Pathways to community transmission of COVID–19 due to rapid evaporation of respiratory virulets

Graphical abstract

The motions of the virus laden respiratory droplets are studied to explore the spreading of viral infection. The role of evaporation guided bioaerosol formation for community transmission has been analyzed for the prediction of the waves of a viral pandemic.

Abstract

Hypothesis

The formation of virus-laden colloidal respiratory microdroplets – the sneeze or cough virulets and their evaporation driven miniaturization in the open air are found to have a significant impact on the community transmission of COVID–19 pandemic.

Simulation details

We simulate the motions and trajectories of virulets by employing laminar fluid flow coupled with droplet tracing physics. A force field analysis has been included considering the gravity, drag, and inertial forces to unleash some of the finer features of virulet trajectories leading to the droplet and airborne transmissions of the virus. Furthermore, an analytical model corroborates temperature (T) and relative humidity (RH) controlled droplet miniaturization.

Results

The study elucidates that the tiny (1–50 µm) and intermediate (60–100 µm) size ranged virulets tend to form bioaerosol and facilitate an airborne transmission while the virulets of larger dimensions (300 to 500 µm) are more prone to gravity dominated droplet transmission. Subsequently, the mapping between the T and RH guided miniaturization of virulets with the COVID–19 cases for six different cities across the globe justifies the significant contribution of miniaturization-based bioaerosol formation for community transmission of the pandemic.

1. Introduction

The recent outbreak of the deadly Severe-Acute-Respiratory-Syndrome Coronavirus 2 (SARS-CoV-2) pandemic has severely affected the health and economic conditions of the countries across the globe [1], [2]. The use of face masks, maintenance of a healthy hand or face hygiene, and the use of a strict physical distancing protocol have become important practices for every individual to minimize the spreading of this highly contagious disease [3], [4]. Importantly, it has now been scientifically established [5] that the pandemic spreads more rapidly through the micro/nanoscale airborne respiratory droplets, which emerge during the coughing, sneezing, breathing, or quiet/loud speeches of an infected individual [6], [7]. This is because the viruses of ∼100 nm diameter create a finely dispersed suspension with the mucosal fluid of respiratory tracts [8] and forceful breathing of air shears through the mucosal-air interface to generate the finely dispersed colloidal bioaerosol [9]. The size distribution of the colloidal microdroplets loaded with viruses is found to be in the range of 500 nm to a few hundred microns under varied conditions [10]. In that, the smaller ones have the tendency to spread the airborne infections while the bigger ones settle on the surfaces under gravity to spread the droplet-borne infections [13].

About a century ago, Flügge [11] in his seminal contribution demonstrated the concept of droplet transmission of infectious diseases such as malaria, tuberculosis, and cholera, which has been instrumental in the advocacy of Mikulicz-Radecki's for the surgical gauze masks [12]. In another path breaking contribution, Wells classified the airborne transmission after dichotomizing the respiratory droplets into larger and smaller varieties for tuberculosis [13]. Importantly, the transmission of other contagious diseases like tuberculosis, pneumonic plague, bronchitis, chicken pox, measles, or influenza also follow a similar transmission mechanism [6], [7], [5], [14]. The droplet mode of transmission is possible when the virus loaded respiratory mucosal droplets are transported from the infected person directly to the exposed person or due to an indirect mode like a contact with the fomites of the infected person [15]. On the other hand, the airborne mode spreads through the exposure of virus-containing respiratory droplets in the air for a longer distance and for a longer period [15]. According to a recent report, the SARS-CoV-2 laden mucosal droplets oozing out from an individual forming bioaerosol [15] are among the prime reasons behind the community spread of the virus. The virus laden droplets then find their way to the healthy host through the angiotensin-converting enzyme-2 – ACE2 receptor protein located abundantly in the mucosae of conjunctiva and nose-epithelia causing serious respiratory disorders [16].

Interestingly, such fluid mechanical events are also very engaging from the perspectives of micro/nano droplet formation at the air-mucus interface and their transport through the air [17]. For example, smaller droplets may float in the air for a longer duration while even a mild air movement may transport the droplets further than usual [18]. Likewise, a smaller droplet may feel a much lesser gravitational pull while a bigger one may immediately precipitate after ensuing out of the nasal or mouth cavity. Importantly, the variation in the environmental temperature (T) and relative humidity (RH) in different geographical regions can play major roles in redefining the dynamics of the ejected droplets owing to the rapid evaporation and subsequent miniaturization. The dynamic change in the droplet diameter with these environmental parameters may transform a gravity-bound bigger droplet into an air bound floating droplet. Such miniaturized droplets are benefitted from deficit pull from gravity and can easily travel to longer distances packed with infectious viruses, which may trigger the community spread of the disease, as discussed in this article.

However, thus far, all the analyses and inferences related to the droplet and airborne modes of transmissions have been rather ad-hoc owing to a few important scientific and technological limitations: (i) the dynamic experimental imaging of the suspended coronavirus nanoparticle inside a moving droplet is a challenging task to achieve [19]; (ii) formation and transport of the mesoscale droplets of different size ranges from the nasal or mouth cavities are very difficult to characterize morphologically even with the state-of-art capacity of high-speed and high-resolution imaging [20]; (iii) the complexity is further elevated while tracking the viruses inside such droplet transport [21]; (iv) the relative humidity, temperature, level of pollutants, speed, and currents in air play crucial roles in the transport of the droplets [22]; and (v) size and angle of nose outlets, height of an individual, and the force of breathing, sneezing, and coughing are some of the parameters, which are very diverse across the globe [23]. Certainly, deeper scientific analysis and understanding of the fluid dynamics of the trajectory of the sneezed or coughed droplets is crucial to establishing a framework related to the physical distancing norms.

In this direction, one of the major scientific tools to explore the finer features of such a problem has been computational fluid dynamics (CFD) [24]. A number of prior seminal contributions have focused on exploiting the capacities of CFD to unearth the details of the respiratory droplet motion [25]. For example, Gupta et al. [26] have performed a benchmark to explore the probability distribution function mimicking the cough jets ejected by patients. Mittal et al. has summarized the generation mechanism of respiratory droplets and its parabolic flow physics governing the transmission of COVID-19 [27]. Xie et al. has reported a wide study of flow dynamics of respiratory droplets considering the turbulent flow of the cough jet considering effect of evaporation, relative humidity, air speed in a stagnant airflow for both water droplet and saline droplets [28]. Further, the effect of turbulence on the dispersion and deposition of the droplets in cough jets [29] and its exposure to close contact [30] has been explored in stagnant airflow conditions. The transmission of the respiratory droplets has also been analyzed in various closed environments like an elevator, supermarkets, classrooms equipped with partition, indoor conditions like an air-conditioned room or ventilated room, Intensive Care Units (ICU) as reported in various studies [28], [31], [32], [33], [34], [35].

Apart from focusing only on the flow dynamics, the effects of various external factors affecting the transmission of respiratory droplets have also been explored broadly. The effects of atmospheric pollution [36], the viability of the 2 m of physical distancing norm [16], [22], [37], flow dynamics of a turbulent dry cough jet/puff having contagious pathogen [21], drying time of the respiratory droplets [19], [38], the effect of evaporation time of droplets on infection transmission [39], [40], [41], and the effect of relative humidity and breakup of cough droplets on transmission intensity [22], [16] have also been explored widely. Also, recently, the role of the size of water droplets [42] and the effect of evaporation of the water droplets [43] on the survival of the airborne viruses has been discussed. A very recent study uncover the flow dynamics of the ejected cough or sneeze droplets considering the movement of the head while sneezing [23]. The detailed study of Pendar [44] show a comprehensive examination of mild cough to intense sneezes considering various initial velocity, size distributions, injection angles of saliva micro-droplets and mouth-opening area along with different environmental factors that drive the transmission airflow inside a room. Also, Zhao et al. has shown that the droplets can travel three times farther in a wide range of temperature and humidity combination [45]. All these factors significantly characterize the aerosol based transmission of the disease attributing the second wave of COVID–19 [46].

