Research ArticleA Model for SARS-CoV-2 Infection with Treatment

Amar Nath Chatterjee 1 and Fahad Al Basir 2

1Department of Mathematics, K.L.S. College, Nawada, Magadh University, Bodh Gaya, Bihar 805110, India2Department of Mathematics, Asansol Girls’ College, Asansol-4, West Bengal 713304, India

Correspondence should be addressed to Fahad Al Basir; fahadalbasir@yahoo.com

Received 2 April 2020; Revised 6 July 2020; Accepted 22 July 2020; Published 1 September 2020

Academic Editor: Raul Alcaraz

Copyright © 2020 Amar Nath Chatterjee and Fahad Al Basir. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.

The current emergence of coronavirus (SARS-CoV-2) puts the world in threat. The structural research on the receptor recognitionby SARS-CoV-2 has identified the key interactions between SARS-CoV-2 spike protein and its host (epithelial cell) receptor, alsoknown as angiotensin-converting enzyme 2 (ACE2). It controls both the cross-species and human-to-human transmissions ofSARS-CoV-2. In view of this, we propose and analyze a mathematical model for investigating the effect of CTL responses overthe viral mutation to control the viral infection when a postinfection immunostimulant drug (pidotimod) is administered atregular 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.

1. Introduction

A novel coronavirus named SARS-CoV-2 (an interim nameproposed by WHO (World Health Organization)) became apandemic since December 2019. The first infectious respira-tory syndrome was recognized in Wuhan, Hubei province ofChina. Dedicated virologists identified and recognized thevirus within a short time [1]. The SARS-CoV-2 is a single-stranded RNA virus genome which is closely related tosevere acute respiratory syndrome- (SARS-) CoV [2]. Theinfection of SARS-CoV-2 is associated with a SARS-CoV-like a disease with a fatality rate of 3.4% [3]. The WorldHealth Organization (WHO) have named the disease asCOVID-19 and declared it as a public health emergencyworldwide [4].

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

may clear within 18 days [5] depending on the immunesystem. Thus, appropriate quarantine measure for a mini-mum of two weeks is taken by the public health authoritiesfor inhibiting community spread [6].

In [1], Zhou et al. identified the respiratory tract as theprincipal infection site for COVID-19 infection. SARS-CoV-2 infects primary human airway epithelial cells. Theangiotensin-converting enzyme 2 (ACE2) receptor of epi-thelial cells plays an important role in cellular entry [1, 7].It has been observed that ACE2 could be expressed in theoral cavity. ACE2 receptors are higher in the tongue thanbuccal and gingival tissues. These findings imply that themucosa of the oral cavity may be a potentially high-riskroute of COVID-19 infection. Thus, epithelial cells of thetongue are the major routes of entry for COVID-19. Zhouet al. [1] also reported that SARS-CoV-2 spikes (S) bindwith the ACE2 receptor of epithelial cells with high affinity.The bonding between S (spikes) of SARS-CoV-2 and ACE2[7] results from the fusion between the viral envelope andthe target cell membrane, and the epithelial cells becomeinfected. The S protein plays a major role in the inductionof protective immunity during the infection of SARS-CoV-

HindawiComputational and Mathematical Methods in MedicineVolume 2020, Article ID 1352982, 11 pageshttps://doi.org/10.1155/2020/1352982

2 by eliciting neutralization antibody and T cell responses[8]. The S protein is not only capable of neutralizing anti-body, but it also contains several immunogenic T cell epi-topes. Some of the epitopes are found in either the S1 orS2 domain. These proteins are useful for SARS-CoV-2 drugdevelopment [9].

We know that virus clearance after acute infection isassociated with strong antibody responses. Antibodyresponses have the potential to control the infection [10].Also, CTL responses help to resolve infection and viruspersistence caused by weak CTL responses [11]. Antibodyresponses against SARS-CoV-2 play an important role inpreventing the viral entry process [8]. Hsueh et al. [2] foundthat antibodies block viral entry by binding to the Sglycoprotein of SARS-CoV-2. To fight against the pathogenSARS-CoV-2, the body requires SARS-CoV-2-specific CD4+

T helper cells for developing this specific antibody [8].Antibody-mediated immunity protection helps the anti-SARS-CoV serum to neutralize COVID-19 infection.Besides that, the role of T cell responses in COVID-19 infec-tion is very much important. Cytotoxic T lymphocyte (CTL)responses are important for recognizing and killing infectedcells, particularly in the lungs [8]. But the kinetics of theCTL responses and antibody responses during SARS-CoV-2 infection is yet to be explored. Our study will focus onthe role of CTL and its possible implication on treatmentand drug development. The drug that stimulates the CTLresponses represents the best hope for control of COVID-19. Here, we have modeled the situation where CTLs caneffectively control the viral infection when the postinfectiondrug is administered at regular intervals.

Mathematical modeling with real data can help in pre-dicting the dynamics and control of an infectious disease[12, 13]. A four-dimensional dynamical model for a viralinfection is proposed by Tang et al. [14] for MERS-CoVmediated by DPP4 receptors. In the case of SARS-CoV-2,the infection process is almost similar with MERS-CoV andSARS-CoV. For SARS-CoV-2 infection, the ACE2 receptorsof epithelial cells are the major target area.

