Quantifying the role of airborne transmission in the spread of COVID-19

Matthew Hayden; Bryce Morrow; Wesley Yang; Jin Wang; Matthew Hayden; Bryce Morrow; Wesley Yang; Jin Wang

doi:10.3934/mbe.2023027

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 1: 587-612. doi: 10.3934/mbe.2023027

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Research article Special Issues

Quantifying the role of airborne transmission in the spread of COVID-19

1.
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
2.
Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717, USA
3.
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA
4.
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA

Academic Editor: Yang Kuang

Received: 29 August 2022 Revised: 04 October 2022 Accepted: 08 October 2022 Published: 12 October 2022

There is an ongoing debate on the different transmission modes of SARS-CoV-2 and their relative contributions to the pandemic. In this paper, we employ a simple mathematical model, which incorporates both the human-to-human and environment-to-human transmission routes, to study the transmission dynamics of COVID-19. We focus our attention on the role of airborne transmission in the spread of the disease in a university campus setting. We conduct both mathematical analysis and numerical simulation, and incorporate published experimental data for the viral concentration in the air to fit model parameters. Meanwhile, we compare the outcome to that of the standard SIR model, utilizing a perturbation analysis in the presence of multiple time scales. Our data fitting and numerical simulation results show that the risk of airborne transmission for SARS-CoV-2 strongly depends on how long the virus can remain viable in the air. If the time for this viability is short, the airborne transmission route would be inconsequential in shaping the overall transmission risk and the total infection size. On the other hand, if the infectious virus can persist in aerosols beyond a few hours, then airborne transmission could play a much more significant role in the spread of COVID-19.

Keywords:

Citation: Matthew Hayden, Bryce Morrow, Wesley Yang, Jin Wang. Quantifying the role of airborne transmission in the spread of COVID-19[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 587-612. doi: 10.3934/mbe.2023027

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Abstract

1. Introduction

A mathematical model is a helpful tool to recognize the conduct of an infection when it starts to affect the community and it is useful to analyze under what conditions it can be screened out or to be continued [1]. A virus is known as infectious when any disease is transferred from one person to another via different ways of transmission like droplets generated when an infected person coughs, sneezes, or exhales, or direct contact with another human, water, or any physical product. To analyze this type of transmission we need some authentic mathematical tools in which few of them are difference equations, initial conditions, working parameters, and statistical estimation. In this new era, new mathematical techniques give us more updated and reliable tools to understand many diseases or infections in epidemiology and even give us updated strategies to control disease or infection in different and suitable conditions [2].

From all of the viruses, the COVID-19 is gradually becoming a watershed pandemic in the antiquity of the planet. COVID-19 is an abbreviation of Coronavirus disease which started in 2019. In December of 2019, the first case of COVID-19 was observed in Wuhan, the city of China [3]. The common symptoms of COVID-19 are loss of smell and taste, fever, dry cough, shortening of breath, fatigue, muscle, and joint pain, phlegm production, sore throat, headache, and chills, these symptoms vary from person to person. The most common incubation period ranges from 1 to 12 days. COVID-19 spreads by physical interaction between individuals. Use of masks, sanitizer, and having a distance of 2 m between individuals results in minimizing the spread of the virus up to much extent [4]. These vaccines played a bold role in minimizing the spread of COVID-19. The main focused area of this spread is the working area, schools, offices, markets, and other open circles [5],[6].

Fractional derivative was originated in 1695. If we describe the list of fractional derivatives then it is divided into two types. Caputo, Riemann-Liouville, and Katugampola [7] are fractional derivatives with the singular kernel. Caputo-Fabrizio(exponential) [8] and ABC(Mittag-Leffler) [9] are fractional derivatives without singular kernels. Fractional calculus has very vast application properties in our daily life. It is being used in chemical, biological, physical, finance, pharmaceutical, engineering [10],[11], and many other fields [12]–[14]. FFD is mostly used because it gives a realistic way of representation of our model and hence we have used this same for representing our COVID-19 epidemics [15]–[18]. A time-fractional compartmental model for the COVID-19 pandemic [21] and classical SIR model for COVID-19 in United States is study in [22]. The COVID-19 pandemic (caused by SARS-CoV-2) has introduced significant challenges for accurate prediction of population morbidity and mortality by traditional variable-based methods of estimation. Challenges to modeling include inadequate viral physiology comprehension and fluctuating definitions of positivity between national-to-international data. This paper proposes that accurate forecasting of COVID-19 caseload may be best preformed non-perimetrically, by vector autoregressive (VAR) of verifiable data regionally [23]. Fundamental properties of the new generalized fractional derivatives in the sense of Caputo and RiemannLiouville are rigorously studied and its related work [24]–[26]. COVID-19 Decision-Making System (CDMS) was developed to study disease transmission in [27]. The change in atmospheric pollution from a public lockdown in Greece introduced to curb the spread of the COVID-19 is examined based on ground-based and satellite observations and some related issues in [28]–[30].

