key: cord-0468064-8yeu5zgi
authors: Bale, Rahul; Iida, Akiyoshi; Yamakawa, Masashi; Li, ChungGang; Tsubokura, Makoto
title: Quantifying the COVID19 infection risk due to droplet/aerosol inhalation
date: 2021-10-06
journal: nan
DOI: nan
sha: 1faaa6fb4daed122086d8b131aa7bf0afbc6121f
doc_id: 468064
cord_uid: 8yeu5zgi

The dose-response model has been widely used for quantifying the risk of infection of airborne diseases like COVID-19. The model has been used in the room-average analysis of infection risk and analysis using passive scalars as a proxy for aerosol transport. However, it has not been employed for risk estimation in numerical simulations of droplet dispersion. In this work, we develop a framework for the evaluation of the probability of infection in droplet dispersion simulations using the dose-response model. We introduce a version of the model that can incorporate the higher transmissibility of variant strains of SARS-CoV2 and the effect of vaccination in evaluating the probability of infection. Numerical simulations of droplet dispersion during speech are carried out to investigate the infection risk over space and time using the model. The advantage of droplet dispersion simulations for risk evaluation is demonstrated through the analysis of the effect of humidity on infection risk.

Dispersal of infectious pathogens such as viruses through airborne sputum droplets can lead to the rapid transmission of diseases in the general population posing a great risk to public health. There are several minor and major infectious diseases that are transmitted through virus-carrying sputum droplets. A benign example of airborne disease is the common cold, while more severe examples include H1N1 influenza, severe acute respiratory syndrome (SARS), middle east respiratory syndrome (MERS), and coronavirus disease 2019 (COVID-19) [1, 2] . Since its outbreak in 2019, COVID-19 has transformed into the most destructive pandemic in over a century. The evidence on COVID-19 so far suggests that the possible modes of transmission of SARS-CoV-2 include respiratory droplets and aerosols, direct person-to-person contact and contact with surfaces (fomite mode of transmission). Direct person-to-person contact transmission and fomite transmission can be controlled by appropriate hygiene practices. However, controlling airborne transmission is far more challenging. Therefore, airborne transmission is likely the primary reason for turning COVID-19 into a global pandemic [3, 4] . The viruses carrying sputum droplets are generated not only during violent expiratory events like coughing and sneezing, but they are also generated during routine respiratory activities like speaking, singing, and breathing. This, coupled with the fact that SARS-CoV2 is transmitted during presymptomatic and asymptomatic phases of COVID-19 [5] , increases the infectiousness and the rapid spread of SARS-CoV2. are likely to be within the breathing zone of a susceptible individual can provide a better estimate of infection risk. The focus of the present work is the estimation of infection risk by direct measurement of droplets and aerosols in the breathing zone of susceptible individuals using numerical simulation of droplet dispersion.

To that end, we extend the dose-response model for droplet dispersion simulations for expiratory events like coughing, sneezing, speaking, etc. Using numerical simulations of droplet dispersion, infection risk during face-to-face conversation is investigated in this work. Furthermore, we also investigate the effect of humidity on the infection probability during casual conversations.

The risk of infection due to inhalation of viruses is commonly quantified using the dose-response model [12, 13, 14] . The model assumes that the number of viral particles needed, on average, to infect an individual is N 0 . Assuming that the infection is a Poisson process, the probability of infection can be written as

where N is the total number of virions inhaled. In order to compute the probability of infection P , both pandemic [15] , other similar studies provide estimates of the value that vary between 100 and 1000 [10, 9] .

In this study, we choose a value of 900 which lies within the range of values reported in the literature.

The number of virions inhaled depends on the total duration of exposure, T , the amount of air inhaled by a person breathing at the rate of B, and the local concentration of virions, C(t), in the breathing zone of the person. Therefore, the relationship of N , with T, B & C can be expressed as [17] . For the present study, we choose a value of B = 0.5 m 3 /hr which assumes that the subject at risk of infection is not involved in any strenuous activity.

With the value of B known, the estimation of N depends on the estimation of C(t). There have been two main approaches of estimating C(t) in literature. The first approach is based on a well-mixed room averaged approximation [15, 10] . In this approach, the virions emitted by an infected person are assumed to be well-mixed due to air circulation and mixing within the domain under consideration. The change in virion concentration in the room is tracked over time, but the concentration is assumed to be uniform throughout the domain under investigation. An alternate approach proposed by Yang et al. [19] involves estimation of C(t) over time and space. In this method, assuming that the aerosol-laden air is well-mixed at the point of ejection during expiratory events such as speaking, singing, coughing, etc., a passive scalar is used to model the transport of aerosols in direct numerical simulation or large eddy simulations. Taking passive scalar concentration as a proxy for virion concentration, an expression of C(t) is obtained.