Since the dynamics of the disease transmission ofSARS-CoV-2 in the cellular level is yet to be explored, weinvestigate the system in the light of the previous literatureof [14–18] to formulate the dynamic model which plays asignificant role in describing the interaction between unin-fected cells, free virus, and CTL responses. We propose anovel deterministic model which describes the cell biologi-cal infection of SARS-CoV-2 with epithelial cells and therole of the ACE2 receptor.

We explained the dynamics in the acute infection stage.It has been observed that CTLs proliferate and differentiateantibody production after they encounter antigens. Here,we investigate the effect of CTL responses over the viralmutation to control viral infection when a postinfectiondrug is administered at regular intervals by a mathematicalperspective.

It is clinically evident that immunostimulants play a cru-cial role in the case of respiratory disease. Among the currentlyavailable immunostimulants, pidotimod is the most effectivefor the respiratory disease [19]. Pidotimod increases the level

of immunoglobulins (IgA, IgM, and IgG) and activates theCTL responses to fight against the disease.

In this article, we have considered the infection dynamicsof SARS-CoV-2 infection in the acute stage. We have usedimpulsive differential equations to study the immunostimu-lant drug dynamics and the effects of perfect drug adherence.In recent years, the effects of perfect adherence have beenstudied by using impulsive differential equations in [20–26].With the help of impulsive differential equations, the effectof maximal acceptable drug holidays and optimal dosagecan be found more precisely [20, 26].

The article is organized as follows. The very next sectioncontains the formulation of the impulsive mathematicalmodel. Dynamics of the system without impulses has beenprovided in Section 3. The system with impulses has beenanalyzed in Section 4. Numerical simulations, on the basisof the outcomes of Sections 3 and 4, have been included inSection 5. Discussion in Section 6 concludes the paper.

2. Model Formulation

As discussed in the previous section, we propose a modelconsidering the interaction between epithelial cells andSARS-CoV-2 virus along with lytic CTL responses overthe infected cells. We consider five populations, namely,the uninfected epithelial cells TðtÞ, infected cells IðtÞ,ACE2 receptor of the epithelial cells EðtÞ, SARS-CoV-2virus V(t) and CTLs against the pathogen CðtÞ.

In this model, we consider which represents the concen-tration of ACE2 on the surface of uninfected cells, whichcan be recognized by the surface spike (S) protein ofSARS-CoV-2 [27].

It is assumed that the susceptible cells are produced at arate λ1 from the precursor cells and die at a rate dT . Thesusceptible cells become infected at a rate βEðtÞVðtÞTðtÞ.The constant dI is the death rate of the infected cells.Infected cells are also cleared by the body’s defensive CTLsat a rate p.

The infected cells produce new viruses at the rate mdIduring their life, and dV is the death rate of new virions,where m is any positive integer. It is also assumed thatACE2 is produced from the surface of uninfected cells atthe constant rate λ2 and the ACE2 is destroyed, when freeviruses try to infect uninfected cells, at the rate θβEðtÞVðtÞTðtÞ and is hydrolyzed at the rate dEE.

CTL proliferation in the presence of infected cells isdescribed by the term

αIC 1 − CCmax

� �, ð1Þ

which shows the antigen-dependent proliferation. Here, weconsider the logistic growth of CTL with Cmax as the maxi-mum concentration of CTL, and dc is its rate of decay.

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

2 Computational and Mathematical Methods in Medicine

dTdt

= λ1 − βEVT − dTT ,

dIdt

= βEVT − dII − pIC,

dVdt

=mdII − dVV ,

dEdt

= λ2 − θβEVT − dEE,

dCdt

= αIC 1 − CCmax

� �− dcC:

ð2Þ

A short description of the model parameters and theirvalues is shown in Table 1. We now modify the above modelby incorporating pulse periodic drug dosing using impulsivedifferential equations [28, 29].

We consider the perfect adherence behavior of theimmunostimulant drug for SARS-CoV-2-infected patientsat fixed drug dosing times tk, k ∈ℕ.

We assume that CTL cells increase by a fixed amount ω,which is proportional to the total number of CTLs that thedrug can stimulate. Thus, the above model takes the follow-ing form:

dTdt

= λ1 − βEVT − dTT ,

dIdt

= βEVT − dII − pIC,

dVdt

=mdII − dVV ,

dEdt

= λ2 − θβEVT − dEE,

dCdt

= αIC 1 − CCmax

� �− dcC, t ≠ tk,

C t+kð Þ = ω + C t−kð Þ, t = tk:

ð3Þ

Here, Cðt−k Þ denotes the CTL cell concentration immedi-ately before the impulse, Cðt+k Þ denotes the concentrationafter the impulse, and ω is the fixed amount which is propor-tional to the total number of CTLs the drug stimulates at eachimpulse time tk, k ∈ℕ.

Remark 1. It can be noted that when there is no drug applica-tion in the system, model (3) becomes model (2).

3. Analysis of the System without the Drug

In this section, we analyze the dynamics of the system with-out impulses, i.e., system (1). We have derived the basicreproduction number for the system. Stability of equilibriais discussed using the number.