The present study focuses on a rather unexplored topic wherein the viral loading and infection potency of the respiratory droplets has been explored in detail to assess the risk of community transmission of SARS–CoV–2 infection. These respiratory droplets are the colloidal microdroplets of biofluid infused viral bodies – the sneeze or cough virulets which atomize into tiny bioaerosol to carry the loaded viruses faraway. Here, the spatiotemporal trajectories of the exhaled virus loaded droplets having various diameters have been simulated employing a complete set of fluid momentum equations and coupled fluid particle tracing physics. Moreover, the analyses of fluid flow fields and relative contributions of gravitational, drag, and inertial forces are quantitatively reported to strengthen the fundamental understanding of virulet dynamics. Furthermore, the effect of several sneezing or coughing frequencies and outdoor breeze conditions on the enhancement of dispersion of droplet transmission has also been reported.

Furthermore, remarkable observations have been unveiled to confirm the effect of T and RH mediated evaporation-driven on-the-fly miniaturization of the various sized virulets on the transmission of the infectious disease. Based on the T and RH data, the rate of miniaturization of the virulets has been analytically calculated for all six regions and has been mapped with the reported infection cases for the first wave of COVID-19 at that particular region. Interestingly, it was found that the trend in the rate of recurrence of the reported COVID-19 cases and their time resembles the mode of miniaturization of the virulets for the entire period for all the regions. In view of these observations, the evaporation model has been employed to predict the future occurrences of the infection waves and then compared with the reported infection cases including the second wave of COVID-19. The study shows that the predicted and the reported COVID-19 infection graphs proportionate well with a compromise of the intensity of the infection which may be attributed to the non-consideration of the additional causes of COVID-19 infection. Thus, the study unveils a strong relationship of the transmission of the disease in airborne mode with the rapidly evaporating microdroplets highlighting the fruitful novelty of the work. The most important part of the present work is drawing nonlinear correlations with statistical predictive models in R programming [47] for all the six cities. The correlation curves led to satisfactory p values indicating strong dependence of droplet miniaturization and COVID-19 infection. Further to this, the role of droplet flying, floating, and fleeting mediated by specific T and RH of that particular zone has been justified by CFD simulations with varying T and RH. For each zone, a demarcation line for T and RH has been drawn that signifies the onset of the high risk of COVID-19 transmission.

2. Theory

2.1. Problem formulation

Fig. 1 schematically shows the motion of respiratory droplets, ejected from an individual in the open air. The rectangular section in the figure shows the solution space, representing aerial premise of 3 m height (h) and 5 m length (l) in the proximity of the individual. The respiratory droplets enter into the solution space from inlets 1 and 2, which emulate the nose and mouth openings, respectively. In this regard, inlets 3 and 4 show the open-air inlets. Walls 1 and 2 represent the top and bottom boundaries taken sufficiently far from the nose inlet to represent the open air above and ground, respectively, while the outlet is assumed to be far away from the individual at a distance of 5 m, to study the extent of air-borne transmission. The position of the nose and mouth openings are placed at a certain height emulating the average height of an individual. The nose inlet has been tilted by an angle of 27°, which represents a downward motion while sneezing, as shown in the inset of the figure.

Fig. 1.

Schematically shows the 2D geometry considered for the simulation study with the dimensions and axes. The nose and mouth openings are indicated as inlets 1 and 2, respectively while inlets 3 and 4 show the open-air inlets. The walls 1 and 2, at the top and bottom show the upper boundary and ground of the solution space, respectively while the outlet in the geometry is shown towards the right side. The respiratory virulets of varied diameter $d$ have been indicated by different color code. The inset schematically shows the SARS-CoV-2 loaded respiratory droplets expelled from an infected person.

For the numerical modeling, a two-dimensional (2D) Cartesian coordinate system has been employed, which considers the effect of external airflow from inlets 3 and 4, on the motion of the expelled respiratory droplets of varied diameter $\left(d\right)$. The fluid (air), in this model, is assumed to be isothermal, Newtonian, laminar, and incompressible. Further, for the droplets, the effect of evaporation has not been considered in the simulation. Additionally, the effect of the evaporated droplets has been incorporated by considering droplets of varying sizes. The horizontal and vertical directions are represented as x and y, respectively. A bold variable represents a vector and a variable in italics represents a scalar quantity. The scalar variables placed in the parenthesis of a vector indicate the components along the x and y directions. The velocity vectors of the fluid and droplets are symbolized as ${\mathit{u}}_f$(u, v) and ${\mathit{u}}_d$(u, v). The governing equations for the fluid flow are solved using the following continuity and Naiver-Stokes equations,

$$\mathrm{\nabla }·{\mathit{u}}_f=0$$ (1)

$${\rho }_f\frac{D{\mathit{u}}_f}{\mathit{Dt}}=-\mathrm{\nabla }{p}_f+{\mu }_f{\mathrm{\nabla }}^2{\mathit{u}}_f+{\rho }_f\mathit{g}$$ (2)

where $\mathrm{\nabla }$, D, $p_f$, ${\mu }_f$, ${\rho }_f$, and g are the gradient operator, material derivative, air pressure, dynamic viscosity of air, density of air, and acceleration due to gravity vector, respectively. The motion of the droplets along with the external airflow has been modeled with the help of the following conservation of momentum equation,

$$\frac{\text{d}\left(m_d{\mathit{u}}_d\right)}{\text{d}t}={\mathit{F}}_d^D+{\mathit{F}}_d^g$$ (3)

where $m_d$ is the mass of the droplet, ${\mathit{F}}_d^D$ is the drag force, and ${\mathit{F}}_d^g$ is the gravitational force experienced by the droplet. It has been observed that the sudden expelling of cough or sneeze droplets causes a transient high Reynolds number flow, which is impulsive [26]. However, the velocity of the droplets decreases with time and during its passage, it merges to a low Reynolds number laminar flow [48]. Hence, for this study, Stokes’ drag [49], as shown in Eq. (4), has been employed for the droplets because the droplet Reynolds number for the different range are: 0.07 (1–50 μm), 1.50 (60–80 μm), and 5.56 (80–200 μm). For 300–500 μm, the Reynolds number is higher than 10, although well below the transition particle Reynolds number which is 500 [50]. However, the virulets in this range are significantly dominated by gravity, and do not tend to follow the fluid profile as discussed later.

$${\mathit{F}}_d^D=\left(\frac{18{\mu }_d}{{\rho }_d{d}^2}\right)m_d{\mathit{u}}_r$$ (4)

Here, the notations, ${\rho }_d$, $d$, and ${\mu }_d$ are density, diameter, and dynamic viscosity of the droplet. Further, ${\mathit{u}}_r={\mathit{u}}_d-{\mathit{u}}_f$ represents the relative velocity of the moving droplet with respect to the velocity of the surrounding air. In addition, the gravitational force for the droplets has been considered employing the following expression,

$${\mathit{F}}_d^g=m_d\frac{\left({\rho }_d-{\rho }_f\right)}{{\rho }_d}\mathit{g}$$ (5)

2.2. Boundary conditions

To solve the Eqs. (1), (2), no-slip and impermeable, ${\mathit{u}}_f=0$, boundary conditions are enforced on walls 1 and 2. The sneezing and coughing conditions are emulated using a Gaussian velocity distribution function at the inlets corresponding to nose and mouth with maximum speed, $u_{\text{m}}$, same as the speed of the sneezing or coughing, respectively. Table 1 depicts the function and the typical values that are employed for the simulations in the present study. The velocity boundary condition for the inflows at inlets 3 and 4 has been enforced as $\mathit{u}=-u_0\mathit{n}$. The atmospheric pressure boundary condition $p=p_0$ is maintained at the outlet while the droplets are considered to be immovable after colliding the walls ${\mathit{u}}_d=0$.

Table 1.

Shows the typical values for the parameters employed in the simulations.