Breathing zone Infected subject A B Figure 1 : A schematic of droplet dispersion (A) during face to face conversation between two subjects, (B) between a group of people seated at a table. The region highlighted by a blue box around the mouth and nose of a susceptible subject, termed the breathing zone, is used to track droplets that are likely to be inhaled.

In contrast to the two methods available in the literature, in this work, we develop a new model to estimate virion inhalation as a function of space and time in droplet dispersion simulations. As opposed to the wellmixed approximation of aerosol-laden air in the passive scalar approach, in simulations modeling direct droplet dispersion, the dynamics of aerosol/droplet transport and evaporation are considered. In contrast to aerosol transport, the dynamics of transport and mixing of droplets can be different due to the larger inertia of droplets. Aerosolization of small and medium droplets, depending on the ambient temperature and humidity, can significantly alter local aerosol and virion concentration. Therefore, the results of droplet dispersion simulations can be qualitatively different from those of the passive scalar approach.

For the evaluation of infection probability in droplet dispersion simulations, we estimate N by tracking the number of droplets in a spatial region defined such that the air within this region is likely to be inhaled by a susceptible subject. Henceforth, we shall refer to this region as the breathing zone. A schematic of the breathing zone is shown in Fig. 1 . For simplicity, we have chosen a small rectangular region around the mouth and nose to represent the breathing zone. The shape of the region could in principle be arbitrary.

Example of how the breathing zone is defined in two scenarios is shown in Fig. 1 . The first scenario is a face-to-face conversion between two persons. Second is a conversion between a group of people seated at a Lastly, the breathing zone placement is such that its entire volume is outside the subject's interior.

The number of virions the subject is likely to inhale depends on the local concentration of virions in the breathing zone. The local concentration can be written as

where n(t) is the instantaneous count of virions at the time instant t in the breathing zone and υ B is the volume of the breathing zone. The instantaneous virion count n(t) is the product of the total ejection volume of droplets and aerosols, υ 0 d (t), in the breathing zone and the viral load or viral density λ v (copies/m 3 ). The superscript 0 in υ 0 d (t) is used to imply that the volume of the droplets in question is the volume at the time of ejection from the mouth of the infected subject. Even as a droplet evaporates, its instantaneous volume, υ d , decreases but its injection volume remains unchanged. Therefore, the total virion count of a given droplet remain constant as a droplet evaporates and aerosolizes. The average viral load of SARS-CoV2 in the sputum is 7×10 6 copies/ml and the maximum reported value is 2×10 9 copies/ml [20] . For the analysis in the present work we choose the value 7 × 10 6 copies/ml. The local virion concentration can now be expressed as a function of viral load and droplet volume

Substituting the above expression into Eq. 2 we obtain the following expression for N

In activities like speaking and singing over prolonged periods of time, the rate of droplet generation and dispersion are expected to reach a quasi-steady-state. The injection droplet volume that enters the breathing zone would reach a steady state value υ 0 d . Therefore, under steady-state conditions the above equation can be simplified to

This equation is applicable to quasi-steady processes like singing and speaking and it is not applicable to transient situations like sneezing and coughing. We must resort to Eq. 5 for transient cases.

Risk after vaccination and due to variant strains: Most of the major vaccines for COVID-19 have reported high efficacy in preventing infection and very high efficacy in preventing severe disease and hospitalization. For example, vaccines by Pfizer, Moderna and AstraZeneca have been reported to have vaccine efficacy of 95% [21] , 94.1% [22] and 81.5% [23] , respectively. From the viewpoint of evaluating the infection probability of a vaccinated subject, the effect of a vaccine may be interpreted as an increase in the minimum number of virions needed to infect a person. This assumption implies that for exposure to small doses of virions, the infection probability will be very low or negligible. However, even an infected person, if exposed to very high doses of virions, may be at risk of infection.

