The one-dimensional transport equation for droplets in the modelled LRT can be represented in a dimensionless manner as (Chakravarty et al., 2022) P e d S t a 2 α β N ∂ ϕ d ∂ τ = ∂ ∂ N 2 β α N 1 − α α ln α 2 ∂ ϕ d ∂ N + P e d q τ 1 − α α  ln α ϕ d − L D ′ ϕ d , (1) where τ, ϕ _d , Pe _d and St _a represent dimensionless time, dimensionless droplet concentration in the airways, droplet Peclet number and airway Strouhal number, respectively. N is the generation number representing spatial position within the LRT, while α and β are length-change and area-change factors assumed while simplifying the geometry (see Supplementary Material for more details). The term on the left hand side of Eq. 1 takes into account temporal change in droplet concentration. The first and second terms on the right hand side of Eq. 1 takes into account droplet diffusion and advective transport, respectively, while the last term takes into account droplet deposition. Detailed derivation of Eq. 1 is provided in the Supplementary Material.

While formulating this model, it is assumed that the droplets are monodispersed, do not coagulate, and do not affect airflow in the airways. The only source of droplets is assumed to be in the pharynx which opens into the trachea (N = 0). These droplets may be a combination of the droplets inhaled during respiration (which have not deposited in the URT) or droplets that are formed through ANM. No additional aerosolization of the mucosa or droplet source are considered. The inhaled droplets are either deposited or washed out of the LRT. Major droplet deposition mechanisms viz. diffusion, sedimentation and impaction have been taken into account while calculating droplet deposition in the LRT (see S1 Text of Chakravarty et al. (2022) for details). The quantity L D ′ ϕ d quantifies dimensionless deposition of the droplets.

The following scaling parameters are used to obtain Eq. 1 τ = t T b , ϕ d = c d c d , 0 , T a = L 0 A 0 | Q max | , S t a = T a T b , P e d = | Q max | L 0 A 0 D d (2) where T _b and T _a represents the breathing time period and the airflow timescale, respectively. c _d is the dimensional droplet concentration with c _d,0 representing the initial c _d at N = 0. D _d represents droplet diffusivity in air D d = ( k B T C S ) / ( 3 π μ a d d ) and is calculated using the Stokes-Einstein relation (Chakravarty et al., 2019). k _B , T, C _S , μ _a , and d _a are the Boltzmann constant, air temperature, Cunningham slip correction factor, air viscosity, and droplet diameter, respectively. It is important to note that Pe _d refers to the droplet Peclet number at N = 0 only. Thus, even if Pe _d becomes extremely large, the local droplet Peclet numbers in the deeper generations (higher N) can remain small. Q represents the volume flow rate of air during breathing and is modelled such that Q = Q ₀ q(τ), q(τ) being a sinusoidal function accounting for airflow variation during breathing. L ₀ and A ₀ are the airway length and cross-sectional area at N = 0, respectively.

The only source of viruses in the LRT is the carrier droplets that deposit in the mucosa of the LRT. The deposited viruses diffuse in the mucosa and are also subjected to mucociliary advective transport. The corresponding transport equation of the deposited viruses in the mucosa is formulated considering these transport mechanisms in addition to kinetics of the virus infection (see Supplementary Material for more details). The one-dimensional virus transport equation, thus formulated, is represented in its dimensionless form as P e v 2 α ζ β N S t m ∂ ϕ v ∂ τ = ∂ ∂ N 2 ζ β α N 1 − α α  ln α 2 ∂ ϕ v ∂ N − P e v 2 ε ζ β N ϕ v + L D ′ D d D v ϕ d + 2 α ζ β N p 0 I − c l ϕ v , (3) where ϕ _v , Pe _v and St _m denotes the dimensionless virus concentration in mucus, virus Peclet number and mucus Strouhal number, respectively. The quantity L D ′ D d D v ϕ d takes into account the deposition of virus in mucus through the virus-laden droplets, while the last term takes into account virus infection kinetics using a modified target-cell limited model (Baccam et al., 2006; Quirouette et al., 2020). The first and second terms on the right hand side of Eq. 1 take into account the diffusive and advective transport of viruses in mucus, respectively. Temporal change in virus concentration is taken into account by the term on the left hand side of Eq. 3. It is assumed that the rate at which the uninfected target cells at any spatial location are infected is dependent on the infection rate and the local virus concentration (see Eq. 4). Once infected, the target cells remain in an eclipse phase for a certain time-span before they become infectious (Eq. 5). The infectious cells produce new virus at a specific rate for a certain duration before undergoing apoptosis (Eq. 6). ∂ T ∂ τ = − I r T ϕ v (4) ∂ E ∂ τ = I r T ϕ v − 1 τ E E (5) ∂ I ∂ τ = 1 τ E E − 1 τ I I (6)

