Mathematical modelling of COVID-19 disease dynamics: Interaction between immune system and SARS-CoV-2 within host

Abstract

SARS-COV-2 (Coronavirus) viral growth kinetics within-host become a key fact to understand the COVID-19 disease progression and disease severity since the year 2020. Quantitative analysis of the viral dynamics has not yet been able to provide sufficient information on the disease severity in the host. The SARS-CoV-2 dynamics are therefore important to study in the context of immune surveillance by developing a mathematical model. This paper aims to develop such a mathematical model to analyse the interaction between the immune system and SARS-CoV-2 within the host. The model is developed to explore the viral load dynamics within the host by considering the role of natural killer cells and T-cell. Through analytical simplifications, the model is found well-posed and asymptotically stable at disease-free equilibrium. The numerical results demonstrate that the influx of external natural killer (NK) cells alone or integrating with anti-viral therapy plays a vital role in suppressing the SARS-CoV-2 growth within-host. Also, within the host, the virus can not grow if the virus replication rate is below a threshold limit. The developed model will contribute to understanding the disease dynamics and help to establish various potential treatment strategies against COVID-19.

Full text

http://www.aimspress.com/journal/Math
AIMS Mathematics, 7(2): 2618–2633.
DOI: 10.3934/math.2022147
Received: 23 September 2021
Accepted: 05 November 2021
Published: 17 November 2021
Research article
Mathematical modelling of COVID-19 disease dynamics: Interaction
between immune system and SARS-CoV-2 within host
S. M. E. K. Chowdhury1, J. T. Chowdhury1, Shams Forruque Ahmed2,∗, Praveen Agarwal3,4,5,
Irfan Anjum Badruddin6and Sarfaraz Kamangar6
1Department of Mathematics, University of Chittagong, Chattogram 4331, Bangladesh
2Science and Math Program, Asian University for Women, Chattogram 4000, Bangladesh
3Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India
4Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman AE 346, United Arab
Emirates
5International Center for Basic and Applied Sciences, Jaipur 302029, India
6Mechanical Engineering Department, College of Engineering, King Khalid University, Abha
61421, Saudi Arabia
*Correspondence: Email: shams.ahmed@auw.edu.bd, shams.f.ahmed@gmail.com.
Abstract: SARS-COV-2 (Coronavirus) viral growth kinetics within-host become a key fact to
understand the COVID-19 disease progression and disease severity since the year 2020. Quantitative
analysis of the viral dynamics has not yet been able to provide sufficient information on the disease
severity in the host. The SARS-CoV-2 dynamics are therefore important to study in the context
of immune surveillance by developing a mathematical model. This paper aims to develop such a
mathematical model to analyse the interaction between the immune system and SARS-CoV-2 within
the host. The model is developed to explore the viral load dynamics within the host by considering
the role of natural killer cells and T-cell. Through analytical simplifications, the model is found well-
posed and asymptotically stable at disease-free equilibrium. The numerical results demonstrate that the
influx of external natural killer (NK) cells alone or integrating with anti-viral therapy plays a vital role
in suppressing the SARS-CoV-2 growth within-host. Also, within the host, the virus can not grow if the
virus replication rate is below a threshold limit. The developed model will contribute to understanding
the disease dynamics and help to establish various potential treatment strategies against COVID-19.
Keywords: MERS-CoV; SARS-CoV-2; COVID-19; mathematical model; immune system; basic
reproduction number
Mathematics Subject Classification: 34A12, 34A34, 37C75

