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A semi-continuous model for transmission of SARS-CoV-2 and other respiratory viruses in enclosed spaces via multiple pathways to assess risk of infection and mitigation strategies

The Covid-19 pandemic has taken millions of lives, demonstrating the tragedy and disruption of respiratory diseases, and how difficult they can be to manage. However, there is still significant debate in the scientific community as to which transmission pathways are most significant and how settings and behaviour affect risk of infection, which all have implications for which mitigation strategies are most effective. This study presents a general model to estimate the rate of viral transfer between individuals, objects, and the air. The risk of infection to individuals in a setting is then computed considering the behaviour and interactions of individuals between themselves and the environment in the setting, survival times of the virus on different surface types and in the air, and mitigating interventions (ventilation, hand disinfection, surface cleaning, etc.). The model includes discrete events such as touch events, individuals entering/leaving the setting, and cleaning events. We demonstrate the model capabilities on three case studies to quantify and understand the relative risk associated with the different transmission pathways and the effectiveness of mitigation strategies in different settings. The results show the importance of considering all transmission pathways and their interactions, with each scenario displaying different dominant pathways depending on the setting and behaviours of individuals therein. The flexible model, which is freely available, can be used to quickly simulate the spread of any respiratory virus via the modelled transmission pathways and the efficacy of potential mitigation strategies in any enclosed setting by making reasonable assumptions regarding the behaviour of its occupants. It is hoped that the model can be used to inform sensible decision-making regarding viral infection mitigations that are targeted to specific settings and pathogens.

place when an infected individual secretes large droplets (> 5 µm) via vocalising, coughing, sneezing etc. that are directly deposited onto susceptible individuals' mucous membranes via close proximity. Early in the COVID-19 pandemic, this was thought to be the main transmission pathway and this is mitigated quite effectively through the use of facemasks 3,4 .
Since, the aerosol/microdroplet pathway has also been identified as a potential respiratory transmission pathway 5 . In this pathway, small airborne particles secreted by infected individuals remain suspended in the air and can be inhaled by susceptible individuals. This has been observed for other viruses such as influenza A 6 and has also been conjectured to have been responsible for SARS-CoV-2 transmission events, such as in a Chinese restaurant 7 , a cruise ship 8 , and the Skagit Valley Chorale superspreading event 9 . Mitigation of this pathway is challenging, but increasing ventilation rates with more outside air, discouraging high concentrations of people indoors and limiting the time spent indoors are most effective 1,10 . It has also been shown in Leung et al. 11 that face coverings can reduce aerosol emissions. However, other studies have found that this effect can be limited due to small aerosol particles either penetrating the mask or leaking through gaps around the cheeks and nose, which results in apparent filtration efficiencies of face coverings from cloth masks to N95-grade respirators in the range of 10-60%. 12,13 Lastly, the contact route is a mechanism in which the surface of an object is contaminated with viral particles, either via deposition from the air (through particle settling or droplets) or from an individuals' contaminated hand. 14 Such a contaminated object is called a fomite. Susceptible individuals can then transfer the virus from the fomite to their mucous membranes through touching the object and then their face. 15 This mechanism has been anecdotally recognised by several hospitals 16 and the WHO and UK governmental advice to frequently wash hands and disinfect surfaces shows that this mechanism is believed to be important. Figure 1 represents the transmission pathways for viral infections.
Most epidemiological studies related to viral disease transmission have focused on macroscopic, large-scale transmission and often use modified classical susceptible-infected-recovered (SIR) modelling frameworks. 18,19 There have been some efforts to use these models to determine which transmission routes are likely to be dominant 20,21 , however they are not able to consider environment-specific risks. Many researchers have also performed detailed transmission studies and simulations to understand and quantify the specific aerosol, [22][23][24] droplet 2,25,26 , and contact routes 27,28 , however few modelling efforts exist for quantifying overall risks of infection based on the combination of transmission routes.
Fomite transmission routes have seen significant study via models and simulations. Zhao et al. 29 developed an Environmental Infection Transmission System model to quantify risks of infection from droplet-contaminated and hand-contaminated routes, concluding that public, large-surface area droplet-contaminated surfaces have the highest transmission potential.
Beamer et al. 30 used micro-activity empirical data to validate their model for fomite pathogen transmission to test different workplace strategies for reducing infection risks from rhinovirus and rotavirus. The effects of increasing hand hygiene to reduce potential infection by SARS-CoV-2 was investigated by Pham et al. 31 and their results show that event-based hand washing (such as after touching an object) is more effective than frequency washing (every 30 minutes, for example). Kraay et al. 15 developed an ordinary differential equation (ODE) model for fomite-mediated virus transmission, with results highlighting that fomites play an important role in virus transmission. The study also simulated cleaning events on hands and surfaces, demonstrating the efficacy of different cleaning strategies.
To understand influenza transmission routes, Xiao et al. 32 used a multi-agent modelling framework to understand a nosocomial outbreak in a Hong Kong hospital. Their detailed spatio-temporal model identified that long-range airborne (94 %) and fomite routes (6 %) were both likely to have played a role in the outbreak. Xiao et al. 33 used a similar multi-agent approach to understand the transmission of a norovirus outbreak in a UK hotel restaurant in 1998 and found that, out of the multiple pathways examined, fomite transmission played the largest role in the outbreak.
Lei et al. 34 developed a multi-route disease transmission model to study in-flight outbreaks of norovirus, SARS-CoV and influenza A H1N1. They used the detailed seat positions of those infected to consider all 3 routes, modelling each of the 3 transmission routes separately. Markov Chains were built to model fomite transmission routes in a surface contamination network via transfer efficiencies and transition matrices. Their study concluded that for H1N1 transmission, the close contact and aerosol routes were more important, but for SARS-CoV, fomites are slightly more important than the other transmission routes, although all 3 play a key role. Finally for norovirus, fomites are the largest contributor to infection risk. Using a similar Markov Chain modelling approaches, Azimi et al. 8 showed that it was likely that multiple transmission routes played a role in the SARS-CoV2 outbreak on the Diamond Princess cruise ship. 35 used a similar approach to attempt to quantify relative contributions of different pathways in healthcare personnel in patient care, determining that all pathways played a part, however, respiratory pathways dominated during the short patient interactions. These studies highlight the importance of developing models to understand the various transmission pathways, quantify the importance of each route, and the environment-specific risks.
Zhang and Li 17 developed a model considering the three different routes of transmission for the Influenza A virus in a student office (see Figure 1). They used observations from camera recordings to track interactions between students and model the risk of infections to individuals. Zhang and Li 17 present two equations to determine the total quantity of virus on a hand V H and on a surface V S at time t:

