Clinical effects of 2-DG drug restraining SARS-CoV-2 infection: A fractional order optimal control study

Abstract

Fractional calculus is very convenient tool in modeling of an emergent infectious disease system comprising previous disease states, memory of disease patterns, profile of genetic variation etc. Significant complex behaviors of a disease system could be calibrated in a proficient manner through fractional order derivatives making the disease system more realistic than integer order model. In this study, a fractional order differential equation model is developed in micro level to gain perceptions regarding the effects of host immunological memory in dynamics of SARS-CoV-2 infection. Additionally, the possible optimal control of the infection with the help of an antiviral drug, viz. 2-DG, has been exemplified here. The fractional order optimal control would enable to employ the proper administration of the drug minimizing its systematic cost which will assist the health policy makers in generating better therapeutic measures against SARS-CoV-2 infection. Numerical simulations have advantages to visualize the dynamical effects of the immunological memory and optimal control inputs in the epidemic system.

Introduction

Infectious diseases have brought devastation to humanity since ancient times. From the last quarter of the year 2019 the world is enduring a pandemic due to SARS-CoV-2 or COVID-19 contagion, which is highly contagious than any other known infectious diseases. The first case of SARS-CoV-2 contagion was reported in Wuhan city in China. According to the Worldometer data, currently about 226 countries and territories have been in the grip of the COVID-19 pandemic situation [1]. World Health Organization (WHO) reported globally around 51.5 Crore confirmed cases including about 6.25 Million deaths due to COVID-19 infection as of May 2022 [2]. To combat the COVID-19 pandemic, vaccine is one of the most convenient equipment; 60% of total population have been administered with full vaccination as of May 2022. Severe acute respiratory syndrome coronavirus 2 or SARS-CoV-2, a progeny of the family Coronaviridae, contains a single-stranded positive sensed RNA genome inducing the outbreak of COVID-19 infection. This infection is also capable to affect other species like dromedary camels, civet cats, and bats. The vertical transmission of SARS-CoV-2 contagion happens through particles released during coughing and sneezing in the form of large respiratory droplets as well as smaller aerosols. The spreading of COVID-19 is also responsible for secondary infections in the crowded environment and poorly ventilated indoor environment through close contact. The SARS-CoV-2 virus substantially targets the lower part of human respiratory tract causing flu-like illness with symptoms such as cough, fever, fatigue, headache, difficulties in breathing, loss of smell and taste, and diarrhea.

SARS-CoV-2 utilizes the membrane protein, angiotensin-converting enzyme 2 (ACE2) receptor to transform the host epithelial cells into more vulnerable ones for its unhindered entry [3–5]. A few of the epithelial cells like myocardial epithelial cells, kidney tubular epithelial cells, and gastrointestinal epithelial cells obstruct the expression of ACE2 to enter in host. ACE2 is enunciated in myocardial cells, proximal tubule cells of the kidney, and bladder urothelial cells, and abundantly in enterocytes of the small intestine (particularly in the ileum). Type II alveolar epithelial cells of lungs contain the better copious expression of ACE2 and thus are taken into consideration as the major target cells of COVID-19 infection [6, 7]. To discern effective strategies in controlling the COVID-19 transmission, it is necessary to understand the intermediate functional relationships among the SARS-CoV-2 strain, host epithelial cells and host immune response. During the activation and differentiation of T cells, the adaptive immunity is set off in host coupled with innate immunity. After the successful entry of the SARS-CoV-2 in host epithelial cells, secretion of different cytokines particularly INF-$\gamma$, IL-6 and IL-10 takes place stimulating the host immune response [8]. Basically $CD4+T$ cells and $CD8+T$ cells impose uttermost influence to confront the SARS-CoV-2 by engendering the virus-specific antibodies along with the activation of B cells (T dependent) and neutralizing the host epithelial cells from spreading infection.

The SARS-CoV-2 Interagency Group (SIG) established by the U.S. Government categorized a new variant of SARS-CoV-2 constructed through mutation, namely Omicron, as a Variant of Concern (VOC) on November 30, 2021. Researches are in progress to detect drug or other approaches applicable to COVID-19 patients with the aim to mitigate the pandemic. A short while ago the Defence Research & Development Organisation (DRDO) under the Government of India provided the license to the Drug firm, Granules India for manufacturing and marketing the drug, 2-Deoxy-D-Glucose (2-DG) in treatment of COVID-19 infection. The clinical trial of this drug exhibited the facts that the molecule assisted the COVID-19 patients to recover quickly and the drug proved to be effective in the reduction to the urgency of supplementary oxygen. Additionally, 2-Deoxy-D-Glucose (2-DG) successfully prevented the growth of SARS-CoV-2 through the termination of viral synthesis and energy production.

Mathematical modeling of the underlying mechanism of the reciprocity between the SARS-CoV-2 and within-host immune system during COVID-19 infection is found to be quite beneficial. A handful of mathematical studies have been accomplished highlighting the dynamical aspects of the transmission of COVID-19 infection [9–16] at the population level. However in developing booster dose or in enhancing the effectiveness of currently available vaccines, proper insight regarding the intrahost viral dynamics and host immune response hindering the contamination of the COVID-19 infection is indispensable; although studies investigating these cellular facts have not been conducted on a large scale yet. Hernandez et al. [17] established a mathematical model examining the cellular kinetics along with T cell responses against the replication of SARS-CoV-2 in COVID-19 infection. Wang et al. [18] explored pathogenic characteristics, anti-inflammatory treatment strategies or combined antiviral drugs in mitigating SARS-CoV-2 infection. Different consequences of both humoral and adaptive immune responses in COVID-19 and possible eradication strategies have been analyzed in [19]. Paul et al. [20] proposed a four-dimensional SEIR model to verify the dynamics of COVID-19 infection in India and Brazil. In Chatterjee et al. [21, 22] conducted studies focusing the lytic and non-lytic role of immune response mutation of SARS-CoV-2 virions to control COVID-19 infection. Mondal et al. [4] constructed a mathematical model highlighting the dynamical behaviors of SARS-CoV-2 virions during COVID-19 infection and essential effects of host immune response to inhibit complicated epidemic states like backward bifurcation and reinfection. The effect of antiviral drug in controlling the COVID-19 and a variable ordered fractional network in host have been studied in [23].

In view of the interdependencies among the immune response of human host and viral kinetics of SARS-CoV-2 accompanying the influences of 2-DG to optimally control the SARS-CoV-2 infection, a mathematical study is delineated here. Sections 2 and 3 contain our proposed compartmental ODE model and its corresponding fractional order model respectively. In Sect. 4, fundamental properties of the fractional order model together with investigation for the possible equilibrium points and stability of the system around them are presented. Section 5 accounts for the optimal strategies to control the SARS-CoV-2 infection and possible eradication of the pandemic in the light of optimal drug influence. In Sect. 6, numerical simulations assist to visualize the dynamics of the epidemic system due to changes in model parameters and implementation of optimal strategies. Section 7 is devoted to discussion and conclusion about the upshots from the study.

Model synthesis

Taking into account the interrelationships between the target cells (epithelial cells), consequences of immune response in human host and the kinesis of SARS-CoV-2 during COVID-19, we are aspired to introduce a compartmental and target cell-limited model consisting of four populations, namely, $E_S$ representing the susceptible target cells of SARS-CoV-2 infection, $E_I$ indicating the infected epithelial cells capable of virions production, A presenting the immune response of human host (combination of innate and adaptive immunity), and V standing for the SARS-CoV-2 virions. Our four-dimensional ODE model which would be beneficial to analyze the dynamical aspects of the infection is given by:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d{E}_S}{\mathit{dt}}& ={\mathrm{\Pi }}_1-(1-{\epsilon }_1)\frac{\beta E_{S}VA}{1+kV}-{\mu }_1{E}_S,\hfill \\ \hfill \frac{d{E}_I}{\mathit{dt}}& =(1-{\epsilon }_1)\frac{\beta E_{S}VA}{1+kV}-{\mu }_2{E}_I,\hfill \\ \hfill \frac{\mathit{dA}}{\mathit{dt}}& ={\mathrm{\Pi }}_2-(1-{\epsilon }_1)\frac{\theta \beta E_{S}VA}{1+kV}-{\mu }_{3}A,\hfill \\ \hfill \frac{\mathit{dV}}{\mathit{dt}}& =(1-{\epsilon }_2)p{E}_{I}V-({\mu }_4+\frac{q{E}_I}{\eta +V})V,\hfill \end{array}\end{array}$$ 1

complemented with biologically realistic non-negative initial conditions (assuming the initial time as $t_0$)

$$\begin{array}{c}\hfill E_S(t_0)=E_{S0},E_I(t_0)=E_{I0},A(t_0)=A_0,V(t_0)=V_0.\end{array}$$ 2