As SARS-CoV2 has spread through populations, it has mutated into several variants [24] . It has been reported that the transmissibility of some of the variants of SARS-CoV2 is higher than that of the original strain [25, 26] . For example, the B. [26] . Within the framework of the infection risk model considered in this work, the higher transmissibility of a given variant could be interpreted as due to two reasons. First, the viral load of the variant strains could be higher. Second, the minimum virion dose needed for infection N 0 for a variant could be lower. The higher transmissibility could be due to one or a combination of these two factors. Under this assumption, the effect of vaccines and variant strains on the probability of infection can be incorporated into the risk model given by Eq. 1 as follows.

where α is factor that accounts for higher transmissibility of variant strains, lower risk of infection for vaccinated individuals, and with α = 1 the above equation falls back to the original form in Eq. 1. For the case where a person is vaccinated with a vaccine of efficacy η vc , it can be shown that α = 1 − η vc . The details of derivation of α can be found in the Appendix section. The effect of vaccination may be interpreted as an increase in the minimum number of virions N 0 needed to infect a vaccinated person. If the efficacy of a vaccine is 100% then the P = 0. But, when vaccine efficacy is 0, Eq. 7 returns to the original form of P in Eq. 1. For a variant strain with a higher transmissibility factor τ , we can write α = τ . For the alpha, strain the transmissibility factor is τ = 1.29 [25] , and for the delta strain it is τ = 2.45 [26] (assuming 90% higher transmissibility compared to alpha variant). For the original strain the transmissibility factor is τ = 1, in which case Eq.7 return to the original form in Eq.1.

The flow solver used in the present work for carrying out the numerical simulation of droplet dispersion is made of an Eulerian reference frame for solving the fluid flow and species transport equations and a

Lagrangian frame for solving the droplet dynamics model. The equations of motion of mass, momentum, energy and species transport can be expressed in compact notation as

Here, U, F and F represent the primitive flow variables, the combined convective and diffusive terms, and the source terms, respectively [27] . The primitive variables vector and the flux vector are expanded below.

where the density and viscosity are represented by ρ and µ, respectively. u, e and P are the velocity, the total specific energy and the pressure, respectively. The components of the velocity along the principle directions 1, 2, 3 are given by (u 1 , u 2 , u 3 ). The vapor phase of water from the liquid sputum is modeled as passive scalar species along with O 2 and N 2 . The mass fraction of the species indexed k is represented by Y k and u k i is corresponding diffusion velocity of the k th species. q = −λ∇T, is the heat flux where T &λ represent temperature and thermal diffusivity, respectively. The density and pressure are constrained together by the state equation P = ρRT , in which R is the gas constant and T is the temperature. The total specific energy is given by

where γ is the ratio of the gas specific heat capacities. The diffusion velocityû k Y k of the k th species is defined in terms of the species diffusivity D k , the relationship is given bŷ

The contribution to the source comes from the bouyancy term and the weak two-way coupling between the droplet-model and flow equations. The source term vector is given by

ρ and ρ 0 are the local and far field ambient density, respectively and g is the acceleration due to gravity (eg. g = (0, 0, −9.81)m/s 2 ) Of the species source terms S ρY k , the non-droplet vapor species are zero.

The widely used single droplet model is adopted in this work for modeling the sputum droplet dynamics.

The droplets are modeled as discrete Lagrangian entities which are coupled with the Eulerian fluid flow equa-tions for a weak-two-way coupling. The droplet transport and evaporation are influenced by the conditions of the ambient air, but the flow field is not affected by the droplets except the species of the vapour phase of the liquid droplet. The transport of the droplets is modeled by and temperature, is

The temperature 

A multi-physics solver known as CUBE [30, 31] has been used for all the numerical simulations presented in this work. CUBE is a finite volume solver based on a hierarchical meshing framework known as the building cube method (BCM) [32] . The meshing framework allows local mesh refinement enabling the high resolution in regions of interest while limiting the overall cell count. The supercomputer Fugaku has been used for carrying out the numerical simulation presented in this work. Fugaku comprises 158,976 nodes. Each node is equipped with a Fujitsu A64FX processor, which consists of 48 compute cores and 4 additional cores, and a memory of 32 GiB. The nodes are interconnected with 28Gbps, 2 lanes, 10 port TofuD interconnect. report a value as high as 50 µm. As there is so much variation in the number and distribution of droplet size in this work we adopt a combination of distribution Duguid [37] and Xie et al. [36] . The distribution of droplet diameter adopted in this work is shown in Fig. 2 . 

The simulation setup involves a human model standing in an upright position similar to the infected subject shown in Fig.3 . The numerical mesh employed in this work is also shown in the figure. A mesh spacing of 4 mm is allocated to the region immediately in front of the mouth, which is the source of speaking flow and droplets, approximately up to a distance of 1 m. The numerical dissipation of the Roe-scheme [38] used for the convective fluxes in our solver enable us to carryout implicit large eddy simulations (ILES) [39] .