Mathematical model
In the following, we will be considering a generic enclosed setting such as an office, a classroom, a retail outlet, a gym or public transport (e.g., train carriage, bus, etc.) which individuals can enter at different times and remain therein for different periods of time.
The way individuals interact with each other and objects in the setting is parameterised in the model since it is dependent on setting type and associated behavioural patterns. For example, in an office environment each individual interacts primarily with few objects on and around their desk ('private surfaces' following the terminology of Zhang and Li 17 ) and only occasionally touches shared objects (or 'public surfaces' 17 ) such as door handles, light switches, water fountains, etc. On the other hand, people visiting retail outlets or gyms or using public transport interact mainly with shared (public) objects, and the associated touching behaviour is markedly different from that in a typical office.
The model tracks the time-dependent spread of virus in the setting from one or more 'sources' (infected individuals) to mucous membranes and respiratory tracts of susceptible individuals through the transmission routes shown in Figure 1  Consider an enclosed space (setting) characterised by volume Vol a that is visited by N p individuals over a period of T hours. Individuals can arrive at different times t 0 j and remain in the setting for the duration ∆t j . The presence of an individual in a setting can be described by the following indicator function: where Θ(·) is the Heaviside step function. This formalism allows modelling people getting on and off public transport, entering and leaving shops, offices and other enclosed settings, and can be easily extended to multiple visits by an individual to the same setting.
The status of an individual is represented by a binary parameter ψ j , j = 1, . . . , N p which takes the value 1 if they are infected or 0 if they are susceptible: We assume that at the time of entry susceptible individuals are free of the virus, i.e., their viral loads on the hands, V H j (t), and mucous membranes, V M j (t) are zero at t = 0, while infected individuals carry a significant amount of the virus on their mucous membranes that is assumed to remain constant throughout the period of interest, and the viral load on their hands at t = 0 is assumed to be in equilibrium with their mucous membranes (see The setting is assumed to contain N S objects that can be touched or handled by people which depends on the dose response parameter k j .
We note that the latter measure may not be reliable due to the high uncertainty in the value of the dose response parameter, k j , measured in the units of virus quantity −1 .
Therefore both the risk of infection and the cumulative viral load, V M j , are considered as important outputs of the model presented.
The above-mentioned transmission mechanisms along with the underlying modelling assumptions are described mathematically below. In the following subsections we are devel-oping a continuous model formulation relying on average frequencies of discrete events such as touching fomite surfaces, breathing, coughing, etc., unless otherwise stated.