2619
1. Introduction
Human civilization has been facing different infectious pathogen-caused epidemics for a long time.
Severe Acute Respiratory Syndrome (SARS-CoV), a coronavirus class, was first identified in 2003 in
Asia, with 8422 cases and an 11% fatality rate [1]. Another class of coronavirus, Middle East
Respiratory Syndrome (MERS-CoV) was reported with approximately 1572 human cases in 2012 in
Arabian Peninsula [2]. SARS-CoV-2, the new class of novel coronavirus has now become a
serious global threat due to its first advent in Wuhan, Hubei province, China in the year 2019 [3]. In
spite of taking different social interventions and pharmaceutical measures by different countries,
SARS-CoV-2 spreads over the globe within a very short time and tossed the biggest challenge ever to
global health [4]. Every morning the world gets introduced with record-breaking cases in both
infection and death caused by SARS-CoV-2. To pull down the race of the SARS-CoV-2 pandemic,
researchers are attempting their best to discover strategies to explore the virus-protein structure of the
SARS-CoV-2 [5], phylodynamics of the SARS-CoV-2 [6], and build up the antibody against
SARS-CoV-2 infection by developing vaccines [7].
Mathematical models are often useful in characterizing the infection dynamics and forecast disease
severity. Many pathogens were studied using a simple model including target cell, infected cell, and a
virus. The target cell limited model was used by many researchers to study the in-host dynamics of
HIV [8–10], Hepatitis [11], Ebola [12, 13], and Influenza viruses [14, 15]. Many mathematical
models [16–19] focused on the transmission dynamics of the SARS-CoV-2 virus and involved
different pharmacological, non-pharmacological interventions to reduce the spread of the COVID-19
disease. However, only a few models explained the replication cycle of SARS-CoV-2 within the host
and interactions between this virus and the host immune system [20–22]. A brief study [22] on
SARS-CoV-2 virus dynamics within-host discussed the target cell model with an eclipse phase, and
secondary infection in presence of lymphocytes. The study found a rapid rise in viral load to reach the
peak at the initial stage pointing to a plateau phase caused by lymphocytes. Also, an adaptive immune
response results in decreasing the viral load at the last stage. Hernandez-Vargas et al. [21] investigated
the interaction between the SARS-CoV-2 virus and the immune system within-host. Though the
authors discussed a couple of models explaining the SARS-CoV-2 dynamic within the host, the
immune cell (T-cell) response model performed better than others to fit the data indicating a slow
immuno-response reaching the peak in 5–10 days after symptom onset.
Some researchers [23–27] studied the COVID-19 disease severity and mortality based on the data
collected from the categorized patients. The viral load of SARS-CoV-2 in a group of patients with
divergent SARS-CoV-2 serotypes was measured in the study of Fajnzylber et al. [23]. The authors
stated that the spectrum of SARS-CoV-2 viremia is linked to inflammation markers and disease
severity, such as elevated C-reactive protein (CRP), IL-6, and a low lymphocyte count. Through a
virological study, Wolfel et al. [26] found sequence-distinct virus replication in the throat, lung tissue,
and upper respiratory. The pulmonary cell affinity with the COVID-19 disease intensity was identified
by computer tomography (CT-scan) and radio-graph of the patients’ chest [25]. Later, Li et al. [24]
used chest radiograph scores to determine the lung cell destruction due to the SARS-CoV-2 virus
infection. They proposed a model describing the kinetics of the interactions between the virus
particles and the lung epithelial cells within-host. Using the chest radiograph data, the authors
determined the model parameters and found the basic reproduction number around 3.79 for the
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SARS-CoV-2 growth within the host.
Against the SARS-CoV-2 virus infection, no therapeutic and prophylactic medications are yet
approved. Patients with acute symptoms are administered oxygen therapy, ventilation, antibiotics, and
corticosteroids to support the respiratory functions and to control inflammatory reactions. These
instructions allow time for activation of the adaptive immune response [28, 29]. Different possible
medical schemes include the use of antiviral drugs inhibiting the replications of the virus cycle.
Repurposing existing anti-viral drugs can be an effective drug strategy against the SARS-CoV-2 that
could reduce the time and limit the cost compared to the de novo drug discovery. In a study of drug
re-purpose designed for SARS-CoV-2, Zhou et al. [30] proposed sixteen repurposable drugs based on
