Progression of SARS-CoV-2 From Successful Intake to Establishment of Infection in a Tissue

By Dharmendra Tripathi, Dinesh Singh Bhandari, and Anuj Mubayi

The course of COVID-19 in infected patients seems to encompass a wide spectrum of clinical manifestations, ranging from mild to severe disease symptoms. Soon after a susceptible individual uptakes an infectious dose, the infection travels through cells in the nasal cavity and respiratory tract and eventually establishes itself in different parts of the organs [9, 12]. Virus particle transportation—from initial entry to ultimate establishment in various parts of the body—is a complex process. Therefore, a thorough comprehension of the mechanisms along the pathways can help clinical scientists address medical therapeutic challenges. As viruses progress through different pathways, they experience varied levels of resistance from cells, blood, and membrane walls that can lead to differential pathological signatures of infection [4]. Here we attempt to use well-established mathematical tools from fluid dynamics to understand individuals’ widely varied responses to COVID-19 infection, from no symptoms to death. Our analyses also partly explain the differential rate of clinical symptoms from time since infection within severe cases [3].

Historical Importance of Fluid Dynamics 

Fluid dynamics is a subdiscipline of fluid mechanics, which comprises the study of fluids and the impact of forces on their transport phenomena. Over the last 75 years, fluid dynamics research has experienced tremendous development — especially in areas like thermodynamics, biofluid dynamics, electrohydrodynamics, magnetohydrodynamics, microfluidics/nanofluidics, and computational fluid dynamics (CFD). The field finds applications in physiology, thermal conduction, laser therapy, and chemical kinetics. Conceptual fluid dynamics is primarily based on the three conservation laws: conservation of mass, conservation of momentum (Navier-Stokes equations), and conservation of energy, also known as the first law of thermodynamics; see equations \((1)\) to \((3)\) below, respectively. Researchers assume that properties such as viscosity, elasticity, plasticity, density, pressure, temperature, and flow velocity of fluids/air are defined at infinitesimally small points in space and vary continuously from one point to another. The equations of fluid dynamics are part of continuum mechanics and prove difficult to solve, but one can simplify them in several ways — all of which make them easier to analyze mathematically.

Our research aims to (i) describe virus propagation phenomena within different types of patients using fluid dynamics, (ii) identify conditions under which one can determine the trajectory of a COVID-19 patient, (iii) identify the data that will parameterize and evaluate such models, and (iv) estimate the potential range for specific parameters that capture virus propagation in different pathway geometries.

Methodology 

In order to model the movement of pathogens in the bloodstream, we need three equations that describe the following items: (i) the structure of the vessel in which blood flows, and blood’s movement within it (fluid flow), (ii) the flow of viruses in a blood or air medium (virus transportation), and (iii) the appropriate patient-level data and estimated parameters that are associated with the establishment of infection in various parts of localized organs (relevant data). In fluid dynamics, solving the momentum equations both analytically and numerically is quite challenging. Therefore, many researchers use various assumptions and approximations to simplify the equations. They employ the lubrication approach for approximate solutions; the perturbation method for series solutions; and the finite difference method, finite element method, finite volume method, mesh-free method, and CFD tools to reach numerical solutions and simulate the results.

Describing the Medium: Fluid Flow

One can analyze fluid prospective and thermal effects on virus transmission with the following continuity, momentum, and energy equations: 

\[\nabla \cdot \boldsymbol{V}=0 \tag1\]

\[\underbrace{\rho \frac{\partial \boldsymbol{V}}{\partial t}}_{ \begin{array}{c}
the\ local\ change \\
\ with\ time \end{array}
}+\underbrace{\rho \left(\boldsymbol{V}\cdot \nabla \right)\boldsymbol{V}}_{ \begin{array}{c}
momentum\  \\
convection \end{array}
}=\underbrace{-\nabla p}_{surface\ force}+\underbrace{\mu {\nabla }^2\boldsymbol{V}}_{ \begin{array}{c}
momentum \\
\ diffusion \end{array}
}+\underbrace{\boldsymbol{Fx}}_{\ external\ force} \tag2\]