Aerosol pathway
During their stay in the setting, individuals are assumed to expel droplets of mucus of various sizes via their respiratory activities (including breathing, coughing and sneezing) and vocalisation. The corresponding rate of continuous viral shedding through these activities by individual j is denoted in the following as r a j . While this rate should clearly be dependent on the viral load on an individual's mucous membranes 40 the form of this dependence is unknown. There are, however, experimental measurements of the rate of virus shedding by infected individuals through this pathway. 41 Therefore, r a j is treated as a parameter in our model, the values of which are presented in Table 3 of Appendix A.
Following the widely-adopted terminology, we will classify droplets expelled through respiratory activities and vocalisation into large and small. 17,34 This subdivision is based on the propensity of small droplets to remain aerosolised for prolonged periods of time in that they are either small enough when expelled or rapidly lose their water content through evaporation before hitting the ground leaving aerosolised nuclei. By contrast, large droplets do not reduce in size sufficiently so as to become aerosolised and land on nearby surfaces, which may lead to their contamination.
In the model, the fraction of large droplets expelled by individual j is denoted as j . These droplets are involved in the close contact transmission pathway which will be discussed in Section 3.2.
Using this notation, the rate at which an individual j expels small droplets, which contribute to the viral load suspended in the air and can subsequently be inhaled by other individuals, can be expressed as Note that we do not consider individual respiratory events, so the shedding of droplets of mucus is treated as a continuous process for the purpose of modelling.
We assume that the air in the setting is perfectly mixed such that aerosol generated anywhere in the room is instantly dispersed equally throughout the room. This approximation is often used in modelling aerosol distribution in enclosed settings since the time scale of air homogenisation is often significantly shorter than other time scales, particularly the duration of time individuals spend in the setting.
The deposition of small virus-laden droplets/aerosols onto fomite surfaces as well as on human skin has also been included in the model. The dynamics between airborne and fomite virus concentrations are discussed in Zhang and Li 17 , including loss of the virus from surfaces through resuspension. The rate of virus deposition onto surface i is proportional to its surface area and the viral concentration in air so that where k d is the rate constant for small droplet deposition.
While we include in our model, for the sake of generality, the terms describing the rates of deposition of small droplets, r SD i , and resuspension of the virus from surface i back into the air, r resp S i , there is no sufficient evidence in the literature to quantify these phenomena.
Thus, although the corresponding terms are included in the formulation, they are kept zero in all the simulations reported below.

Close contact pathway
Large droplets expelled by an infected individual may travel only a short distance from the source before landing on nearby surfaces, other people's skin, or mucous membranes. The viral load deposited on an object through this pathway depends on the following: • The rate at which large droplets are emitted by the infected individual, j r a j (see Section 3.1).
• The amount of time the infected individual j spends in close proximity to the object or other individual x. This is represented in the model by the fraction of the time, θ jx , that the pairs spend in close proximity.
• The fraction of large droplets emitted by the infected individual j that land on object x, π jx , which depends on the relative positions and the distance between the source of the droplets and the acceptor surface. 37 The latter can also change with time, so this parameter should be interpreted as the average fraction of large droplets transferred while in close proximity.
Thus, the rate of viral deposition from source j onto surface x through the close contact pathway is given by while both the individual j and acceptor x are present in the setting. As with the aerosol generation, the deposition of large droplets is treated as a continuous process, and the above rate can be considered as the average rate of large droplet deposition throughout the duration of close contact between the individual j and acceptor surface x.
Viral particles deposited onto surfaces with large droplets can then be spread further by individuals through touch (fomite-mediated transmission), which is discussed in 3.3.