network proximity analyses (NPA) of drug targets within the human host. The authors also proposed
three prospective drug combinations such as mercaptopurine and melatonin, sirolimus and
dactinomycin, toremifene and emodin, against the SARS-CoV-2 virus infection.
In any infections within the host, natural killer (NK) cells serve as first-line protection. NK cell
and T-cell are necessary for a strong confrontation and clearance of the virus from the host. Among
various anti-viral activities, these cells are accountable for directly killing the target cells and the
release of immunomodulatory cytokines [31]. After conducting a thorough literature review, it is
found that only a few research [21, 22, 24] have been carried out reporting the SARS-CoV-2 viral
growth in the host under the impact of T-cells. However, no studies have been found which analyses
the interaction between the immune system and SARS-CoV-2 within the host considering T -cells and
NK-cells together.
A deterministic mathematical model based on physiologically plausible assumptions performs
better in forecasting disease severity over time in the absence of relevant experimental data. A
deterministic model reflecting the short-term expected antiviral effect during SARS-CoV-2
development within the host is required in addition to forecasting illness severity. As a result,
developing a mathematical model under the effect of both T -cells and NK-cells becomes essential for
qualitative analysis of the viral load within the host. To comprehend the SARS-CoV-2 growth
dynamics within the host under immune surveillance, a mathematical model is therefore developed in
the present study by analysing the interaction between the immune system and SARS-CoV-2 within
the host integrating the function of NK cells and T-cells. In essence, the model looks into ways to
control the pace of SARS-CoV-2 viral replication within a host, because a quicker viral replication
rate inside the host is directly related to community transmission. This research will help physicians,
scientists, and policymakers to understand how patients’ immunity interact against COVID-19
disease, as well as to devise disease treatment strategies.
2. Model formulation
The mathematical framework of the proposed model is based on the interaction between virus
particles and different lymphocytes within-host including the most common lymphocytes, NK cells,
and T-cells. T-cells are in control of both adaptive and innate immune responses while NK cells,
which are part of the innate immune system, are the first immune effectors against virus infection
before any specific immunity arrives. Furthermore, in the absence of specific antibodies, the NK cell
recognizes and kills the distressed cell [32].
The target cell of the SARS-CoV-2 virus in the respiratory system is not completely known. Though
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there is a dispute in current literature about the primary compartments of SARS-CoV-2 infection, it is
usually believed that the primary infection succeeds in respiratory epithelial cells [33]. Based on the
results came out from the studies [33–35], epithelial cell in the respiratory system has been considered
in the present model as a primary target of the SARS-CoV-2 virus. The infected epithelial cells are
assumed to be capable of producing new virus particles at a constant rate. Taking the interactions of
the virus-immune system and virus-epithelial cells, the proposed model is developed as follows:
dE
dt =a1−a2E−a3EV,
dEi
dt =a3EV −b1Ei,
dV
dt =c1Ei−c2V−c3V N −c4V L,
dN
dt =d1−d2N,
dL
dt =e1LV
(n+V)−e2L+e3,
(2.1)
where E,Ei,V,N,Lrepresent the number of susceptible epithelial cells, infected epithelial cells, viral
load, natural killer cells, and T-lymphocytes respectively; a3is the infection rate of the virus; c1is
the production rate; and the terms a2,b1,c2,d2, and e2, are the natural death rates corresponding to
E,Ei,V,N,and Lrespectively.
The regeneration of the epithelial cells is considered as a1=a2×E(0) [21], where a2is the natural
death rate of the epithelial cells and E(0) represents the number of initial epithelial cells. The term
a3EV in the first equation denotes virus particles infecting a healthy epithelial cell at an infection rate
of a3. The growth of infected epithelial cells is denoted by a3EV in the second equation. The first
term c1Eiin the third equation implies the reproduction of new SARS-CoV-2 virus from an infected
cell at a rate c1. Also, c3VN and c4V L represent the local interaction dynamics of the virus (V) with
natural killer cells (N) and T- lymphocytes (L). Due to these interactions, virus particles are reduced
at the rates c3and c4respectively. Similarly, d1implies the constant external source of the natural killer