\[\underbrace{\rho c_p\frac{\partial T}{\partial t}}_{ \begin{array}{c}
local\ energy\ change \\
\ with\ time \end{array}
}+\underbrace{\rho c_p\boldsymbol{V}\frac{\partial T}{\partial x_i}}_{ \begin{array}{c}
energy\  \\
convection \end{array}
}=\mathrm{\ }\underbrace{p\nabla \boldsymbol{V}}_{pressure\ work}+\underbrace{K{\nabla }^2T}_{ \begin{array}{c}
heat\ flux \\
\left(diffusion\right) \end{array}
}+\underbrace{\Phi }_{ \begin{array}{c}
heat\ source\ sink \\
parameter \end{array}.} \tag3\]

These equations are subject to the subsequent boundary conditions:

\[{\frac{\partial u}{\partial y}}_{|y=0\ }=0,\ \ \ \ \ \ v_{|y=0}=0\ ,\ \ \ \ \ \ u_{|y\mathrm{=h}}\mathrm{=0,}{\mathrm{\ \ \ \ \ \ }v}_{\mathrm{|}y\mathrm{=h}}\mathrm{=}\frac{\partial h}{\partial t}.\]

Here, \(\boldsymbol{Fx}\) acts as a magnetic force \((\overrightarrow{J}\times \overrightarrow{B})\), \(\overrightarrow{J}\) is the electric current density, and \(\overrightarrow{B}\) is the magnetic field. The other components are electric force \(({\rho }_eE_x)\), Darcy resistance \(\left(\frac{\mu \boldsymbol{V}}{\boldsymbol{\kappa }}\right)\) for porus media (where \(\boldsymbol{\kappa}\) is the permeability), and buoyancy force \((\rho g\beta (T-T_0))\). Moreover, \(\rho\) is the fluid density, \(\boldsymbol{V}=(u,\ v,\ 0)\) is the fluid’s velocity vector field, \(p\) is the pressure, \(\mu\) is the fluid kinematic viscosity, \(C_p\) is the specific heat capacity, \(K\) is the heat conductivity, and \(T\) is the temperature.

The rhythmic propagation of muscles (compression and expansion) governs most of the physiological fluids’ internal movements within the vessels (blow flow in capillaries); see Figure 1 for a visual depiction. Here, \(h(x,t)\) represents transverse vibration of the blood vessel and \(\phi\) is the wave’s amplitude. The parameters \(t\), \(\lambda\), and \(a_0\) respectively denote the time, wavelength, and half width of the channel.

Figure 1. Transmission of SARS-CoV-2 through a blood vessel. Figure courtesy of the authors.

Particle Propagation: Virus Transportation Within the Medium

We consider viruses to be spherical particles that move through the viscous fluid flow, which is governed by \((1)\) to \((3)\). Previous researchers have addressed the peristaltic flow of bacteria in the urinary system with the Basset-Boussinesq-Oseen (BBO) equation [7]. This equation (equation \((4)\) below) represents the sum of the steady state drag force, pressure, buoyancy force, virtual mass force, and body force — the latter of which equates to mass times the acceleration of an isolated particle. Using a similar BBO approach, we modeled the movement of viruses and their position inside the vessels. Doing so describes the motion of a small particle in unsteady flow at low Reynolds numbers \((\textrm{Re}_p=|v-v_p|(\rho a_0)/\mu <<1)\) via the particle velocity vector (\(\boldsymbol{V}_{\boldsymbol{p}}= (u_p,v_p,0)\), where \(u_p\) and \(v_p\) are axial and transverse velocity components, respectively):

