key: cord-0726700-6fpogc50
authors: Bavi, O.; Hosseininia, M.; Heydari, M.H.; Bavi, N.
title: SARS-CoV-2 rate of spread in and across tissue, groundwater and soil: A meshless algorithm for the fractional diffusion equation
date: 2022-02-09
journal: Eng Anal Bound Elem
DOI: 10.1016/j.enganabound.2022.01.018
sha: 4c45b0ff7739defa238df48cc26e67a8687633da
doc_id: 726700
cord_uid: 6fpogc50

The epidemiological aspects of the viral dynamic of the SARS-CoV-2 have become increasingly crucial due to major questions and uncertainties around the unaddressed issues of how corpse burial or the disposal of contaminated waste impacts nearby soil and groundwater. Here, a theoretical framework base on a meshless algorithm using the moving least squares (MLS) shape functions is adopted for solving the time-fractional model of the viral diffusion in and across three different environments including water, tissue, and soil. Our computations predict that by considering the [Formula: see text] (order of fractional derivative) best fit to experimental data, the virus has a traveling distance of [Formula: see text] in water after 22, regardless of the source of contamination (e.g., from tissue or soil). The outcomes and extrapolations of our study are fundamental for providing valuable benchmarks for future experimentation on this topic and ultimately for the accurate description of viral spread across different environments. In addition to COVID-19 relief efforts, our methodology can be adapted for a wide range of applications such as studying virus ecology and genomic reservoirs in freshwater and marine environments.

SARS-CoV-2 virus, responsible for covid-19 pandemic, caught the whole world rapidly and by surprise. Since its initial reported positive cases in 2019, the global effort has been put in three major directions: implementation of effective mitigation and prevention protocols, finding effective vaccinations and treatments and finally predicting the importance as even in a developed country such as USA, nearly 65 % of the outbreak of aquatic diseases stems from the entry and transmission of the virus into the groundwater aquifers [12] .

The World Health Organization (WHO) guidelines for burying deaths from Ebola include 12 steps to maximize safety and prevent transmission of the virus [13] . Although researchers have designed to improve the condition of current corpse bags to prevent the spread of the Coronavirus to the soil [14] , due to the likelihood of oversight, it is still imperative to have information about the pattern, rate, and rate of virus transmission from a contaminated corpse to the water or soil that transmits the virus to locations far beyond the burial site. The same possibility applies to the accessories and disposals from those infected with corvid with or without symptoms. As a result, he recent focus has been on developing computational and experimental framework to determine the pattern of virus spread in different media.

While most studies are currently focused on how Covid-19 is transmitted through the air and airborne particles [15] [16] [17] [18] , the epidemiological aspect of the bulk virus spread through other environments has been left relatively under studied. Here, we describe a novel framework based on moving least squares (MLS) method for solving the time fractional equation of viral diffusion. In this methodology, we provide a precise and flexible mathematical model based on fractional space to simultaneously consider different virus diffusion coefficients and spread rate in 4 different environments, includingtissue→water, tissue→soil, tissue→solid→water and finally tissue→water→soil. This allows us to investigate how the virus is spread and the contamination patterns across different environments.

The origin of fractional derivatives and integrals (i.e. the fractional calculus) as an extension of the ordinary derivatives and integrals (i.e. the classical calculus) can be attributed to Leibniz in 1965 [19] . After Leibniz, from 1730 to 1976, fractional calculus came to the attention of many mathematicians, including Euler, Lagrange, Laplace, Fourier, Liouville, Riemann, Grunwald, Letnikov and Caputo. The Grunwald-Letnikov, Riemann-Liouville and Caputo definitions of the fractional derivative are the well-known fractional derivatives [19] . Fractional derivatives are able to model dynamical systems more accurately due to their non-local property and a greater degree of freedom versus the classical derivatives. Recently, applications of fractional derivatives have been reported in various problems, including anomalous diffusion [20] , diffusion-reaction problems [21] , electrochemistry [22] , chaotic problems [23] , viscoelasticity [24] , image processing [25] , fluid mechanics [26] , teletraffic [27] , oscillators problems [28] , Gaussian noise [29] , vibrators [30] , biology [31] and medicinal modelling [32] . Generally, analytical and numerical methods are two classes of methods for solving fractional differential equations. The analytical methods include Laplace transform, Fourier transform, Adomain decomposition method, Mellin transform, Green's function method, etc. For more details, see [33] and references therein. However, in most cases it is often impossible to analytical solve fractional differential equations. In recent years, scholars have developed various numerical methods for solving fractional differential equations including finite difference/spectral method [34] , Petrov-Galerkin method [35] Sinc-Chebyshev collocation 2 J o u r n a l P r e -p r o o f Journal Pre-proof method [36] , etc.