3.1. Existence of Equilibria. Model (2) has three steady states,namely, (i) the disease-free equilibrium E1ðλ1/dT , 0, 0, λ2/dE,

0Þ; (ii) with �E > dTdV /βλ1m, there is a CTL response-freeequilibrium, E2ð�T ,�I, �V , �E, 0Þ, where

�T = dVβm�E

,

�I = βλ1m�E − dTdVβdIm�E

,

�V = βλ1m�E − dTdVβdV�E

,

�E =− θβλ1m − βλ2mð Þ +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiθβλ1m − βλ2mð Þ2 + 4βmdTdVdEθ

q2βdEm

;

ð4Þ

and (iii) the endemic equilibrium E∗ which is given by

T∗ = λ1α − dIαI∗ − pdc

dTα,

V∗ = dImI∗

dV,

E∗ = λ2α − θαdII∗ − θpdc

dEα,

ð5Þ

C∗ = αI∗ − dcð ÞCmaxαI∗

, ð6Þ

where I∗ is the positive root of the cubic equation

L0I3 + L1I

2 + L2I + L3 = 0, ð7Þ

Table 1: Set of parameter values used of numerical simulations.

Parameter Explanation Assigned value

λ1 Production rate of uninfected cell 5

λ2 Production rate of ACE2 1

β Disease transmission rate 0.0001

θ Bonding rate of ACE2 0.3

dT Death rate of uninfected cells 0.1

dI Death rate of infected cells 0.1

dV Removal rate of virus 0.1

dE Hydrolyzing rate of epithelial cells 0.1

dc Decay rate of CTL 0.1

p Killing rate of infected cells by CTL 0.01

m Number of new virions produced 10-100

α Proliferation rate of CTL 0.22

Cmax Maximum proliferation of CTL 100

3Computational and Mathematical Methods in Medicine

with

L0 = −α2θβd3I m,L1 = −2αθβd2I dcmp + α2θβd2Iλ1m + α2βd2Iλ2m,L2 = α2dTdIdVdE + αθβdIdcλ1mp + αβdIdcλ2mp

− α2βdIλ1λ2m − θβdId2cmp2,

L3 = αdTdVdEdcp:

ð8Þ

Remark 2. Note that L0 < 0 and L3 > 0. Thus, equation (7) hasat least one positive real root. If L1 > 0 and L2 < 0, then (3) canhave two positive roots. For a feasible endemic equilibrium, wealso need

min λ1α − pdcdIα

, λ2α − θpdcθαdI

� �> I∗ > dc

α: ð9Þ

3.2. Stability of Equilibria. In this section, the characteristicequation at any equilibrium is determined for the local stabilityof system (2). Linearizing system (2) at any equilibriumEðT , I, V , E, CÞ yields the characteristic equation

Δ ξð Þ = ∣ξIn −A∣ = 0, ð10Þ

where In is the identity matrix and A = ½aij� is the following5 × 5 matrix given by

A =

−βEV − dT 0 −βET −βVT 0

βEV −dI − pC βET βVT −pI

0 dIm −dv 0 0

−θβEV 0 −θβET −θβVT − dE 0

0 αC 1 − CCmax

� �0 0 a55

2666666666666666666666664

3777777777777777777777775

,

ð11Þ

with a55 = αIð1 − 2C/CmaxÞ − dc. We finally get the charac-teristic equation as

ψ ξð Þ = ξ5 + A1ξ4 + A2ξ

3 + A3ξ2 + A4ξ + A5 = 0: ð12Þ

The coefficients Ai, i = 1, 2,⋯, 5, are given in theappendix.

Looking at stability of any equilibrium E, the Routh-Hurwitz criterion gives that all roots of this characteristic

equation (12) have negative real parts, provided the followingconditions hold

A5 > 0,A1A2 − A3 > 0,

A3 A1A2 − A3ð Þ − A1 A1A4 − A5ð Þ > 0,A1A2 − A3ð Þ A3A4 − A2A5ð Þ − A1A4 − A5ð Þ2 > 0:

ð13Þ

Let us define the basic reproduction number as

R0 =mβλ1λ2dTdEdV

: ð14Þ

Then, using (5), we can derived the following result.

Theorem 3. Disease-free equilibrium E1ðλ1/dT , 0, 0, λ2/dE , 0Þof model (2) is stable for R0 < 1 and unstable for R0 > 1.

At E2, one eigenvalue is −dc and the rest of the eigen-values satisfy the following equation:

ξ4 + B1ξ3 + B2ξ

2 + B3ξ + B4 = 0: ð15Þ

The coefficients Bi, i = 1, 2,⋯, 5, are given in theappendix.

Using the Routh-Hurwitz criterion, we have the follow-ing theorem:

Theorem 4. The CTL-free equilibrium, E2ð�T ,�I, �V , �E, 0Þ, isasymptotically stable if and only if the following conditionsare satisfied:

B1 > 0,B2 > 0,B3 > 0,B4 > 0,

B1B2 − B3 > 0,B1B2 − B3ð ÞB3 − B2

1B4 > 0:

ð16Þ

DenotingA∗i = AiðE∗Þ and using (5), we have the following

theorem establishing the stability of coexisting equilibrium E∗.