The fraction of uninfected target cells, infected cells in the eclipse phase and infectious cells are represented as T, E and I, respectively. I _r represents the dimensionless infection rate, while τ _E and τ _I are the dimensionless time-period of the eclipse phase and infectious phase, respectively. p ₀ is the dimensionless replication rate of virus from the infectious cells and c _l is the dimensionless virus clearance rate due to various non-specific clearance mechanisms (Baccam et al., 2006; Quirouette et al., 2020). The following scaling parameters are used to obtain Eqs. 3–6 (see Supplementary Material for detailed derivation) τ = t T b , ϕ v = c v c v , 0 , S t m = T m T b , P e v = | V m , 0 | L 0 D v , p 0 = L 0 2 D v p c v , 0 , c l = L 0 2 D v c , I r = β c v , 0 T b , τ E = T E T b , τ I = T I T b (7) where c _v is the dimensional virus concentration and c _v,0 is the initial c _v at N = 0 due to droplet deposition. V _m,0 is the mucociliary advective velocity at N = 0. T _m and D _v are the time-scale for mucociliary advection and virus diffusivity, respectively. p, c _l and β represents the dimensional virus replication rate, virus clearance rate and infection rate, respectively. T _E and T _I are the time-scales for the eclipse phase and the infectious phase, respectively. c _v,0, T _m and D _v are determined as c v , 0 = ϕ l c d , 0 A 0 A m , 0 , T m = L 0 | V m , 0 | , D v = k B T 3 π μ m d v (8)

The impact of a human body’s immune response to viral infections in the LRT is taken into account through a simplified immune response model, following Quirouette et al. Quirouette et al. (2020), which considers innate immune response due to interferons, humoral immune response through antibodies and cellular immune response due to cytotoxic T-lymphocytes. The mathematical models used for these immune responses are discussed in the following sections.

The LRT is assumed to be initially devoid of droplets and viruses, i.e., ϕ _d | _τ=0 = ϕ _v | _τ=0 = 0. Fractions of target cell and infected cells are also assumed to be initially zero (T| _τ=0 = E| _τ=0 = I| _τ=0 = 0). The trachea (N = 0) is assumed to be exposed to virus-laden droplets, presumably a combination of inhaled droplets (which have not deposited in the URT) or those formed due to ANM, for a specific inhalation period (τ _inh ). The droplets are transported deeper into the LRT, along with the airflow, during inhalation (Eq. 16) and washed out during exhalation (Eq. 17). In contrast, the viruses are always assumed to be washed out of the trachea (N = 0) along with mucus, irrespective of inhalation/exhalation (Eq. 18), due to the nature of mucociliary transport. At the distal end of the lungs (N = 23), the total advection-diffusion flux of both droplets and viruses is assumed to be zero (Eq. 19). Mathematically, these conditions are expressed as follows ϕ d N = 0 = 1 , τ ≤ τ inh , = 0 , τ > τ inh , (16) ∂ F d ∂ N N = 0 = 0 , τ > 0 , (17) ∂ F v ∂ N N = 0 = 0 , τ > 0 , (18) F d | N = 23 = F v | N = 23 = 0 , τ > 0 , (19) where F _d and F _v are the total advection-diffusion flux in the droplet transport (Eq. 1) and virus transport equation (Eq. 3), respectively.