The human model is modeled with the immersed boundary condition [40] to impose no-slip and isothermal boundary conditions. The temperature of the human body surface is set to 300K to include the effects of buoyancy-driven flow by the human model, although the effect is not expected to be significant. The outer boundaries of the computational domain are treated with the slip boundary condition. The initial conditions for the simulation were set to the STP conditions. The temperature, pressure and relative humidity were 

A numerical simulation of the evaporation of a single isolated droplet was carried out to validate the droplet model. The experiment of Ranz and Marshall [41, 42] , in which the evaporation dynamics of a Fig. 4 where it can be seen that there is excellent agreement between the simulation results and the experimental data.

The infection risk model presented in this work is adopted to investigate the probability of infection of a susceptible subject who is in a face-to-face conversation with an infected individual. For this we carried out the numerical simulation of dispersion of droplets ejected by an infected subject in a standing pose assumed to be continuously speaking for the duration of the simulation. In the simulation, we model only the geometry of the infected subject, however, the geometry of the susceptible subject is not modeled. The risk of infection of a virtual subject, whose dimensions are assumed to be identical to that of the infected subject, is evaluated at varying distances from the infected subject and also a function of time. The numerical simulation is carried out in a computational domain measuring 32 × 32 × 16 m 3 along x, y and z-axis, respectively. The human geometry is placed with the base of its feet at the centre of the computational domain along x and y-axis and at the bottom along the z-axis. The details of the boundary and initial conditions of the setup can be found in the Methods section. The details of the speaking model employed in this work are provided in the section Droplet modeling parameters. One cycle of the speaking model involves counting from 1 to 10 with two inhalation phases after the words '5' and '10', respectively, for mass balance (see Fig. 2 ). The speaking is modeled for the duration of the simulation by indefinitely looping the cycle of the speaking model.

A visualization of instantaneous states of the dispersion of droplets at two time instants is presented in Fig. 5 . The size of the droplets is indicated by the coloring scheme of the droplets. The largest droplets are colored red and the smallest blue. It is evident that many of the droplets larger than 20 µm quickly settle on the ground under the influence of gravity. As the initial condition for droplet velocity is 0, gravity contribution dominates the velocity of the larger droplets. Therefore, the horizontal distance traversed by larger droplets is not significant when compared to smaller droplets. The influence of gravity on droplets smaller than 10µ m is negligible because of aerosolization due to very short evaporation timescales and consequently the flowinduced drag forces dominate the small droplet and aerosol velocity. As a consequence, the smaller droplets and a small number of medium-sized droplets remain airborne and they are carried by flow generated by the speech. The dispersion of the aerosolized droplets in the horizontal direction over two instants of time is shown in Fig. 5 .

We next move on the investigation of infection probability of a virtual subject placed at different distances in front of the infected subject. The probability of infection at different distances in front of the infected subject for an exposure duration of T = 15 min is plotted in Fig. 6a . As for the infectious dose N 0 , we have chosen a value of 900 which lies with the ranges values reported in the literature [15, 10, 9] . The variation of infection probability over distance exhibits a decaying profile. At distances less than 0.5m the probability of infection is greater than 70%, which rapidly decays to less than 20% as the distance is increased to 2 m. It can be noted from Fig. 5 that the droplet concentration rapidly decays due to its dispersion in the vertical (z-axis) and in-plane direction (x-axis) as the droplets are advected away from the infection source thereby lowering the infection probability with distance. The shaded region in the figure depicts the change in the infection risk if the infectious dose is changed from 300 to 2000. The infection risk at a given distance is lower for larger values of N 0 and vice-versa. It is interesting to note that the shaded region narrows as the distance from the infected person increases from 0.25 to 2 m. As the distance from the infection source increases, due to the dispersion of droplets, the virion concentration in the inhalation zone decreases which in turn decreases the number of inhaled virions. When the inhaled virion count is small enough, the magnitude of the infectious dose N 0 becomes less important resulting in the narrowing of the shaded region.

The variation of infection probability overtime at distances D = (0.5, 1.0, 2.0) m is plotted in Fig. 6b .