Theorem 5. The coexisting equilibrium E∗ is asymptoticallystable if and only if the following conditions are satisfied:

A∗5 > 0,

A∗1A

∗2 − A∗

3 > 0,A∗3 A∗

1A∗2 − A∗

3ð Þ − A∗1 A∗

1A∗4 − A∗

5ð Þ > 0,A∗1A

∗2 − A∗

3ð Þ A∗3A

∗4 − A∗

2A∗5ð Þ − A∗

1A∗4 − A∗

5ð Þ2 > 0:

ð17Þ

4 Computational and Mathematical Methods in Medicine

4. Dynamics of the System with ImpulsiveDrug Dosing

In this section, we consider the model system (3). Before ana-lyzing the system, we first discuss the one-dimensionalimpulse system as follows:

dCdt

= αIC 1 − CCmax

� �− dcC, t ≠ tk,

C t+kð Þ = ω + C t−kð Þ, t = tk:

ð18Þ

Cðt−k Þ denotes the CTL responses immediately before theimpulse drug dosing, Cðt+k Þ denotes the concentration afterthe impulse, and ω is the dose that is taken at each impulsetime tk, k ∈ℕ.

We now consider the following linear system:

dCdt

= −dcC, t ≠ tk,

ΔC = ω, t = tk,ð19Þ

where Δ = Cðt+k Þ − Cðt−k Þ. Let τ = tk+1 − tk be the period of thecampaign. The solution of system (10) is

C tð Þ = C t+kð Þe−dc t−tkð Þ, for tk < t ≤ tk+1: ð20Þ

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

C t+kð Þ = C t−kð Þ + ω: ð21Þ

Thus, the amount of CTL before and after the impulse isobtained as

C t+kð Þ = ω 1 − e−kτdc� �1 − e−τdc

,

C t−k+1ð Þ = ω 1 − e−kτdc� �

e−τdc

1 − e−τdc:

ð22Þ

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

limk→∞

C t+kð Þ = ω

1 − e−τdc,

limk→∞

C t−k+1ð Þ = ωe−τdc

1 − e−τdc,

C t+k+1ð Þ = ωe−τdc

1 − e−τdc+ ω = ω

1 − e−τdc:

ð23Þ

Definition 6. Let Λ ≡ ðSu, Sa, I, CÞ and B0 = ½B : R4+ → R+�;

then, we say that B belong to class B0 if the following condi-tions hold:

(i) B is continuous on ðtk, tk+1� × R3+, n ∈N , and for all

Λ ∈ R4, limðt,μÞ→ðt+k ,ΛÞBðt, μÞ = Bðt+k ,ΛÞ exists

(ii) B is locally Lipschitzian in Λ

We now recall some results for our analysis from [28, 29].

Lemma 7. Let ZðtÞ be a solution of system (9) with Zð0+Þ ≥ 0.Then, ZiðtÞ ≥ 0, i = 1,⋯, 4, for all t ≥ 0. Moreover, ZiðtÞ > 0,i = 1,⋯, 4, for all t > 0 if Zið0+Þ > 0, i = 1,⋯, 4.

Lemma 8. There exists a constant γ such that TðtÞ ≤ γ, IðtÞ≤ γ, VðtÞ ≤ γ EðtÞ ≤ γ, and CðtÞ ≤ γ for each and everysolution ZðtÞ of system (9) for all sufficiently large t.

Lemma 9. Let B ∈ B0 and also consider that

D+B t, Zð Þ ≤ j t, B t, Z tð Þð Þð Þ, t ≠ tk,B t, Z t+ð Þð Þ ≤Φn B t, Z tð Þð Þð Þ, t = tk,

ð24Þ

where j : R+ × R+ → R is continuous in ðtk, tk+1� for e ∈ R2+,

n ∈N , the limit limðt,VÞ→ðt+k Þ jðt, gÞ = jðt+k , xÞ exists, and Φin

ði = 1, 2Þ: R+ → R+ is nondecreasing. Let yðtÞ be a maximalsolution of the following impulsive differential equation:

dx tð Þdt

= j t, x tð Þð Þ, t ≠ tk,

x t+ð Þ =Φn x tð Þð Þ, t = tk, x 0+ð Þ = x0,ð25Þ

existing on ð0+,∞Þ. Then, Bð0+, Z0Þ ≤ x0 implies that Bðt, ZðtÞÞ ≤ yðtÞ, t ≥ 0, for any solution ZðtÞ of system (9). If jsatisfies additional smoothness conditions to ensure the exis-tence and uniqueness of solutions for (12), then yðtÞ is theunique solution of (12).

We now consider the following subsystem:

dC tð Þdt

= −dcC, t ≠ tk, C t+kð Þ = C tkð Þ + ω, C 0+ð Þ = C0:

ð26Þ

The lemma provided above gives the following result.