2. Basic concepts

The fractional-order derivative of AB in Reimann Liouville-Caputo sense (ABC) [19] is given by

$_{γ}^{A B C} D_{t}^{γ} {f (t)} = \frac{A B (γ)}{m - γ} \int_{γ}^{t} \frac{d^{m}}{d w^{m}} f (w) E_{γ} [- γ \frac{{(t - w)}^{γ}}{m - γ}] d w, m - 1 < γ < m$ (1)

where E_γ is the Mittag-Leffler function and AB(γ) is a normalization function and AB(0) = AB(1) = 1. The Laplace transform of above is given by

$[_{γ}^{A B C} D_{t}^{γ} f (t)] (s) = \frac{A B (γ)}{1 - γ} \frac{s^{γ} L [f (t)] (s) - s^{γ - 1} f (0)}{s^{γ} + \frac{γ}{1 - γ}}$ (2)

with the aid of sumudu transformation, we get

$S T [_{γ}^{A B C} D_{t}^{γ} f (t)] (s) = \frac{B (γ)}{1 - γ} (γ Γ (γ + 1) E_{γ} (- \frac{1}{1 - γ} ν^{γ})) \times [S T (f (t)) - f (0)]$ (3)

The ABC fractional integral of order γ of a function f(t) is given by

$_{γ}^{A B C} I_{t}^{γ} {f (t)} = \frac{1 - (γ)}{B - γ} f (t) + \frac{(γ)}{B (γ) Γ (γ)} \int_{γ}^{t} f (s) (t - s)^{γ - 1} d s$ (4)

3. Materials and methods

In this section, consider the improved SEIR model given in [20] having compartments SEIQRPD, where S represents the number of uninfected individuals, E represents infected individuals at the time t but still in incubation period (without clinical symptoms and low infectivity), I represents the number of infected individuals at the time t (with obvious clinical symptoms), Q represents the number of individuals who have been diagnosed and isolated at the time t, R represents the number of recovered individuals at the time t, P represents the number of susceptible individuals who are not exposed to the external environment at the time t and D represents the number of death cases at time t.

$\begin{matrix} _{0}^{A B C} D_{t}^{γ} S (t) = - β_{1} (t) a_{1} (S (t)) b_{1} (I (t)) - β_{2} a_{2} (S (t)) b_{2} (E (t)) - ρ S (t), \\ _{0}^{A B C} D_{t}^{γ} E (t) = β_{1} (t) a_{1} (S (t)) b_{1} (I (t)) + β_{2} a_{2} (S (t)) b_{2} (E (t)) - ϵ E (t), \\ _{0}^{A B C} D_{t}^{γ} I (t) = ϵ E (t) - δ I (t), \\ _{0}^{A B C} D_{t}^{γ} Q (t) = δ I (t) - (λ (t) + κ (t)) Q (t), \\ _{0}^{A B C} D_{t}^{γ} R (t) = λ (t) Q (t), \\ _{0}^{A B C} D_{t}^{γ} P (t) = ρ S (t) \\ _{0}^{A B C} D_{t}^{γ} D (t) = κ (t) Q (t) \end{matrix}$ (5)

here $β_{1} (t) = σ_{1} e x p (- σ_{2} t), λ (t) = λ_{1} (1 - e x p (λ_{2} t))$ and $κ (t) = κ_{1} e x p (- κ_{2} t)$ $σ_{1}, σ_{2}, λ_{1}, λ_{2}, κ_{1}$ and κ₂ are the parameters which are all positive, here simulation is used by the σ affect of government control. It should be emphasized that the protection rate ρ for susceptible individuals also reflects the intensity of government control [18]. $_{0}^{A B C} D_{t}^{γ}$ , is the ABC sense fractional derivative with $0 < γ \leq 1$ . The initial conditions of the system Eq 5 are:

$\begin{matrix} S_{0} (t) = S (0), E_{0} (t) = E (0), I_{0} (t) = I (0), Q_{0} (t) = Q (0) \\ R_{0} (t) = R (0), P_{0} (t) = P (0), D_{0} (t) = D (0) \end{matrix}$ (6)

applying ST operator on both sides, we get

$\begin{matrix} O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ}) [S T (S (t)) - S (0)] \\ = S T [- β_{1} (t) a_{1} (S (t)) b_{1} (I (t)) - β_{2} a_{2} (S (t)) b_{2} (E (t)) - ρ S (t)], \\ O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ}) [S T (E (t)) - E (0)] \\ = S T [β_{1} (t) a_{1} (S (t)) b_{1} (I (t)) + β_{2} a_{2} (S (t)) b_{2} (E (t)) - ϵ E (t)], \\ O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ}) [S T (I (t)) - I (0)] \\ = S T [ϵ E (t) - δ I (t)], \\ O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ}) [S T (Q (t)) - Q (0)] \\ = S T [δ I (t) - (λ (t) + κ (t)) Q (t)], \\ O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ}) [S T (R (t)) - R (0)] \\ = S T [λ (t) Q (t)], \\ O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ}) [S T (P (t)) - P (0)] \\ = S T [ρ S (t)], \\ O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ}) [S T (D (t)) - D (0)] \\ = S T [κ (t) Q (t)] \end{matrix}$ (7)

where $O_{γ} = \frac{B (γ) γ Γ (γ + 1)}{1 - γ}$ system Eq 7 becomes

$\begin{matrix} S T [S (t)] = S (0) + \frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [- β_{1} (t) a_{1} (S (t)) b_{1} (I (t)) - β_{2} a_{2} (S (t)) b_{2} (E (t)) - ρ S (t)], \\ S T [E (t)] = E (0) + \frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [β_{1} (t) a_{1} (S (t)) b_{1} (I (t)) + β_{2} a_{2} (S (t)) b_{2} (E (t)) - ϵ E (t)], \\ S T [I (t)] = I (0) + \frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [ϵ E (t) - δ I (t)], \\ S T [Q (t)] = Q (0) + \frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [δ I (t) - (λ (t) + κ (t)) Q (t)], \\ S T [R (t)] = R (0) + \frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [λ (t) Q (t)], \\ S T [P (t)] = P (0) + \frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [ρ S (t)], \\ S T [D (t)] = D (0) + \frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [κ (t) Q (t)] \end{matrix}$ (8)

taking inverse Sumudu Transform on both sides, we get

$\begin{matrix} S (t) = S (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [- β_{1} (t) a_{1} (S (t)) b_{1} (I (t)) - β_{2} a_{2} (S (t)) b_{2} (E (t)) - ρ S (t)]}, \\ E (t) = E (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [β_{1} (t) a_{1} (S (t)) b_{1} (I (t)) + β_{2} a_{2} (S (t)) b_{2} (E (t)) - ϵ E (t)]}, \\ I (t) = I (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [ϵ E (t) - δ I (t)]}, \\ Q (t) = Q (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [δ I (t) - (λ (t) + κ (t)) Q (t)]}, \\ R (t) = R (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [λ (t) Q (t)]}, \\ P (t) = P (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [ρ S (t)]}, \\ D (t) = D (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [κ (t) Q (t)]} \end{matrix}$ (9)

Therefore, the following is obtained

$\begin{matrix} S_{(m + 1)} (t) = S_{m} (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [- β_{1} a_{1} (S_{m} (t)) b_{1} (I_{m} (t)) - β_{2} a_{2} (S_{m} (t)) b_{2} (E_{m} (t)) - ρ S_{m} (t)]}, \\ E_{(m + 1)} (t) = E_{m} (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [β_{1} a_{1} (S_{m} (t)) b_{1} (I_{m} (t)) + β_{2} a_{2} (S_{m} (t)) b_{2} (E_{m} (t)) - ϵ E_{m} (t)]}, \\ I_{(m + 1)} (t) = I_{m} (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [ϵ E_{m} (t) - δ I_{m} (t)]}, \\ Q_{(m + 1)} (t) = Q_{m} (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [δ I_{m} (t) - (λ + κ) Q_{m} (t)]}, \\ R_{(m + 1)} (t) = R_{m} (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [λ Q_{m} (t)]}, \\ P_{(m + 1)} (t) = P_{m} (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [ρ S_{m} (t)]}, \\ D_{(m + 1)} (t) = D_{m} (0) + S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [κ Q_{m} (t)]} \end{matrix}$ (10)

Let's consider Eq 10, and then we get

$\begin{array}{l} S (t) = \underset{m \to \infty}{l i m} S_{m} (t); E (t) = \underset{m \to \infty}{l i m} E_{m} (t); I (t) = \underset{m \to \infty}{l i m} I_{m} (t); \\ Q (t) = \underset{m \to \infty}{l i m} Q_{m} (t); R (t) = \underset{m \to \infty}{l i m} R_{m} (t); P (t) = \underset{m \to \infty}{l i m} P_{m} (t)); D (t) = \underset{m \to \infty}{l i m} D_{m} (t) \end{array}$ (11)

Theorem 3.1: Let $(X, | . |)$ be a Banach space and H a self-map of X satisfying

$∥ H_{r} - H_{x} ∥ \leq θ ∥ X - H_{r} ∥ + θ ∥ r - x ∥$ (12)

for all $r, x \in X$ , where $0 \leq θ < 1$ . Assume that H is Pichard H-stable

Let us consider Eq 10, and we obtain

$\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})}$ (13)

the above equation is associated with the fractional Lagrange multiplier.