Parameter/Expression	Values
Density of air, ${\rho }_f$	1 kg/m3
Dynamic viscosity of fluid, ${\mu }_f$	1.8 × 10−5 Pa s
Velocities at nose and mouth inlets, $u_{\text{m}}{\left(2\pi {\sigma }^2\right)}^{-1/2}{e}^{-\left[{\left(t-0.3\right)}^2/\left(2s^2{\sigma }^2\right)\right]}$	$u_{\text{m}}$ = 9.2 m/s for sneezing, $u_{\text{m}}$ = 6.2 m/s for coughing, and $\sigma =$ 0.1
Average air inflow, $u_0$	0, 1, 2 m/s
Droplet density, ${\rho }_d$	1200 kg/m3
Droplet diameter, $d_d$	1–500 µm
Dynamic viscosity of droplet, ${\mu }_d$	10−3 Pa s

3. Numerical simulation and validation

COMSOL^TM Multiphysics, a finite element based CFD software, has been utilized for this work to solve the Eqs. (1), (2), (3), (4), (5) after enforcing the aforesaid boundary conditions. The software employs the Galerkin least-square method for the discretization of the equations with the boundary conditions and the fluid-particle-tracing (FPT) module has been employed for the droplet tracking in the surrounding air. In particular, the physics controlled unstructured triangular grids are used as the mesh elements during the simulations. For time marching, a backward difference scheme has been employed with an optimum time step of ∼10⁻⁴ s to obtain an accurate solution. Since the solution space is significantly large, 3 m × 5 m, as compared to the size of the droplets, 1–500 µm, a grid independence study is performed to satisfy the adequate mesh required for an accurate numerical solution. The Table S1 of the Electronic Supporting Information (ESI) shows the details of the meshes, in particular number and size range, for four different cases, which are considered for the comparison. Further, the accuracy of the FPT module has been validated against a simple analytical solution and the simulation results obtained by Dbouk [22] are contrasted with the simulations from the proposed framework. The details of the mesh independence study and validation are shown in Fig. S1 of the ESI.

4. Results and discussion

4.1. Viral load of Sneeze/Cough droplets

The respiratory processes like sneezing and coughing eject numerous water droplets with varying sizes ranging from less than 1 µm to 1000 µm in diameter or sometimes even larger [20]. These droplets are enriched with various elements like water, microbes, proteins, carbohydrates, or salts, apart from having different disease-causing agents like the SARS-CoV-2 virus. Since the focus of this work is to gauge the potency of these droplets in the transmission of viral diseases, these water droplets are termed as ‘virulets’ in this script. The viral load $\left(V_L\right)$ of a droplet ejected from a SARS-CoV-2 infected patient depends on his/her viral concentration ($C_v$) in the mucus. For example, asymptomatic to weakly infected patients have $C_v$ in the range of ∼${10}^4$ copies/mL while severely infected patients may have as high as ${10}^9$ copies/mL [51]. Thus, a droplet ejected from a patient can have a varied amount of viral load based on its size and the number of virus particles present inside the same. The probability of total viruses loaded inside a virulet for different $C_v$, can be approximated using Poisson’s distribution [52] as,

$$f\left(n\right)=e^{-V_{L}d\frac{{\left(V_{L}d\right)}^n}{n!}}$$ (6)

Here, $f\left(n\right)$ is the probability of finding n number of viruses inside a virulet of diameter $d$, and $V_L\left(d\right)$ is the viral load,

$$V_L\left(d\right)=\frac{\pi }{6}\left(d^3{C}_v\right)$$ (7)

In this study, $d$ have been varied from 1 to 500 µm to consider the entire range of virulet sizes ejected during the respiratory activities like sneezing and coughing, and the size of the SARS-CoV-2 virus has been considered to be 100 nm in diameter, as reported elsewhere [53]. Fig. 2 a shows the maximum number of viruses ($n_{\mathit{max}}$) that can coexist in a virulet of size $d$ ranging from 1-500 µm for different $C_v$ ranging from ${10}^4$ to ${10}^9$ copies/mL. The plots suggest that the $n_{\mathit{max}}$ of an evicted droplet increases progressively with $C_v$. Considering a specific case, a 100 µm droplet can hold ∼10 viruses for $C_v={10}^4$ copies/mL, however, the same droplet can hold ∼6000 viruses when ejected from a patient with $C_v={10}^9$ copies/mL. In addition, a higher load of the virus is expected inside the virulets with a higher $d$. On the contrary, a bigger droplet with more viral load is expected to settle on the surface much faster while the smaller ones with a less viral load have the capacity to become airborne, as discussed later. The plots in the figure indicate that the infection potency of the virulets is less for smaller d (e.g. less than 100 μm) and lower $C_v$ cases. However, such virulets may be of concern when expelled from a highly infected person.

Fig. 2.

Plot (a) shows the maximum number of viruses $n_{\mathit{max}}$ loaded inside a droplet of size $d$ ejected from an infected person, and plot (b) shows the fraction of $E_v$ among the total droplets ejected of size $d$, for $C_v$ ranging from ${10}^4$ copies/mL to ${10}^9$ copies/mL and $d$ ranging from 1 to 500 µm.

The other important factor to note here is that all the virulet sneezed or coughed out may not be infectious. Fig. 2b shows the fraction of the effective number of virulets ($E_v$) among total number droplets ejected during sneezing or coughing. $E_v$ can be defined as those specific droplets among all the expelled droplets which are loaded with viruses. Here, the total number of the ejected droplets is the total number of droplets of a specific size $d$ that can be accommodated in 1 mL volume of water. The detailed calculation of $E_v$ is given in Section S2 of the ESI. The plots in Fig. 2 signify that for a lower $C_v$, a small fraction of the total ejected droplets are loaded with the viruses, whereas for a higher $C_v$, nearly all the ejected droplets can be filled with a significant number of viruses. For example, if 1 mL of water is transformed into its corresponding number of say $d$ of 200 µm, then for $C_v$ = ${10}^4$ copies/mL, only ∼22% of droplets host at least one virus. Conversely, for $C_v$ = ${10}^9$ copies/mL, each droplet (100%) may carry more than one virus during the ejection. The figure shows that for lower $C_v$, not all the droplets ejected are infectious. However, the majority of the droplets ejected from higher $C_v$ patients are capable of spreading the disease. The plots also help in inferring that although the risk of infection of a healthy exposed person is high from droplets ejected from a higher $C_v$ valued patient owing to higher $n_{\mathit{max}}$ and $E_v$, the risk is not zero from asymptomatic or weakly infected patients.

4.2. Motion of respiratory virulets

In addition to the infection potency of the virulets, the infection risk of an exposed person also depends on the spatial distance of exposure from the infected person, i.e., the spreading distance of the virulets. Fig. 1 schematically shows the motion of the virulets ejected during a sneeze in different pathways for $d$ of 1–500 µm – (i) floating in the air and traveling far away, (ii) falling to the ground at a shorter distance, and (iii) fleeting virulets traveling to an intermediate distance before sinking. Such different modes of transmission of the virulets are governed by their characteristic flow dynamics. Here, we have numerically simulated the micro virulets ejection of different size ranges through a human sneezing process and its subsequent spreading in the open air, as described in Fig. 3 . In order to mimic the normal sneezing airflow, we consider a Gaussian velocity distribution function, as described in the previous section. In accordance with the common sneezing conditions reported by the previous study [54], the time of ejection of the droplets and maximum amplitude of the function are taken as 300 ms and 9.2 m/s, respectively. The Reynolds number calculated for the present system is found to be ∼10⁵ for 1 m/s external air condition and ∼10⁴ for 0.5 m/s, which is well below the turbulent transition limit of the open channel boundary layer system [55]. Also, we have compared the flow profile of the virulets for both 0.5 m/s and 1 m/s velocity conditions, as shown in Fig. S2 of Section S3 of the ESI, and it illustrates no change in the motion of the virulets except the distance of spreading of the virulets which varies with the outdoor air velocity. Hence, the consideration of laminar flow is scientifically justified in the present system. Therefore, for all the calculations of the transmission of the droplets, as discussed later, the maximum air limit of laminar of 1 m/s has been imposed to study the spreading of the droplets.