Uninfected epithelial cells, that is, the target cells of SARS-CoV-2 infection are recruited in the system (1) at the constant rate ${\mathrm{\Pi }}_1$. Thereafter SARS-CoV-2 infects the target cells with the rate $\beta$ agitating the T cells to expand as well as differentiated and the expansion of T cells would be saturated by the function $(1+kV)$. The uninfected epithelial cells are naturally cleared at a rate ${\mu }_1$ owing to apoptosis. Infected cells are capable to persuade the replication of SARS-CoV-2 at the rate p and wipe out at the rate ${\mu }_2$ from the system due to cytolytic effects of immune response. Cytokine triggers particularly the adaptive immunity to produce virion-specific antibodies (like IgM and IgG antibodies) at the rate ${\mathrm{\Pi }}_2$ by neutralization. SARS-CoV-2 utilizes the JaK/STAT pathway in diminishing the production of antibodies [4]. The SARS-CoV-2 has the innate immunity suppressing attribute which is represented in the functional form as $\frac{\theta \beta E_{S}VA}{1+kV}$ resulting deficiency in cytokines production, where $\theta$ denotes the pathogenicity of SARS-CoV-2. At the rate ${\mu }_3$, the efficiency of antibody response fades away from the human host due to functional exhaustion of T cells. The destruction of the virions by host immune response is stated through functional response $\frac{q{E}_I}{\eta +V}$, where q stands for the immune destruction of the virions and at the rate ${\mu }_4$ the virions be washed off from the system. Here k and $\eta$ denote saturation and half-maximal saturation constants respectively. In the system (1), we introduce $0\le {ϵ}_i,i=1,2\le 1$ to represent effectiveness of 2-DG (${ϵ}_1$ denotes the efficacy of the drug in blocking transmission of COVID-19 and ${ϵ}_2$ stands for the efficacy of the drug in blocking production of new virions). Note that ${ϵ}_1={ϵ}_2=0$ represents no antiviral drug effect while ${ϵ}_1={ϵ}_2=1$ represents 100% efficacy of the drug. In our proposed study, all of the model parameters and the state variables are non-negative (their source values are listed in tabular form in Table 1).

Table 1.

Values and sources of the model parameters associated with the fractional order system (3)

Parameters	Mean value (unit)	Sources
${\mathrm{\Pi }}_1$	5 $(cellsm{l}^{-1}da{y}^{-1})$	[18]
${\epsilon }_1$	$0\le {\epsilon }_1\le 1$	-
$\beta$	0.0001 $(ml{(RNAcopies)}^{-1}da{y}^{-1})$	[17, 18]
k	0.1	assumed
${\mu }_1$	0.2 $(da{y}^{-1})$	estimated
${\mu }_2$	0.189 $(da{y}^{-1})$	[18]
${\mathrm{\Pi }}_2$	5 $(cellsm{l}^{-1}da{y}^{-1})$	estimated
$\theta$	$0\le \theta \le 1$	-
${\mu }_3$	0.1 $(da{y}^{-1})$	estimated
${\epsilon }_2$	$0\le {\epsilon }_2\le 1$	-
p	0.65 $(da{y}^{-1})$	[17, 18]
${\mu }_4$	0.1 $(da{y}^{-1})$	[18]
q	0.4 $(ml{(RNAcopies)}^{-1}da{y}^{-1})$	estimated
$\eta$	0.7	[17, 18]

Formulation of fractional order model

Intending to interpret the natural phenomena associated to nonlocality, fractional order differential equations are used as an excellent tool in epidemiology. Specifically to deal with some special epidemic behaviors like memory and hereditary properties and to obtain sufficient accurate results from the real data of a disease outbreak, fractional order model would be more adequate than its integer order counterpart. Naik et al. [24] constructed a COVID-19 model with Caputo and Atangana-Baleanu fractional derivative operators to estimate the parameter and they have carried out the qualitative analysis. With the help of fixed point theory, Chen et al. [25] verified that Caputo-Fabrizio type fractional order COVID-19 model has a unique solution. Mahata et al. [26] proposed and studied an SEIRV epidemic model of COVID-19 with optimal control in the context of the Caputo fractional derivative. Paul et al. [27] proposed an Adam-Bashforth-Moulton predictor-corrector scheme for the SIQR model. A SEIR model of COVID-19 has been studied by Paul et al. [28] with the help of the fractional order derivatives employing Caputo operator. Mahata et al. [29] considered the Caputo derivative to study the spread of COVID-19.

Caputo derivative operator has been widely used in the study of the SARS-CoV-2 infection [21, 22, 26, 30, 31]. It is one of the useful derivative operators to define more effectively memory effect dynamics that exist in real-world phenomena [30]. Thus we use the well-known and reliable Caputo derivative operator in fractional calculus to our proposed model (1) and thus the system (1) is transformed into the following form:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_t^{\alpha }{E}_S& ={\mathrm{\Pi }}_1-(1-{\epsilon }_1)\frac{\beta E_{S}VA}{1+kV}-{\mu }_1{E}_S,\hfill \\ \hfill D_t^{\alpha }{E}_I& =(1-{\epsilon }_1)\frac{\beta E_{S}VA}{1+kV}-{\mu }_2{E}_I,\hfill \\ \hfill D_t^{\alpha }A& ={\mathrm{\Pi }}_2-(1-{\epsilon }_1)\frac{\theta \beta E_{S}VA}{1+kV}-{\mu }_{3}A,\hfill \\ \hfill D_t^{\alpha }V& =(1-{\epsilon }_2)p{E}_{I}V-({\mu }_4+\frac{q{E}_I}{\eta +V})V,\hfill \end{array}\end{array}$$ 3

supplemented with non-negative initial conditions (2). In system (3), the left-Caputo fractional derivative of order $\alpha (0<\alpha \le 1)$ is denoted by $D_t^{\alpha }$ and it might be noticeable that for $\alpha =1$, the system (3) will shrink to the system (1).

Qualitative analysis of the fractional model

In this section, the fundamental characteristics of the fractional order system (3), specifically the existence, uniqueness and non-negativity of the solutions of the system (3), possible equilibrium points and the stability of the system (3) around these equilibrium points would be addressed.

First we express the left-Caputo differential equations system (3) of differentiation order $\alpha$ together with non-negative initial conditions (2) as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_t^{\alpha }\vartheta (t)& =\varrho (t,\vartheta (t)),0<\alpha \le 1,\hfill \\ \hfill \vartheta (t_0)& ={\vartheta }_0,t_0>0,\hfill \end{array}\end{array}$$ 4

where $\varrho (t,\vartheta (t))={({\varrho }_1,{\varrho }_2,{\varrho }_3,{\varrho }_4)}^T$, $\vartheta (t)={(E_S,E_I,A,V)}^T\in {\mathbb{R}}_{+}^4$ and ${\vartheta }_0={(E_{S0},E_{I0},A_0,V_0)}^T\in {\mathbb{R}}_{+}^4$. We consider the region ${\mathcal{R}}_{+}=\{(E_S,E_I,A,V)\in {\mathbb{R}}_{+}^4:max(|E_S|,|E_I|,|A|,|V|)\le K^{{}^{\prime }}\}\subset {\mathbb{R}}^n$, $n\ge 1$ and $K^{{}^{\prime }}$ is a finite positive real number. Next we define the function $\varrho (t,\vartheta (t))$ such that $\varrho (t,\vartheta (t)):[t_0,\infty )\times {\mathcal{R}}_{+}\to {\mathbb{R}}^n$ forms a vector field where ${\varrho }_1$, ${\varrho }_2$, ${\varrho }_3$ and ${\varrho }_4$ are designated as

$$\begin{array}{cc}\hfill {\varrho }_1& ={\mathrm{\Pi }}_1-(1-{\epsilon }_1)\frac{\beta E_{S}VA}{1+kV}-{\mu }_1{E}_S,\hfill \\ \hfill {\varrho }_2& =(1-{\epsilon }_1)\frac{\beta E_{S}VA}{1+kV}-{\mu }_2{E}_I,\hfill \\ \hfill {\varrho }_3& ={\mathrm{\Pi }}_2-(1-{\epsilon }_1)\frac{\theta \beta E_{S}VA}{1+kV}-{\mu }_{3}A,\hfill \\ \hfill {\varrho }_4& =(1-{\epsilon }_2)p{E}_{I}V-({\mu }_4+\frac{q{E}_I}{\eta +V})V.\hfill \end{array}$$

Positiveness of the solutions

In this subsection, we study the criteria for positiveness of the solution trajectories of the system (3). First we state the generalized mean value theorem [32] and a corollary.

Theorem 1

Let $\varrho (t),D_a^{\alpha }\varrho (t)\in C[a,b]$, for $0<\alpha \le 1$. Then $[\varrho (\vartheta )=\varrho (a)+\frac{1}{\mathrm{\Gamma }(\alpha )}(D_a^{\alpha }\varrho )(\varphi ){(\vartheta -a)}^{\alpha }]$ holds for all $\vartheta \in (a,b]$, where $a\le \varphi \le \vartheta$.

Corollary 1

Let $\varrho (x),D_a^{\alpha }\varrho (\vartheta )\in C[a,b]$, for $0<\alpha \le 1$. If the fractional derivatives $D_a^{\alpha }\varrho (\vartheta )\ge 0$ for all $\vartheta \in [a,b]$, then $\varrho (\vartheta )$ is non-decreasing for each $\vartheta \in [a,b]$. On the other hand, if $D_a^{\alpha }\varrho (\vartheta )\le 0$, then, for all $\vartheta \in [a,b]$, then, $\varrho (\vartheta )$ is non-increasing for every $\vartheta \in [a,b]$.