Fractional diffusion equations are one of the most important categories of fractional partial differential equations which have been widely studied in recent years. These equations have been utilized in modeling viscoelastic materials [21] , turbulent flow [34] , biology [37] , chaotic dynamics of classical conservative systems [38] and other problems [39] . We recall that analytical solution of such problems is extremely difficult and in most cases is impossible. This caused that in recent years, several numerical methods have been developed for solving fractional diffusion equations.

In [40] , a multigrid method developed for spatial fractional diffusion equations with variable coefficients. Iterated fractional Tikhonov regularization method used in [41] for the spherically symmetric backward time fractional diffusion equation. In [33] , neural networks based on Legendre polynomials have been developed to solve space and time fractional diffusion equations. Non-standard finite difference and Chebyshev collocation methods have been used in [42] for solving time fractional diffusion equation. In [43] , Bayrak et al. have proposed a Chebyshev collocation scheme to solve time fractional diffusion equation. The authors of [44] have applied a numerical method for solving time fractional nonlinear reaction-diffusion equations. An analytic algorithm has been utilized in [45] for finding approximation solution of nonlinear time fractional reaction-diffusion equation. In [46] , Cheng et al. used a novel linearized compact alternating direction implicit scheme for Riesz space fractional nonlinear reaction-diffusion equations.

Meshless (or meshfree) approaches are the most important numerical methods for finding the solution of high dimensional (in most cases with complex geometries) differential equations [47] . During last years, meshless approaches provided using the shape functions of moving least squares (MLS) have been extensively applied for divers problems, such as 2D elliptic interface problems [48] , fractional telegraph problem [49] , fractional version of advection-diffusion problem [50] , fractional form of reaction-diffusion equation [51] and integral equations systems [52] .

In this study, a meshlfree algorithm regarding the shape functions of MLS is developed for finding the solution of the time fractional diffusion equation of the coronavirus with different diffusion coefficients in tissue, soil and water environments. The presented algorithm contains these steps: Applying the finite differences technique (accomplished with θ weight) for approximating the fractional derivative and consequently making a recurrence algorithm. Next, approximating the problem solution, as well as its partial derivatives using the MLS functions, and inserting them into the main equation. Eventually, extracting an algebraic system of equations which its solution should be found at each time level.

The rest of this work is as follows: We review the MLS approximation in Section 2. Sections 3 and 4 explain the proposed meshless method and materials, respectively. The obtained results and related discussion are provided in Section 5. Eventually, in Section 6, the conclusion of this work is provided. Using the shape functions of MLS [51] , we able to represent any real function Θ(x) (x = (x, y)) as follows:

where p(x) and a(x) are respectively the vector of basis functions and coefficients. The following cases of p(x) are often utilized for two dimensional problems [53] :

The vector a(x) in relation (2.1) is as follows:

The vector a(x) is evaluated by minimizing the relation

where Θ i = Θ(x i ) andÑ is the number of the nodes in the neighborhood of x, which satisfy the condition ω(x−x i ) ω i (x) = 0. Despite different weight functions, in this study, we apply the Gaussian weight functions [54] 

where d i = x − x i 2 and h i is the radius of the influence domain of the node x i . By considering relation (2.3) and 6) and the matrix B is as follow:

Regarding relation (2.5), we conclude that a(x) has the following structure:

where the functions ψ i (x) (1 ≤ i ≤Ñ ) are called the shape functions, and can be computed as

So, the shape functions vector can be defined as follows:

The second order partial derivatives of the vector Ψ T (x) can be expressed as follows [51, 53] :

Different versions of the diffusion equation have been adopted for modeling the transmission of viruses, cells, biomolecules [1, 4] , modeling of tumor growth [16] , formation and diffusion of amyloid-beta plaques in Alzheimer's disease [17] , and transmission of viruses in unsaturated sand columns [18] . In this study, in order to have a more compatibility of real viral behavior of virus, the time fractional diffusion equation (expressed in relation (3.1)) is considered and solved by using the MLS meshless method. Since in the case of disposal of the infected corpse or personal protective equipment (PPE) in soil or water, it is assumed that there are no nutritional resource of the host (or living cell) available for reproducing the virus, the time fractional diffusion equation is considered without the source term as follows:

accompanied by the initial and boundary conditions

in which Θ is the unknown solution and η Θ is a given constant. Moreover, g 1 , g 2 , g 3 and h are given functions. Herein, we have used the Caputo fractional derivative, which is given in [19] as follows:

(3.5)

