In addition, every
higher-order perturbation contributes less energy as compared to the
energy contribution by lower-order perturbation.

The size match of SRAS-CoV-2 with the range where quantum phenomena
appear makes it possible to study various scenarios of COVID-19 using
Quantum Perturbation Theory. In this work, the energy corrections
provided by the Quantum Perturbation Theory are applied to the corona
various attack under various conditions.

\textbf{RESULTS AND DISCUSSION}

To apply Quantum Perturbation Theory to a person affected by COVID-19,
we consider the following cases.

\textbf{Case 1: SARS-CoV-2 attack on a healthy person.}

Upon SARS-CoV-2 attacks on a healthy person with no pre-existing
condition, the total strength of the virus attack on this healthy person
is represented by the unperturbed Hamiltonian \textbf{H 0} and its
maximum effect on the patient is given by the total energy, given by

\selectlanguage{greek}\textbf{E\textsubscript{n}\textsuperscript{(0)} = \textless{}
Ψ\textsubscript{n}\textsuperscript{(0)}}\selectlanguage{english}
|\textbf{H\textsubscript{0}|Ψn\textsuperscript{(0)}\textgreater{}}\selectlanguage{english}

This equation shows that the energy is directly proportional to the
Hamiltonian. It concludes that the stronger the viral attack on a
healthy person, the sicker the person becomes and the longer the person
remains a COVID-19 patient.

Figure 2 depicts a comparison of healthy lungs and lungs damaged by
COVID-19 during its initial onslaught. The graphic, which is an x-ray
imaging of a patient's lungs, depicts how COVID-19 affects and damages
the lungs of a healthy individual.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{(b)}

\textbf{Figure 2.} Comparison of (a) healthy lungs, and (b) lungs
affected by Covid-19 in its first attack.

\textbf{Case 2: SARS-CoV-2 attack on a healthy person with pre-existing
conditions.}

If SARS-CoV-2 infects a person who has a pre-existing condition like
heart disease or Asthma, it will make the person sicker and affect them
more than if they did not have a pre-existing condition. Quantum
Perturbation Theory describes that in this case, the total effect of
COVID-19 on the person attacked by the virus will be the sum of
unperturbed energy and the first-order correction of energy. Thus,
mathematically

Strength of Covid-19 attack on a person with pre-existing
disease\textbf{E} = \textbf{E\textsubscript{n}\textsuperscript{(0)} +
E\textsubscript{n}\textsuperscript{(1)}}

\selectlanguage{greek}\textbf{E = \textless{} Ψ\textsubscript{n}\textsuperscript{(0)}}\selectlanguage{english}
|\textbf{H\textsubscript{0}|Ψn\textsuperscript{(0)}\textgreater{} +
\textless{} Ψ\textsubscript{n}\textsuperscript{(0)}|H
p|Ψ\textsubscript{n}\textsuperscript{(0)} \textgreater{} / \textless{}
Ψ\textsubscript{n}\textsuperscript{(0)}|Ψ\textsubscript{n}\textsuperscript{(0)}\textgreater{}}\selectlanguage{english}
(13)

Similarly, if a person has two chronic conditions and is infected with
coronavirus, the viral infectivity with the two pre-existing diseases
will be,

\textbf{E =} \textbf{E\textsubscript{n}\textsuperscript{(0)} +
E\textsubscript{n}\textsuperscript{(1)} +
E\textsubscript{n}\textsuperscript{(2)}}

Which gives,

\selectlanguage{greek}\textbf{E = \textless{} Ψ\textsubscript{n}\textsuperscript{(0)}}\selectlanguage{english}
|\textbf{H\textsubscript{0}|Ψn\textsuperscript{(0)}\textgreater{} +
\textless{} Ψ\textsubscript{n}\textsuperscript{(0)}|H
p|Ψ\textsubscript{n}\textsuperscript{(0)} \textgreater{} / \textless{}
Ψ\textsubscript{n}\textsuperscript{(0)}|Ψ\textsubscript{n}\textsuperscript{(0)}\textgreater{}}\selectlanguage{english}

\textbf{+} [?] \textbf{\{|\textless{}
Ψ\textsubscript{n}\textsuperscript{(0)}|H
p|Ψ\textsubscript{j}\textsuperscript{(0)}\textgreater{}|\textsuperscript{2}
/ ( E\textsubscript{j}\textsuperscript{(0)} --
E\textsubscript{n}\textsuperscript{(0)} )\}} (14)

Thus, the existence of any number of pre-existing diseases will increase
the effect of COVID-19 infection and the extra risk is proportional to
the order of perturbation.

An important point worth noting here is the fact that every higher-order
perturbation contributes less energy than the lower-order perturbation.
Each additional disease is represented by a higher order of
perturbation. This results in a significant outcome that, if an
individual is affected by the COVID-19 and has one pre-existing disease,
then the effect is represented mathematically by the addition of
first-order correction to the energy. Similarly, if another person has
two existing diseases, then the effect of COVID-19 will be increased by
the sum of first-order energy correction and second-order energy
correction. Since second-order energy is less than the first order that
means doubling the pre-existing disease in a person will not double the
COVID-19 effect but the effect on the person will be less than double.
This could be most probably attributed to the fact that in the presence
of two pre-existing diseases as compared to one when coronavirus attacks
then the immunity system of the person will be more active and will
fight stronger. As a result, the COVID-19 effect will be less than
double. The effect of COVID-19 on a healthy person and on patients with
one or more pre-existing chronic diseases is given in Figure 3.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Figure 3.} Increase in the chances of death by COVID-19 with
pre-existing diseases.

\textbf{Case 3: Role of age on the first-time effect of COVID-19.}

It has also been reported and is widely accepted that people of old age
are affected more severely by the COVID-19 as compared to the younger
population. This situation is also modeled by the Quantum Perturbation
theory. However, the reverse Quantum Perturbation effect applies to this
scenario in which the unperturbed term represents the effect of COVID-19
on the old people above the age of 80. The first-order correction to
energy term represents the next lower age population and so on. The
lower effect of COVID-19 on young age people could be attributed to
their immunity, which decreases in strength with age, keeping all other
factors the same. Mathematically, the varying effect with age is given
by the following equations.

The effect of COVID-19 on very old age people is represented by the
unperturbed energy term\textbf{E\textsubscript{n}\textsuperscript{(0)}.}
The effect on the next lower age group is modeled by the first-order
energy correction term given as

\selectlanguage{greek}\textbf{E\textsubscript{n}\textsuperscript{(1)} = \textless{}
Ψ\textsubscript{n}\textsuperscript{(0)}|H
p|Ψ\textsubscript{n}\textsuperscript{(0)} \textgreater{} / \textless{}
Ψ\textsubscript{n}\textsuperscript{(0)}|Ψ\textsubscript{n}\textsuperscript{(0)}\textgreater{}}\selectlanguage{english}

The decreasing effect on the next lower ag population is given by the
second-order energy correction term given below.

\textbf{E\textsubscript{n}\textsuperscript{(2)} =} [?]

\textbf{\{|\textless{} Ψ\textsubscript{n}\textsuperscript{(0)}|H
p|Ψ\textsubscript{j}\textsuperscript{(0)}\textgreater{}|\textsuperscript{2}
/ ( E\textsubscript{j}\textsuperscript{(0)} --
E\textsubscript{n}\textsuperscript{(0)} )\}}

Figure 4 represents the effect of COVID-19 on the increasing age of the
affected population. Age effect has been observed in many case studies
and is one of the leading causes of early vaccination of elderly people
in the USA.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Figure 4.} The effect of age of a patient on the severity of the
impact of COVID-19 attack.

\textbf{Case 4: SARS-CoV-2 attack on a vaccinated person for the first
time or reinfecting a patient for the second time.}

What would happen if a person has recovered from COVID-19 and received a
vaccine against the virus? Another significant finding, according to
Quantum Perturbation Theory, is a link between repeated COVID-19 attacks
and immunity responses.

Assume that a person has recovered from the first attack of coronavirus
and is suffering for a certain time from COVID-19. Quantum Perturbation
Theory prescribes that this person can be affected by the COVID-19
again. However, this time the attack may not be as severe as the first
time because the body has developed an affinity and the immunity system
has got the strength and has already recognized the virus to resist
strongly. The reduced effect of COVID-19 on this person is given by the
first-order correction to the energy of the perturbed system. There
would not be an unperturbed energy effect this time. Since the
first-order correction to the energy is very small, therefore, the
second attack on a person will be very weak as well and the person will
recover soon. Mathematically, the effect of COVID-19 on a person being
suffered the second time is given mathematically by,

Strength of second attack of COVID-19 attack on a
person\textbf{E\textsubscript{n}\textsuperscript{(1)}} =
\selectlanguage{greek}\textbf{\textless{} Ψ\textsubscript{n}\textsuperscript{(0)}|H
p|Ψ\textsubscript{n}\textsuperscript{(0)} \textgreater{} / \textless{}
Ψ\textsubscript{n}\textsuperscript{(0)}|Ψ\textsubscript{n}\textsuperscript{(0)}\textgreater{}}\selectlanguage{english}

If a person got a vaccine against coronavirus, then this person can
still be suffered from COVID-19. In this situation, like the above case
of second attack, the effect of COVID-19 will be weak and is represented
by the first-order correction to the
energy\textbf{E\textsubscript{n}\textsuperscript{(1)}} =
\selectlanguage{greek}\textbf{\textless{} Ψ\textsubscript{n}\textsuperscript{(0)}|H
p|Ψ\textsubscript{n}\textsuperscript{(0)} \textgreater{} / \textless{}
Ψ\textsubscript{n}\textsuperscript{(0)}|Ψ\textsubscript{n}\textsuperscript{(0)}\textgreater{}}\selectlanguage{english}

Figure 5 shows how the effect of Covid-19 is tremendously reduced after
vaccination.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

\textbf{(b)}

\textbf{Figure 5.} Comparison of (a) Covid-19 effect on lungs of a
non-vaccinated healthy person, to (b) the effect of Covid-19 on the
lungs of a vaccinated person.