Fomite transmission pathway
The fomite transmission pathway involves the transfer of viral particles between an individual's hand and a fomite surface or their own mucous membranes (mouth, nose or eyes).  In the following, we will be treating the surface touch behaviour as a continuous process as opposed to individual touch events. We assume that the rate of transfer from a donor object and y, f xy , and contact area, A c xy , specific to the way objects x and y come into contact. V x is the viral load on x and A x is its surface area. Using this notation, the rate r touch xy of viral transfer from object x to object y can be described using the following formalism: where R xy is the fraction of the viral load on surface x within the area of contact A c xy that is transferred to y upon one contact, and m xy = R xy f xy A c xy /A x is the overall transfer rate constant from x to y. Note that f xy = f yx and A c xy = A c yx for any x and y.
In our case studies presented in Section 4 we assume that there are no direct contacts between individuals (i.e., the hands and mucous membranes of one individual do not come into contact with either hands or mucous membranes of another), which corresponds to physical distancing being enforced. Therefore, contacts between fomites are also not considered, so R S i S j = 0 for any i and j.

Removal and inactivation of the virus 3.4.1 Continuous formulation
Natural inactivation of the virus is assumed to be exponential in air and on all surfaces, including skin, so that the absolute rates of inactivation can be described as The latter parameter is dependent on the environmental conditions (temperature and relative humidity) and, for fomites, on their material type (e.g., stainless steel, copper, plastic, paper, etc.) and structure of the surface (porous or non-porous). These dependencies can be readily accounted for in the model using an Arrhenius-type inactivation model such as the one proposed by Yap et al. 42 .
In the continuous formulation, the rates of any washing/cleaning interventions are assumed to be described by continuous rates which, on average, correspond to the frequency of cleaning events.
The absolute rate of decrease of viral load on the hands due to washing (or using hand sanitiser gel, wipes, etc.) can be described as: where α j is the 'efficiency' of virus removal during hand-washing, and f wash H j is the frequency of hand washing. If the fraction of the virus removed in a hand-washing event,α j , is given the two parameters are related through 1 and ifα j is represented using the log 10 reduction value (LRV),α j = 10 LRV j , (10) A similar expression can be used for the cleaning of fomite surfaces: where β i is the virus deactivation efficiency on surface i, and f clean S i is the frequency of cleaning 1 The rate of viral load reduction due to hand washing expressed using the continuous formulation is The relative reduction of the viral load between hand washing events is then If the fraction of the virus removed in a single intervention,β i , is given, the two parameters are related through β i = ln β i and ifβ i is represented using log 10 reduction value (LRV),β i = 10 LRV i ,

Mixed continuous-discrete formulation
Some or all of the events such as touching surfaces, washing hands and cleaning of surfaces can be modelled as discrete occurrences. This can be combined with continuous formulations for other transmission rates, however the form of some of the terms would change.
If the frequency of contacts between the hands of individuals and various surfaces is much higher than the rate of handwashing and cleaning, it is justified to describe only the latter two using a discrete formulation. Hand washing (or using hand sanitiser gel, wipes, etc.) at discrete times t w k is assumed to lead to an instantaneous drop in virus concentration on the hands by a factorα j (see the footnote on page 15) so that the absolute rate of viral removal is: where N w j is the total number of times individual j washes their hands, and δ is the Dirac delta-function.
A similar expression can be used for the cleaning of fomite surfaces: where N c i is the number of times surface i is cleaned, andβ i is the fraction of the virus deactivated by each of the interventions taking place at times t c k .