Proof

Define K be a self-map is given by

$\begin{matrix} K [S_{(m + 1)} (t)] = S_{(m + 1)} (t) = S_{m} (0) + S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [- β_{1} a_{1} (S_{m} (t)) b_{1} (I_{m} (t)) - β_{2} a_{2} (S_{m} (t)) b_{2} (E_{m} (t)) - ρ S_{m} (t)] \\ K [E_{(m + 1)} (t)] = E_{(m + 1)} (t) = E_{m} (0) + S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [β_{1} a_{1} (S_{m} (t)) b_{1} (I_{m} (t)) + β_{2} a_{2} (S_{m} (t)) b_{2} (E_{m} (t)) - ϵ E_{m} (t)] \\ K [I_{(m + 1)} (t)] = I_{(m + 1)} (t) = I_{m} (0) + S T^{- 1} [\frac{1 -}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [ϵ E_{m} (t) - δ I_{m} (t)] \\ K [Q_{(m + 1)} (t)] = Q_{(m + 1)} (t) = Q_{m} (0) + S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [δ I_{m} (t) - (λ + κ) Q_{m} (t)] \\ K [R_{(m + 1)} (t)] = R_{(m + 1)} (t) = R_{m} (0) + S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [λ Q_{m} (t)] \\ K [P_{(m + 1)} (t)] = P_{(m + 1)} (t) = P_{m} (0) + S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [ρ S_{m} (t)] \\ K [D_{(m + 1)} (t)] = D_{(m + 1)} (t) = D_{m} (0) + S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [κ Q_{m} (t)] \end{matrix}$ (14)

Applying the properties of the norm and triangular inequality, we get

$\begin{matrix} ∥ K [S_{m} (t)] - K [S_{n} (t)] ∥ \leq ∥ S_{m} (t) - S_{n} (t) ∥ + ∥ S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [- β_{1} a_{1} S_{m} (t) b_{1} (I_{m} (t)) - β_{2} a_{2} (S_{m} (t)) b_{2} (E_{m} (t)) - ρ S_{m} (t)]} \\ - S T^{- 1} {\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [- β_{1} a_{1} (S_{n} (t)) b_{1} (I_{n} (t)) - β_{2} a_{2} (S_{n} (t)) b_{2} (E_{n} (t)) - ρ S_{n} (t)]} ∥, \\ ∥ K [E_{m} (t)] - K [E_{n} (t)] ∥ \leq ∥ E_{m} (t) - E_{n} (t) ∥ + ∥ S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [β_{1} a_{1} (S_{m} (t)) b_{1} (I_{m} (t)) + β_{2} a_{2} (S_{m} (t)) b_{2} (E_{m} (t)) - ϵ E_{m} (t)]} \\ - S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [β_{1} a_{1} (S_{n} (t)) b_{1} (I_{n} (t)) + β_{2} a_{2} (S_{n} (t)) b_{2} (E_{n} (t)) - ϵ E_{n} (t)]} ∥, \\ ∥ K [I_{m} (t)] - K [I_{n} (t)] ∥ \leq ∥ I_{m} (t) - I_{n} (t) ∥ + ∥ S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [ϵ (E_{m} (t)) - δ (I_{m} (t)))]} - S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [ϵ E_{n} (t) - δ I_{n} (t)]} ∥, \\ ∥ K [Q_{m} (t)] - K [Q_{n} (t)] ∥ \leq ∥ Q_{m} (t) - Q_{n} (t) ∥ + ∥ S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [δ I_{m} (t) - (λ + κ) Q_{m} (t))]} - S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [δ I_{n} (t) - (λ + κ) Q_{n} (t)]} ∥ \\ ∥ K [R_{m} (t)] - K [R_{n} (t)] ∥ \leq ∥ R_{m} (t) - R_{n} (t) ∥ + ∥ S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [λ (Q_{m} (t))]} - S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [λ Q_{n} (t)]} ∥, \\ ∥ K [P_{m} (t)] - K [P_{n} (t)] ∥ \leq ∥ P_{m} (t) - P_{n} (t) ∥ + ∥ S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [ρ S_{m} (t)]} - S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [ρ S_{n} (t)]} ∥, \\ ∥ K [D_{m} (t)] - K [D_{n} (t)] ∥ \leq ∥ D_{m} (t) - D_{n} (t) ∥ + ∥ S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [κ Q_{m} (t)]} - S T^{- 1} [\frac{1}{O_{γ} E_{γ} (- \frac{1}{1 - γ} ω^{γ})} \\ \times S T [κ Q_{n} (t)]} ∥ \end{matrix}$ (15)