Fig. 3.

Plots (a) – (d) show the flight trajectory and (e) – (h) shows the corresponding velocity profiles of the virulets expelled from the sneeze of a SARS-CoV-2 infected person, taking droplet size ranges of 1–50 µm, 60$-80$ µm, 80–200 µm, 300–500 µm, respectively. The color bar in subplot (d) shows the dimension range of the simulated virulets, the terms ‘min’ and ‘max’ indicate minimum virulet diameter and the maximum virulet diameter of that droplet range, as considered in subplots (a–d).

For each range, 5000 number of virulets are taken to emulate the actual concentration during a human sneeze, and the same is at par with a previous study [54]. For all the simulation studies, external airflow is maintained at a light breeze of 1 m/s, moving from the left of the solution space to the right, maintaining a laminar flow. The flow profiles of such systems have been simulated for different-sized droplets for a time period of 5 s. A set of four droplet size ranges is considered to review the landing trajectory and flow velocities at different time periods until 5 s. The typical droplet size ranges taken are, 1–50 µm, 60–80 µm, 80–200 µm, and 300–500 µm, as shown in the Fig. 3a–d and Supporting Video 1 of ESI. The color bar in Fig. 3d represents the corresponding size of the virulets considered for the study. Fig. 3e–h show temporal variations of the x and y-directional velocities of 1–50 µm, 60$-80$ µm, 80–200 µm, and 300–500 µm sized virulets, respectively.

Fig. 3a shows the flow trajectory of 1–50 µm sized virulets, which suggests that these virulets are likely to travel a longer distance, larger than 5 m in 5 s time following a re-circulatory flow pattern, as can be found in the image set of Fig. 4 a. Interestingly, the virulets form trajectories resembling the rotating fluid vortices, when they float in the air. Fig. 3e suggests that the x-directional velocity, $u$, of the virulets is maximum at the nose inlet (at 0.3 s), which reduces progressively with the time and distance traveled. On the other hand, the y-directional velocity of the virulets, $v$, is towards the ground (less than 0) during the ejection. With the progress in time, the virulets become air bound as $v$ increases with time. The figures together suggest that these tiny virulets drift in a forward direction from the inlet and drive upwards during its pathway with an increase in $v$, and do not settle to the ground easily. Therefore, the virulets in this range remain suspended in the air for a long time and can transit to a far distance with the airflow. The ejection of fluid into the computation domain develops local counter rotating twin vortices in the nearby space, as shown in Fig. 4.

Fig. 4.

Image sets show the velocity contour of the virulets sized (a) 1–50 µm, (b) 60$-80$ µm, (c) 80–200 µm, (d) 300–500 µm.

Such local wakes infused with virulets are convected further by the laminar flow of the surrounding fluid. However, the virulets to follow the fluid vortices are decided by the forces acting on it – inertia, drag, and gravity force. The smaller droplets by virtue of inertia tend to float along with the fluid vortices mimicking the circulations, which lifts the tiny droplets to travel to long distances, as shown in Fig. 4a. The detailed analysis of the forces acting on the virulets is discussed further in the script.

Fig. 3b shows the trajectory of the 60$-$80 µm sized virulets and the corresponding temporal variations of the x- and y-directional velocities are shown in Fig. 3f. Fig. 3b suggests that the spreading distance of these virulets after 5 s is relatively smaller than in 3a. The figure also shows that the bigger virulets in the range settle on the ground at a much shorter distance while the smaller ones can still become air borne within 5 m. Again, Fig. 3f shows a positive $v$ at the downstream indicating an upward drift for the smaller virulets in the range, following the rotation of the fluid vortices, as portrayed in Fig. 4b. Fig. 4b also illustrates that the bigger virulets in the 60$-$80 µm range depart from the vortex street, after a while, towards the downstream. This is due to the dominance of gravity on the bigger virulets governing its trajectory over the course of time. Similarly, the motions of 80–200 µm sized virulets, as shown in Fig. 3c, illustrate that the virulets settle to the ground at a much shorter time as compared to the prior two cases.

The corresponding velocity profiles in Fig. 3g also suggest a substantial downward motion of the virulets. Unlike the prior two cases, these virulets shed from vortex street and move away to the downstream rapidly, as shown in Fig. 4c, whereas the smaller ones followed the fluid vortices. Interestingly, with a further increase in the size to 300–500 µm, the virulets settle to the ground within a second of the ejection, as can be found in Fig. 3d. Fig. 4d also shows that the virulets escape the fluid vortices and settle to the ground immediately after ejection from the nose. The corresponding velocity profiles in Fig. 3h also confirm the sharp decline in $u$ and $v$ within a short time.

Concisely, the simulations suggest that, with an increase in the virulet size from 1–50 µm to 300–500 µm, the settling distance decreases from more than ∼5 m in 5 s to less than 1.5 m in ∼1 s, respectively. The flow trajectory of the virulets ejected during coughing is illustrated in Fig. S3 in section S4 and Supporting video 2 of the ESI. The simulations help in differentiating the floating virulets to be with $d<70\mathrm{\mu }\mathrm{m}$, fleeting to be with $70\mathrm{\mu }\mathrm{m}\le d\le 200\mathrm{\mu }\mathrm{m}$, and falling to be $d>200\mathrm{\mu }\mathrm{m}$. Remarkably, the simulations suggest that they can travel a much larger distance than the standard protocol of 2 m set for physical distancing and can be one of the major reasons behind the community transmission with the high viral potency. In this regard, one may easily envision that the floating virulets may exclusively adopt the airborne mode of disease transmission while the falling ones may facilitate droplet or fomites mode of disease transmission. The fleeting mode may contribute towards both the modes. The images in Fig. 3 also illustrate that an exposed person at a similar height or more than the infected one is vulnerable to the passing virulets even at a distance of greater than 2 m when the virulets are airborne. However, the children are more susceptible to the virulets even at the close vicinity of the infected person through the falling virulets. Moreover, the floating respiratory droplets ejected from an infected child can be sufficiently active to transmit infections to the adults through the air borne recirculation of the virulets.

4.3. Analyses of force fields acting on virulets

Detailed understanding of the droplet dynamics coming out of patients’ nose or mouth is crucial to foresee the infectivity potential of these virulets. The virulets ejected during a sneeze, form a conical jet near the nose inlet and are mostly affected by the inertia force and thus travel along the direction of the initial velocity. As the virulets travel far from the nose inlet, the droplet velocity, ${\mathit{u}}_d$, coupled to the surrounding fluid velocity, ${\mathit{u}}_f$, decreases progressively from its maxima, as described earlier. The virulets in this regime experience a swirling flow during the motion due to the shearing of the ejected conical jet in the calm air around the inlet of the solution space [56]. As the particles move further, the size of the virulets determines their destination, as depicted in Fig. 5 . The component-wise force analyses along the normal directions of the Cartesian reference frame are done to explore the finer features of virulet dynamics. The trajectory of the virulets is mostly governed by three major factors – ejection inertia, downward gravitational pull, and viscous drag. Since the study is done employing virulets of a varying size range, the contribution of different forces alters with subsequent changes in its trajectories.

Fig. 5.

Plots (a) to (e) show the x – directional drag force (${\mathrm{F}}_d^{\mathit{Dx}}$), y – directional drag force (${\mathrm{F}}_d^{\mathit{Dy}}$), x – directional inertia force (${\mathrm{F}}_d^{\mathit{Ix}}$), and negative y – directional inertia force (${\mathrm{F}}_d^{\mathit{Iy}}$), and negative gravitational force (${\mathrm{F}}_d^g$) acting on the particles of size ranges of 1–50 µm, 60–80 µm, 80–200 µm, 300–500 µm, ejected via sneezing.