Now we consider a time instant $t_1$ with $t_0\le t\le t_1$ such that

$$\begin{array}{cc}\hfill E_S(t)& =0,\mathrm{for}0\le t\le t_1,\hfill \\ \hfill & =0,\mathrm{for}t=t_1,\hfill \\ \hfill & <t_1\mathrm{for}t=t_1^{+}.\hfill \end{array}$$

From the system (3), note that $D_t^{\alpha }{E}_S{|}_{E_{S(t_1)}}=0$. Applying Corollary 1, it is observed that $E_S(t_1)=0$ contradicting our assumption. Thus we have $E_S(t)\ge 0$, at any instant t. In a similar manner, it can be shown that $E_I(t),A(t),V(t)\ge 0$ implying that all the solutions of the system (3) are non-negative for any time $t\in [t_0,\infty )$.

Uniform boundedness of the solutions

Here we determine whether the solutions of the system (3) are bounded. For that, we construct the function $h(t):R_{0,+}⟶R_{0,+}$ defined by

$$\begin{array}{c}\hfill h(t)={\varrho }_1+{\varrho }_2+{\varrho }_3+{\varrho }_4,\forall t\ge t_0.\end{array}$$ 5

Thus placing the values of ${\varrho }_i$, $i=1,2,3,4$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill h(t)& =({\mathrm{\Pi }}_1+{\mathrm{\Pi }}_2)-((1-{\epsilon }_1)\frac{\theta \beta E_{S}A}{1+kV}+\frac{q{E}_I}{\eta +V}-(1-{\epsilon }_2)p{E}_I)V\hfill \\ \hfill & -({\mu }_1{E}_S+{\mu }_2{E}_I+{\mu }_{3}A+{\mu }_{4}V)\hfill \\ \hfill & \le ({\mathrm{\Pi }}_1+{\mathrm{\Pi }}_2)-\mu (E_S+E_I+A+V),\hfill \end{array}\end{array}$$ 6

where $\mu =min\{{\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4\}$ and $(1-{\epsilon }_1)\frac{\theta \beta E_{S}A}{1+kV}+\frac{q{E}_I}{\eta +V}<(1-{\epsilon }_2)p{E}_I$. Thus we get,

$$\begin{array}{cc}\hfill h(t)\le & ({\mathrm{\Pi }}_1+{\mathrm{\Pi }}_2)-\mu \vartheta (t),\forall t\ge t_0.\hfill \end{array}$$ 7

Therefore using Lemma 3 presented in [33], and for $t\to \infty$, it can be concluded that all the solutions trajectories of the system (3) initiating from ${\mathcal{R}}_{+}$ are uniformly bounded in the region

$$\begin{array}{c}\hfill \mathrm{\Omega }=\{(E_S,E_I,A,V)\in {\mathcal{R}}_{+}:E_S+E_I+A+V\le \frac{{\mathrm{\Pi }}_1+{\mathrm{\Pi }}_2}{\mu }\}.\end{array}$$ 8

Local existence and uniqueness of solutions

In this subsection, we will investigate the local existence and uniqueness of solution trajectories $(E_S,E_I,A,V)$ of the system (3). The existence criteria for the solutions of fractional order system (3) would be studied using the theorem proposed in [34] which is presented below:

Theorem 2

Let

$$\begin{array}{cc}\hfill \mathcal{P}& =[t_0-a,t_0+a],\mathcal{B}=\{\vartheta \in {\mathcal{R}}_{+}:\Vert \vartheta -{\vartheta }_0\Vert \le b\},\hfill \\ \hfill \mathcal{D}& =\{(t,\vartheta )\in [t_0,\infty )\times {\mathcal{R}}_{+}:t\in \mathcal{P},\vartheta \in \mathcal{B}\}.\hfill \end{array}$$

Further we assume that the function $\varrho :\mathcal{D}⟶{\mathbb{R}}_{+}^4$ satisfies the following conditions:

• (i)$\varrho (t,\vartheta (t))$ is Lebesgue measurable for $t\in \mathcal{P}$;
• (ii)$\varrho (t,\vartheta (t))$ is continuous where $\vartheta$ belongs to $\mathcal{B}$;
• (iii)there is a real-valued function $w(t)\in {\mathcal{L}}^2(\mathcal{P})$ such that $\Vert \varrho (t,\vartheta (t))\Vert \le w(t)$, for almost every $t\in \mathcal{P}$ and for all $\vartheta (t)$ belongs to $\mathcal{B}$.

Then for $\alpha$ be chosen as $\alpha >1/2$, there exists at least one solution to the initial value problem (4) in the interval $[t_0-h,t_0+h]$, for some $h>0$ [34].

With the help of Theorem 2, we discuss the uniqueness of solutions of the system (3).

Theorem 3

Suppose that the assumptions (i)–(iii) of Theorem 2 hold and there exists a real-valued function $\mathrm{\Lambda }(t)\in {\mathcal{L}}^4(\mathcal{P})$ such that

$$\begin{array}{cc}\hfill \Vert \varrho (t,\vartheta )-\varrho (t,\nu )\Vert \le & \mathrm{\Lambda }(t)\Vert \vartheta -\nu \Vert ,\hfill \end{array}$$ 9

for almost every $t\in \mathcal{P}$ and all $\vartheta ,\nu \in \mathcal{B}$. Then there exists a unique solution of the initial value problem (4) on the interval $[t_0,\infty )\times {\mathcal{R}}_{+}$ [34].

Proof

In order to prove the uniqueness of the solution trajectories of the system (3), following the method proposed by Li et al. [33], we construct the function $\varrho (t):{\mathcal{R}}_{+}\to {\mathbb{R}}_{+}^4$ as $\varrho (t,\vartheta )=({\varrho }_1(t,\vartheta ),{\varrho }_2(t,\vartheta ),{\varrho }_3(t,\vartheta ),{\varrho }_4(t,\vartheta ))$, where ${\varrho }_1$, ${\varrho }_2$, ${\varrho }_3$ and ${\varrho }_1$ are defined previously.

Here we utilize the norm $\Vert \varrho (t,\vartheta )\Vert =|{\varrho }_1(t,\vartheta )|+|{\varrho }_2(t,\vartheta )|+|{\varrho }_3(t,\vartheta )|+|{\varrho }_4(t,\vartheta )|$, for $\varrho (t)\in {\mathcal{R}}_{+}$. Note that ${\mathcal{R}}_{+}$ is endowed with the proper norm $\Vert .\Vert$ and $[t_0,\infty )\times {\mathcal{R}}_{+}$ is a Banach space with respect to this norm [34]. Let us choose any two points $\vartheta (t)=(E_S,E_I,A,V)$ and $\nu (t)=(E_S^{{}^{\prime }},E_I^{{}^{\prime }},A^{{}^{\prime }},V^{{}^{\prime }})$ belonging to the region ${\mathcal{R}}_{+}$ and for these two points $\vartheta (t)$ and $\nu (t)$,

$$\begin{array}{cc}\hfill \Vert \varrho (t,\vartheta )-\varrho (t,\nu )\Vert =& |{\varrho }_1(t,\vartheta )-{\varrho }_1(t,\nu )|+|{\varrho }_2(t,\vartheta )-{\varrho }_2(t,\nu )|+\hfill \\ \hfill & |{\varrho }_3(t,\vartheta )-{\varrho }_3(t,\nu )|+|{\varrho }_4(t,\vartheta )-{\varrho }_4(t,\nu )|.\hfill \end{array}$$

Using the definitions of ${\varrho }_1,{\varrho }_2,{\varrho }_3$, and ${\varrho }_4$, we can write

$$\begin{array}{cc}\hfill \Vert \varrho (t,\vartheta )-\varrho (t,\nu )\Vert & =|{\mu }_1(E_S-E_S^{{}^{\prime }})+(1-{\epsilon }_1)\beta (\frac{E_{S}VA}{1+kV}-\frac{E_S^{{}^{\prime }}{V}^{{}^{\prime }}{A}^{{}^{\prime }}}{1+k{V}^{{}^{\prime }}})|\hfill \end{array}$$