K fulfills the conditions associated with theorem 3.1 when

$θ = (0,0,0,0,0,0,0), θ = (\begin{array}{l} ∥ S_{m} (t) - S_{n} (t) ∥ \times ∥ - S_{m} (t) + S_{n} (t) ∥ \\ - ∥ β_{1} a_{1} (S_{m} (t) - S_{n} (t)) b_{1} (I_{m} (t) - I_{n} (t)) ∥ \\ - ∥ β_{2} a_{2} (S_{m} (t) - S_{n} (t)) b_{2} (E_{m} (t) - E_{n} (t)) ∥ \\ - ∥ ρ (S_{m} (t) - S_{n} (t)) ∥ \\ \times ∥ E_{m} (t) - E_{n} (t) ∥ \times ∥ - E_{m} (t) + E_{n} (t) ∥ \\ + ∥ β_{1} a_{1} (S_{m} (t) - S_{n} (t)) b_{2} (E_{m} (t) - E_{n} (t)) ∥ \\ - ∥ ϵ (E_{m} (t) - E_{n} (t)) ∥ \\ \times ∥ I_{m} (t) - I_{n} (t) ∥ \times ∥ - I_{m} (t) + I_{n} (t) ∥ \\ + ∥ ϵ (E_{m} (t) - E_{n} (t)) ∥ - δ ∥ I_{m} (t) - I_{n} (t) ∥ \\ \times ∥ Q_{m} (t) - Q_{n} (t) ∥ \times ∥ - Q_{m} (t) + Q_{n} (t) ∥ \\ + ∥ δ (I_{m} (t) - I_{n} (t)) ∥ - (λ (t) + κ (t)) ∥ Q_{m} (t) - Q_{n} (t) ∥ \\ \times ∥ R_{m} (t) - R_{n} (t) ∥ \times ∥ - R_{m} (t) + R_{n} (t) ∥ \\ + ∥ λ (Q_{m} (t) - Q_{n} (t)) ∥ \\ \times ∥ P_{m} (t) - P_{n} (t) ∥ - ∥ - P_{m} (t) + P_{n} (t) ∥ \\ + ∥ ρ (S_{m} (t) - S_{n} (t)) ∥ \\ \times ∥ D_{m} (t) - D_{n} (t) ∥ - ∥ - D_{m} (t) + D_{n} (t) ∥ \\ + ∥ κ (Q_{m} (t) - Q_{n} (t)) ∥ \end{array}$ (16)

and we add that K is Picard K-stable.

4. Numerical results and discussion

In this section, consider the numerical simulations of the proposed scheme using the ABC technique for the fractional-order COVID-19 model. Figure 1 shows the simulation S(t) represents the number of uninfected individuals. Shows a deep decreasing curve till point (20, 0.25) and then becomes constant and reduced to zero at (100, 0). Figure 2 shows the simulation E(t) of infected individuals but still is in the incubation period (without clinical symptoms and low infectivity). Figure 3 I(t) which represents the number of infected individuals. Here the graph shows a rapid increase (10, 9) and then decrease rapidly with the same rate and then it becomes constant at (100, 0). Figure 4 represents the number of individuals who have been diagnosed and isolated. Figures 5 and 6 shows the simulation of recovered individuals and those not exposed to the external environment respectively. Figure 7 shows the simulation D(t), which represents the death due to increasing or decreasing the infection rate of COVID-19 in society. It can be easily observed from all figures the solution will converge to steady-state and lie in the bounded domain by decreasing the fractional value. Moreover, it has been demonstrated that physical processes are better well described using the derivatives of fractional order which are more accurate and reliable in comparison with the classical-order derivatives. Moreover, it can be seen from all figures that tell that all infected individual comes zero after a few days due to the quarantine effect. The behavior of the dynamics obtained for different instances of fractional-order was shown in the form of numerical results has been reported.

Figure 1. Simulation of S(t) at the time t with parametric value of γ with ABC.

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Figure 2. Simulation of E(t) at the time t with parametric value of γ with ABC.