For large virion concentrations at closer distances like D = 0.5m, the P rapidly increases and saturates to the maximum value (1). On the other hand, at a further distance, due to lower virion concentration, the rate of change of P is more gradual. At any given instant of time, say 10, 20, 30 min, etc., the plot provides the relative risk of maintaining different distances from a likely infected person. Focusing on the horizontal grid line corresponding to P = 0.2., it can be seen that the exposure time required for 20% probability of infection is approximately 3, 10 and 21 min for distances of 0.5, 1 and 2 m, respectively. 

The probability of infection can change significantly due to variant strains of SARS-CoV-2 and vaccination. To account for such factors a generalized form of the equation for infection probability was introduced through Eq. 7. By altering the parameter α, the equation can be adapted for variant strains of higher transmissibility factor τ , and vaccinated individuals. The transmissibility of B.1.1.7 strain (alpha variant) and the B.1.617.2 strain (delta variant) is 29% and 145% higher than the original strain, i.e. τ = 1.29 and τ = 2.45 for the alpha and the delta variant, respectively. Evidently, the value τ = 1 corresponds to the original strain. A comparison of how the P , evaluated at a distance of 1 m, increases with time for the original strain, the alpha and the delta variants is plotted in Fig. 7a . For the duration of exposure of less than 20 min, we find that there is a significant difference between the risk of infection of the original strain and the delta strain. This difference gradually reduces over time as P saturates to its maximum value.

The probability of infection for the delta variant is approximately 2 times greater than the original strain. However, the difference between the infection probability of the alpha and the original strain is not very significant.

Through the risk evaluation model presented in this work, it is also possible to incorporate the effect of vaccination on the probability of infection. As discussed in the previous section, the generalized form of Eq. 7 can be applied to vaccinated cases by using the expression α = 1 − η vc . For unvaccinated cases, we can set η vc = 0. To evaluate how the infection risk changes due to vaccination, we choose two values for η vc , 80% and 95%, which approximately correspond to the efficacies of the AstraZeneca and Pfizer vaccines, respectively. The comparison of P evaluated at a distance of 1m from the infected person for the different values of η vc is presented in Fig. 7b . The infection probability remains below 20% for exposure periods of up to 40 mins for η vc = 0.85 and it remains well below 10% for exposure period plotted in the figure.

Direct simulation of droplet dispersion, as opposed to simulations of scalar transport as a proxy for aerosols, enables the investigation of the effect of environmental factors like temperature and humidity on droplet aerosolization and dispersion, and consequently on the risk of infection. We extend the numerical simulation of droplet dispersion during speech presented in the previous section to study the effect of humidity on infection risk. The relative humidity of the ambient environment could significantly affect the evaporation of medium and large droplets. This leads to the prevention or aerosolization of medium droplets that have a higher virion count compared to that of smaller droplets. As a consequence, the concentration of virions could be significantly altered by humidity, thereby influencing the probability of infection. In order to investigate role of humidity on infection risk, we carried out a numerical simulation of droplet dispersion during speech under three different humidity conditions 10%, 50% and 100%. The numerical setup and the boundary conditions are identical to those of the simulation in the previous section. The only parameter varied is the relative humidity (RH). Three separate simulations were carried out in which the relative humidity was set to 10%, 50% and 100%.

In Fig. 8 , a snapshot of the droplet dispersion at t = 24 s for RH = 50% and RH = 100% is presented.

For the RH = 100% case, lack of evaporation prevents the aerosolization of medium-sized droplets. Droplets smaller than 5 µm remain airborne for prolonged periods. The velocity of some of the droplets between the sizes of 5 and 50 µ, whose timing in ejection matches the peak velocity of the speaking model, is initially dominated by the fluid velocity. However, as the flow velocity decreases away from the mouth due to dissipation, the influence of gravity dominates the velocity of these droplets. This results in the droplets settling to the ground under the influence of gravity. In contrast, at lower RH values, the medium droplets in question can get partially or completely aerosolized and remain airborne. As a result, the concentration of aerosols and consequently the virions depends directly on the relative humidity of the ambient environment.

This effect is quantified through the evaluation of the infection probability which directly depends on the local virion concentration. The variation of infection probability over distance from the infection source of the different humidity cases are compared in Fig. 9a . As the local droplet concentration is strongly influenced by the evaporation of medium droplets with higher virion count, the number of virions likely to be inhaled decreases with increasing humidity. The consequence of which can be seen in Fig. 9a and Fig. 9b . The trend of P decreasing with distance is consistent across all the humidity cases. However, for a given distance from the infection source, the probability of infection is lower for higher humidity. Fig. 9b plots the evolution of P over time evaluated at a distance of 1 m from the infection source for the different humidity cases. From these plots, it could be inferred that for a given temperature higher humidity lowers the risk of infection.

However, the magnitude of reduction in the probability of infection could be strongly influenced by the 

In this work, we have developed a framework for quantifying the risk of infection due to airborne diseases like COVID-19 using the dose-response model in droplet-dispersion simulations. We have detailed in this work a methodology for estimating the inhalation dose, by measuring the volume of droplets/aerosols (proxies for virion count) in the breathing zone, directly from droplet dispersion simulations. We have adopted this framework to estimate the risk of infection from a person ejecting droplets while speaking. For this, we have carried out a numerical simulation of droplet dispersion from a single person standing and speaking continuously in an isolated environment. The infection probability was found to decrease with distance from the infected person. The magnitude of infection probability is strongly influenced by the minimum infection dose N 0 which can be seen in Fig 6a. If the minimum infection dose is assumed to be small then the infection risk remains relatively large even at distances as far as 2m from the infected person. On the other hand, if the infection dose is large then the infection probability is small even at a distance of 1 m. Furthermore, as the simulations have been carried out in a quiescent or poorly ventilated environment, the infection probability is likely to be high. This prediction can drastically change depending on the ambient environment's flow conditions. Therefore, we would not like to make any specific recommendations on social distancing based on these results. The main purpose of this work is to demonstrate the applicability of the framework of risk estimation using the dose-response model in droplet dispersion simulations.

A generalized form of the dose-response model that can incorporate the effect of increased transmissibility of various strains of COVID-19 and the effect of vaccination has been presented in this work. A comparison of the infection risk due to variant strains of higher transmissibility was with the standard strain was presented.

The results show that for exposure duration the infection risk of variant strain can be significantly higher than the standard strain. The difference in infection between strains decreases as exposure duration increases.

Similarly, we also presented a comparison of the infection risk for a vaccinated person with that for an unvaccinated person. The probability of infection for vaccinated persons is very low for short and medium exposure duration. However, the infection risk can increase significantly provided the exposure duration is very long.

One of the main advantages of droplet dispersion simulations is their ability to investigate the effect of environmental factors, keeping all other variables fixed, such as temperature and humidity on the infection risk. To demonstrate this point, an investigation of the effect of humidity of the ambient environment on the infection risk was carried out. This is an aspect that cannot be analyzed using well-mixed room average analysis or passive scale based aerosol transport models. The results of our analysis show that the infection risk is strongly influenced by humidity due to its effect on the evaporation of medium and small droplets. The infection risk at a given distance from the infected person has an inverse dependence on humidity. Lowering the ambient humidity increases the risk and vice-versa. This relationship between infection risk and humidity depends, to some extent, on the droplet diameter distribution adopted in the simulation. For example, a hypothetical droplet diameter profile that includes only aerosols may not result in a strong dependence of infection risk on humidity as the results of this study indicate.

The probability of infection can be written as (15) where N c is the virion dose per unit time under steady state conditions of exposure such as speaking, singing etc. The exact form of N c is not pertinent to the present derivation, it can deduced from Eq. 6. The probability of no infection can be defined asP = 1 − P . The net probability of no infection is given bŷ

Assuming that the vaccination leads to reduction in infection probability due to increase in N 0 , the net probability of no infection for a vaccinated individual can be expressed aŝ

Let η vc be the efficacy of a given vaccine. The probability of no infection with vaccination decreases by a factor ofP vc net η vc if the vaccination is not administered. Therefore, we can express a relationship between P net andP vc net as followsP net =P vc net −P vc net η vc .

Rearranging the terms we getP 

18

Assuming an exposure to the same average virion dose per unit time, we can obtain an expression for N vc 0 from the above two equation as

The interpretation of the above two equations is straightforward. If the vaccine efficacy the 100% then the minimum number of virions needed for infection will be ∞. Consequently,P = 1 at all time instants t and P = 0 and the net probability of no infection will be ∞. On the other hand, if the vaccine efficacy is 0 then the N vc 0 = N 0 . Therefore, probability of infection remains unchanged. The general form of the probability of infection under vaccinated and unvaccinated situations can be expressed as

where α = 1 for no vaccination cases and α = 1 − η vc for vaccinated cases.

The net probability of safety decreases for the variants strains due to higher transmissibility τ compared to the original strain. WithP vr net representing the probability of no infection, its relationship withP net can be written asP 

Along the same lines of derivation of α for the vaccination case, it can be shown that α = τ .

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