Fig. 5a and Fig. 5b show drag force acting in horizontal (x–axis) and vertical (y–axis) directions, respectively, for different sized particles. In addition, Fig. 5c and Fig. 5d show the inertial forces in horizontal (x–axis) and vertical (y–axis) directions, respectively. The domination of gravitational pull with size can be substantiated from Fig. 5e. For 1–50 µm sized virulets, on comparing the two x-directional forces, i.e. ${\mathrm{F}}_d^{\mathit{Dx}}$ acting along the negative x-direction and ${\mathrm{F}}_d^{\mathit{Ix}}$ acting along the positive x-direction, it can be observed that the inertia of the virulets is relatively much higher than its drag force. Thus, the act of inertia is responsible for the faraway forward motion of these tiny virulets. Comparing all the y-directional forces, i.e. ${\mathrm{F}}_d^{\mathit{Dy}}$ acting in the upward y-direction, and ${\mathrm{F}}_d^{\mathit{Iy}}$ and ${\mathrm{F}}_d^g$ acting along the negative y-direction, it can be inferred that the act of gravity is relatively very small due to smaller size, and an act of inertia is very less due to lesser mass as compared to the drag force. Thus, the effective drag force in positive y-direction causes the lift of the virulets in the upward direction.

Similar observations can be made for 60 $-$ 80 µm sized virulets, which causes the lift and forward motion. However, the effectiveness of the resultant x and y directional forces for 60–80 µm sized virulets are relatively minor than for 1–50 µm, which causes them to settle earlier. Comparing the x-directional forces for 80–200 µm and 300–500 µm sized virulets, the inertial force is comparatively smaller than its drag force along the x-direction. Also, the combined forces of gravity and inertia along negative y-direction are higher than the upward drag force. Due to higher gravitational and inertial act along the downward direction, these virulets tend to settle much faster as compared to the 1–50 µm and 60–80 µm sized virulets. Of course, the resultant action of the x and y directional forces is higher for 300–500 µm sized virulets owing to larger size and mass than for 80–200 µm, which explains the immediate fall of the 300–500 µm sized virulets after ejection as for 80–200 µm virulets.

In order to further explore the flow dynamics of the virulets, Froude’s number (Fr), Reynolds number (Re_p), Weber number (We), and Capillary number (Ca) has been calculated for the virulets of size ranges of 1–50 µm, 60–80 µm, 80–200 µm, 300–500 µm, ejected via sneezing, for the time period of 5 s, as shown in Fig. 6 a–6d. Fig. 6a shows that the smallest virulets of 1–50 µm size experience higher Fr owing to its high inertia and low gravity and with an increase in virulet size to 300–500 µm, the Fr decreases gradually. This is because the gravitational force acting on the virulets of larger virulets increases with an increase in size as shown in Fig. 5. Here, the floating virulets of 1–50 µm size shows an oscillatory behavior of the Fr during its course of the flow which is mainly because of the variation in the velocity of the virulets at a particular time owing to various forces acting on it as described in Fig. 3 and Fig. 5. Fig. 6b shows the Re_p increases with an increase in the diameter and velocity of the virulets at a given location. However, Re_p for all the virulets size is well below the turbulent limit of 500 [50] which shows the consideration of laminar is valid in the present study. Similarly, We and Ca shown in Fig. 6c and Fig. 6d, respectively, show that the spherical liquid bubbles do not deform during their motion in the external wind.

Fig. 6.

Plots (a–d) show the Froude number, Reynolds number, Weber number, and Capillary number for the virulets of size ranges of 1–50 µm, 60–80 µm, 80–200 µm, 300–500 µm, ejected via sneezing, respectively.

4.4. Effects of multiple ejections and wind flow

Further, the infectivity risk from the traversing virulets can also augment by frequent medical factors like simultaneous sneezing and coughing by the infected person, heightened coughing or sneezing frequency, or by external influences like a windy day. The effect of these factors on the spreading of the virulets is depicted in Fig. 7 . The motion of infected respiratory virulets in case of simultaneous sneezing and coughing from an infected person is shown in Fig. 7a and Fig. 7b and in Supporting Video 3 of the ESI. A range of 1–500 µm sized droplets, expelled from the nose or mouth of an infected person with velocities same as for single sneezing or coughing processes has been considered for the study. It can be easily inferred from the figure that the density of the virulets in the given area is more during simultaneous sneezing and coughing process as compared to single sneezing or single coughing processes with an increase in the time lag between the two processes from 0.1 s to 0.4 s, although the trajectory of motion remains the same. The more the virulets are expelled by the infected person, the more is the risk of the spread of the infection. Another such event of enrichment of infected droplets in a given area could be multiple ejections of the infected respiratory droplets, by an infected person.

Fig. 7.

Shows different events that enhances the transmission of SARS-CoV-2 virulets, enhancing the community spreading of the disease. Plot (a) shows the spreading of virulets ejected during simultaneous sneezing and coughing from a SARS – CoV – 2 infected patient with time lag between the two events be 0.1 s and (b) 0.4 s. Plot (c) shows the distribution of virulets during single sneezing and (d) sneezing for 4 times. Plot (e) shows the dispersal of the virulets during external wind conditions of light breeze of 1 m/s from left to right, (f) moderate breeze of 2 m/s blowing from left to right , and (g) light breeze of 1 m/s from left to right and botton to up of the solution space, respectively. The time frames from 0.3 s to 5 s has been combined in one image for all the cases.

Fig. 7c andFig. 7d and SupportingVideo 4 of the ESI highlight the droplet trajectories and the increase in density of the virulets in case of multiple sneezing from a single sneeze to four times simultaneous sneezing. It can be depicted from the figure that with the increase in the sneezing frequency, the density of the virulets also increases in the given area. Consequently, anyone, irrespective of the height, exposed to the infected person's vicinity is at higher risk of inhalation of the expelled virulets and thus getting infected with the disease.

Additionally, the external wind conditions can also augment the spreading of the virulets. The effect of the external wind speed on the spreading distance of the respiratory virulets is illustrated in Fig. 7e–g and in Supporting Video 5 of the ESI. Three different external wind speed conditions, light breeze (1 m/s) and moderate breeze (2 m/s) from left to right of the solution space, and an angular light breeze (1 m/s) from left to right and bottom to top are considered for the analyses for 1 s time for virulets ranging from 1–500 µm. It can be observed that under light breeze conditions of 1 m/s, Fig. 7e, the virulets do not traverse more than 2 m distance in 1 s time whereas it is 3 m for 2 m/s, shown in Fig. 7f. In the case of angular breeze condition, Fig. 7g, the aerial transmission of the ejected virulets is amplified by the external breeze. Consequently, the natural pathways may also play an important role in the community transmission of the disease.

4.5. Effect of temperature and relative humidity on virulet evaporation

The spreading of the virulets is a major factor in the community transmission of SARS – CoV – 2, which depends on how remote the respiratory droplets can travel carrying the infection causing agents. The time of fall of the virulets termed as settling time ${(t}_s)$ decides how long it can survive freely before dropping to the ground. $t_s$ of a virulet can be obtained by balancing the Stroke’s friction force with the ${\mathit{F}}_d^g$ experienced by the virulets during its flow [41], as given in Eq (8),

$$t_s=\frac{h}{V_s}$$ (8)

Here $h$ is the height at which virulets are ejected which is the height of the infected person – 1.5 m in this case, and $V_s$ is the velocity of the virulet of radius $R_0$ at the time of fall – the settling velocity, given by.

$$V_s=1.2\times {10}^8\left(R_0^2\right)$$ (9)

Remarkably, $t_s$ of the virulets alters during its flight due to the modification in its size ($R_0$) owing to surrounding weather conditions such as $T$ and $\mathit{RH}$. The variations in $T$ and $\mathit{RH}$ stimulate the miniaturization of the traversing virulets by virtue of evaporation. Accordingly, from Eq. (8), the reduction in $R_0$ enhances the $t_s$ and hence the virulets can flow with air for a longer time before falling to the ground and spread faraway aiding the community transmission of the disease.