Lemma 10. System (13) has a unique positive periodic solution~CðtÞ with period τ and given by

~C tð Þ = ω exp −dc t − tkð Þð Þ1 − exp −τdcð Þ , tk < t ≤ tk+1, ~C 0+ð Þ

= dc1 − exp −τdcð Þ :

ð27Þ

We use this result to derive the following theorem.

Theorem 11. The disease-free periodic orbit ð~T , 0, 0, ~E, ~CÞ ofsystem (2) is locally asymptotically stable if

~R0 < 1, ð28Þ

5Computational and Mathematical Methods in Medicine

where

~R0 =mdIβ

dTdEdVτ

ðτ0

~T~E

dI + p~Cdt: ð29Þ

Proof. Let the solution of system (9) without infected peoplebe denoted by ð~T , 0, 0, ~E, ~CÞ, where

~C tð Þ = ω exp −dc t − tkð Þð Þ1 − exp −τdcð Þ , tk < t ≤ tk+1, ð30Þ

with initial condition Cð0+Þ as in Lemma 10. We nowtest the stability of the equilibria. The variational matrixat ð~T , 0, 0, ~E, ~CÞ is given by

M tð Þ = mij

=

−dT 0 m13 0 0

0 − dI + p~C� �

β~E~T 0 0

0 mdI −dv 0 0

0 0 m43 −dE 0

0 m52 0 0 −dc

0BBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCA

: ð31Þ

The monodromy matrix ℙ of the variational matrixMðtÞ is

ℙ τð Þ = In expðτ0M tð Þdt

� �, ð32Þ

where In is the identity matrix. Note that m13, m43, andm52 are not required for this analysis; therefore, we havenot mentioned their expressions.

We can write ℙðτÞ = diag ðσ1, σ2, σ3, σ4, σ5Þ, where σi,i = 1, 2, 3, 4, 5, are the Floquet multipliers and they aredetermined as

σ1 = exp −dTτð Þ,

σ2,3 = expðτ0

12 −A ±

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 − 4B

ph idt

� �,

σ4 = exp −dEτð Þ,σ5 = exp −dcτð Þ:

ð33Þ

Here, A = dI + dV + p~C and B = dVðdI + p~CÞ −mdIβ~E~T.Clearly, λ1,4,5 < 1. It is easy to check that A2 − 4B > 0, and if B≥ 0 and hold, then we have λ2,3 < 1. Thus, according to Floquettheory, the periodic solution ð~T , 0, 0, ~E, ~CÞ of system (9) islocally asymptotically stable if the conditions given in (14) hold.

5. Numerical Results and Discussion

In this section, we have observed the dynamical behaviors ofthe system without the drug (Figures 1 and 2) and with impul-sive effect of the drug dose (Figures 3 and 4) through numericalsimulations taking the parameters mainly from [14, 19, 30].

We have mainly focused on the role of CTL and its possibleimplication on the treatment and drug development. The drugthat stimulates the CTL responses represents the best hope forcontrol of COVID-19. Here, we have determined the situationwhere CTLs can effectively control the viral infection when thepostinfection drug is administered at regular intervals.

Existence of equilibria of the system without the drug doseis shown for different values of basic reproduction number R0.In plotting Figure 1, we have varied the value of infection rateβ. It is observed that for the lower infection rate (that corre-sponds to R0 < 1), disease-free equilibrium E1 is stable(corroborated with Theorem 3). It becomes unstable andensures the existence of the CTL-free equilibrium E2 which

R0

0

5

10

15

20

25

I

0 1 2 3 4 5

StableE1

Stable E2

Unstable E2

Stable E⁎

(a)

R0

0 1 2 3 4 50

50

100

150

200

250

V

Stable E⁎

StableE1

Stable E2

Unstable E2

(b)

Figure 1: Existence and stability of equilibria is shown with respect to R0: Parameter values used in this figure are taken from Table 1 andm = 10. We have varied the value of β in ð0:00001,0:0001Þ.

6 Computational and Mathematical Methods in Medicine

is stable if R0 < 2:957 (which corresponds to β = 0:00005963)and unstable otherwise. (This satisfies Theorem 4.) Again,we see that when E2 is unstable, E

∗ is feasible. Also, wheneverE∗ exists, it is stable which verified the Theorem 5.

The effect of the immune response rate α is plotted inFigure 2. We observe that in the absence of the drug, the CTLcount and ACE2 increase with increasing value of α. Thesteady-state value of infected cell I∗ and virusV∗ decreases sig-nificantly as α increases.

Due to the impulsive nature of the drugs, there are noequilibria of the system; i.e., population does not reach

towards the equilibrium point, rather approach a periodicorbit. Hence, we evaluate equilibrium-like periodic orbits.There are two periodic orbits of system (3), namely, thedisease-free periodic orbit and endemic periodic orbit. Here,our aim is to find the stability of the disease-free periodic orbit.