The implemented mathematical model is validated with respect to droplet deposition and virus transport within the lungs as well as viral infection characteristics. Droplet deposition computed using the present model is compared with the experimental data of Heyder et al. (1986) for whole lung deposition as well as alveolar deposition (see Supplementary Figure S2).

The predictions of the coupled virus transport and infection kinetics model are compared with the computational results of (Quirouette et al., 2020) for Influenza A infection (see Figures 1A, B. While only diffusive transport is considered in the comparison shown in Figure 1A, both diffusive and advective transport is taken into account for the comparison shown in Figure 1B. The immunity model is not considered in the above comparison. The corresponding model parameters are listed in Supplementary Table S3.

The infection kinetics-immunity model is additionally fitted against viral load data for SARS-CoV-2 infections. The best fits of the model are shown in Figures 1C, D for two different sets of clinically-obtained data from hospitalised patients (Wang et al., 2020; Néant et al., 2021). The best-fit parameters are listed in Supplementary Table S3. These parameters are used in this study for predicting SARS-CoV-2 viral progression. A comparison, similar to that carried out for Influenza-A, could not be carried out for SARS-CoV-2 infection due to lack of sufficient data on SARS-CoV-2 virus transport within the lungs.

It can, thus, be observed from these comparisons that the present model can appreciably determine droplet transport and deposition within the lungs, and simultaneously predict virus transport within the lungs considering the effects of infection kinetics.

The effects of inhaled droplet size and airflow rate are studied by varying the droplet Peclet number (Pe _d ). Pe _d is defined as the ratio of advective airflow in the lung to diffusive transport of the droplets in air (see Eq. 2). As such, a stronger advective airflow or a larger inhaled droplet size results in a larger Pe _d and vice versa. Although it is expected that a larger Pe _d would make the inhaled droplets reach deeper regions of the lung, analyses have established that droplet deposition in the deep lung is non-monotonic (Hofmann, 2011; Chakravarty et al., 2022). The initial ϕ _v (and subsequently, ϕ _v, max) in the deep lung follows the droplet deposition characteristics (see Figures 3A, B). However, the infection progression remains qualitatively similar for all Pe _d with no significant difference in τ _inf,p . The time taken for infection resolution also remains unaffected since Pe _d does not influence virus clearance from the lung. This suggests the inevitability of an infection (which often becomes severe) even if a small fraction of the inhaled droplets reach the deep lung. Transport of viruses to the deep lung, thus, needs to be minimised.

In contrast, τ _exp is observed to significantly influence infection progression in the lungs although it does not influence the deposition location. A longer τ _exp results in the droplets being inhaled into the LRT for a longer duration resulting in larger droplet deposition in the mucus (Chakravarty et al., 2022) and in a higher initial ϕ _v . This leads to faster infection progression (shorter τ _inf,p ) and a substantially larger ϕ _v, max (see Figure 4). Thus, minimizing the inhalation duration can help control the severity of the infection.

The impact of virus size, mucociliary advection and mucus viscosity is studied by varying the virus Peclet number (Pe _v ). Pe _v is defined as the ratio of advective mucociliary transport and diffusive transport of the deposited viruses in the mucus layer (see Eq. 7). A larger Pe _v , as such, indicates stronger mucociliary advection (or weaker diffusive transport) and vice versa. Weaker diffusive transport can be the result of a larger virus or more viscous mucus.

A larger Pe _v (stronger mucociliary advection) leads to relatively faster washout of viruses from the upper airways of the lung where mucociliary clearance is substantial (see Figure 5, N = 12). Faster washout reduces the residence time of the viruses in the upper airways. This inhibits virus replication and reduces ϕ _v, max in the upper airways. Similar effects are observed in the upper airways for the entire range of Pe _d considered in this analysis. In contrast, only a minor impact is observed in the deep lung (N = 23) for the range of Pe _v considered. This can be primarily attributed to the negligible mucociliary clearance from the deep lung. A change in Pe _v in the deep lung, thus, implies modification of virus diffusivity. However, the time-scale of virus diffusion is too long for it to have any substantial impact on the infection time-course in the deep lung.