Fig. 8a schematically shows the miniaturization of the ejected virulets to the evaporated virulets at $T=25{}^{°}\mathrm{C}$ and $\mathit{RH}=50\%$. It can be observed from the figure that the diameter of the ejected droplets can change from the falling to the floating ones owing to the evaporation followed by miniaturization of the size. Such miniaturization increases the infection potency because, as discussed in Fig. 2, the bigger droplets always carry a larger number of viruses, although they tend to settle easily on the surface under gravitational pull. However, if the T and RH of the surroundings facilitate the floating of such falling virulets due to rapid evaporation, the droplets with more virus loading may become air borne with a more potency towards the viral transmission. Further, the miniaturization of the ejected droplets also depends on the number of viruses loaded inside that droplet. If we assume the complete evaporation of an ejected droplet to be a state at which all the water content of the ejected droplet vanishes, then the miniaturized size of that ejected droplet is equivalent to the diameter corresponding to the total volume of all the viruses contained in that particular droplet.

Fig. 8.

Shows the effect of evaporation on the virulets. Plot (a) schematically shows the miniaturization of the virulets at $T=25{}^{°}\mathrm{C}$ and $\mathit{RH}=50\%$. Plot (b) shows the minimum possible $d_e$ that an ejected droplet $d$ can achieve after complete evaporation for different $C_v$ ranging from ${10}^6$ to ${10}^9$ copies/mL. Plot (c) shows the comparison of $t_s$, $t_{\mathit{eva}}$, and $t_e$ for $C_v$${=10}^6$ copies/mL.

Fig. 8b shows the minimum possible diameter of the virulets after complete evaporation, $d_e$, for different $C_v$ ranging from ${10}^6$ to ${10}^9$ copies/mL. For example, say for $C_v=$ ${10}^6$ copies/mL, a droplet of 100 µm can hold a maximum of 400 viruses, as can be found in Fig. 2. After complete evaporation, the minimum possible miniaturized droplet size, $d_e=$ 155 nm, which is the equivalent diameter to the volume of 400 number of 100 nm sized viruses loaded inside a 100 µm sized ejected droplet for $C_v=$ ${10}^6$ copies/mL. However, the evaporation and subsequent state of miniaturization of $d$ to $d_e$ depends on $t_s$, as described in Fig. 8c.

Fig. 8c shows a comparison of the evaporation time $(t_{\mathit{eva}})$ of $d$,$t_s$ and the settling time of the miniaturized droplets ${(t}_e)$ after evaporation. It can be observed from the plot that for $d$ less than 80 µm, $t_s$ is substantially larger than the $t_{\mathit{eva}}$. This means the virulets get adequate time to evaporate before settling to the ground and thus these virulets are capable to reach its $d_e$ at its corresponding $t_{\mathit{eva}}$. For $d$ ranging from 80 µm to 200 µm, $t_s$ is comparable to $t_{\mathit{eva}}$ and thus the evaporation fate of these virulets to reach $d_e$ depends on the external T and RH conditions. However, for $d$ greater than 200 µm, $t_s$ is very less than $t_{\mathit{eva}}$, which means that the virulets will not get sufficient time to evaporate much and will settle to the ground immediately after ejection. In addition, $t_e$ is found to be in the order of several days as shown in Fig. 8c, which emphasizes that the evaporated virulets that form the floating miniaturized droplets are the bioaerosols that can suspend and circulate in air for days before falling down. The calculations for determining $t_{\mathit{eva}}$ and $d_e$ is given in Section S5 of the ESI.

Seemingly, it is evident from the results shown above that the effect of evaporation plays a significant role in the spreading of viral infections via virulets, which are also highlighted in the past literature [41]. In a tropical country like India, the climatic conditions relative to $T$ and $\mathit{RH}$ varies across the geography. The western regions experience high temperatures up to $T=50$ °C whereas the eastern regions experience cold weather throughout the year, and the coastal regions experience high humidity as compared to the other states. Similarly, the climatic conditions of different countries across the globe also vary during different seasons throughout the year. These variations in the $T$ and $\mathit{RH}$ decides the rate of the evaporation of the virulets in a particular region and at a particular time of the year. Thus, the rate of the miniaturization of the virulets to evolve into bioaerosol also varies with geographic location and seasonal changes.

Fig. 9 represents schematically the effect of change in the $T$ and $\mathit{RH}$ on the miniaturization of 100 μm sized virulet for various $T$ and $\mathit{RH}$ conditions. The figure shows that for $T=40$ °C and $\mathit{RH}=10\%$, the 100 μm can miniaturize to 1.1 μm in just 2 s whereas, for $T=40$ °C and $\mathit{RH}=40\%$, the virulet evolve to 6.24 μm in 4.5 s. Similarly, for cold conditions, like for $T=10$ °C and $\mathit{RH}=10\%$, the virulet evolves to approximately half of its size in 4.5 s and for $T=27$ °C and $\mathit{RH}=95\%$, the miniaturization is minimum i.e. 77.43 μm in 5 s. Thus, the figure illustrates that the hottest and low to moderate humid conditions suffer the fastest bioaerosol formation.

Fig. 9.

Schematically shows the effect of evaporation due to variation in $T$ (in °C) and $\mathit{RH}$ (in %) conditions on a 100 μm sized virulet.

4.6. Comparison of COVID-19 infection and virulet evaporation

The transmission of COVID–19 in a particular region depends on a number of crucial factors like the density of population [57], genetic makeup [58] and age of the people [59], metabolic co-morbidities [60], healthcare facilities [61], lifestyle factors of hygiene [62] and diet [63], susceptibility of the exposed person [64], administrative control [65], climatic conditions of the region [66], mutation of the virus [67], vaccination rate [68] and among others. Of all the factors listed one of the major causes of the bioaerosol formation is the meteorological conditions of a location that has been accredited to the occurrence of the highly infectious second wave of SARS-CoV-2 pandemic. A number of recent studies have been performed to study the possibility of the airborne transmission of the infection. In one study, authors have detected the presence of SARS-CoV-2 ribonucleic acid in the air samples collected from various regions, with successfully cultured virus from a few samples. Along with that, they have also reported that SARS-CoV-2 is viable in artificially generated aerosols [69]. Another study also reports the presence of SARS-CoV-2 virus in air collected from both indoor and outdoor environments [70]. In another study, the evidence of airborne transmission of the disease has been proved in a church singing event in Australia. It was found that the secondary case patients were seated in approximately 15 m proximity of the primary patients without close physical contacts, suggesting air as the mode of the transmission of the disease [71]. Apart from these, epidemiologists have also reported the airborne transmission of the virus and the low air humidity as the most possible reason for the rapid spread of the disease in the Amazonian states [72]. Moreover, other than SARS-CoV-2, air borne transmission of other diseases like tuberculosis, streptococci, pneumococci, coliform organisms, and influenza virus has already been reported for long time [73], [74], [75], [76]. In view of this, we have voluntarily opted to map the effect of evaporation of the ejected respiratory droplets as an effect of external $T$ and $\mathit{RH}$ with the trend in the increase in infection rate at different geographical regions.