Figure 3 compares the system without and with impulsedrug effect. In the absence of the drug, we observe that theCTL count approaches a stable equilibrium. Under regulardrug dosing, the CTL count oscillates in an impulsiveperiodic orbit. Assuming perfect adherence, if the drug issufficiently strong, both infected cell and virus populationapproach towards extinction. In this case, the total number

0 0.2 0.4 0.60

2

4

6

T⁎

𝛼

(a)

0 0.2 0.4 0.60

5

10

𝛼

I⁎

(b)

0 0.2 0.4 0.60

5

10

𝛼

V⁎

(c)

0 0.2 0.4 0.67

7.5

8

8.5

𝛼

E⁎

(d)

0 0.2 0.4 0.60

20

40

60

80

𝛼

C⁎

(e)

Figure 2: In the absence of the drug, the effect of the growth rate of CTL, i.e., α on the steady-state values of model population, is shown.Parameter values used in this figure are the same as Figure 1 except α.

7Computational and Mathematical Methods in Medicine

of uninfected cells reaches its maximum level which impliesthat the system approaches towards its infection-free state(Theorem 11).

If we take sufficiently large impulsive interval τ = 5 days(keeping rate ω = 50 fixed, as in Figure 3) or lower dosageeffect ω = 20 (keeping interval τ = 2 fixed, as in Figure 3), inboth the cases, infection remains present in the system. Thus,the proper dosage of drug and optimal dosing interval areimportant for infection management.

6. Conclusion

In this article, the role of the immunostimulant drug (mainlypidotimod) during interactions between SARS-CoV-2 spikeprotein and epithelial cell receptor ACE2 in COVID-19 infec-tion has been studied as a possible drug dosing policy. To reac-tivate the CTL responses during the acute infection period,immune activator drugs are delivered to the host system inan impulsive mode.

0 50 100 Time (days)

0

5

10

Susc

eptib

le ce

ll

With impulse, 𝜏 = 2,𝜔 = 50Without impulse

(a)

0 50 100 Time (days)

0

2

4

6

8

Infe

cted

cell

10 15 20

0.4

0.8

With impulse, 𝜏 = 2,𝜔 = 50Without impulse

(b)

0 50 1000

2

4

6

8

Viru

s

Time (days)

With impulse, 𝜏 = 2,𝜔 = 50Without impulse

(c)

0 50 1002

4

6

8

10

ACE

2

Time (days)

With impulse, 𝜏 = 2,𝜔 = 50Without impulse

(d)

0 50 1000

100

200

300

CTL

resp

onse

0 50 1000

5

10

Time (days)

With impulse, 𝜏 = 2,𝜔 = 50Without impulse

(e)

Figure 3: Numerical solution of the model system with and without the drug dose is shown taking parameters as in Figure 1. In this figure,τ = 2 and ω = 50.

8 Computational and Mathematical Methods in Medicine

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

From this study, it has been observed that when the basicreproduction ratio lies below one, we expect the system to

attain its disease-free state. However, the system switchesfrom the disease-free state to the CTL-free equilibrium statewhen 1 < R0 < 2:957. If R0 > 2:957, the CTL-free equilibriummoves to an endemic state (Figure 1).

Here, we have explored the immunostimulant drugdynamics by the help of impulsive differential equations.With the help of impulsive differential equations, we havestudied how the effect of the maximal acceptable optimal

0 50 100Time (days)

0

2

4

6

8

Susc

eptib

le ce

ll

𝜏 = 2,𝜔 = 20𝜏 = 5,𝜔 = 50

(a)

0 50 1000

2

4

6

8

Infe

cted

cell

Time (days)

𝜏 = 2,𝜔 = 20𝜏 = 5,𝜔 = 50

(b)

0 50 1000

2

4

6

Viru

s

Time (days)

𝜏 = 2,𝜔 = 20𝜏 = 5,𝜔 = 50

(c)

2

4

6

8

10

ACE

2

0 50 100Time (days)

𝜏 = 2,𝜔 = 20𝜏 = 5,𝜔 = 50

(d)

0

50

100

150

CTL

resp

onse

0 50 100Time (days)

𝜏 = 2,𝜔 = 20𝜏 = 5,𝜔 = 50

(e)

Figure 4: Numerical solution of the model system for different rates of drug dosing and different intervals of impulses.

9Computational and Mathematical Methods in Medicine

dosage can be found more precisely. The impulsive systemshows that the proper dosage and dosing intervals are impor-tant for the eradication of the infected cells and virus popula-tion which results in the control of the pandemic (Figure 3).

It has also been observed that the length of the dosinginterval and the drug dose play a very decisive role to controland eradicate the infection. The most interesting prediction ofthis model is that effective therapy can often be achieved, evenfor low adherence, if the dosing regimen is adjusted appropri-ately (Figure 4). Also, if the treatment regimen is not adjustedproperly, the therapy is not effective at all. This approachmight also be applicable to a combination of antiviral therapy.

Future extension work of the combination of drug ther-apy should also include more realistic patterns of nonadher-ence (random drug holidays, imperfect timing of successivedoses) and more accurate intracellular pharmacokineticswhich leads towards better estimates of drug dosage and drugdosing intervals.

We end the paper with the quotation: “This outbreak is atest of political, financial and scientific solidarity for the worldto fight a common enemy that does not respect borders..., whatmatters now is stopping the outbreak and saving lives,” by Dr.Tedros, Director General, WHO [31].

Appendix

Analysis of the System without the Drug

A1 = − a11 + a22 + a33 + a44 + a55ð Þ,

A2 = a11 a22 + a33ð Þ + a23a32 + a22a33 − a14a41

+ a11 + a22 + a33ð Þa44 − a25a52 + a11 + a22 + a33 + a44ð Þa55,

A3 = a32 a11a23 − a13a21 − a24a43ð Þ − a11a22 a44 + a33ð Þ

+ a14a33a41 + a44 a23a32 − a11a33 − a22a33ð Þ

+ a25a52 a33 + a44 + a11ð Þ − a11a55 a22 + a33ð Þ

+ a23a32a55 − a22a33a55 − a44a55

� a11 + a14a41a55 + a22 + a33ð Þ + a14a22a41,

A4 = a32a41 a23a14 − a13a24ð Þ − a22a33 a14a41 − a44a55ð Þ

− a14a21a32a43 + a13a21a32a44 − a11 a23a32a44 + a25a33a52ð Þ

+ a52 a14a25a41 − a11a25a44 − a25a33a44ð Þ + a13a21a32a55

− a11a55 a23a32 − a22a33ð Þ − a41a55 a14a22 + a14a33ð Þ

+ a55 a24a32a43 + a11a22a44 − a23a32a44ð Þ

+ a11a33 a22a44 + a44a55ð Þ + a11a24a32a43,

A5 = a25a33 a11a44a52 − a14a41a52ð Þ + a14a21a32 − a11a24a32ð Þa43a55+ a41a55 a13a24a32 − a14a23a32 + a14a22a33ð Þ

+ a44a55 a11a23a32 − a13a21a32 − a11a22a33ð Þ:

ðA:1Þ

B1 = − b11 + b22 + b33 + b44ð Þ,

B2 = b11b22 − b23b32 + b33 b11 + b22ð Þ − b14b41 + b44 b11 + b22 + b33ð Þ,

B3 = b32 b11b23 − b13b21ð Þ − b22 b11b33 − b14b41ð Þ

+ b14b33b41 − b24b32b43 + b23b32 − b11b22ð Þb44 − b33b44 b11 + b22ð Þ,

B4 = b14b41 b23b32 − b22b33ð Þ − b13b24b32b41 − b14b21b32b43

+ b32 b11b24b43 + b13b21b44ð Þ − b44b11 b23b32 − b22b33ð Þ:

ðA:2Þ

Data Availability

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

Conflicts of Interest

The authors declare that there is no conflict of interest.

Authors’ Contributions

Both authors contributed equally to this work.

References

[1] P. Zhou, X. L. Yang, X. G.Wang et al., “A pneumonia outbreakassociated with a new coronavirus of probable bat origin,”Nature, vol. 579, no. 7798, pp. 270–273, 2020.

[2] P. R. Hsueh, L. M. Huang, P. J. Chen, C. L. Kao, and P. C. Yang,“Chronological evolution of IgM, IgA, IgG and neutralisationantibodies after infection with SARS-associated coronavirus,”Clinical Microbiology and Infection, vol. 10, no. 12, pp. 1062–1066, 2004.

[3] N. Zhu, D. Zhang, W. Wang et al., “A novel coronavirus frompatients with pneumonia in China, 2019,” The New EnglandJournal of Medicine, vol. 382, no. 8, pp. 727–733, 2020.

[4] World Health Organization, “WHO Director-General's open-ing remarks at the media briefing on COVID-19-11 March2020,” Switzerland, Geneva, 2020.

[5] L. Zou, F. Ruan, M. Huang et al., “SARS-CoV-2 viral load inupper respiratory specimens of infected patients,” NewEngland Journal of Medicine, vol. 382, no. 12, pp. 1177–1179,2020.

[6] World Health Organization, “Considerations for quarantine ofindividuals in the context of containment for coronavirus dis-ease (COVID-19): interim guidance,” 2020, 29 February 2020.

[7] Y. Wan, J. Shang, R. Graham, R. S. Baric, and F. Li, “Receptorrecognition by the novel coronavirus fromWuhan: an analysisbased on decade-long structural studies of SARS coronavirus,”Journal of Virology, vol. 94, no. 7, 2020.

[8] H.-L. J. Oh, S. K.-E. Gan, A. Bertoletti, and Y.-J. Tan, “Under-standing the T cell immune response in SARS coronavirusinfection,” Emerging Microbes & Infections, vol. 1, no. 1,pp. 1–6, 2012.

[9] G. Li and E. De Clercq, “Therapeutic options for the 2019novel coronavirus (2019-nCoV),” Nature Reviews Drug Dis-covery, vol. 19, no. 3, pp. 149-150, 2020.

10 Computational and Mathematical Methods in Medicine

[10] H. A. Rothan and S. N. Byrareddy, “The epidemiology andpathogenesis of coronavirus disease (COVID-19) outbreak,”Journal of Autoimmunity, vol. 109, 2020.

[11] M. Zheng, Y. Gao, G. Wang et al., “Functional exhaustion ofantiviral lymphocytes in COVID-19 patients,” Cellular &Molecular Immunology, vol. 17, no. 5, pp. 533–535, 2020.

[12] S. Qureshi and A. Yusuf, “Fractional derivatives applied toMSEIR problems: comparative study with real world data,”The European Physical Journal Plus, vol. 134, no. 4, p. 171,2019.

[13] S. Qureshi and A. Yusuf, “Transmission dynamics of varicellazoster virus modeled by classical and novel fractional opera-tors using real statistical data,” Physica A: Statistical Mechanicsand its Applications, vol. 534, p. 122149, 2019.

[14] S. Tang, W. Ma, and P. Bai, “A novel dynamic model describ-ing the spread of the MERS-CoV and the expression of dipep-tidyl peptidase 4,” Computational and mathematical methodsin medicine, vol. 2017, Article ID 5285810, 6 pages, 2017.

[15] M. A. Nowak and R. M. May, Virus Dynamics: MathematicsPrinciples of Immunology and Virology, Oxford UniversityPress, London, UK, 2000.

[16] A. S. Perelson and P. W. Nelson, “Mathematical analysis ofHIV-1 dynamics in vivo,” SIAM Review, vol. 41, no. 1, pp. 3–44, 1999.

[17] A. N. Chatterjee, S. Saha, and P. K. Roy, “Human immunode-ficiency virus/acquired immune deficiency syndrome: usingdrug from mathematical perceptive,” World Journal of Virol-ogy, vol. 4, no. 4, pp. 356–364, 2015.

[18] P. K. Roy, A. N. Chatterjee, D. Greenhalgh, and Q. J. A. Khan,“Long term dynamics in a mathematical model of HIV-1infection with delay in different variants of the basic drug ther-apy model,” Nonlinear Analysis: Real World Applications,vol. 14, no. 3, pp. 1621–1633, 2013.

[19] F. Puggioni, M. Alves-Correia, M. F. Mohamed et al., “Immu-nostimulants in respiratory diseases: focus on pidotimod,”Multidisciplinary Respiratory Medicine, vol. 14, no. 1, pp. 1–10, 2019.

[20] R. E. Miron and R. J. Smith, “Modelling imperfect adherenceto HIV induction therapy,” BMC Infectious Diseases, vol. 10,no. 1, p. 6, 2010.

[21] J. Lou and R. J. Smith, “Modelling the effects of adherence tothe HIV fusion inhibitor enfuvirtide,” Journal of TheoreticalBiology, vol. 268, no. 1, pp. 1–13, 2011.

[22] R. J. Smith and B. D. Aggarwala, “Can the viral reservoir oflatently infected CD4(+) T cells be eradicated with antiretrovi-ral HIV drugs?,” Journal of Mathematical Biology, vol. 59,no. 5, pp. 697–715, 2009.

[23] R. J. Smith, “Explicitly accounting for antiretroviral druguptake in theoretical HIV models predicts long-term failureof protease-only therapy,” Journal of Theoretical Biology,vol. 251, no. 2, pp. 227–237, 2008.

[24] R. J. Smith and L. M. Wahl, “Drug resistance in an immuno-logical model of HIV-1 infection with impulsive drug effects,”Bulletin of Mathematical Biology, vol. 67, no. 4, pp. 783–813,2005.

[25] J. LOU, L. CHEN, and T. RUGGERI, “An impulsive differen-tial model on post exposure prophylaxis to HIV-1 exposedindividual,” Journal of Biological Systems, vol. 17, no. 4,pp. 659–683, 2011.

[26] P. K. Roy, A. N. Chatterjee, and X. Z. Li, “The effect of vaccina-tion to dendritic cell and immune cell interaction in HIV dis-

ease progression,” International Journal of Biomathematics,vol. 9, no. 1, 2015.

[27] E. Qing and T. Gallagher, “SARS coronavirus redux,” Trendsin Immunology, vol. 41, no. 4, pp. 271–273, 2020.

[28] H. Yu, S. Zhong, and R. P. Agarwal, “Mathematics analysis andchaos in an ecological model with an impulsive control strat-egy,” Communications in Nonlinear Science and NumericalSimulation, vol. 16, no. 2, pp. 776–786, 2011.

[29] V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, “The-ory of impulsive differential equations,” in Series in ModernAppliedMathematics, vol. 6, World Scientific Publishing, 1989.

[30] C. Wu, Y. Liu, Y. Yang et al., “Analysis of therapeutic targetsfor SARS-CoV-2 and discovery of potential drugs by computa-tional methods,” Acta Pharmaceutica Sinica B, vol. 10, no. 5,pp. 766–788, 2020.

[31] World Health Organization, “Laboratory testing for coronavi-rus disease 2019 (COVID-19) in suspected human cases:interim guidance,” World Health Organization, 2020.

[32] H. Xu, L. Zhong, J. Deng et al., “High expression of ACE2receptor of 2019-nCoV on the epithelial cells of oral mucosa,”International Journal of Oral Science, vol. 12, no. 1, p. 8, 2020.

[33] J. Chen, C. Hu, L. Chen et al., “Clinical study of mesenchymalstem cell treatment for acute respiratory distress syndromeinduced by epidemic influenza A (H7N9) infection: a hintfor COVID-19 treatment,” Engineering, 2020.

11Computational and Mathematical Methods in Medicine