Regarding relation (3.5) and the finite difference scheme, we can discretize the fractional derivative expressed in relation (3.1) as follows (more details are given in [55] ):

where t n = nδt, Θ n (x) = Θ (x, t n ) for n = 0, 1, . . . ,N , δt = T /N and

Regarding the finite difference technique (accomplished with θ weight), we can write the following relation for So, a recursive relation is obtained by putting relation (3.6) (without error term) into relation (3.7) as follows:

(3.8)

Utilizing the shape functions of MLS, the function Θ n (x) expressed in relation (3.8) can be represented as follows:

where c n l (l = 1, 2, . . . ,Ñ ) are expansion coefficients which should be computed. To find the values of the coefficients c n l , we should compute the function Θ n (x) in the points x i = (x i , y i ) as follows:

Relation (3.10) can be rewritten by

(3.13)

Regarding relations (2.12) and (3.9), we approximate Θ n xx (x) and Θ n yy (x) as follows:

(3.14)

So, regarding the above relation, we get 

we rewritten relation (3.15) as follow:

Using the internal points, the matrices D xxd and Q yyd are defined as follows: 

Here Moreover, all simulations have been performed using MATLAB R2019b software.

The governing equation is solved for exploring the diffusion of the virus of four cases with different combinations of tissue, soil, and water domains which are introduced in Fig. 1 schematically. In all cases under the study, it is [18] . These values are calculated for environment temperature (298K) and

presented in Table 1 .

The governing equations distances can also be monitored over time. In this case, the virus concentration in the remote areas is first zero and then gradually increases over time until it reaches its peak value and after that, the concentration of the virus in the area decreases (see Fig. 4 for case I and case III and also see Fig. S3 for case II and case IV).

The percentage of infected areas over time is also shown in Fig. 5 for all the cases. According to this graph, the rate of environmental pollution increases nonlinearly over time. Therefore, it is not possible to consider a constant velocity for the spread of the virus and predict a linear relationship across different areas.

Given the values of diffusion coefficients for the tissue, soil, and water medium (Table 1) , it is clear that the rate of virus progression in the soil environment is higher than the other one. For example, in Case 4, as the virus reaches the soil area, the spread rate increases relative to the same area in other cases, and this increase can be easily seen from the slope at the tail-end of the curve in Fig. 5 .

It should be noted that the virus diffusion coefficients for the aquatic environment are considered without considering the dynamics or mass transform of the water molecules. Therefore, the coefficient obtained for porous soil is reported to be higher than static water mass (see Table 1 ). Hence the data of figures, which are plotted for different radial distances from the central infected site, it can be seen that in the early times of propagation (1 µm distance from the infection site), the diffusion rate of all derivatives is close to each other.

However, at longer distances, the difference in equation behavior becomes more apparent for the order of the fractional derivatives. For example, the required time to reach the peak of contamination to a radius of 40 µm for α = 0.1 is approximately 2.2, 3.0, and 6.5 times to that of α = 0.9, α = 0.7 and α = 0.5, respectively. Therefore, the rate of virus propagation strongly depends on the order of the derivative of the governing equation.

It is important to note that although the rate of virus spread depends on the derivative order, the results indicate that the maximum concentration in this region is the same for all the fractional derivative orders. In other words, the maximum concentration of the virus at different distances will not be a function of the fractional derivative order. In Fig. 10 , we compared our computational predictions with available values of virus spread for different alpha values. We compared them when the virus reaches its maximum concentration at distances of 1 µm, 10 µm, and 1mm

for case I (tissue→water) [58] . For ease of display, the distance traveled by the virus (horizontal axis) is illustrated logarithmically. As can be seen from the comparison of our computational results with the experimental results ( Fig.   10 ), it is obvious that the virus diffusion behavior in the transition from tissue to water at the intervals of diffusion is completely dependent on the order of the fractional derivative. In such a way that in initial propagation times (at distances < 10µm) choosing of lower orders and in times of more than one hour (distance > 10µm) assigning higher order of the fractional derivative (close to α = 1) will produce more compatible results with the experimental data.

It is worth noting that in the experimental study, the dynamics and mass transfer of water molecules have certainly The data obtained in Fig. 10 show that for distances of 10 to 1000 µm, the order of the fractional derivative that is most consistent with the experimental data is α = 0.9. Fig. 11 shows the results of the relative concentration of the virus in radius of 40µm for all cases with α = 0.9. One extrapolation of our data can be used to estimate the diffusion time of the virus from a contaminated body until it reaches one meter from the center of the contamination. According corps burial at such distances as the chances that the virus does not biologically survive due to various reasons such as temperature, humidity, and UV light.