In this section, we focus on comparing two sets of parameters based on the $T$ and $\mathit{RH}$ conditions to understand such community transmission of the virulets. In the first set, we investigate the correlation between the evaporation driven rates of miniaturization of virulent $\left(\text{d}{m}^R/\text{d}t\right)$ with the rate of occurrence of confirmed cases of COVID–19 $\left(\text{d}{i}^R/\text{d}t\right)$. As a second case study, we show the correlation of evaporation rate (m) and the absolute number of confirmed cases (i). The entire comparison analyses are made considering the three most COVID–19 infected places, Maharashtra, Karnataka, and Delhi, from India and three major cities across the globe, Los Angeles, São Paulo, and Paris. The study is envisioned to predict the future occurrence of COVID–19 waves based on the T and RH of the various places across the globe. The data for the number of confirmed cases of COVID–19 infection have been collected from two authentic websites; (i) https://www.covid19india.org/ for Indian regions, and (ii) https://www.worldometers.info/ for the world data. The rate of miniaturization of the virulet in the air due to evaporation has been calculated from Eq. (10), given by Williamson and Threadgill [77],

$$\frac{\text{d}d}{\text{d}t}=\frac{-4M_L{D}_{v}f}{d{\rho }_{L}RT}\mathrm{\Delta }P\left(1+0.276R_e^{0.5}{S}_c^{1/3}\right)$$ (10)

Here, $M_L$, ${\rho }_L$, $T$, $D_{v,f}$, $R$ , $\mathrm{\Delta }p$, $\mathit{Re}\mathrm{a}\mathrm{n}\mathrm{d}Sc$ are the molecular weight of water, the density of water, average absolute temperature, the average diffusion coefficient for vapor molecules at $T$, gas constant, the difference between the vapor pressure near $d$ and that in the ambient atmosphere, Reynolds number, and Schmidt number, respectively. The T and RH data for the calculation are collected from the meteorological website, https://www.timeanddate.com/, for the respective date on which confirmed cases have been taken for the study. The T and RH data used for the study are provided date-wise in section S6 of the ESI.

It may be noted here that the parameters we are exploring in this study are not the only factors for the spreading of the pandemic. There are other multiple scientific, mutational, physiological, environmental, and administrative factors that determine the spreading of the infection. However, despite having the influence of such parameters, we observe a very faithful correlation between $\left(\text{d}{i}^R/\text{d}t\right)$ and $\left(\text{d}{m}^R/\text{d}t\right)$ .

Fig. 10a–f show the case study for three major COVID–19 affected regions in India – Maharashtra, Karnataka, and New Delhi, respectively. Fig. 10a shows the rate of miniaturization of the ejected droplets and the rate of change of confirmed COVID–19 cases against different dates counted from June 2020 to June 2021 to study the first and second waves. It should be noted here that the dates considered for the study are in accordance with the rising of first and second COVID–19waves at that particular location. It is evident from the figure that the trend of matches fairly with. The plots also show a time lag between the rate of infection and the rate of miniaturization of the droplets. Intuitively, one can easily envisage that this lag is correlated to the formation of bioaerosol first and then the spreading of the infection leading to the increase in the cases. In a way, such an observation justifies the possibility of the formation of bioaerosol due to the variation in T and RH at different times of the year and in different regions.

Fig. 10.

Shows the comparison of rate of miniaturization $\left(\text{d}{m}^R/\text{d}t\right)$ of 100 µm sized virulet with the rate of change of the confirmed COVID cases$\left(\text{d}{i}^R/\text{d}t\right)$, and the evaporation rate m and confirmed cases i to predict the expected further occurrence of covid cases in near future. Subplots (a)$-$(f) show the trend for the three most covid affected regions in India– Maharashtra, Karnataka, and Delhi, respectively and subplots (g)$-$(l) show three most covid affected locations in world– Los Angeles, São Paulo, and Paris, respectively.

In a different case analysis, Fig. 10b shows the comparison of occurrence of COVID–19 cases i and the evaporation rate of the droplets m. It can be seen from the figure that in this case too, the trend in the increase in COVID–19 infection is comparable with the evaporation rate curve. The dotted curves are the extrapolated arches of the rise in COVID–19 cases in accordance with the evaporation rate calculated from July 2021 to December 2021, as discussed later. Fig. 10c and Fig. 10d show similar analyses for Karnataka, and Fig. 10e and Fig. 10f show analyses for Delhi. These cities show a very similar trend of infection profile against evaporation rate and droplet miniaturization. For both Karnataka, shown in Fig. 10c, and New Delhi, shown in Fig. 10e, the rate of infection $\left(\text{d}{i}^R/\text{d}t\right)$ follows the rate of miniaturization of the droplets $\left(\text{d}{m}^R/\text{d}t\right)$ and these observations substantiate that the miniaturization of the virulet size and formation of bioaerosol directs the infection rate in the region. Fig. 10d and Fig. 10f show a good agreement of COVID–19 cases i with the evaporation rate of the droplets m for Karnataka and New Delhi. A small deviation is observed for the case of Karnataka and the same might have happened for other causes that have not been considered for the present study.

Now since the results established the faithfulness of the model for the community spreading of the infections during the waves of the pandemic, we attempt to predict the number of COVID–19 cases in the near future. For the prediction algorithm, we follow a higher order polynomial curve fitting analysis of the number of absolute cases i vs. days. The data of absolute COVID–19 cases farthest from May 2020 to June 2021 are taken as a reference for developing the polynomial fitting and the same is subsequently extrapolated for the prediction of future COVID–19 cases. The broken line portion in the plots of Fig. 10b, Fig. 10d, Fig. 10f, Fig. 10h, Fig. 10i, and Fig. 10l show the prediction data for upcoming possible COVID–19 cases. The values for the T and RH from July 2021 to December 2021 are taken as an average of the last three years' data for that specific region.

We also investigate cases of three foreign cities Los Angles, São Paulo, and Paris, since the weather conditions are markedly different in the Indian cities and abroad. The plots suggest that despite a significant demographical change there is a direct correlation of the rate of infection $\left(\text{d}{i}^R/\text{d}t\right)$ with the miniaturization of the droplets $\left(\text{d}{m}^R/\text{d}t\right)$. A similar correlation is also observed for absolute infection i against the rate of evaporation m. Fig. 10g, Fig. 10i, and Fig. 10k show a comparison of the rate of infection $\left(\text{d}{i}^R/\text{d}t\right)$ with the miniaturization of the droplets $\left(\text{d}{m}^R/\text{d}t\right)$ for São Paulo, Los Angles, and Paris, respectively. These plots show a close match with occasional deflection, which may be attributed to other pandemic causes. Fig. 10h, Fig. 10j, and Fig. 10l show a correlation between the trend in the increase in COVID–19 cases i and evaporation rate m. In this absolute case analysis too, we find a close mapping between the curves.

In order to evaluate the degree of correlation between $\left(\text{d}{i}^R/\text{d}t\right)$ and $\left(\text{d}{m}^R/\text{d}t\right)$, we plotted the non-linear correlation for all the six places for the entire time period of infection. Fig. 11 shows the non-linear correlation plot for the six places. The non-linear correlation was evaluated using the non-linear estimator ‘nlcor’ package of the R programming language [78]. It implements a heuristic approach to evaluate the non-linear correlation between the two given parameters and returns a p-value as a factor to signify the degree of correlation. It identifies piecewise linear correlations of the local regions to estimate the overall non-linear correlation. The correlation estimate (nlcor value) and p-value obtained for the six selected regions are listed in Table 2 . The correlation plot and Table 2 shows that the rate of infection and rate of miniaturization of the virulets for all six locations are significantly correlated as the p-value is less than the threshold (p < 0.005) [79]. Therefore, it is justified to claim that the rate of occurrence of infection cases in a particular location is also a factor of the rate of miniaturization of the ejected virulets driven by the geographical conditions of T and RH of the corresponding location.

Fig. 11.

Plot shows the non-linear correlation plot of di^R/dt against dm^R/dt for the six selected places.

Table 2.

Shows the nlcor value and p value for the non-linear correlation of di^R/dt against dm^R/dt for the six selected places.

Location	nlcor value	p-value
Maharashtra	0.363	0.00
Karnataka	0.255	0.00
New Delhi	0.163	0.02
Los Angeles	0.401	0.00
São Paulo	0.469	0.00
Paris	0.158	0.03

The above results enlighten the role of T and RH on the evaporation driven miniaturization of the virulets and its significant correlation with the trend in the occurrence of the infection cases in the six different geographical locations considered in the present study. In view of that, we have done CFD simulations considering T and RH for six different places considered in the study, as shown in Fig. 12 , to elucidate the transmission of the miniaturized virulets and to mark the optimal whether conditions that favor the maximal transmission of the virulets to far distances. Here, we simulated the spreading distance of the ejected virulets on certain weather conditions for six different places, as shown in Fig. 12.

Fig. 12.