cells as expressed in the fourth equation of the model. The term e3is the natural recruitment rate of
T-lymphocytes and e1LV/(n+V) is the proliferation in response to the presence of virus particles. This
functional proliferation form is employed here as the model assumes the recruitment of T-lymphocytes
due to the signals to release different cytokines (prompting molecule that regulates and mediates the
immune system). The half-saturation constant nis theT-lymphocytes proliferation by a virus which is
estimated as 1.26 ×105for a sigmoidal function to the best data fit [21].
The model is considered under the initial conditions E(t)≥0,Ei(t)≥0,V(t)≥0,N(t)≥0 and
L(t)≥0 with non-negative value of the model parameters. It is obvious that the right-hand parts of
the system of Eq (2.1) are continuous and satisfy the Lipschitz condition [36]. The uniqueness and
existence criterion of the ordinary differential equation confirms that model (2.1) has a unique solution
with the non-negative initial conditions [36]. The first equation of the model (2.1), is considered to
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explore the non-negativity property of the solution curves as follows:
dE
dt +a2E≥0.(2.2)
The solution trajectory of inequality (2.2) is obtained as
E(t)≥E(0)e−a2t.(2.3)
The inequality (2.3) gives E(t)≥0 for t→ ∞ which indicates that for every t∈[0,∞) the solution
trajectory E(t) will be entirely in the positive orthant for the given non-negative initial condition and
parameter values. In a similar approach, it is easy to show Ei(t)≥0,V(t)≥0,N(t)≥0,L(t)≥0 for
t→ ∞.
The first equation of the model (2.1) also reveals that
E(t)≤a1
a2
+ E(0)−a1
a2!e−a2t,(2.4)
i.e., for long term behaviour t→ ∞,E(t) is bounded by a1
a2= Π1(say)∈R.
Moreover, the solution trajectories of Ei(t),V(t), and L(t) of the system (2.1) give
Ei(t)=p
b1
+ Ei(0) −p(0)
b1!e−b1t;p=a3EV,
V(t)=q1
q2
+ Ei(0)−q1(0)
q2(0) !e−q2t;q1=c1Ei,q2=c2+c3N+c4L,
L(t)=r1
r2
+ L(0)−r1(0)
r2!e−r2t;r1=−e1V
(n+V)+e2;r2=e3,
(2.5)
where p(0),q1(0),q2(0),and r1(0) represent the initial values of the corresponding functions in terms of
state variables E,V,Nand L. Since Eand Vremain constant in the second equation of the model (2.1),
the supremum of Ei(t) is given by
lim
t→ ∞ sup Ei(t)=p
b1
≤Π2(say)∈R.
Defining a function F(t)=E(t)+Ei(t)+V(t)+N(t)+L(t) and its derivative along the trajectories
yields,
dF
dt =dE
dt +dEi
dt +dV
dt +dN
dt +dL
dt
=β−a2E−b1Ei+c1Ei−c2V−d2N−e2L−c3VN −c4V L +µ
≤(−a2E−b1Ei−c2V−d2N−e2L)+β+µ+c1Π2
≤ −hF +H,
(2.6)
where H=β+µ+c1Π2,h=min {a2,b1,c2,d2,e2},β=(a1+d1+e3), and µ=e1LV/(n+V).
Inequality (2.6) reveals that F(t) is bounded by H/h, whereas E(t) and Ei(t) have bounds Π1and
Π2respectively. Therefore, N(t),V(t) and L(t) are eventually bounded with an other bound Π∈R.
Denoting M=max {Π,Π1,Π2}, for sufficiently large time scale t, it reveals that E(t)≤M,Ei(t)≤M,
V(t)≤M,N(t)≤Mand L(t)≤M.
Therefore, under the non-negative initial conditions with the positive parameter values, the solution
space of the system (2.1) gives
Ω = (E,Ei,V,N,L)∈R5; 0 ≤E(t),Ei(t),V(t),N(t),L(t)≤M.
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3. Model analysis
Evaluation of the basic reproduction number R0at equilibrium points is the simplest way to
investigate the disease dynamics described by a mathematical model. Setting the right-hand side of
the model equations to zero, the first equilibrium point η(E0,Ei
0,V0,N0,L0) in absence of the
SARS-CoV-2 virus of the model (2.1) is obtained as
(E0,Ei
0,V0,N0,L0)=(a1
a2
,0,0,d1
d2
,e3
e2
).(3.1)
Further, at endemic equilibrium, none of the state variables is identically zero in presence of the
virus within-host. In that case, the second equilibrium point η∗(E∗
0,Ei∗
0,V∗
0,N∗
0,L∗
0) for model (2.1) is
given by:
E∗=a1
a2+a3V∗
Ei∗=a1a3V∗
(a2+a3V∗)b1
V∗=c1Ei∗
c2+c3N∗+c4L∗
N∗=d1
d2
L∗=e3(n+V∗)
e2(n+V∗)−e1V∗
(3.2)
The infection ability of a pathogen within-host is quantified by determining R0for the corresponding
model [37]. This number is employed to evaluate the average secondary infections resulting from an
infected cell. Mathematically, R0<1 implies that an infected cell (pathogen) might infect less than one
cell, resulting in an invasion of the disease from the host. In contrast, R0>1 suggests a suppression of
the host cells resulting in the progress of the disease within-host.
The concept of the next-generation matrix (NG M) [38] is used to determine the basic reproduction
number R0. The system (2.1) has three infected states E,Eiand V, and two uninfected states, Nand L.
Linearising the Eqs (1)–(3) of the system (2.1) as a subsystem of infection, the following transmission
matrix (T) and transition matrix (Q) are obtained:
T=















