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The current emergence of coronavirus (SARS-CoV-2) puts the world in threat. The structural research on the receptor recognition by SARS-CoV-2 has identified the key interactions between SARS-CoV-2 spike protein and its host (epithelial cell) receptor, also known as angiotensin-converting enzyme 2 (ACE2). It controls both the cross-species and human-to-human transmissions of SARS-CoV-2. In view of this, we propose and analyze a mathematical model for investigating the effect of CTL responses over the viral mutation to control the viral infection when a postinfection immunostimulant drug (pidotimod) is administered at regular intervals. Dynamics of the system with and without impulses have been analyzed using the basic reproduction number. This study shows that the proper dosing interval and drug dose both are important to eradicate the viral infection.

A novel coronavirus named SARS-CoV-2 (an interim name proposed by WHO (World Health Organization)) became a pandemic since December 2019. The first infectious respiratory syndrome was recognized in Wuhan, Hubei province of China. Dedicated virologists identified and recognized the virus within a short time [

The common symptoms of COVID-19 are fever, fatigue, dry cough, and myalgia. Also, some patients suffer from headaches, abdominal pain, diarrhea, nausea, and vomiting. In the acute phase of infection, the disease may lead to respiratory failure which leads to death also. From clinical observation, within 1-2 days after patient symptoms, the patient becomes morbid after 4-6 days and the infection may clear within 18 days [

In [

We know that virus clearance after acute infection is associated with strong antibody responses. Antibody responses have the potential to control the infection [^{+} T helper cells for developing this specific antibody [

Mathematical modeling with real data can help in predicting the dynamics and control of an infectious disease [

Since the dynamics of the disease transmission of SARS-CoV-2 in the cellular level is yet to be explored, we investigate the system in the light of the previous literature of [

We explained the dynamics in the acute infection stage. It has been observed that CTLs proliferate and differentiate antibody production after they encounter antigens. Here, we investigate the effect of CTL responses over the viral mutation to control viral infection when a postinfection drug is administered at regular intervals by a mathematical perspective.

It is clinically evident that immunostimulants play a crucial role in the case of respiratory disease. Among the currently available immunostimulants, pidotimod is the most effective for the respiratory disease [

In this article, we have considered the infection dynamics of SARS-CoV-2 infection in the acute stage. We have used impulsive differential equations to study the immunostimulant drug dynamics and the effects of perfect drug adherence. In recent years, the effects of perfect adherence have been studied by using impulsive differential equations in [

The article is organized as follows. The very next section contains the formulation of the impulsive mathematical model. Dynamics of the system without impulses has been provided in Section

As discussed in the previous section, we propose a model considering the interaction between epithelial cells and SARS-CoV-2 virus along with lytic CTL responses over the infected cells. We consider five populations, namely, the uninfected epithelial cells

In this model, we consider which represents the concentration of ACE2 on the surface of uninfected cells, which can be recognized by the surface spike (S) protein of SARS-CoV-2 [

It is assumed that the susceptible cells are produced at a rate

The infected cells produce new viruses at the rate

CTL proliferation in the presence of infected cells is described by the term

With the above assumptions, we have the following mathematical model characterizing the SARS-CoV-2 dynamics:

A short description of the model parameters and their values is shown in Table

Set of parameter values used of numerical simulations.

Parameter | Explanation | Assigned value |
---|---|---|

Production rate of uninfected cell | 5 | |

Production rate of ACE2 | 1 | |

Disease transmission rate | 0.0001 | |

Bonding rate of ACE2 | 0.3 | |

Death rate of uninfected cells | 0.1 | |

Death rate of infected cells | 0.1 | |

Removal rate of virus | 0.1 | |

Hydrolyzing rate of epithelial cells | 0.1 | |

Decay rate of CTL | 0.1 | |

Killing rate of infected cells by CTL | 0.01 | |

Number of new virions produced | 10-100 | |

Proliferation rate of CTL | 0.22 | |

Maximum proliferation of CTL | 100 |

We consider the perfect adherence behavior of the immunostimulant drug for SARS-CoV-2-infected patients at fixed drug dosing times

We assume that CTL cells increase by a fixed amount

Here,

It can be noted that when there is no drug application in the system, model (

In this section, we analyze the dynamics of the system without impulses, i.e., system (

Model (

Note that

In this section, the characteristic equation at any equilibrium is determined for the local stability of system (

The coefficients

Looking at stability of any equilibrium

Let us define the basic reproduction number as

Then, using (

Disease-free equilibrium

At

The coefficients

Using the Routh-Hurwitz criterion, we have the following theorem:

The CTL-free equilibrium,

Denoting

The coexisting equilibrium

In this section, we consider the model system (

We now consider the following linear system:

In presence of impulsive dosing, we can get the recursion relation at the moments of impulse as

Thus, the amount of CTL before and after the impulse is obtained as

Thus, the limiting case of the CTL amount before and after one cycle is as follows:

Let

We now recall some results for our analysis from [

Let

There exists a constant

Let

We now consider the following subsystem:

The lemma provided above gives the following result.

System (

We use this result to derive the following theorem.

The disease-free periodic orbit

Let the solution of system (

The monodromy matrix

We can write

Here,

In this section, we have observed the dynamical behaviors of the system without the drug (Figures

Existence and stability of equilibria is shown with respect to

In the absence of the drug, the effect of the growth rate of CTL, i.e.,

Numerical solution of the model system with and without the drug dose is shown taking parameters as in Figure

Numerical solution of the model system for different rates of drug dosing and different intervals of impulses.

We have mainly focused on the role of CTL and its possible implication on the treatment and drug development. The drug that stimulates the CTL responses represents the best hope for control of COVID-19. Here, we have determined the situation where CTLs can effectively control the viral infection when the postinfection drug is administered at regular intervals.

Existence of equilibria of the system without the drug dose is shown for different values of basic reproduction number

The effect of the immune response rate

Due to the impulsive nature of the drugs, there are no equilibria of the system; i.e., population does not reach towards the equilibrium point, rather approach a periodic orbit. Hence, we evaluate equilibrium-like periodic orbits. There are two periodic orbits of system (

Figure

If we take sufficiently large impulsive interval

In this article, the role of the immunostimulant drug (mainly pidotimod) during interactions between SARS-CoV-2 spike protein and epithelial cell receptor ACE2 in COVID-19 infection has been studied as a possible drug dosing policy. To reactivate the CTL responses during the acute infection period, immune activator drugs are delivered to the host system in an impulsive mode.

When the immunostimulant drug is administered, the best possible CTL responses can act against the infected or virus-producing cells to neutralize infection. This particular situation can keep the infected cell population at a very low level. In the proposed mathematical model, we have analyzed the optimal dosing regimen for which infection can be controlled.

From this study, it has been observed that when the basic reproduction ratio lies below one, we expect the system to attain its disease-free state. However, the system switches from the disease-free state to the CTL-free equilibrium state when

Here, we have explored the immunostimulant drug dynamics by the help of impulsive differential equations. With the help of impulsive differential equations, we have studied how the effect of the maximal acceptable optimal dosage can be found more precisely. The impulsive system shows that the proper dosage and dosing intervals are important for the eradication of the infected cells and virus population which results in the control of the pandemic (Figure

It has also been observed that the length of the dosing interval and the drug dose play a very decisive role to control and eradicate the infection. The most interesting prediction of this model is that effective therapy can often be achieved, even for low adherence, if the dosing regimen is adjusted appropriately (Figure

Future extension work of the combination of drug therapy should also include more realistic patterns of nonadherence (random drug holidays, imperfect timing of successive doses) and more accurate intracellular pharmacokinetics which leads towards better estimates of drug dosage and drug dosing intervals.

We end the paper with the quotation: “

The data used for supporting the findings are included within the article.

The authors declare that there is no conflict of interest.

Both authors contributed equally to this work.