The plot shows the high and low-risk weather conditions at which maximum transmission of the virulets is probable at different geographical locations.

The farther reaching virulets can be more potent to spread the infection to more people, however, if the virulets are settling at a shorter distance, the infection risk can be low but not absolutely zero. For the simulations, we kept the T of a particular region to be constant to its minimal and maximum limit and varied the RH from its lowest to the highest value, as reported on meteorological websites. In the subplots (a)$-$(l) of Fig. 12, the high-risk RH has been marked in red and the low-risk RH is marked in green color. If we consider the case of Maharashtra, it can be seen that for the low $T={25}^{°}C$, if $\mathit{RH}<50\%$, the virulets could float in the air and reach to farther distances in less time as compared to the settling of the virulets as observed for $\mathit{RH}>50\%$, as shown in Fig. 12(a). However, with the increase in $T=34{}^{°}\mathrm{C}$, the virulets could float even for larger RH, as shown in Fig. 12(b). Thus, there is a high probability that if the $\mathit{RH}$ reaches lower than 50% in Maharashtra at any time of the year, there is a higher possibility of an increase in infectivity ratio.

Similarly, in Karnataka, there is a high risk of an increase in infectivity whenever the RH condition reaches below the semi-humid level, as shown in Fig. 12(c) and (d). Comparatively, if we consider New Delhi, the high-risk RH limit in colder days is found to be very lower than in warmer days, as shown in Fig. 12(d) and (e). Similarly, in Los Angeles, the high-risk RH limit is found to be very lower on colder days and it reaches semi-humid conditions with an increase in temperature, as shown in Fig. 12(e) and (f). In Sao Paulo, the high RH limit is found to vary minimally cold and warm, as shown in Fig. 12(g) and (h). In Paris, it was found that the lower the temperature, the risk level reduces which probably reduces the infectivity ratio, however, the limit increases to the semi-humid limit with an increase in temperature, as shown in Fig. 12(i) and (j).

Conclusively, it can be illustrated from all the subplots in Fig. 12 that both cold-dry and warm-semi humid conditions could be attributed to the major contributor of virulet miniaturization and transmission based mediator of high infectivity ratio. Though, the risk of infection transmission in other weather patterns at different geographical locations is absolutely not zero. Thus, the present work shows a satisfactory resemblance to the weather patterns for influenza flu transmission as reported in the prior studies [80], [81].

Further, we have shown the authenticity of the claimed evaporation model by comparing the predicted COVID–19 infection cases (i, showed in the broken blue line in Fig. 10) with the reported COVID–19 infection cases ($i_R$) for the first and second waves of the infection in the last two years for six different geographical regions. Fig. 13 shows the match between the predicted and reported COVID–19 infection cases for the mentioned time period. It can be inferred from the figure that the trend in the occurrence of the reported infection waves in several regions matches well with the COVID–19 infection waves predicted in the present study.

Fig. 13.

Plot shows the similarity between the predicted COVID–19 infection cases (i, broken blue line) and the reported COVID–19 infection cases ($i_R$, solid green line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Precisely, the predicted model shows exactly the same number of waves, including both weak and intense peaks, occurring at a similar time of the year as it is reported during the first and second waves of COVID-19 infection. Although, there is a lag in the intensity of the real infection cases as compared to the predicted one, which may be attributed to other additional carriers of transmission like viral seeding, population susceptibility, person-to-person contact, vaccination rate, face and mask hygiene, and others, which varies over the year and has not been added to the present study as the aim of the work is to understand the effect of T and RH only, on the miniaturization of the virulets and to predict its transmission with CFD study. Hence, witnessing Fig. 10, Fig. 11, Fig. 12, Fig. 13, it is worth inferring that like with other parameters, evaporation is obviously one of the major factors in the transmission of the COVID–19 via miniaturization of the respiratory virulets.

Concisely, despite having multiple deciding factors for COVID community transmission, the present study unveils that T, RH, and subsequent kinetics of droplet evaporation leading to bioaerosol formation can be one of the very effective tools in predicting the future waves of a viral pandemic.

5. Conclusions

The present work compiles the various possible fluid dynamical causes of the community transmission of the highly contagious SARS–CoV–2 infection. The detailed calculations of the viral content inside the colloidal biofluid microdroplets of varying sizes, the virulets, ejected during the respiratory activities like sneezing and coughing gives an idea of airborne and/or droplet modes of the transmission of the disease. Further, the flow trajectories of the respiratory droplets have been simulated employing the fluid particle tracing physics, to manifest the settling distance of the virulets and to understand the role of bioaerosol transmission in the spreading of the disease. In relevance, it is to be mentioned that since the inception of the pandemic a few reports on droplet or bioaerosol transmission are published from last year. Most of the reported articles explore the droplet movement in the air considering only Newton’s second law of motion [23], [22] neglecting finer features of fluid dynamics, as described in the present study. The flow profiles of the virulets shown in Fig. 3 prove that the prescription of 2 m of physical distance to control the SARS–CoV–2 infections are only applicable to the large virulets whereas the bioaerosol is effective enough in spreading the disease to longer distances also. Furthermore, this work explicitly addresses the effect of gravity, drag, and inertial forces on the virulets of various sizes, which clearly explains the extent of spread of a virulet of interest. Likewise, the events like increased frequency of virulet ejection or a windy day which further enhances the spread of the disease are also discussed precisely.

Additionally, the work reveals the influence of geographical conditions on the transmission of the virulets at a given location. Here, the effect of T and RH has been examined [82], [83], [41] to measure the rate of bioaerosol generation owing to the effect of evaporation, attributed to the occurrence of the second wave of SARS–CoV–2 infection. In Fig. 10, reported infection cases of six different locations – Maharashtra, Karnataka, New Delhi, Los Angeles, São Paulo, and Paris have been mapped with the evaporation driven miniaturization rate of the virulets of that location for the time period of the first wave of COVID-19 infection. Interestingly, the reported infection waves show a similar trend in occurrence as that of the rate of miniaturization for a given location. To further justify the quantitative relation, in Fig. 11 and Table 2, a non-linear correlation analysis of the curves shows the p-value is nearly zero for all the considered locations, which proves the significant relationship between the two. The CFD studies that reported in Fig. 12 prove that both the cold-dry and warm-semi humid condition enhances the transmission of the virulets to long distances, attributed to the increase in infectivity ratio for that location. Importantly, the CFD simulation with varying T and RH prameters for all the cities predicted a critical T and RH value that varies from one city to another. This CFD information, can further be used for studying similar disease spreading. We extended the study to predict the future infection waves based on the T and RH mediated miniaturization of the virulets and later matched with the reported infection cases of the second wave of COVID-19 infection. The curves matched well in terms of frequency and time period of occurrence of the second infection wave but with a compromise of the intensity, which could be attributed to the non-consideration of various other factors of the infection at that region.

In a way, the study establishes the importance of T and RH of a given location on the spread of the infection. The study also paves the pathway to future research on explaining the most accurate method of bioaerosol formation from sneeze/cough droplets considering the various other environmental conditions and effects of elements of the colloidal microdroplets. This will also help in explaining the most realistic dynamics of bioaerosol motion and subsequent community transmission.

Mitali Basak and Shirsendu Mitra performed all of the numerical investigations, data analysis, and drafted the initial version of the manuscript. All the authors contributed in conceptualizing, writing, editing, and analyzing the results of the manuscript. The manuscript has been written through the contributions of all authors. All authors have approved the final version of the manuscript.

CRediT authorship contribution statement

Mitali Basak: Conceptualization, Methodology, Software, Investigation, Writing – original draft, Writing – review & editing, Validation, Formal analysis, Visualization. Shirsendu Mitra: Conceptualization, Methodology, Software, Investigation, Writing – original draft, Writing – review & editing, Validation, Formal analysis, Visualization. Dipankar Bandyopadhyay: Resources, Conceptualization, Methodology, Funding acquisition, Supervision, Project administration, Formal analysis, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary material

The following are the Supplementary data to this article: