Figure 1. SARS-CoV-2 and perpetual immunity: A significant number of high-affinity antibodies are created, which outwit the virus.
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 and the state of a system are represented by a wave function ψ 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 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 ψ as given below (Griffith, 1994; Liboff, 2002).
HΨ = EΨ (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),
H0 Ψn(0) = En(0)Ψn(0) (2)
Where H0 is the Hamiltonian (total strength of the coronavirus) and Ψn(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).
En(0)= < Ψn(0)H0│Ψn(0)> (3)
Then small perturbation is applied to find the approximate solution for the same system under a slight perturbation.
H Ψn = En Ψn(4)
To see the effect of perturbation, we expand the Hamiltonian into a modified form,
H = H 0 + λH p (5)
Where λ is a dimensionless parameter meant to keep track of the degree of “smallness”. Hp is the perturbed Hamiltonian. When λ → 0, H0 → H.
Incorporating equation (4) into equation (3), we get,
(H 0 + λH p) Ψn = EnΨn (6)
We can easily the perturbed wave function in terms of the unperturbed wave function and the strength of perturbation λ. And the energy of the perturbed system in terms of the energy of the unperturbed system and λ. Therefore,
Ψn = Ψn(0) + λ Ψn(1) + λ2Ψn(2) + λ3Ψn(3) (7)
And En = En(0)+ λ En(1) + λ2En(2) + λ3En(3) (8)
Putting equation (7) and equation (8) in equation (4) and collecting terms with the same exponent of λ, we can get variation or correction to the energy and wave function of the system.