\textbf{Case 5: SARS-CoV-2 second-time attack on a vaccinated person.}

What would be the case of a person who has recovered from SARS-CoV-2 and
got a vaccine against the virus? Quantum Perturbation Theory describes
that it is still possible that the person may be affected by the
SARS-CoV-2 again. However, this time the attack will be very weak, even
weaker than the person described in case 3. Mathematically, the effect
of SARS-CoV-2 on this person is given by the second-order correction to
the energy of the perturbed system. Thus, the strength of coronavirus
attack on a vaccinated person for the second time is given by,

\textbf{E\textsubscript{n}\textsuperscript{(2)}=} [?]
\textbf{\{|\textless{} Ψ\textsubscript{n}\textsuperscript{(0)}|H
p|Ψ\textsubscript{j}\textsuperscript{(0)}\textgreater{}|\textsuperscript{2}
/ ( E\textsubscript{j}\textsuperscript{(0)} --
E\textsubscript{n}\textsuperscript{(0)} )\}}

Since En\textsuperscript{\textbf{(2)}} \textbf{\textless{}
E\textsubscript{n}\textsuperscript{(1)},} therefore, the person will get
better very quickly. This reduction in the COVID-19 attack could be
attributed to the fact that the body has developed much higher affinity
than the previous time and the immunity system has got even higher
strength because of two factors: the vaccine and the second attack of
SARS-CoV-2. Similarly, by adding any additional factor which can further
boost the affinity and the resistance of the immunity system, the
strength of coronavirus will further decrease and will be given by the
very small 3\textsuperscript{rd} order correction to the perturbed
energy. Thus, Quantum Perturbation Theory suggests that multiple times
suffering from SARS-CoV-2 is not impossible but every additional time or
if any extra factor which boosts the body or organ affinity (like a
vaccine) the strength of the attack will decline sharply. Figure 6
provides a systematic decrease in the effect of SARS-CoV-2 in such
circumstances.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}

\textbf{Figure 6.} The role of affinity factors like vaccine and
recurrence on the effect of SARS-CoV-2.

The present work provides a quantum mechanical approach to variations in
the Covid-19 effect. Any person who shows a different path of variation
in the Covid-19 effect has either got another underlying disease that is
not diagnosed yet or has another affinity factor that is yet to be
detected. It is also worth noting that we have not looked into the
effect of a time interval between two attacks in the event of multiple
SARS-CoV-2 attacks. In the present work, we have used time-independent
quantum perturbation theory. To account for the time interval between
two attacks, time-dependent quantum perturbation theory must be
considered.

Another important fact and information this model provides is an
approach to underlying disease, which is either not detected, or ignored
during Covid-19 patients' studies. Assume that a patient reports one
pre-existing disease when affected by Covid-19 then the first-order
energy perturbation term should be applied to the patient. If it is
found that the effect of Covid-19 is severe than this model predicts
then that indicates that the patient has a second underlying disease as
well which is either unreported or ignored. Therefore, the quantum
perturbation model of Covid-19 can also predict a hidden underlying
disease, which can affect the body's immunity to Covid-19.

\textbf{CONCLUSION}

To enter human cells, SARS-CoV-2 uses the angiotensin-converting enzyme
2 (ACE2) receptor. The virus replicates inside the cell by releasing its
single-strand RNA genome into the cytoplasmic compartment. B cells
initiate an early response by generating anti-SARS-CoV-2 specific IgA,
IgG, and IgM antibodies. The specificity of these antibodies increases
with reinfection, immunization, and exposure to new variants. The
antibody maturation cycle is repeated indefinitely, with each cycle
producing new and higher affinity antibodies. When a larger number of
people are re-infected or vaccinated and survive, herd immunity
develops.~ Age and pre-existing conditions have an impact on this
process as well. Being a new pandemic with very little understood, It is
very important to put it in a standard model based on universal ideas
and scope. Quantum Perturbation Theory is used in the present work to
successfully model the effect of SARS-CoV-2 and its variation as a
function of various factors. Images of Covid-19 patients at various
stages are included. The Quantum Modelling provides a strong base and
interprets mathematically why the effect of Covid-19 does not change
linearly and follows exponential variation with each factor developing
affinity against the virus. The Quantum Model will open a unique way of
understanding the effect, prevention, and control of pandemic diseases.

\textbf{Acknowledgment:} We are very thankful to Dr. Muhammad Kamal, In
Charge Medical Officer, General Hospital, Rustam, Mardan, Pakistan, for
providing lungs images of healthy adults and Covid-19 patients he
treated in his hospital.

\textbf{Conflict of Interest Statement:} All authors declare no conflict
of interest.

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