\[\overbrace{\frac{d{\boldsymbol{V}}_{\boldsymbol{p}}}{dt}}^{Acceleration~} =\overbrace{\frac{1}{S_N}\left(\frac{2S}{2S+1}\right)\left(\boldsymbol{V}-{\boldsymbol{V}}_{\boldsymbol{p}}\right)}^{Stokes~Drag~}+\overbrace{\frac{3}{2S+1}\frac{d\boldsymbol{V}}{dt}}^{Virtual~Mass-I~}+\overbrace{\left(\frac{{\alpha }^2}{40\left(2S+1\right)}\right)\frac{d}{dt}\left(~{\nabla }^2\boldsymbol{V}\right)}^{Virtual~Mass-II~} + \\ \underbrace{\frac{\alpha }{{24S}_N}\left(\frac{2S}{2S+1}\right){\nabla }^2\boldsymbol{V}}_{Faxén} +\underbrace{\sqrt{\frac{9}{2\pi SS_N}}\left(\frac{2S}{2S+1}\right)\left[\int^t_0{ \begin{array}{c}
\frac{\left(\frac{d}{d\overline{t}}\left(\boldsymbol{V}-{\boldsymbol{V}}_{\boldsymbol{p}}\right)\right)}{\sqrt{t-t^*}}dt^* \\
~ \end{array} }+\frac{\left({\boldsymbol{V}}_{\boldsymbol{0}}-{\boldsymbol{V}}_{\boldsymbol{p}\boldsymbol{0}}\right)}{\sqrt{t}}\right]}_{Basset~}+\\ \underbrace{\frac{2\left(S-1\right)}{2S+1}{\frac{{\tau }_c}{c}g}_0g}_{Gravity~}. \tag4\]

This equation is subjected to the initial condition \(\boldsymbol{V}_{\boldsymbol{p}}(\boldsymbol{x},\boldsymbol{t}\boldsymbol{=}\boldsymbol{0})=0\). Additionally, \(g=\frac{\overline{g}}{g_0}\) is the gravity vector, \(\alpha =d_p/a_0\) is the aspect ratio of particle diameter and channel width, \(S_N\) is the stokes number, \(S\) is the density ratio \((\rho_p/\rho)\), \(c\) is the wave velocity, \(\tau_c\) is the particle relaxation time, and \(d_p\) is the virus diameter. The subscript \(\boldsymbol{p}\) designates particle-related parameters/variables. 

Relevant Data and Parameter Estimation

The first species of coronavirus was discovered in 1966 [2]. The discovery and ensuing studies showed that the diameter of these virus particles ranges from 80 to 200 nanometers (nm); moreover, particles with diameters that are below five microns generally do not settle and instead remain airborne [10]. One can analyze the motion of a single aerosol particle in a fluid medium [5] and calculate the required parameters for the virus—such as Stokes drag, virtual mass, Basset, and gravity—based on available data about the virus’ structure, density, blood viscosity, and diameter [6, 8]. Drag force is highly responsible for the extremely large value of the axial particle velocity; in other words, drag force heavily drives the movement of the coronavirus.

Figure 2. Spatial velocity distribution of fluid and virus particles. 2a. The contour plot of the stream function of fluid at a fixed value of \(\phi=0.4\) and time \(t=0.1\). 2b. Axial velocity of coronavirus for different values of virus diameter and blood viscosity. Note that the lower and upper estimates of the three coloring scales are different and capture three different scenarios.

Computational Results

To analyze the behavior of viral spread in a viscous fluid medium (i.e., a bloodstream), one must first compute the fluid velocity \((\boldsymbol{V})\), stream function \(\left(\psi =-\int vdx+\int udy+c\right)\), and particle velocities \((\boldsymbol{V}_{\boldsymbol{p}})\) in the axial and transverse directions. We systematically present the stream lines of the fluid flow field in Figure 2a, where we attain the maximum stream lines at the blood vessel’s expansion region (red and blue areas of the figure represent regions with maximum stream lines, but in different directions). This observation means that the difference between two successive stream lines represents the volumetric flow rate. In other words, contraction phases generate the force; this realization implies that the maximum flow occurs during the contraction phase, because the distance between two stream lines is greater during this point. We simulate the particle’s velocity under the various applicable forces (Stokes’ drag, virtual mass, Basset, and gravity).