The can be limited to a few centimeters. Regardless of the exact estimate of the diffusion coefficient, the focus of the present study is to provide a methodological framework to solve the fractional problem, and comparing them with the results obtained using the conventional solution of the diffusion equation. We showed the effect of the fractional order derivative on the actual propagation behavior is considerable. Therefore, we believe that considering a time-varying alpha (variable-order fractional derivatives) seems to be the best solution for accurately predicting the behavior of known viruses over large time and dimensional scales. These observations reshape our understanding of the dynamics of COVID-19 spread in and across various environments and ultimately, our work should be used to take precautions to avoid environmental cross-contamination. In addition to pandemic relief efforts, our methodology can be adopted for a wide range of applications, such as studying virus ecology and genomic reservoirs in freshwater and marine environments.

Modeling influenza viral dynamics in tissue. International Conference on Artificial Immune Systems

The Sars-Cov-2 nucleocapsid protein is dynamic, disordered, and phase separates with rna. bioRxiv

Cryo-Em structure of the Sars-Cov-2 3a ion channel in lipid nanodiscs

A reaction-diffusion model for viral infection and immune response

Silico screening of potential spike Glycoprotein inhibitors of Sars-Cov-2 with drug repurposing strategy

Dynamics of respiratory droplets carrying Sars-Cov-2 virus in closed atmosphere

Progress and trends in mathematical modelling of influenza a virus infections

A mathematical model describing the localization and spread of influenza a virus infection within the human respiratory tract

Diffusion-reaction compartmental models formulated in a continuum mechanics framework: application to Covid-19, mathematical analysis, and numerical study

Simulating the spread of Covid-19 via spatially-resolved susceptible-exposed-infected-recovereddeceased (Seird) model with heterogeneous diffusion

Virus persistence in groundwater

How to conduct safe and dignified burial of a patient who has died from suspected or confirmed Ebola or Marburg virus disease

Conceptual design of a body bag for preventing infections and safe disposal of deceased from Covid-19 virus

Viral dynamics: a model of the effects of size, shape, motion and abundance of single-celled planktonic organisms and other particles

A novel in silico platform for a fully automatic personalized brain tumor growth

Exploring the formation of Alzheimer's disease senile plaques in silico

Transport of viruses through saturated and unsaturated columns packed with sand

Fractional Differential Equations

A new fractional operator of variable order: application in the description of anomalous diffusion

Semi-implicit Galerkin-Legendre spectral schemes for nonlinear time-space fractional diffusion-reaction equations with smooth and nonsmooth solutions

Fractional differential equations in electrochemistry

On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems

General fractional derivatives with applications in viscoelasticity

The use of a Riesz fractional differential based approach for texture enhancement in image processing

Application of fractional calculus to fluid mechanics

Multi-fractional generalized Cauchy process and its application to teletraffic

Three classes of fractional oscillators, symmetry-Basel

Modified multifractional Gaussian noise and its application

Theory of vibrators with variable-order fractional forces

Novel dynamic structures of 2019-ncov with nonlocal operator via powerful computational technique

New approach for the model describing the deathly disease in pregnant women using Mittag-Leffler function

Neural network method for solving fractional diffusion equations

Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations

Fractional-order Genocchi-Petrov-Galerkin method for solving time-space fractional Fokker-Planck equations arising from the physical phenomenon

Numerical solution of time fractional nonlinear Klein-Gordon equation using sinc-Chebyshev collocation method

Fractional calculus models of complex dynamics in biological tissues

Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids

On fractional calculus and fractional multipoles in electromagnetism

An efficient multigrid solver for two-dimensional spatial fractional diffusion equations with variable coefficients

Iterated fractional tikhonov regularization method for solving the spherically symmetric backward time-fractional diffusion equation

Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation

Numerical solution of fractional diffusion equation by Chebyshev collocation method and residual power series method

A new approach and solution technique to solve time fractional nonlinear reactiondiffusion equations

An analytic algorithm for time fractional nonlinear reaction-diffusion equation based on a new iterative method

A novel compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equations

A greedy meshless local Petrov-Galerkin method based on radial basis functions

Interpolating stabilized moving least squares (MLS) approximation for 2D elliptic interface problems

Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag-Leffler non-singular kernel

Time-dependent fractional advection-diffusion equations by an implicit MLS meshless method

A meshless method to solve nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag-Leffler kernel

Modified moving least squares method for two-dimensional linear and nonlinear systems of integral equations

Classification and overview of meshfree methods

Simulating free surface flows with sph

Two-dimensional Legendre wavelets for solving variable-order fractional nonlinear advection-diffusion equation with variable coefficients

Fenner's veterinary virology

The molecular weight and other biophysical properties of bromegrass mosaic virus

Cell movements: from molecules to motility