While the size of inhaled viruses cannot be controlled, the rate of mucociliary clearance and mucus viscosity can be therapeutically modified. This provides a viable approach for controlling infection progression in the upper airways of the lung. A similar viable approach for deep lung infection is, however, not feasible.

The impact of breathing time period is studied by varying two parameters—the airway Strouhal number (St _a ) and the mucus Strouhal number (St _m ). St _a is defined as the ratio of time scale of advective airflow in the LRT to the breathing time period (see Eq. 2). St _m is defined as the ratio between time scale of mucociliary advection and the breathing time period (see Eq. 7). While St _a affects the initial deposition of the inhaled virus-laden droplets, St _m influences the washout of the deposited viruses and hence, the infection time-course.

Figure 6A shows the impact of St _a on ϕ _v in the deep lung. A longer breathing time period (smaller St _a ) leads to a larger volume of droplets inhaled into the lung, keeping all other parameters constant. This allows a greater portion of the inhaled droplets to reach the deep lung (Chakravarty et al., 2022) which, in turn, results in increased deposition of viruses. A larger ϕ _v at the end of τ _inh causes faster infection progression in the lung (shorter τ _inf,p ) and a corresponding higher ϕ _v, max (see Figure 6A). However, the time required for infection resolution remains similar. It is to be noted that there is no substantial difference between the results obtained in the deep lung for St _a ≥ .05 since droplet deposition in the deep lung remains almost invariant beyond this limit (Chakravarty et al., 2022).

A longer breathing period also reduces St _m suggesting greater mucociliary clearance from the upper airways in a breathing cycle, keeping all other parameters constant. In other words, less number of breathing cycles are required to achieve equivalent mucus clearance (and hence, virus washout) from the upper airways of the lung at lower St _m . This can be observed from Figure 6B. The actual time taken for virus washout is, however, much longer since St _m is inversely dependent on T _b . Thus, faster virus washout takes place from the upper airways at larger St _m and vice versa. This leads to a steeper concentration gradient between the deep lung and the upper airways resulting in faster diffusive transport in the deep lung. Thus, infections get resolved relatively faster in the deep lung at larger St _m . In addition, a larger T _b (for lower St _m ) also allows the deposited viruses to replicate more leading to relatively higher ϕ _v, max at lower St _m (see Figure 6B).

In summary, longer breaths result in a larger ϕ _v, max, and slower infection resolution, both of which are bad from a clinical perspective. ϕ _v, max beyond a certain limit in the deep lung (see Figure 6B) can cause pneumonia, which may become life threatening. Thus, shorter breaths, which lowers the viral load in the lung and relatively shortens the infection time-course, are more beneficial.

The impact of virus growth and clearance is studied by varying p ₀ and c _l , respectively, other parameters remaining fixed. It is observed that a higher p ₀ (higher virus replication) leads to larger ϕ _v in the deep lung throughout the time-course of infection (see Figure 7A). ϕ _v, max in the deep lung, as such, increases as p ₀ becomes larger and vice versa. However, no significant change is observed in τ _inf,p .

In contrast, the virus clearance rate (c _L ) is observed to affect ϕ _v as well as τ _inf,p (see Figure 7B). Greater c _l results in larger virus clearance from the lung which reduces ϕ _v (and hence, ϕ _v, max) and also delays attainment of the peak infectious state. The reverse happens when c _l is reduced.

The simplified immune response considered in the present study is a combined model of humoral response (antibodies), innate response (interferons) and cellular response (cytotoxic T-lymphocytes). As such, it becomes necessary to distinguish between these responses in order to identify the key effects of the individual responses on virus concentration in the lung and infection time-course. This is achieved by varying the parameters of a specific response model, while keeping the other response models constant.