Figure 2b (i and ii) illustrates the behavior of the coronavirus particle’s axial velocity in the blood vessel. The influence of the coronavirus particle’s diameter on the axial velocity component is evident. Our results suggest that the small-diameter particle rapidly moves through the blood vessel via the blood flow. In addition, Figure 2b (ii and iii) depicts blood viscosity’s impact at a fixed value of the virus diameter. This outcome indicates that high viscosity reduces the virus particle’s transmission.

Figure 3 presents the corresponding two-dimensional plot of the axial particle velocity profile. Figure 3a demonstrates that the axial particle velocity’s distribution along the y-axis is high for virus diameter \(d_p=80\) nm, as compared to \(d_p=160\) nm and \(200\) nm. Furthermore, the virus particles may circulate and move in the fluid flow medium, where the rhythmic propagation governs the flow. More circulations and movement of the virus ensue in the contracted region than in the relaxed region, and Figure 3b portrays blood viscosity effects for the transmission of virus flow. The axial particle velocity component slightly enhances blood viscosity, i.e., high viscosity increments the axial flow in the blood vessel. The rheological effects on virus transmission may also play a significant role in movement, and researchers can even analyze thermal effects’ impact on virus transmission by incorporating the energy equation into the proposed model. Such analysis may define the importance of fluid dynamics in preventing the spread of COVID-19 and other similar viruses. 

Figure 3. Variation of the axial particle velocity for different values of (3a) the virus diameter and (3b) blood viscosity. Figure courtesy of the authors.

Discussion 

Here we outline several implications from our study:

1. The estimated range of viscosity for human blood is roughly \((3.5-5.5)\times {10}^{-3}\). The large variation in viscosity may yield a greater chance for the SARS CoV-2 virus to spread either rapidly or extremely slowly in the blood vessel, which might damage patient blood vessels or result in a wide spectrum of disease severity.
2. Different strains of the COVID-19 virus have varying sizes. Our results not only suggest that the diameter of a virus strain impacts the amount of virus flow within a blood vessel, but also that it quantifies the virus’ impact. The particle’s lesser velocity flow leads to a reduced spread of infection within the body but more localized infection. Moreover, the size and load of virus particles largely determine individuals’ abilities to protect themselves and those around them from SARS-CoV-2. 
3. Some researchers have indicated that one can control the flow of physiological fluid (blood) by applying external forces. One such study suggested that a magnetic field can reduce blood’s axial velocity [11], while another revealed that an electric field has the capability to alter the motion of an aqueous electrolyte’s peristaltic transport [1]. 
4. If we can control the velocity of particles in a patient, the patient has the potential to recover more quickly with a lesser level of internal damage.

Ultimately, our study addressed the propagation of virus particles in the physiological system through the viscous fluid medium. Because the theoretical model is limited to select fluid properties and ignores biophysical interactions with the immune system, it currently does not capture the full effect of viruses in the body and the role of medicine on particles. We thus aim to understand how quickly virus particles can move within the body and how we can control particle velocity to hence manage infection severity. By examining a virus’ spread within the peristaltic flow, doctors and scientists can visualize and track patterns and better understand the resulting disease.

	Dharmendra Tripathi is an associate professor at the National Institute of Technology, Uttarakhand in India. His research focuses on biofluid mechanics.
	Dinesh Singh Bhandari is a Ph.D. student at the National Institute of Technology, Uttarakhand. His research focuses on biofluid mechanics.
	Anuj Mubayi is a principal scientist (Infectious Disease Forecasting Lead) with The Public Health Company. He is also a Distinguished IBA Fellow in the Center for Collaborative Studies in Mathematical Biology at Illinois State University, a senior fellow at the Kalam Institute of Health Technology in India, and an adjunct faculty member in the Department of Mathematics and Computer Science at the Sri Sathya Sai Institute of Higher Learning in India. Mubayi's expertise is in health decision science as well as infectious disease modeling and dynamics.