0 0 −a3a1
a2
0 0 a3a1
a2
0 0 0
















and Q=









−a20 0
0−b10
0c1−α










.
where α=c2+c3d1
d2
+c4e3
e2
.
Since the first two columns of the transmission matrix (T) are zero, the next generation matrix [38]
will be K=ETT Q−1Ewhere the axillary matrix Eis given by:
E=









1 0
0 1
0 0










.
Using the next-generation matrix K, the R0of the model (2.1) can be written as
R0=a1a3c1
a2b1α.(3.3)
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3.1. Steady state analysis
The stability of model (2.1) at the virus-free equilibrium point η, is determined by the following
lemma:
Lemma 1. The disease-free equilibrium of model (2.1) is locally asymptotically stable for R0<1 and
unstable for R0>1.
Proof. The Jacobian matrix of the model at disease-free equilibrium point ηcan be written as
JE0=






























−a20−a1a3
a2
0 0
0−b1
a1a3
a2
0 0
0c1−α0 0
0 0 0−d20
0 0 e1e3
ne2
0−e2






























.
The characteristic equation of this Jacobean matrix is given by
(−λ−a2) (−λ−d2) (−λ−e2)λ2+k1λ+k2=0,(3.4)
where k1=(b1+α)>0 and k2=b1α−(a1a3c1)/(a2).
It is clear that the first three eigen-values obtained from Eq (3.4) are negative as a2>0, d2>0 and
e2>0. As per the Descarte’s sign rule [39], the rest of the eigen-values are negative for k1>0 and
k2>0. Rewriting the condition k2>0 in R0, it gives k2=1−R0>0, that means, k2>0 if only if
R0<1. Thus all of the eigenvalues are negative. On the contrary, k2becomes negative if R0>1 which
provides at least one of the eigenvalues is positive. Hence, Lemma 1 follows.
The analytical computation of the characteristic equation at endemic equilibrium point η∗produced
a large expression and therefore, the determination of the sign of eigenvalues at η∗becomes very
complicated. Rather, the qualitative behaviour of the solution trajectories at η∗was investigated through
a series of numerical explorations.
4. Numerical simulation and discussion
A series of numerical simulations were performed using Python (V. 3.7) SciPy.integrate package
and odeint library to solve the ordinary non-linear differential equations of the present model. The
model equations comprised 14 parameters and assumed 5 initial conditions. The parameter values were
estimated based on some homeostatic process, and available statistical data of SARS-CoV-2 infected
patients [21, 24]. Considering the physical significance of the parameters, numerical investigations
corresponding to the effects of external NK cell influx, anti-viral therapy, and their combined effect over
the SARS-CoV-2 growth projection have been analyzed under immune surveillance in section 4.1–4.3
respectively.
The initial values for healthy and infected epithelial cells, SARS-CoV-2 virus, and T-cell are based
on COVID-19 patient data analysis [21, 24], which may provide realistic results for numerical
simulations. The model assumes the initial conditions as follows: Ei(0) =2.59 (score) [24],
E(0) =25 −Ei(0) =22.41 [24]), V(0) =0.061 (score) [24], L(0) =106 [21], N(0) =105 [40]. The
virus clearance rate c2was fixed and characterized as a process that is not linked with the immune
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system. The parameter a1was estimated as [24]: a1=a2×E(0) =0.02241, and the other simulation
parameters are tabulated in Table 1. It is noted that the parameter values for epithelial cells and
SARS-CoV-2 were chosen from published literature based on chest radiograph score which is an
approximation, not actual infected data for pulmonary cells [24].
Table 1. Model parameters used in the simulations.
Parameters Description Value Reference
a1Constant regeneration of E.C 0.0224 (estimated) Li et al. [24]
a2Death rate of E.C (day−1) 10−3
a3Infection rate of E.C by virus 0.55
(day−1score−1))
b1Death rate of infected E.C.(day−1) 0.11
c1Production rate of virus from per 0.24
infected E.C.(day−1)
c2Death rate of virus (day−1) 5.36
c4Killing rate of virus by T-lymp. (day−1) 1.89 ×10−6Hernandez et al. [21]
e1Proliferation rate of T-lymp. 0.194
e2Death rate of T-lymp. (day−1) 0.1
e3Constant regeneration of T-lymp. 105
nHalf saturation constant of T-lymp. 1.26 ×105
c3Killing rate of virus by N.K (day−1) 1.5×10−6(assumed) Present study
d1External influx of N.K. cell (day−1cells) 3.2×103Kuznetsov et al. [41]
d2Death rate of N.K. cell (day−1cells) 4.12 ×10−2de Pillis et al. [42]
4.1. External influx of natural killer (NK) cell