$$\begin{array}{cc}& +|{\mu }_2(E_I-E_I^{{}^{\prime }})+(1-{\epsilon }_1)\beta (\frac{E_{S}VA}{1+kV}-\frac{E_S^{{}^{\prime }}{V}^{{}^{\prime }}{A}^{{}^{\prime }}}{1+k{V}^{{}^{\prime }}})|\hfill \\ \hfill & +|{\mu }_3(A-A^{{}^{\prime }})+(1-{\epsilon }_1)\theta \beta (\frac{E_{S}VA}{1+kV}-\frac{E_S^{{}^{\prime }}{V}^{{}^{\prime }}{A}^{{}^{\prime }}}{1+k{V}^{{}^{\prime }}})|\hfill \\ \hfill & +|{\mu }_4(V-V^{{}^{\prime }})-(1-{\epsilon }_2)p(E_{I}V-E_I^{{}^{\prime }}{V}^{{}^{\prime }})+q(\frac{E_{I}V}{\eta +V}-\hfill \\ \hfill & \frac{E_I^{{}^{\prime }}{V}^{{}^{\prime }}}{\eta +V^{{}^{\prime }}})|\hfill \\ \hfill & \le {\mu }_1|(E_S-E_S^{{}^{\prime }})|+3(1-{\epsilon }_1)\beta K^{{}^{\prime }}(|E_S-E_S^{{}^{\prime }}|+|V-V^{{}^{\prime }}|\hfill \\ \hfill & +|A-A^{{}^{\prime }}|)+{\mu }_2|(E_I-E_I^{{}^{\prime }})|+3(1-{\epsilon }_1)\beta K^{{}^{\prime }}(|E_S-E_S^{{}^{\prime }}|\hfill \\ \hfill & +|V-V^{{}^{\prime }}|+|A-A^{{}^{\prime }}|)+{\mu }_3|(A-A^{{}^{\prime }})|+3(1-{\epsilon }_1)\theta \beta K^{{}^{\prime }}(|E_S-E_S^{{}^{\prime }}|\hfill \\ \hfill & +|V-V^{{}^{\prime }}|+|A-A^{{}^{\prime }}|)+{\mu }_4|(V-V^{{}^{\prime }})|+2(1-{\epsilon }_2)p{K}^{{}^{\prime }}(|E_I-E_I^{{}^{\prime }}|\hfill \\ \hfill & +|V-V^{{}^{\prime }}|)+2q\frac{K^{{}^{\prime }}}{{\eta }^2}(|E_I-E_I^{{}^{\prime }})+|V-V^{{}^{\prime }}|)\hfill \\ \hfill & =({\mu }_1+6(1-{\epsilon }_1)\beta K^{{}^{\prime }}+3(1-{\epsilon }_1)\theta \beta K^{{}^{\prime }})|E_S-E_S^{{}^{\prime }}|\hfill \\ \hfill & +({\mu }_2+2(1-{\epsilon }_2)p{K}^{{}^{\prime }}+2q\frac{K^{{}^{\prime }}}{{\eta }^2})|E_I-E_I^{{}^{\prime }}|\hfill \\ \hfill & +({\mu }_3+3(1-{\epsilon }_1)\beta K^{{}^{\prime }}+3(1-{\epsilon }_1)\theta \beta K^{{}^{\prime }})|A-A^{{}^{\prime }}|\hfill \\ \hfill & +({\mu }_4+6(1-{\epsilon }_1)\beta K^{{}^{\prime }}+3(1-{\epsilon }_1)\theta \beta K^{{}^{\prime }}+2(1-{\epsilon }_2)p{K}^{{}^{\prime }}\hfill \\ \hfill & +2q\frac{K^{{}^{\prime }}}{{\eta }^2})|A-A^{{}^{\prime }}|\hfill \\ \hfill & \le L^{{}^{\prime }}\Vert \vartheta -\nu \Vert ,\hfill \end{array}$$

considering L' = max (${\mu }_1+6(1-{\epsilon }_1)\beta K^{{}^{\prime }}+3(1-{\epsilon }_1)\theta \beta K^{{}^{\prime }}$, ${\mu }_2+2(1-{\epsilon }_2)p{K}^{{}^{\prime }}+2q\frac{K^{{}^{\prime }}}{{\eta }^2}$, ${\mu }_3+3(1-{\epsilon }_1)\beta K^{{}^{\prime }}+3(1-{\epsilon }_1)\theta \beta K^{{}^{\prime }}$, ${\mu }_4+6(1-{\epsilon }_1)\beta K^{{}^{\prime }}+3(1-{\epsilon }_1)\theta \beta K^{{}^{\prime }}+2(1-{\epsilon }_2)p{K}^{{}^{\prime }}+2q\frac{K^{{}^{\prime }}}{{\eta }^2}$. Thus $\varrho (t,\vartheta )$ satisfies Lipschitz’s condition for $\vartheta \in {\mathcal{R}}_{+}$ [33]. Now using the Banach Contractive Mapping Principle [34], it can be concluded that a unique real-valued function $\mathrm{\Lambda }(t)\in {\mathcal{L}}^4(\mathcal{P})$ must exist such that

$$\begin{array}{c}\hfill \mathrm{\Lambda }(t)={\vartheta }_0+\frac{L^{{}^{\prime }}}{\mathrm{\Gamma }(\alpha )}{\int }_{t_0}^t{(t-s)}^{\alpha -1}\varrho (s,\mathrm{\Lambda }(s))ds.\end{array}$$

Consequently, the initial value problem (4) exhibits unique solution on the interval $[t_0,\infty )\times {\mathcal{R}}_{+}$. Hence the proof is completed.

Global existence of the solutions

In this subsection, the global existence of the solution trajectories of the system (3) is studied using Theorem 3.1 [34].

Theorem 4

Let the assumptions (i)–(ii) of Theorem 2 and the following condition: $\Vert \varrho (t,\vartheta )\Vert \le {\kappa }_1+{\kappa }_2\Vert \vartheta \Vert$ hold in global space, where ${\kappa }_1>0$ and ${\kappa }_2>0$ are two constants, for almost each $t\in [t_0,\infty )$ and all $\vartheta (t)\in {\mathcal{R}}_{+}$. Then there exists a function $\vartheta (t)\in [0,+\infty )$ which is a solution of the initial value problem (4) [34].

Proof

With respect to certain $t_0\in [t_0,\infty )$ and ${\vartheta }_0\in {\mathcal{R}}_{+}$ and using the assumptions of the above statement, it can be observed that $\varrho (t,\vartheta (t))$ is locally bounded in the region $\mathcal{D}$. Again the weak singularity of $\varrho (t,\vartheta (t))$ indicates the existence of the solution $\vartheta (t)$ of the initial value problem (4) specified on the interval $[t_0,\infty )\times {\mathcal{R}}_{+}$.

Let $\vartheta (t)$ possesses a maximal existence interval $[0,l^{{}^{\prime }})\subset [0,+\infty )$ such that $l^{{}^{\prime }}<+\infty .$ According to [34], the solution $\vartheta (t)$ can be expressed as

$$\begin{array}{c}\hfill \vartheta (t)={\vartheta }_0+\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_{t_0}^t{(t-s)}^{\alpha -1}\varrho (s,\vartheta (s))ds.\end{array}$$

Thus, employing the assumptions of the statement, the Eq. (9) is transformed into the following form:

$$\begin{array}{c}\hfill \Vert \vartheta (t)\Vert \le \Vert {\vartheta }_0\Vert +\frac{{\kappa }_1}{\mathrm{\Gamma }(\alpha +1)}{|l^{{}^{\prime }}-t_0|}^{\alpha }+{\kappa }_2{\int }_{t_0}^t{(t-s)}^{\alpha -1}\Vert \vartheta (s)\Vert ds,\end{array}$$

for $t_0,t\in [0,l^{{}^{\prime }})$ together with $t_0\le t$. With the help of the generalized Gronwall inequality, we can find a constant z such that $\Vert \vartheta (t)\Vert <z$ on $[t_0,l^{{}^{\prime }})$ [34]. Consequently, according to Theorem 3.1 of [34], the global existence and uniqueness of solution $\vartheta (t)$ of the initial value problem (4) can be ensured on the interval $[0,+\infty )$. Hence the proof is completed.

Steady states and stability of the fractional order system

This subsection is concerned with the steady states executed by the epidemic system (3) and the stability of the fractional order system (3) around these steady states.

Disease-free equilibrium (DFE)

The infected compartments of our proposed epidemic fractional order system (3) are $E_I(t)$ and V(t). Thus solving the system (3) by letting $E_I(t)=0$ and $V(t)=0$, we find that the system (3) executes the disease-free equilibrium (DFE) ${\mathcal{E}}_0=({\mathrm{\Pi }}_1/{\mu }_1,0,{\mathrm{\Pi }}_2/{\mu }_3,0)$. Next we compute the Jacobian matrix ${\mathcal{J}}_{{\mathcal{E}}_0}$ of the system (3) around the DFE ${\mathcal{E}}_0$ to study the stability of the system (3) which is given by

$$\begin{array}{c}\hfill {\mathcal{J}}_{{\mathcal{E}}_0}=\left(\begin{array}{cccc}-{\mu }_1& 0& 0& -\frac{(1-{ϵ}_1)\beta {\mathrm{\Pi }}_1{\mathrm{\Pi }}_2}{{\mu }_1{\mu }_3}\\ 0& -{\mu }_2& 0& \frac{(1-{ϵ}_1)\beta {\mathrm{\Pi }}_1{\mathrm{\Pi }}_2}{{\mu }_1{\mu }_3}\\ 0& 0& -{\mu }_3& -\frac{(1-{ϵ}_1)\beta \theta {\mathrm{\Pi }}_1{\mathrm{\Pi }}_2}{{\mu }_1{\mu }_3}\\ 0& 0& 0& -{\mu }_4\end{array}\right).\end{array}$$

With the aid of the following theorem [35], we would study the local stability of the fractional order system (3) around the DFE.

Theorem 5

Let the fractional order system (3) together with the initial condition (2) be expressed as $D_t^{\alpha }\vartheta (t)=\mathcal{M}\vartheta (t)$, with the same definition of $\vartheta (t)$ as stated previously and $\mathcal{M}\in {\mathbb{R}}^{4\times 4}$. The system (3) is locally asymptotically stable iff all the four eigenvalues ${\lambda }_i$, $i=1,2,3,4$, of the matrix $\mathcal{M}$ satisfy the relation $|arg({\lambda }_i)|>\frac{\pi \alpha }{2}$ and the system is stable iff the eigenvalues would hold the relation $|arg({\lambda }_i)|\ge \frac{\pi \alpha }{2}$ satisfying the critical condition $|arg({\lambda }_i)|=\frac{\pi \alpha }{2}$ along with the geometric multiplicity equal to one [35].

Proof

Obviously the eigenvalues of Jacobian matrix ${\mathcal{J}}_{{\mathcal{E}}_0}$ are $-{\mu }_1$, ${\mu }_2$, ${\mu }_3$ and $-{\mu }_4$ which are strictly real and negative. Hence $|arg({\lambda }_i)|=\pi >\frac{\pi \alpha }{2}$, since $0<\alpha <1$ ($i=1,2,3,4$). This shows the fact that the system (3) is locally asymptotically stable around the DFE.

Endemic equilibrium (EE)

The epidemic system (3) possesses a unique endemic equilibrium $E^{\ast }(E_S^{\ast },E_I^{\ast },A^{\ast },V^{\ast })$. The components of EE are obtained by solving equations $D_t^{\alpha }{E}_S(t)=0$, $D_t^{\alpha }{E}_I(t)=0$, $D_t^{\alpha }A(t)=0$, $D_t^{\alpha }V(t)=0$ and are given by

$$\begin{array}{c}\hfill E_S^{\ast }=\frac{{\pi }_1-{\mu }_2{E}_I^{\ast }}{{\mu }_1},A^{\ast }=\frac{{\pi }_2-\theta {\mu }_2{E}_I^{\ast }}{{\mu }_3},V^{\ast }=\frac{\eta {\mu }_4-\{\eta p(1-{ϵ}_2)-q\}{E}_I^{\ast }}{(1-{ϵ}_2)p{E}_I^{\ast }-{\mu }_4},\end{array}$$

whereas the value of $E_I^{\ast }$ is derived from the cubic equation

$$\begin{array}{c}\hfill a_1{E_I^{\ast }}^3+a_2{E_I^{\ast }}^2+a_3{E}_I^{\ast }+a_4=0,\end{array}$$ 10

$$\begin{array}{cc}\hfill \mathrm{where},a_1& =\theta {\mu }_2^2\beta (1-{ϵ}_1)\{(1-{ϵ}_2)\eta p-q\},\hfill \\ \hfill a_2& ={\mu }_2[(1-{ϵ}_2)p-k\{\eta p(1-{ϵ}_2)-q\}]-(1-{ϵ}_1)\beta \{({\pi }_1\theta +{\pi }_2)\hfill \\ \hfill & (\eta p(1-{ϵ}_2)-q)+\eta \theta {\mu }_2^2{\mu }_4\},\hfill \\ \hfill a_3& =(1-{ϵ}_1)\beta [\{\eta p(1-{ϵ}_2)-q\}{\pi }_1{\pi }_2-({\pi }_1\theta +{\pi }_2)\eta {\mu }_2{\mu }_4+{\mu }_2{\mu }_4(k\eta -1)],\hfill \\ \hfill a_4& =-(1-{ϵ}_1)\beta \eta {\pi }_1{\pi }_2{\mu }_4.\hfill \end{array}$$

Next we calculate the Jacobian matrix of the system (3) about the EE which is given by

$$\begin{array}{c}\hfill {\mathcal{J}}_{{\mathcal{E}}^{\ast }}=\left(\begin{array}{cccc}-j_{11}& 0& -j_{13}& -j_{14}\\ j_{21}& -j_{22}& j_{13}& j_{14}\\ -\theta j_{21}& 0& -j_{33}& -\theta j_{14}\\ 0& j_{42}& 0& -j_{44}\end{array}\right),\end{array}$$

$$\begin{array}{cc}\hfill \mathrm{where},j_{11}& ={\mu }_1+\frac{(1-{ϵ}_1)\beta A^{\ast }{V}^{\ast }}{1+k{V}^{\ast }},j_{21}=\frac{(1-{ϵ}_1)\beta A^{\ast }{V}^{\ast }}{1+k{V}^{\ast }},\hfill \\ \hfill j_{22}& ={\mu }_2,j_{42}=(1-{ϵ}_2)p{V}^{\ast }-\frac{q{V}^{\ast }}{\eta +V^{\ast }},\hfill \\ \hfill j_{13}& =\frac{(1-{ϵ}_1)\beta E_S^{\ast }{V}^{\ast }}{1+k{V}^{\ast }},j_{33}=\frac{(1-{ϵ}_1)\beta \theta E_S^{\ast }{V}^{\ast }}{1+k{V}^{\ast }}+{\mu }_3,\hfill \\ \hfill j_{14}& =\frac{(1-{ϵ}_1)\beta E_S^{\ast }{A}^{\ast }}{{(1+k{V}^{\ast })}^2},j_{44}=\frac{q{E}_I^{\ast }\eta }{{(\eta +V^{\ast })}^2}-(1-{ϵ}_2)p{E}_I^{\ast }+{\mu }_4.\hfill \end{array}$$

The characteristic equation of the Jacobian matrix ${\mathcal{J}}_{{\mathcal{E}}^{\ast }}$ corresponding to the eigenvalue $\lambda$ can be written as

$$\begin{array}{c}\hfill {\lambda }^4+{\zeta }_1{\lambda }^3+{\zeta }_2{\lambda }^2+{\zeta }_3\lambda +{\zeta }_4=0,\end{array}$$ 11

$$\begin{array}{cc}\hfill \mathrm{where},{\zeta }_1& =j_{11}+j_{22}+j_{33}+j_{44},\hfill \\ \hfill {\zeta }_2& =(j_{11}+j_{33})(j_{22}+j_{44})+j_{11}{j}_{33}+j_{22}{j}_{44}-j_{14}{j}_{42}+j_{13}{j}_{21},\hfill \\ \hfill {\zeta }_3& =j_{11}{j}_{33}(j_{22}+j_{44})+j_{22}{j}_{44}(j_{11}+j_{33})-j_{14}{j}_{42}(j_{11}+j_{33})\hfill \\ \hfill & +j_{13}{j}_{21}(j_{22}+j_{44})\theta +j_{13}{j}_{14}{j}_{42}\theta +j_{14}{j}_{21}{j}_{42},\hfill \\ \hfill {\zeta }_4& =j_{11}{j}_{33}(j_{22}{j}_{44}-j_{14}{j}_{42})+j_{13}{j}_{21}{j}_{22}{j}_{42}\theta +j_{14}{j}_{42}\theta (j_{21}{j}_{33}-j_{13}{j}_{21}).\hfill \end{array}$$

Next we present two propositions to study the local asymptotic stability of the system (3) around the EE.

Proposition 1

If each eigenvalue ${\lambda }_i,i=1,2,3,4$ of the Jacobian matrix ${\mathcal{J}}_{{\mathcal{E}}^{\ast }}$ satisfies the condition $|arg(\lambda )|>\frac{\lambda \pi }{2}$. Then the epidemic system (3) is locally asymptotically stable around the EE (using Theorem 5).

Next, we determine the discriminant of the characteristic Eq. (11) as:

$$\mathrm{{\rm Y}}(\phi )=\left|\begin{array}{ccccccc}1& {\zeta }_1& {\zeta }_2& {\zeta }_3& {\zeta }_4& 0& 0\\ 0& 1& {\zeta }_1& {\zeta }_2& {\zeta }_3& {\zeta }_4& 0\\ 0& 0& 1& {\zeta }_1& {\zeta }_2& {\zeta }_3& {\zeta }_4\\ 4& 3{\zeta }_1& 2{\zeta }_2& {\zeta }_3& 0& 0& 0\\ 0& 4& 3{\zeta }_1& 2{\zeta }_2& {\zeta }_3& 0& 0\\ 0& 0& 4& 3{\zeta }_1& 2{\zeta }_2& {\zeta }_3& 0\\ 0& 0& 0& 4& 3{\zeta }_1& 2{\zeta }_2& {\zeta }_3\end{array}\right|,$$

$$\begin{array}{c}\hfill \begin{array}{cc}& ={\zeta }_1^2{\zeta }_2^2{\zeta }_3^2-4{\zeta }_1^2{\zeta }_2^3{\zeta }_4-4{\zeta }_1^3{\zeta }_3^2+18{\zeta }_1^3{\zeta }_2{\zeta }_3{\zeta }_{-}27{\zeta }_1^4{\zeta }_2^2+4{\zeta }_2^3{\zeta }_3^2+16{\zeta }_2^4{\zeta }_4\hfill \\ \hfill & +18{\zeta }_1{\zeta }_2{\zeta }_3^3-80{\zeta }_1{\zeta }_2^2{\zeta }_3{\zeta }_4-6{\zeta }_1^2{\zeta }_3^2{\zeta }_4+144{\zeta }_1^2{\zeta }_2{\zeta }_4^2-27{\zeta }_3^4+144{\zeta }_2{\zeta }_3^2{\zeta }_4\hfill \\ \hfill & -128{\zeta }_2^2{\zeta }_4^2-192{\zeta }_1{\zeta }_3{\zeta }_4^2+256{\zeta }_4^3.\hfill \end{array}\end{array}$$ 12

In terms of the discriminant $\mathrm{{\rm Y}}(\phi )$, we construct the following proposition to study the local asymptotic stability of the system (3) around the endemic equilibrium point.

Proposition 2

[22] (a) The epidemic system (3) is locally asymptotically stable around the EE if $\mathrm{{\rm Y}}(\phi )>0$, in addition to the conditions (i)  ${\zeta }_1>0$, (ii)  ${\zeta }_1{\zeta }_2>{\zeta }_3$, and (iii)  ${\zeta }_1{\zeta }_2{\zeta }_3-{\zeta }_1^2{\zeta }_4-{\zeta }_3>0$.

(b) The epidemic system (3) is locally asymptotically stable around the EE for $\alpha \in (0.5,1)$, if $\mathrm{{\rm Y}}(\phi )<0$, in addition to the conditions (i)  ${\zeta }_1>0$, (ii)  ${\zeta }_2>0$, (iii)  ${\zeta }_1{\zeta }_2>{\zeta }_3$, and (iv)  ${\zeta }_1{\zeta }_2{\zeta }_3-{\zeta }_1^2{\zeta }_4-{\zeta }_3=0$.

(c) The epidemic system (3) is unstable around the EE for $\alpha >2/3$, if $\mathrm{{\rm Y}}(\phi )<0$ together with the conditions (i)  ${\zeta }_1<0$, (ii)  ${\zeta }_2<0$, and (iii)  ${\zeta }_3<0$.

Optimal control of the fractional order model

In this section, we discuss the possible diminishment of the SARS-CoV-2 infection through optimal control strategy using 2-DG. In our study, we implement the robust control method - output feedback $H_{\infty }$ control where discrete time frame is stipulated and is based on Pontryagin’s Minimum Principle [36].

Optimal strategies and fractional order control model

In Epidemiology, optimal control technique is applied to a mathematical model attributed to biological or biomedical state of a disease system. Several researches have been conducted to study the dynamical effects of optimal control in fractional order mathematical modeling [37–44]. At first, we determine optimal control strategies based on Pontryagin’s Minimum Principle [36] and use the explicit expression of the stability conditions. Two control variables $u_1(t)$ and $u_2(t)$ are introduced in the fractional order system (3) satisfying $0\le u_i(t)\le 1,i=1,2$. Particularly $u_i=0,i=1,2$ indicate that no drug is implemented in the epidemic system and $u_i(t)=1,i=1,2$ indicate maximal use of the drug 2-DG and its influence to combat SRAS-CoV-2 infection. The drug 2-DG is mainly prescribed to human host for the purpose of turning on the host immune system and to mediate in stimulating the epithelial cells proliferation in a controlled manner (since extravagant proliferation of epithelial cells may cause carcinoma).

Therefore, the fractional order model (3) is converted into the following fractional order control model:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_t^{\alpha }{E}_S& ={\mathrm{\Pi }}_1-(1-{\epsilon }_1)\frac{\beta E_{S}VA}{1+kV}-{\mu }_1{E}_S+\frac{u_1(t)E_S}{1+{\rho }_1{E}_S},\hfill \\ \hfill D_t^{\alpha }{E}_I& =(1-{\epsilon }_1)\frac{\beta E_{S}VA}{1+kV}-{\mu }_2{E}_I,\hfill \\ \hfill D_t^{\alpha }A& ={\mathrm{\Pi }}_2-(1-{\epsilon }_1)\frac{\theta \beta E_{S}VA}{1+kV}-{\mu }_{3}A+\frac{u_2(t)A}{1+{\rho }_{2}A},\hfill \\ \hfill D_t^{\alpha }V& =(1-{\epsilon }_2)p{E}_{I}V-({\mu }_4+\frac{q{E}_I}{\eta +V})V,\hfill \end{array}\end{array}$$ 13

with the same initial conditions (2). The constants ${\rho }_1$ and ${\rho }_2$ represent the half-maximal simulations for host immune response. The control system (13) can be expressed in the following form:

$$\begin{array}{c}\hfill D_t^{\alpha }y=f(y(t),u(t),t),y(0)=y_0,\end{array}$$ 14

where $y={(E_S,E_I,A,V)}^T$ and $y_0={(E_{S0},E_{I0},A_0,V_0)}^T$. To minimize the infected epithelial cells load and viral load, we define the following objective functional

$$\begin{array}{c}\hfill J(u_1(t),u_2(t))={\int }_{T_0}^{T_f}[w_1{u}_1^2(t)+w_2{u}_2^2(t)+w_3{E}_I^2+w_4{V}^2]dt,\end{array}$$ 15

where $w_1{u}_1^2(t)$ and $w_2{u}_2^2(t)$ stand for the cost-benefit of 2-DG in inducing the epithelial cells proliferation and agitating host immune response respectively. Here $w_1$ and $w_2$ are weights that assist in regularizing the optimal controls; $w_3$ and $w_4$ keep balance in the load of the infected epithelial cells and virus population. Our aim is to find the optimal controls ($u_1^{\ast }(t),u_2^{\ast }(t)$) implemented in the system (13) to minimize the objective functional (15) over the control set $\mathcal{U}$ such that

$$\begin{array}{c}\hfill \underset{(u_1(.),u_2(.))\in \mathcal{U}}{inf}J(u_1(.),u_2(.))=J(u_1^{\ast }(.),u_2^{\ast }(.)),\end{array}$$ 16

where $\mathcal{U}$ is defined as $\mathcal{U}={\mathcal{U}}_1\times {\mathcal{U}}_2=\{(u_1(t),u_2(t)):0\le u_i(t)\le 1,i=1,2;u_i\text{are}\text{measurable};t\in [T_0,T_f]\}$.

Derivation of optimal conditions for the FOCP

In this subsection, we frame the general formulation and derivation of fractional order control problem (FOCP) described through the control induced fractional order system (13) [22]. We rewrite the objective functional (15) subjected to the system (14) as

$$\begin{array}{c}\hfill J(u)={\int }_{T_0}^{T_f}f(y(t),u(t),t)dt.\end{array}$$

In view of the system (14), we define the FOCP as

$$\begin{array}{c}\hfill infJ(u)={\int }_{T_0}^{T_f}f(y(t),u^{\ast }(t),t)dt.\end{array}$$ 17

Accordingly, we can rewrite the FOCP in compact form as

$$\begin{array}{c}\hfill \underset{u(.)\in \mathcal{U}}{inf}J(u(.))=J(u^{\ast }(.)),\end{array}$$ 18

subjected to the state system

$$\begin{array}{c}\hfill D_t^{\alpha }y=g(y(t),u^{\ast }(t),\sigma ),y(0)=y_0,\end{array}$$ 19

where the co-state vector $\sigma$ satisfies the following condition:

$$\begin{array}{c}\hfill D_{t_g}^{\alpha }\sigma =\frac{\partial f}{\partial y}+{\sigma }^T\frac{\partial g}{\partial y},\sigma (t_f)=0.\end{array}$$ 20

Using the algorithm presented in [45], we conclude the optimal control $u^{\ast }(t)$ is the solution of the equation

$$\begin{array}{c}\hfill \frac{\partial f}{\partial u^{\ast }}+{\sigma }^T\frac{\partial g}{\partial u^{\ast }}=0.\end{array}$$ 21

Note that the Eqs. (19), (20), and (21) represent the Euler-Lagrange optimal conditions for the FOCP and with the help of these conditions, we determine the existence conditions so that controls $u_1^{\ast }(t)$ and $u_2^{\ast }(t)$ would be optimal. In this context, we construct a Hamiltonian function for the FOCP as

$$\begin{array}{c}\hfill \mathcal{H}=f+{\sigma }_i^T{g}_i,i=1,2,3,4,\end{array}$$ 22

where $f=w_1{u}_1^2(t)+w_2{u}_2^2(t)+w_3{E}_I^2+w_4{V}^2$, $\sigma ={({\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4)}^T$, $g_i={(g_1,g_2,g_3,g_4)}^T$. The following theorems provide the means to minimize the objective functional (15).

Theorem 6

There exist a pair of optimal controls $u_i^{\ast }(t),i=1,2$ to the FOCP (13) subjected to the state system (19).

We consider a Hamiltonian for the FOCP (13) subjected to the state system (19) as

$$\begin{array}{c}\hfill \mathcal{H}=w_1{u}_1^2(t)+w_2{u}_2^2(t)+w_3{E}_I^2+w_4{V}^2+\sum _{i=1}^4{\sigma }_i{g}_i.\end{array}$$ 23

Next we prove the existence of the optimal control and verify the following conditions:

1. The state system (13) possesses bounded coefficient and hence the controls $u_i^{\ast }(t),i=1,2$ and the corresponding state variables are non-empty.

2. As the state system (13) is bounded, the control set $\mathcal{U}$ is convex and closed.

3. Since the state system (13) is bilinear in $u_1(t)$ and $u_2(t)$, the R.H.S. of the system (13) is bounded by a linear function associated with state and control variables.

4. The integrand $w_1{u}_1^2(t)+w_2{u}_2^2(t)+w_3{E}_I^2+w_4{V}^2$ is convex on $\mathcal{U}$.

5. There exist ${\delta }_1>0$, ${\delta }_2>0$ and $p>1$ such that the integrand of the objective functional (15) satisfies

   $$\begin{array}{c}\hfill w_1{u}_1^2(t)+w_2{u}_2^2(t)+w_3{E}_I^2+w_4{V}^2\ge {\delta }_1(|u_1{|}^2+|u_2{{|}^2)}^{p/2}-{\delta }_2,\end{array}$$

   where ${\delta }_1$ and ${\delta }_2$ depend on the boundedness of $E_I$ and V, since $w_1>0$ and $w_2>0$.

Now, applying Pontryagin’s Minimum Principle [36] to the Hamiltonian (23), we prove the following theorem.

Theorem 7

Associated with the optimal control pair $(u_1^{\ast }(t),u_2^{\ast }(t))$ and solution of the state system (13), there exists adjoint variables ${\sigma }_i,i=1,2,3,4$ satisfying the adjoint system of equations

$$\begin{array}{c}\hfill D_{t_g}^{\alpha }{\sigma }_1=-\frac{\partial \mathcal{H}}{\partial E_S},D_{t_g}^{\alpha }{\sigma }_2=-\frac{\partial \mathcal{H}}{\partial E_I},D_{t_g}^{\alpha }{\sigma }_3=-\frac{\partial \mathcal{H}}{\partial A},D_{t_g}^{\alpha }{\sigma }_4=-\frac{\partial \mathcal{H}}{\partial V},\end{array}$$ 24

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{that}\mathrm{is},D_{t_g}^{\alpha }{\sigma }_1& =(1-{\epsilon }_1)\frac{\beta VA}{1+kV}({\sigma }_1-{\sigma }_2+\theta {\sigma }_3)+{\sigma }_1{\mu }_1-\frac{{\sigma }_1{u}_1(t)}{{(1+{\rho }_1{E}_S)}^2},\hfill \\ \hfill D_{t_g}^{\alpha }{\sigma }_2& =-2w_3{E}_I+{\sigma }_2{\mu }_2-{\sigma }_4(1-{\epsilon }_2)pV-\frac{{\sigma }_{4}qV}{\eta +V},\hfill \\ \hfill D_{t_g}^{\alpha }{\sigma }_3& =(1-{\epsilon }_1)\frac{\beta E_{S}V}{1+kV}({\sigma }_1-{\sigma }_2+\theta {\sigma }_3)+{\sigma }_3{\mu }_3-\frac{{\sigma }_3{u}_2(t)}{{(1+{\rho }_{2}A)}^2},\hfill \\ \hfill D_{t_g}^{\alpha }{\sigma }_4& =-2w_4{E}_I+(1-{\epsilon }_1)\frac{\beta E_{S}A}{{(1+kV)}^2}({\sigma }_1-{\sigma }_2+\theta {\sigma }_3)-{\sigma }_4(1-{\epsilon }_2)p{E}_I\hfill \\ \hfill & +{\mu }_4{\sigma }_4+\frac{{\sigma }_4\eta q{E}_I}{{(\eta +V)}^2},\hfill \end{array}\end{array}$$ 25

with the boundary conditions ${\sigma }_i(t_g)=0$, for $i=1,2,3,4$.

Furthermore, the expressions for optimal control pair $(u_1^{\ast }(t),u_2^{\ast }(t))$ are determined through the relation (21) characterized by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_1^{\ast }(t)=-\frac{{\sigma }_1{E}_S}{2w_1(1+{\rho }_1{E}_S)},\\ u_2^{\ast }(t)=-\frac{{\sigma }_{3}A}{2w_2(1+{\rho }_{2}A)}.\end{array}\right)\end{array}$$ 26

Hence, the boundedness of the optimal control pair $(u_1^{\ast }(t),u_2^{\ast }(t))$ could be defined as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_1^{\ast }(t)=min\{max\{-\frac{{\sigma }_1{E}_S}{2w_1(1+{\rho }_1{E}_S)},0\},1\},\\ u_2^{\ast }(t)=min\{max\{-\frac{{\sigma }_{3}A}{2w_2(1+{\rho }_{2}A)},0\},1\}.\end{array}\right)\end{array}$$ 27

Proof

Using standard results of Pontryagin’s Minimum Principle [36], the expressions of adjoint variables and boundary conditions can be derived. By partially differentiating the Hamiltonian (23) with respect to the corresponding states, the adjoint equations system can be expressed through (25) together with the boundary conditions ${\sigma }_i(t_g)=0$, for $i=1,2,3,4$. With the help of Pontryagin’s Minimum Principle [36], it can be observed that the unrestricted optimal controls pair $(u_1^{\ast }(t),u_2^{\ast }(t))$ must satisfy

$$\begin{array}{c}\hfill \frac{\partial H}{\partial u_1^{\ast }(t)}=0,\frac{\partial H}{\partial u_2^{\ast }(t)}=0.\end{array}$$

We observe that

$$\begin{array}{cc}\hfill \mathcal{H}& =w_1{u}_1^2(t)+w_2{u}_2^2(t)+{\sigma }_1\frac{u_1(t)E_S}{1+{\rho }_1{E}_S}+{\sigma }_3\frac{u_2(t)A}{1+{\rho }_{2}A}\hfill \\ \hfill & +\text{other}\text{terms}\text{excluding}{u}_1(t)\text{and}{u}_2(t),\hfill \end{array}$$

which leads to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{\partial H}{\partial u_1(t)}& =2w_1{u}_1+{\sigma }_1\frac{E_S}{1+{\rho }_1{E}_S},\hfill \\ \hfill \frac{\partial H}{\partial u_2(t)}& =2w_2{u}_2+{\sigma }_3\frac{A}{1+{\rho }_{2}A}.\hfill \end{array}\end{array}$$ 28

Consequently, solving Eq. (28) we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_1^{\ast }(t)=-\frac{{\sigma }_1{E}_S}{2w_1(1+{\rho }_1{E}_S)},\\ u_2^{\ast }(t)=-\frac{{\sigma }_{3}A}{2w_2(1+{\rho }_{2}A)}.\end{array}\right)\end{array}$$ 29

Thus boundedness of the optimal controls pair assists us to derive the control functions $u_1^{\ast }(t)$ and $u_2^{\ast }(t)$ in the following form:

$$\begin{array}{cc}\hfill u_1^{\ast }(t)& =\left\{\begin{array}{cc}0,& -\frac{{\sigma }_1{E}_S}{2w_1(1+{\zeta }_1{E}_S)}\le 0;\\ -\frac{{\sigma }_1{E}_S}{2w_1(1+{\zeta }_1{E}_S)},& 0<-\frac{{\sigma }_1{E}_S}{2w_1(1+{\zeta }_1{E}_S)}<1;\\ 1,& -\frac{{\sigma }_1{E}_S}{2w_1(1+{\zeta }_1{E}_S)}\ge 1.\end{array}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill u_2^{\ast }(t)& =\left\{\begin{array}{cc}0,& -\frac{{\sigma }_{3}A}{2w_2(1+{\zeta }_{2}A)}\le 0;\\ -\frac{{\sigma }_{3}A}{2w_2(1+{\zeta }_{2}A)},& 0<-\frac{{\sigma }_{3}A}{2w_2(1+{\zeta }_{2}A)}<1;\\ 1,& -\frac{{\sigma }_{3}A}{2w_2(1+{\zeta }_{2}A)}\ge 1.\end{array}\right)\hfill \end{array}$$

The optimal controls pair $u_1^{\ast }(t)$ and $u_2^{\ast }(t)$ in the compact form can be written as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_1^{\ast }(t)=min\{max\{-\frac{{\sigma }_1{E}_S}{2w_1(1+{\zeta }_1{E}_S)},0\},1\},\\ u_2^{\ast }(t)=min\{max\{-\frac{{\sigma }_{3}A}{2w_2(1+{\zeta }_{2}A)},0\},1\}.\end{array}\right)\end{array}$$ 30

Therefore it is notable that the solution of the FOCP could be obtained by replacing $u_1(t)$ and $u_2(t)$ in place of the optimal controls $u_1^{\ast }(t)$ and $u_2^{\ast }(t)$ in the system (13).

Accordingly, the optimality of the FOCP establishes a two-point boundary value problem with reference to a system of fractional order differential equations.

Remark 1

The optimality of the controlled system (13) referred to the adjoint system (25) together with the defined initial and boundary conditions and the optimal controls pair $(u_1^{\ast }(t),u_2^{\ast }(t))$ characterized by (29) could be presented as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill D_t^{\alpha }{E}_S& ={\mathrm{\Pi }}_1-(1-{\epsilon }_1)\frac{\beta E_{S}VA}{1+kV}-{\mu }_1{E}_S+\frac{u_1(t)E_S}{1+{\zeta }_1{E}_S},\hfill \\ \hfill D_t^{\alpha }{E}_I& =(1-{\epsilon }_1)\frac{\beta E_{S}VA}{1+kV}-{\mu }_2{E}_I,\hfill \\ \hfill D_t^{\alpha }A& ={\mathrm{\Pi }}_2-(1-{\epsilon }_1)\frac{\theta \beta E_{S}VA}{1+kV}-{\mu }_{3}A+\frac{u_2(t)A}{1+{\zeta }_{2}A},\hfill \\ \hfill D_t^{\alpha }V& =(1-{\epsilon }_2)p{E}_{I}V-({\mu }_4+\frac{q{E}_I}{\eta +V})V,\hfill \\ \hfill D_{T_f}^{\alpha }{\sigma }_1& =(1-{\epsilon }_1)\frac{\beta VA}{1+kV}({\sigma }_1-{\sigma }_2+\theta {\sigma }_3)+{\sigma }_1{\mu }_1-\frac{{\sigma }_1{u}_1(t)}{{(1+{\zeta }_1{E}_S)}^2},\hfill \\ \hfill D_{T_f}^{\alpha }{\sigma }_2& =-2w_3{E}_I+{\sigma }_2{\mu }_2-{\sigma }_4(1-{\epsilon }_2)pV-\frac{{\sigma }_{4}qV}{\eta +V},\hfill \\ \hfill D_{T_f}^{\alpha }{\sigma }_3& =(1-{\epsilon }_1)\frac{\beta E_{S}V}{1+kV}({\sigma }_1-{\sigma }_2+\theta {\sigma }_3)+{\sigma }_3{\mu }_3-\frac{{\sigma }_3{u}_2(t)}{{(1+{\zeta }_{2}A)}^2},\hfill \\ \hfill D_{T_f}^{\alpha }{\sigma }_4& =-2w_4{E}_I+(1-{\epsilon }_1)\frac{\beta E_{S}A}{{(1+kV)}^2}({\sigma }_1-{\sigma }_2+\theta {\sigma }_3)-{\sigma }_4(1-{\epsilon }_2)p{E}_I+{\mu }_4{\sigma }_4\hfill \\ \hfill & +\frac{{\sigma }_4\eta q{E}_I}{{(\eta +V)}^2}.\hfill \end{array}\right)\end{array}$$

The above optimal system reveals that it is required to keep the human host immune system strong enough to control the SARS-CoV-2 infection, which is possible if susceptible epithelial cells proliferation would be high. Implementation of the drug 2-DG benefits to lessen the level of infected epithelial cells and the load of virions in host body.

Numerical findings

To construct a fractional order model, the Caputo fractional derivatives are applied extensively in epidemiology to model infectious diseases taking into account the interactions between host immune response and virus particles in the past by incorporating the characteristic “memory” in the system. In this section, we intend to numerically visualize the kinetic behaviors of our proposed fractional order system (3) and control induced fractional order system (13) for memory $0<\alpha \le 1$. In order to solve our proposed fractional order system (3) numerically, we follow the iterative scheme presented in [46, 47] using MATLAB by taking the baseline parameter values from Table 1. Our proposed fractional order system (3) is fitted with the real-time patient data from Germany [17, 18].

Numerical simulation of epidemic system (3) without control

Figure 1 describes the dynamical behavior of the epidemic system (3) varying the memory effect $\alpha$, where $\alpha =1$ implies the alignment of the fractional order system (3) with its corresponding integer order system. It is observed that the stability of the solution trajectories exhibits periodic nature for $\alpha =1$, but in case of fractional order parameter values with $\alpha =0.90$ and $\alpha =0.80$, the disease system converges to its endemic steady state in shorter time. Figure 2 portrays the behavioral changes in the system (3) considering different values of q, the destruction rate of SARS-CoV-2 virions through immune response (taking $\alpha =0.8$). The host immune response utilizes its capability to destroy the SARS-CoV-2 virions through immunological memory. Figure 2 shows that increased immune destruction using immunological memory enables to control the COVID-19 infection.

Fig. 1.

Solution trajectories of the system (3) for different values of memory ($\alpha =1.0,0.90,0.80)$ keeping other parameter values same as listed in Table 1

Fig. 2.

Solution trajectories of the system (3) varying the destruction rate q of virions via host immune response ($q=0.3,0.4,0.5$) with memory $\alpha =0.8$ keeping other parameter values same as listed in Table 1

Figure 3 displays the phase portraits of the system (3) in phase spaces $E_S-E_I-A$ (left panel) and $E_I-A-V$ (right panel) exhibiting the dynamical behavior of the system in presence of memory ($\alpha =0.70,0.80,0.90$) and also for $\alpha =1.0$. The figure is showing that infection level could be monitored through immunological memory of healthy immune system.

Fig. 3.

Left Panel: Phase portrait of the system (3) corresponding to the state variables $E_S(t),E_I(t)$ and V(t) for different values of memory ($\alpha =1.0,0.90,0.80,0.70$). Right Panel: Phase portrait of the system (3) corresponding to the state variables $E_I(t),A(t)$ and V(t) for different values of memory ($\alpha =1.0,0.90,0.80,0.70$)

In Figs. 4 and 5, the efficacy of 2-DG in prohibiting the transmission of the infection and replication of SARS-CoV-2 virions are observed, respectively. It is worthwhile to notice that the reduced level of COVID-19 infection proper administration of the drug might be recommended. In this scenario, appropriate policies based on fractional order optimal control would be helpful in monitoring the level of infection and mitigating the infection.

Fig. 4.

Solution trajectories of the system (3) varying the efficacy ${\epsilon }_1$ of the drug 2-DG in blocking transmission of COVID-19 ($0\le {\epsilon }_1,\&{\epsilon }_1=0$) while $\alpha =0.8$ keeping other parameter values as same as listed in Table 1

Fig. 5.

Solution trajectories of the system (3) varying the efficacy ${\epsilon }_2$ of the drug 2-DG in blocking transmission of COVID-19 ($0\le {\epsilon }_2,\&{\epsilon }_2=0$) while $\alpha =0.8$ keeping other parameter values as same as listed in Table 1

Numerical simulation of epidemic system (13) with control

This subsection is concerned with directional behavior of the system (13) when fractional order optimal controls are applied to the disease system. In this regard, we numerically solve the state system (13) as an initial value problem and the co-state system (25) as a boundary value problem. Using an efficient iterative method, we obtain Figs. 6 and 7 whereas forward iterative scheme is applied to solve the state system (13) and backward iterative scheme is applied to solve the co-state system (25) for the values of the fractional order parameter $\alpha =1,0.85,0.75$. In Figs. 6 and 7, varying the weightage of the drug 2-DG, it is observed that the weightage of the drug input increases (in Fig. 7) as it agitates the host immune system in a regular manner. The healthy increase of immunity via antiviral effects of the drug 2-DG works as a stimulant in proliferation of the epithelial cells. Here we consider estimated value of the half-maximal constants as ${\rho }_{1,2}=0.3$.

Fig. 6.

Behaviors of the controlled system (13) varying memory ($\alpha =1,0.85,0.75)$ in presence of the control input $u_1$ (left panel) and the control input $u_2$ (right panel)

Fig. 7.

Behaviors of the controlled system (13) varying memory ($\alpha =1,0.85,0.75)$ in presence of the increased weightage of the control input $u_1$ (left panel) and the control input $u_2$ (right panel)

Discussion

Calibration of the complex multi-scale reciprocity between host and viral particles at micro level for the newly emerged COVID-19 infection is an exigent topic in the present scenario. In this study, a four-dimensional deterministic cell-limited model has been framed delineating the interplay among the host epithelial cells, host immune response and SARS-CoV-2 virions in COVID-19 transmission process. Our proposed epidemic model is perturbed into a Caputo fractional order deterministic system in presence of immunological memory. The apprehension regarding previous status of an epidemic benefits in inhibition of transmission and control of the infection. Caputo fractional differential equations are capable of proving specified and biologically interpretable initial conditions in modeling of a disease system. Additionally, the Caputo fractional order system has advantages in flexible utilization of classical initial conditions leading the non-negativity, uniqueness and local as well as global existence of solutions of the proposed epidemic system. The system is locally asymptotic stable around both the disease-free steady state and endemic steady state executed by the system.

Quantifying the role of 2-DG drug in controlling the SARS-CoV-2 infection via improvement of host immune response, fractional order optimal control problem (FOCP) has presented in the study. FOCP benefits to determine optimal dose of the drug 2-DG and its minimum systemic cost using Pontryagin’s Minimum Principle. Based on real data and some estimated data, the dynamical behaviors of the fractional order system (for both without control and controlled systems) have been studied numerically. We observed from our analytic as well as numerical findings that fractional order model generates better results than its integer order counterpart. These findings will assist the health policy makers for better administration of 2-DG in prevention and control of the SARS-CoV-2 infection on a global basis.

Author contribution

Conceptualization: A. N. C., J. M. Formal analysis: J. M., P. S. Investigation: J. M., P. S., A. N. C. Methodology: B. A., J. M. Project administration: B. A., A. N. C. Supervision: J. M., B. A. Writing — original draft: J. M., P. S. Writing — review and editing: B. A., A. N. C.

Funding

This research has been supported to the first author to pursue her Ph.D. (Swami Vivekananda Merit-cum-Means Scholarship) by the West Bengal Higher Education Department, Govt. of West Bengal, Bikash Bhavan, India with G.O. No. 52-Edn(B)/5B-15/2017 dated 07.06.2017.

Declarations

Ethical approval

The research demonstrated here has adhered to the accepted ethical standards of a genuine research study.

Informed consent

The content of the article is not submitted anywhere yet.

Conflict of interest

The authors declare no competing interests.