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Figure 3. Simulation of I(t) at the time t with parametric value of γ with ABC.

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Figure 4. Simulation of Q(t) at the time t with parametric value of γ with ABC.

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Figure 5. Simulation of R(t) at the time t with parametric value of γ with ABC.

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Figure 6. Simulation of P(t) at the time t with parametric value of γ with ABC.

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Figure 7. Simulation of D(t) at the time t with parametric value of γ with ABC.

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5. Conclusions

We consider the COVID-19 model with fractional operator for this work to check the dynamical behavior of infection of disease in society. In this regard, ABC derivative gave a realistic approach to analyze the effect of disseise during Quarantine which will be helpful for such type of epidemic. The existence and unique solution of the fractional-order model was made with the help of fixed point theory and iterative method. Numerical simulation has been made to check the actual behavior of the COVID-19 effect during quarantine which shows that infected individuals start decreasing after a few days. Such kind of results are very helpful for planning, decision-making, and developing control strategies to overcome the effect of COVID-19 in society.

References

[1]	World Health Organization, Coronavirus disease (COVID-19) pandemic. Available from: https://www.who.int/emergencies/diseases/novel-coronavirus-2019.
[2]	Centers for Disease Control and Prevention, SARS-CoV-2 transmission. Available from: https://www.cdc.gov/coronavirus/2019-ncov/science/science-briefs/sars-cov-2-transmission.html.
[3]	C. Geller, M. Varbanov, R. E. Duval, Human coronaviruses: insights into environmental resistance and its influence on the development of new antiseptic strategies, Viruses, 4 (2012), 3044–3068. https://doi.org/10.3390%2Fv4113044
[4]	G. Kampf, D. Todt, S. Pfaender, E. Steinmann, Persistence of coronaviruses on inanimate surfaces and its inactivation with biocidal agents, J. Hosp. Infect., 104 (2020), 246–251. https://doi.org/10.1016/j.jhin.2020.01.022 doi: 10.1016/j.jhin.2020.01.022
[5]	C. Yeo, S. Kaushal, D. Yeo, Enteric involvement of coronaviruses: Is faecal-oral transmission of SARS-CoV-2 possible? Lancet Gastroenterol. Hepatol., 5 (2020), 335–337. https://doi.org/10.1016/s2468-1253(20)30048-0 doi: 10.1016/s2468-1253(20)30048-0
[6]	N. van Doremalen, T. Bushmaker, D. H. Morris, M. G. Holbrook, A. Gamble, B. R. Williamson, et al., Aerosol and surface stability of SARS-CoV-2 as compared with SARS-CoV-1, N. Engl. J. Med., 382 (2020), 1564–1567. https://doi.org/10.1056/NEJMc2004973 doi: 10.1056/NEJMc2004973
[7]	A. C. Fears, W. B. Klimstra, P. Duprex, A. Hartman, S. C. Weaver, K. C. Plante, et al., Comparative dynamic aerosol efficiencies of three emergent coronaviruses and the unusual persistence of SARS-CoV-2 in aerosol suspensions, medRxiv, 2020 (2020). https://doi.org/10.1101/2020.04.13.20063784 doi: 10.1101/2020.04.13.20063784
[8]	R. Bhardwaj, A. Agrawal, How coronavirus survives for days on surfaces, Phys. Fluids, 32 (2020), 111706. https://doi.org/10.1063/5.0033306 doi: 10.1063/5.0033306
[9]	Z. He, S. Shao, J. Li, S. S. Kumar, J. B. Sokoloff, J. Hong, Droplet evaporation residue indicating SARS-COV-2 survivability on surfaces, Phys. Fluids, 33 (2021), 013309. https://doi.org/10.1063%2F5.0038562
[10]	Y. Shah, J. W. Kurelek, S. D. Peterson, S. Yarusevych, Experimental investigation of indoor aerosol dispersion and accumulation in the context of COVID-19: effects of masks and ventilation, Phys. Fluids, 33 (2021), 073315. https://doi.org/10.1063/5.0057100 doi: 10.1063/5.0057100
[11]	G. A. Somsen, C. J. M. van Rijn, S. Kooij, R. A. Bem, D. Bonn, Measurement of small droplet aerosol concentrations in public spaces using handheld particle counters, Phys. Fluids, 32 (2020), 121707. https://doi.org/10.1063%2F5.0035701
[12]	T. Greenhalgh, J. L. Jimenez, K. A. Prather, Z. Tufekci, D. Fisman, R. Schooley, Ten scientific reasons in support of airborne transmission of SARS-CoV-2, Lancet, 397 (2021), 1603–1605. https://doi.org/10.1016/s0140-6736(21)00869-2 doi: 10.1016/s0140-6736(21)00869-2
[13]	C. C. Wang, K. A. Prather, J. Sznitman, J. L. Jimenez, S. S. Lakdawala, Z. Tufekci, et al., Airborne transmission of respiratory viruses, Science, 373 (2021), eabd9149. https://doi.org/10.1126/science.abd9149 doi: 10.1126/science.abd9149
[14]	D. Lewis, Why the WHO took two years to say COVID is airborne, Nature, 604 (2022), 26–31. https://doi.org/10.1038/d41586-022-00925-7 doi: 10.1038/d41586-022-00925-7
[15]	X. Zhang, J. Wu, L. M. Smith, X. Li, O. Yancey, A. Franzblau, et al., Monitoring SARS-CoV-2 in air and on surfaces and estimating infection risk in buildings and buses on a university campus, J. Exposure Sci. Environ. Epidemiol., 32 (2022), 751–758. https://doi.org/10.1038/s41370-022-00442-9 doi: 10.1038/s41370-022-00442-9
[16]	J. R. Port, C. K. Yinda, I. O. Owusu, M. Holbrook, R. Fischer, T. Bushmaker, et al., SARS-CoV-2 disease severity and transmission efficiency is increased for airborne compared to fomite exposure in Syrian hamsters, Nat. Commun., 12 (2021), 4985. https://doi.org/10.1038/s41467-021-25156-8 doi: 10.1038/s41467-021-25156-8
[17]	J. Bai, X. Wang, J. Wang, An epidemic-economic model for COVID-19, Math. Biosci. Eng., 19 (2022), 9658–9696. https://doi.org/10.3934/mbe.2022449 doi: 10.3934/mbe.2022449
[18]	R. Li, S. Pei, B. Chen, Y. Song, T. Zhang, W. Yang, et al., Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2), Science, 368 (2020), 489–493. https://doi.org/10.1126/science.abb3221 doi: 10.1126/science.abb3221
[19]	K. Liang, Mathematical model of infection kinetics and its analysis for COVID-19, SARS and MERS, Infect., Genet. Evol., 82 (2020), 104306. https://doi.org/10.1016/j.meegid.2020.104306 doi: 10.1016/j.meegid.2020.104306
[20]	B. Tang, X. Wang, Q. Li, N. L. Bragazzi, S. Tang, Y. Xiao, et al., Estimation of the transmission risk of 2019-nCoV and its implication for public health interventions, J. Clin. Med., 9 (2020), 462. https://doi.org/10.3390%2Fjcm9020462
[21]	A. R. Tuite, D. N. Fisman, Reporting, epidemic growth, and reproduction numbers for the 2019 novel coronavirus (2019-nCoV) epidemic, Ann. Intern. Med., 172 (2020), 567–568. https://doi.org/10.7326/m20-0358 doi: 10.7326/m20-0358
[22]	J. Wang, Mathematical models for COVID-19: applications, limitations, and potentials, J. Public Health Emerg., 4 (2020), 9. https://doi.org/10.21037%2Fjphe-2020-05
[23]	J. T. Wu, K. Leung, G. M. Leung, Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study, Lancet, 395 (2020), 689–697. https://doi.org/10.1016/S0140-6736(20)30260-9 doi: 10.1016/S0140-6736(20)30260-9
[24]	C. Yang, J. Wang, A mathematical model for the novel coronavirus epidemic in Wuhan, China, Math. Biosci. Eng., 17 (2020), 2708–2724. https://doi.org/10.3934/mbe.2020148 doi: 10.3934/mbe.2020148
[25]	C. Yang, J. Wang, Modeling the transmission of COVID-19 in the US – A case study, Infect. Dis. Modell., 6 (2021), 195–211. https://doi.org/10.1016/j.idm.2020.12.006 doi: 10.1016/j.idm.2020.12.006
[26]	C. Yang, J. Wang, COVID-19 and underlying health conditions: a modeling investigation, Math. Biosci. Eng., 18 (2021), 3790–3812. https://doi.org/10.3934%2Fmbe.2021191
[27]	K. M. Bubar, K. Reinholt, S. M. Kissler, M. Lipsitch, S. Cobey, Y. H. Grad, et al., Model-informed COVID-19 vaccine prioritization strategies by age and serostatus, Science, 371 (2021), 916–921. https://doi.org/10.1126/science.abe6959 doi: 10.1126/science.abe6959
[28]	B. H. Foy, B. Wahl, K. Mehta, A. Shet, G. I. Menon, C. Britto, Comparing COVID-19 vaccine allocation strategies in India: a mathematical modelling study, Int. J. Infect. Dis., 103 (2021), 431–438. https://doi.org/10.1016/j.ijid.2020.12.075 doi: 10.1016/j.ijid.2020.12.075
[29]	E. Shim, Optimal allocation of the limited COVID-19 vaccine supply in South Korea, J. Clin. Med., 10 (2021), 591. https://doi.org/10.3390%2Fjcm10040591
[30]	M. Hadei, S. R. Mohebbi, P. K. Hopke, A. Shahsavani, S. Bazzazpour, M. Alipour, et al., Presence of SARS-CoV-2 in the air of public places and transportation, Atmos. Pollut. Res., 12 (2021), 302–306. https://doi.org/10.1016/j.apr.2020.12.016 doi: 10.1016/j.apr.2020.12.016
[31]	R. G. Passos, M. B. Silveira, J. S. Abrahão, Exploratory assessment of the occurrence of SARS-CoV-2 in aerosols in hospital facilities and public spaces of a metropolitan center in Brazil, Environ. Res., 195 (2021), 110808. https://doi.org/10.1016/j.envres.2021.110808 doi: 10.1016/j.envres.2021.110808
[32]	A. Robotto, A. Civra, P. Quaglino, D. Polato, E. Brizio, D. Lembo, SARS-CoV-2 airborne transmission: a validated sampling and analytical method, Environ. Res., 200 (2021), 111783. https://doi.org/10.1016%2Fj.envres.2021.111783
[33]	F. Brauer, A singular perturbation approach to epidemics of vector-transmitted diseases, Infect. Dis. Modell., 4 (2019), 115–123. https://doi.org/10.1016/j.idm.2019.04.004 doi: 10.1016/j.idm.2019.04.004
[34]	E. J. Hinch, Perturbation Methods, Cambridge University Press, 1991. https://doi.org/10.1017/CBO9781139172189
[35]	M. H. Holmes, Introduction to Perturbation Methods, Springer, 1995. https://doi.org/10.1007/978-1-4612-5347-1
[36]	G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967. https://doi.org/10.1017/CBO9780511800955
[37]	F. Verhulst, Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics, Springer, 2005. https://doi.org/10.1007/0-387-28313-7
[38]	University of Michigan COVID-19 Data, Campus Maize & Blueprint, 2020. Available from: https://campusblueprint.umich.edu/dashboard.
[39]	G. Chowell, Fitting dynamic models to epidemic outbreaks with quantified uncertainty: a primer for parameter uncertainty, identifiability, and forecasts, Infect. Dis. Modell., 2 (2017), 379–398. https://doi.org/10.1016/j.idm.2017.08.001 doi: 10.1016/j.idm.2017.08.001
[40]	A. Raue, J. Karlsson, M.P. Saccomani, M. Jirstrand, J. Timmer, Comparison of approaches for parameter identifiability analysis of biological systems, Bioinformatics, 30 (2014), 1440–1448. https://doi.org/10.1093/bioinformatics/btu006 doi: 10.1093/bioinformatics/btu006
[41]	Centers for Disease Control and Prevention, COVID-19. Available from: https://www.cdc.gov/coronavirus/2019-ncov.
[42]	J. Dehning, J. Zierenberg, F. P. Spitzner, M. Wibral, J. P. Neto, M. Wilczek, et al., Inferring change points in the spread of COVID-19 reveals the effectiveness of interventions, Science, 369 (2020), eabb9789. https://doi.org/10.1126/science.abb9789 doi: 10.1126/science.abb9789
[43]	L. Refisch, F. Lorenz, T. Riedlinger, H. Taubenbock, M. Fischer, L. Grabenhenrich, et al., Data-driven prediction of COVID-19 cases in Germany for decision making, BMC Med. Res. Methodol., 22 (2022), 116. https://doi.org/10.1186/s12874-022-01579-9 doi: 10.1186/s12874-022-01579-9
[44]	H. A. Aboubakr, T. A. Sharafeldin, S. M. Goyal, Stability of SARS-CoV-2 and other coronaviruses in the environment and on common touch surfaces and the influence of climatic conditions: a review, Transboundary Emerging Dis., 68 (2021), 296–312. https://doi.org/10.1111/tbed.13707 doi: 10.1111/tbed.13707
[45]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[46]	C. Castillo-Chavez, Z. Feng, W. Huang, On the computation of $\mathcal{R}_0$ and its role on global stability, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Springer-Verlag, New York, (2002), 229–250. https://doi.org/10.1007/978-1-4613-0065-6
[47]	Z. Shuai, P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513–1532. https://doi.org/10.1137/120876642 doi: 10.1137/120876642

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