One of the key aspects of the SARS-CoV-2 pandemic has been the consistent mutation of the virus leading to emergence of newer variants as the pandemic progressed. Some of these variants (for e.g., Omicron) caused faster infection spread with milder symptoms in the infected individuals. Other variants (for e.g., Delta) often caused more severe health effects (Puhach et al., 2022; Silva et al., 2022). The time-course of infection also varied from one variant of the virus to the other. Symptoms lasted typically for ∼ 7 days in case of the Omicron variant as compared to ∼ 9 days for the Delta variant (Menni et al., 2022).

Our results indicate that the reason for such difference in symptom severity and infection time-course can be correlated to the relative replication and neutralization of the virus in the host cells (see Figure 7). For a given rate of virus neutralization, the viral load reduced and the infection time-course shortened as the replication rate attenuated. Greater virus neutralization rate, for a given virus replication rate, has a similar impact. A lower viral load suggests occurrence of milder symptoms in the infected individual. Thus, it can be stated that the virus replication rate in the lung becomes attenuated (or virus neutralization is enhanced) in case of the variants causing milder symptoms and shorter infection time-courses (for e.g., Omicron) as compared to the variants causing severe symptoms and longer infection time-courses (for e.g., Delta). A recent study by Silva et al. (2022) corroborates this inference.

Vaccination against SARS-CoV-2 has played a major role in suppressing the severity of the pandemic. Vaccines work by inducing the production of B − lymphocytes and T − lymphocytes (or memory cells) within the body (Scourfield et al., 2021; Zollner et al., 2021). The B − lymphocytes are responsible for the production of antibodies which fight the viruses. T − lymphocytes are cytotoxic in nature and attack the cells infected by the virus. The end outcome is that the body is left with heightened defenses against the virus. Similar effects are true for prior infections as well, although magnitudes of such effects may vary.

Our results confirm that prior presence of T − lymphocytes within the body readily suppresses an infection (see Figure 9B). Prior presence of T − lymphocytes (which reduces the time required for T − lymphocyte concentration to peak) suppress virus replication from the onset of infection itself leading to lower ϕ _v, max and hence, reduces the probability of disease severity. Similarly, a larger initial antibody presence in the body allows targeting (and subsequent neutralisation) of a larger number of viruses. This slows down virus replication and enables other immune effects to act effectively, thereby decreasing disease severity (see Figure 9C).

Effectiveness of the antibodies and T − lymphocytes also play an important role in infection progression. An effective antibody against SARS-CoV-2 neutralizes the viruses at a much faster rate, thereby, restricting its replication and reducing probability of the infection progressing to a severe disease (see Figure 8A). Effectiveness of the T − lymphocytes, however, does not influence infection progression until their concentration becomes large enough, after which an effective T − lymphocyte readily suppress the infection (see Figure 8C).

Results indicate that a certain portion of virus-laden droplets present at the entrance to the trachea inevitably reach the deep lung where they deposit releasing the viruses causing infection in the deep lung. These droplets are a combination of the inhaled droplets (which have not deposited in the nasopharynx) and those formed due to aerosolisation of the mucus layer during respiratory motions (coughs etc.). More the availability of droplets, longer is the duration for which these may be inhaled into the LRT and greater is the probability of these droplets reaching the deep lung causing infection (see Figure 4). Clinical conditions such as bronchitis, asthma, etc. accentuate the probability of formation of aerosolised droplets due to mucus hypersecretion or airway constriction (Fahy and Dickey, 2010). Thus, the probability of contracting deep lung infection remains higher when clinically associated with other upper airway diseases (Lee et al., 2020) which promote aerosolization of nasopharyngeal mucus.

The severity of a deep lung infection is determined by the binding affinity of the virus with cells in the deep lung as well as the replication/neutralization characteristics of the virus. Binding affinity is lower and replication becomes attenuated for the Omicron variant of SARS-CoV-2 in the deep lung cells (Silva et al., 2022) resulting in much lower probability of a severe deep lung infection. This is in contrast to the Delta variant for which the probability of a severe deep lung infection is much more due to its higher binding affinity and larger replication rate in the cells of deep lung (Silva et al., 2022). Greater the severity of deep lung infection, larger is the probability of development of pneumonia and associated respiratory complications.

Onset of pneumonia is a common yardstick used to gauge the severity of any infection since it can progress rapidly to respiratory failure. Clinical studies have found that the time taken for the development of respiratory failure from pneumonia in a human body from a SARS-CoV-2 infection is typically 6–14 days after the initial onset of symptoms (Feng et al., 2020; Grant et al., 2021), primarily after oropharyngeal infection (Wiersinga et al., 2020). There is autopsy-based evidence of the migration of deposited viruses from the pharyngeal region to the lower respiratory tract and the distal alveolar region (Hou et al., 2020; Grant et al., 2021), although the mechanism still remains unclear. Aspiration is a possible mechanism Basu et al. (2022). Another plausible mechanism is through inhalation of infected droplets as considered in the present study. Residual inhaled droplets or those formed by aerosolisation of the infected mucosa (due to interaction of the infected mucosa with airflow) are subsequently transported from the nasopharynx deeper into the lung where they can release the viruses causing further infection (Darquenne et al., 2022).

The results presented above are used to estimate the time taken for the SARS-CoV-2 concentration in the deep lung to reach the critical magnitude (c _v,cr ) required for onset of pneumonia. Virus concentration in the pharyngeal region (N = 0) is assumed to be the clinically observed viral load in mild infections in pharyngeal samples, where SARS-CoV-2 is initially registered (c _v,0 = 10² copies/mL) (Fajnzylber et al., 2020). The critical concentration in the deep lungs (c _v,cr ) is assumed to be the clinically observed median viral load in sputum samples (7 × 10⁶ copies/mL) of symptomatic individuals exhibiting pneumonia (Fajnzylber et al., 2020; Wölfel et al., 2020) (see Supplementary Tables S3, S4 for time-estimates with other c _v,cr ). It is further assumed that the pharyngeal viral load remains constant throughout the inhalation duration. It is calculated that the time required for onset of pneumonia in a SARS-CoV-2 infection can vary from ∼ 2.5 − 7 days, depending on c _v,cr and associated parameters. Table 2 summarises the change in pneumonia onset time with variation in different fluid dynamics and physiological parameters. Table 3 lists the change in pneumonia onset time with variation in different infection parameters.

These estimates suggest no significant change ( < 3.5 % ) in pneumonia onset time with change in virus size (d _v ), volume flow rate of air (Q ₀) and length geometry (L ₀). A relatively larger change ( ∼ 8.9 % ) is observed with variation in droplet diameter (d _d ), while a substantial change is observed when the breathing period ( ∼ 33 % ) and inhalation duration ( ∼ 43 % ) is varied. In contrast, significant changes in the pneumonia onset time are observed with variation in most of the infection parameters. The onset time is observed to change by ∼ 34 % and ∼ 51 % within the studied range of virus replication rate and clearance rate, respectively. No pneumonia is observed to be established below a critical replication rate and above a critical clearance rate (see Table 3). Pneumonia onset is fastest when interferons are absent or take a long time to build up and vice versa, with ∼ 27 % variation in the onset time. A large enough interferon effectiveness sufficiently suppressed virus replication to resist pneumonia onset. A larger initial antibody presence and greater effectiveness of the antibodies at neutralizing the viruses delayed pneumonia onset ( ∼ 10 % and ∼ 20 % variation in onset time, respectively) with no pneumonia occurrence above a critical magnitude. Interestingly, pneumonia onset time did not change significantly with change in effectiveness of the T − lymphocytes. However, a fast enough production of T − lymphocytes resisted pneumonia onset.