The impact of NK cell influx from an external source on the disease progression is assessed to
quantify the viral dynamics within-host. The immunity system works to clear the virus from the day
of symptom onset. However, the immunity strength depends on different factors including age and
co-morbidness of the patients. Figure 1 shows the effect of external NK cell influx over viral load.
As seen in the figure, the viral load increase very fast at the beginning of the infection due to slow
immune response and reaches the peak of 0.35 (chest score) on day 10 after symptom onset where
the viral load is expressed as a chest radiograph score. A small increase in the external influx of NK
cell as d1=d1+j∗1000 (0 ≤j≤5) results in a reduction of viral load about 0.3 to 1.5% in
each loop iteration. The maximum viral load (chest score) for the corresponding loop iterations are
v=0.3501,v=0.3492,v=0.3475,v=0.3450,and v=0.3417 respectively. Because the viral
loads corresponding to d=3200,d=4200,and d=6200 are so similar, Figure 1 contains plots
corresponding to d=3200, d=9200, and d=13200, which illustrate the intensity of viral load relative to
the change in external NK cell influx.
Figure 1(d) shows the basic reproduction number R0which gradually decreases subject to the
external influx of the NK cell. The large influx of NK cells strengthens the immune system, helps to
restrict viral proliferation. Diminishing R0from 3.65 to 3.33 (approx.) implies the invasion of viral
replication within the host during the disease. A simulation was also run with higher initial values of
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the SARS-CoV-2 virus to investigate the initial viral load over the disease progression. A significant
rise was observed in the chest radiograph score (viral load) which indicates a symptom severity in
patients.
Figure 1. Viral load (V) and reproduction number (R0) relative to NK cell influx (d1).
4.2. Anti-viral therapy
The role of possible anti-viral therapy can reduce the SARS-CoV-2 virus replication rate. A high
replication rate c1of the SARS-CoV-2 implies a rapid invasion of the susceptible epithelial lung
cells (E). The SARS-CoV-2 replication rate on viral load dynamics is portrayed in Figure 2. The
maximum viral load of 0.35 (score) occurs in Figure 2(a), while the minimum viral load of 0.003
(score) ensures in Figure 2(e). For each loop decrement (c1=c1−j∗0.02) in the value of c1, viral
load was decreased by approximately 14 to 31%. For a lower replication rate, SARS-CoV-2 growth
dynamics are found less steep than for a higher replication rate (Figure 2), which indicates that an
anti-viral therapy can more effectively combat the virus. In addition, the SARS-CoV-2 virus cannot
grow within-host under immune surveillance if the replication rate is less than or equal to 0.04 as seen
in Figure 2(e). The threshold value of c1was found 0.04 for the defined loop iterations. The actual
threshold value of c1for model (2.1) was determined as 0.09 implying that SARS-CoV-2 cannot grow
in the host with a replication rate less than 0.09.
Figure 2(f) depicts the graph of basic reproduction number R0relative to the SARS-CoV-2 virus
replication rate. The number R0shows a positive linear trend with virus replication rate c1. A higher
replication rate clearly contributes to an increased symptom and disease prevalence in patients,
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resulting in community disease transmission and lung cell destruction. This finding suggests that
prescribing appropriate antiviral drugs to a patient sooner may reduce the symptom.
Figure 2. Viral load (V) and reproduction number (R0) relative to SARS-CoV-2 replication
rate (c1).
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Figure 3. Viral load (V) and reproduction number (R0) relative to the combined effect of (c1)
and (d1).
4.3. Combination of anti-viral therapy and influx of external NK cell
Combinations of different therapeutic treatments have become popular in many viral disease
management protocols. For instance, monoclonal antibodies with other anti-viral medicine showed an
important role in Ebola virus infection [12]. In this section, the combined impact of anti-viral therapy
and the eternal influx of NK over the SARS-CoV-2 virus growth projection has been demonstrated.
The combined effect of possible anti-viral drugs and influx of external NK cells is shown in Figure 3.
The influx of NK cells strengthens the patient’s immune system, while antiviral therapy reduces virus
replication. It is found that administering this combined therapy results in a quick reduction of viral
load from the initial peak. Such a substantial reduction in viral load causes symptom relief and
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disease invasion in the patient.
The basic reproduction number plot Figure 3(f) shows a sharp fall due to the combined effect of
NK cell influx and anti-viral drug. This reflection implies that the combined effect is more effective in
suppressing the viral load within the host under immune surveillance.
5. Sensitivity analysis
The sensitivity of a specific parameter identifies how a small change in numeric value causes a large
impact on the model output. This analysis may assist to establish the possible treatment protocol for
the SARS-CoV-2 disease. The sensitivity indices of the basic reproduction number R0relative to the
parameters is defined as [12]:
Θa1=r
R0
∂Ro
∂r,
where ris the model parameter. The estimated sensitivity indices of the model (2.1) are summarized
in Table 2.
Table 2. Sensitivity of R0with respect to the parameters.
Sensitivity index Index value
Θa11
Θa31
Θc11
Θd20.01581549
Θe20.25656672
Θa2−1
Θb1−1
Θc2−0.72761779
Θc3−0.01581549
Θd1−0.01581549
Θc4−0.25656672
Θe3−0.25656672
From the sensitivity indices, it is obvious that a1,a3,c1,d2, and e2have a positive impact on R0
while the rest of the parameters shows a negative impact. The negative impact implies the invasion of
viral load whereas the positive impact indicates a progression of virus load within-host. The indices
Θa1,Θa3, and Θc1are being equal to 1 (most sensitive) reveals that a 10% increase in the corresponding
parameter values causes a 10% growth projection in R0. Similarly, a2and b1show reverse correlation
to R0, i.e., 10% increase in a2and b1(least sensitive) results in decreasing the value of R0by 10%.
These results fairly agree with the numerical results (Figures 1 and 2). The constant regeneration a1
shows positive sensitivity to R0as in Table 2. It means that a host with higher cells of pulmonary
epithelial is more likely to become infected, and provides clarification why especially babies having
lower epithelial cells are not most expected to get infected.
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6. Conclusions
A mathematical model was developed to explore the interaction between the immune system and
SARS-CoV-2 within the host by integrating the function of NK cells and T-cells. A series of
simulations were performed to investigate the qualitative behaviour of the model. The numerical
results complied with the analytical results of the model where the solution trajectories were found
bounded in the positive quadrant. A sensitivity analysis was also carried out for basic reproduction
number R0to analyse the impact of the parameters on the disease progression. The model is found to
be advantageous in assessing the antiviral medication efficacy and NK cell influx into the host body,
and with the value of R0across the time of infection period, the model can be used in forecasting
disease severity. The numerical analysis showed that the use of anti-viral therapy and influx of NK
cells causes a fall in the value of R0. In addition, SARS-CoV-2 cannot grow within-host if the virus
replication rate (c1) is under a threshold limit of 0.09. The combined use of anti-viral therapy and
influx of external NK cells together played an important role to control the SARS-CoV-2 viral growth
within-host. If the proposed anti-viral combinations are capable to control or reduce the SARS-CoV-2
replication rate, it may be a good attempt to use those from the very beginning of the disease. The
present model can be used further to investigate the role of antibody development and cross-reaction.
Furthermore, the present model can be extended to study the innate immune response to different
SARS-CoV-2 mutations. Each new variety must have its own replication rate, which has an impact on
the severity of the disease and its propagation in the population. This idea would help us to develop a
new mathematical model in the future study.
Acknowledgements
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid
University for funding this work through the research groups program under grant number
RGP.1/327/42.
Conflict of interest
The authors declare that they have no conflict of interest.
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