Full model
Using the above descriptions of the rates of the three viral transmission pathways, we can formulate the following ODEs describing the evolution of the viral loads on fomite surfaces, and the hands and mucous membranes of individuals present/visiting the setting: The right-hand side of eq. (6a) accounts for the fomite pathway (transfer from and to the hands of individuals while they are present in the setting; see eq. (1)), the reduction of the viral load due to natural inactivation, resuspension from the surface and cleaning as well as the deposition of aerosol (small droplets, SD) and large droplets (LD) emitted by any individual in close proximity.
In eqs. (6b) and (6c), the right-hand side is non-zero only when individual j is present in the setting (i.e., then the indicator function eq. (1) is non-zero) and the model is able to track their interactions with the environment. Owing to this formulation, the viral loads on both their hands and mucous membranes remain constant when the individual is outside the setting (either before they enter it or after they leave) since the model is unaware of the individuals' whereabouts and interactions when they are not in the enclosed setting being modelled.
When individual j is in the setting, the viral load on their hands follows eq. (6b) which involves transfer to/from fomite surfaces and the individuals' own mucous membranes, natural deactivation of the virus and its removal through hand washing as well as the deposition onto the hands of small and large droplets.
In eq. (6c) we assume that the viral load on the mucous membranes of infected individuals (for whom ψ j = 1) is so high that no interactions with the surroundings can either increase or reduce it. Therefore, the right-hand side of equation eq. (6c) is multiplied by (1 − ψ j ) to 'freeze' the viral load on the mucosa of infected individuals at its initial level to avoid any artificial decrease in their viral loads due to shedding (also, the timescales considered here are assumed to be shorter than those leading to significant changes in the condition Variations in the viral load within the total volume of air, Vol a , of the setting are described by the following ODE: where the only positive contributions are those due to aerosol produced by infected individuals while they are in the setting and possible resuspension of the virus from fomite surfaces. Aerosolised viral particles can be removed from the air by people inhaling them, via the physical removal of the virus due to ventilation with the rate r vent = q vent V a /Vol a (q vent is the ventilation air flow in m 3 h −1 ), or through its natural inactivation. Note that, although the terms describing the deposition of small droplets, r SD i , and resuspension of the virus back into the air, r resp S i , are included in this formulation, their magnitudes can be assumed to be negligible in comparison with the rate of natural inactivation in air and those of other transmission events. Therefore, their values are set to zero in all the simulations reported below.
We assume that start of the observation period the setting is uncontaminated, so that the initial conditions for the viral loads on fomite surfaces and in the air are zero: Similarly, susceptible individuals entering the setting are assumed to have had no prior exposure to the virus: On the other hand, infected individuals carry significant viral loads V M 0 (see Table 3 of Appendix A) on their mucous membranes which are assumed to remain unchanged throughout the simulated period of T hours: Prior to entering the setting, infected individuals are also assumed not to have touched their faces and mucous membranes with typical frequencies but not any fomites, so that an equilibrium has been established between the viral loads on the individuals' mucous membranes and hands: Now the risk of infection can be defined in the following way: where we are explicitly assuming that the values of the dose response parameters k j associated with the different pathways are the same.

Model implementation
The mathematical model presented in this section was implemented in both MATLAB and Python. Refer to Appendix B for relevant implementation details.

Model Demonstrations and Case Studies
To      Table 1 and Table 3 of Appendix A, the rate of transmission from contaminated hands to the objects should be highest for the document, followed by the door handle and then by the desk. However, the viral loads in    into each other's face. 26 The results show that the viral exposure decreases most rapidly with the increase in the number of cleaning events. When the number of cleaning events is greater than two (e.g., disinfection is performed more often than once every two hours) suppressing the direct transmission through large droplets can bring about a further reduction in the accumulated viral dose and hence the risk of infection.  Table 1 of Appendix A for the other parameter values. The results of simulating the same scenarios as in Case Study 1 are presented in Figure 6.
The change in the way the individuals interact with each other and their environment leads to a different pattern of fomite contamination, shown in Figure 6a, in which the document is less than half as contaminated as during prolonged close contact in Case Study 1 and the desk of the susceptible individual is over 40 times less contaminated than that of the infected person. There is no difference, however, in the contamination of the air (see Figure 6b), since the aerosol pathway is not affected by the changes related to the fomite and close contact pathways.
Compared to Case Study 1, there is a nearly 7-fold decrease in the total viral exposure of the susceptible person at the end of the 8 h-long working day (see Figure 6c). This effect is explained by drastically reduced contributions of the fomite and close contact pathways due to reduced handling of shared objects (i.e., each other's desks and the document) as well as lower fomite contamination through the deposition of large droplets from the infected individual. This results in a different pattern of relative contributions of the three transmission pathways, as shown in Figure 6d, in which the fomite pathway is still dominant overall, while the aerosol pathway dominates over relatively short times between approximately 0.25 h to 2 h. However, it is important to emphasise here that the absolute viral exposure due to the aerosol pathway in this case is exactly the same as in Case Study 1 while the total exposure, and hence the risk of infection, are significantly lower than those predicted for Case Study 1.
Hand and fomite surface disinfection at t = 4 h leads to a significant reduction in the viral loads on surfaces as well as a 19% decrease in the total exposure of the susceptible individual ( Figure 6a,c,d). The relative reduction in exposure is lower, however, than in Case Study 1 because of the lower contribution of the fomite transmission pathway into the total exposure.
An additional cleaning event at 2 h leads to an overall decrease in viral exposure by 30%.
The effect of combining mitigation strategies on the total viral exposure is reported in  Figure 5f shows that ventilation has a more pronounced influence on the total exposure than in Case Study 1 due to a higher relative importance of the aerosol pathway (although the absolute exposure is significantly lower, as mentioned above). At very low ventilation rates (<1 ACH), hand and surface cleaning become ineffective at reducing exposure since the aerosol pathway under such conditions becomes dominant. As the ventilation rate increases, the aerosol pathway becomes less important and disinfection becomes the main mitigation.

Case Study 3: Graduate Student Office
To demonstrate the model's capability to track all transmission pathways while considering a large number of individuals and objects, a test case considering a graduate student office with one infected student and 38 susceptible students is used. It is based on the work and data of Zhang and Li 17 . While some of the features described in the original test case are retained, such as the categorisation of objects into private and public surfaces depending on personal or public use, several modifications have been made to demonstrate the model's wider applicability: • Only desks and chairs are considered as private surfaces; all other personal belongings such as mugs, bags, etc. are ignored.
• Public surfaces include cabinet handles of 3 cabinets, printer, water dispenser, and door handle.
• Each cabinet is used by 13 students, where cabinet 1 is used by students 1 -13, cabinet 2 by students 14 -26, and cabinet 3 by students 27 -39.
• For each public object, the surface area of the surface or part of the object with the highest touch frequency is considered. For example, the considered surfaces are: printer touch screen, water dispenser button, and cabinet handle.
• Average values are used for all contact frequencies, as opposed to individual contact frequencies based on individual behaviour, however, the model is also able to consider discrete touch events. and respective values used in this test case are given in Table 2

Acknowledgements
The authors would like to express their gratitude to all those that participated in the Royal Society's Rapid Action in Modelling the Pandemic (RAMP), particularly those involved in Subgroup 4. The interactions and discussions in these meetings was always thoughtprovoking and inspired much of this work. We would also like to thank Dr Marco-Felipe King for his insights and encouragement during the initial model development. Finally, we are grateful to Dr Nan Zhang for providing us access to the data from their extensive work in Zhang et al. 25 .

A.1 Human behaviour and Objects
The parameters used to describe human behaviour and interactions with objects are described in this section, for all three case studies. Table 1 contains the parameters specific to Case Studies 1 and 2, where two individuals interact with one another and fomites during a one-to-one meeting or while working alongside each other.   Table 3 contains parameters used in all three case studies, primarily those associated with viral transfer, transmission, and inactivation.

Appendix B Implementation notes
Both the continuous and discrete model formulations of the model have been implemented in MATLAB and Python. The resulting system of ODEs is solved numerically using ode15s solver in MATLAB or SciPy's odeint() in the Python implementation.
To avoid discontinuities on the right-hand sides of eqs. (6a) to (6c) and (7) due to the appearance of both the Dirak's delta-function in eq. (4) and eq. (5) and indicator functions eq. (1) we replace them with continuous approximations using a triangular function to approximate Dirak's delta function: , |t| ≤ t and the following piecewise linear approximation of the Heaviside function: The discontinuous indicator function eq. (1) is then replaced in the code by the following trapezoidal function: I j (t) =Θ(t − t 0 j )Θ(t 0 j + ∆t j + t − t), j = 1, . . . , N p One, however, must ensure that the parameter t is much smaller than the shortest duration of occupancy: t << max j=1,...,Np ∆t j and the maximum integration step size used by ode15s or odeint() is significantly less than t .