INTRODUCTION
The risk of acquiring nosocomial infections is a recognized problem in health‐care facilities worldwide (Harbarth, Sax, & Gastmeier, 2003). While the transmission routes for some diseases are well documented, the precise mode of transmission is uncertain for many infections, particularly for those pathogens that cause health‐care–acquired infections (HCAIs). Although it is probable that the majority of transmission occurs via contact routes (Sax et al., 2009), there is increasing recognition that the air plays an important role in disease spread (Li et al., 2007). Understanding the role that ventilation airflow plays in the dispersion of infectious microorganisms is tantamount to assessing exposure to pathogens and hence infection risk. This study aims to provide an analytical link between airborne hospital infection spread, ventilation design, and outbreak management.
Ventilation has been found to have a significant impact on the distribution of infectious material in hospital settings. Examples include Influenza A (e.g. H5N1 and H7N9) (Reed & Kemmerly, 2009), Mycobacterium tuberculosis (Escombe et al., 2010), measles (Atkinson, Chartier, Pessoa‐Silva, Jensen, & Li, 2009), and norovirus (Teunis et al., 2008). One of the most infamous examples occurred in 2003 during the severe acute respiratory syndrome (SARS) outbreak in Hong Kong. Analysis of airflow patterns and outbreak data demonstrated that ventilation routes were critical in the short‐ and long‐range spread of aerosolized coronavirus (Li, Huang, Yu, Wong, & Qian, 2005). Ventilation is recognized as an important infection control approach in health‐care design, with strategies such as mechanical ventilation and pressure zoning set out in international (Atkinson et al., 2009) and national guidance (Department of Health, 2007).
Evaluating the influence of ventilation on infection risk typically applies models such as the Wells–Riley equation (Riley, Murphy, & Riley, 1978) or a dose–response approach (Sze & Chao, 2010) to estimate the influence of ventilation on the number of new cases of an infection. Liao, Chang, and Liang (2005) presented a probabilistic transmission dynamic model to assess indoor airborne infection risks and Ko, Burge, Nardell, and Thompson (2001) and Ko, Thompson, and Nardell (2004) developed models for tuberculosis spread incorporating a zonal ventilation model. A number of authors have also looked at control strategies, including Wein and Atkinson (2009), who modeled infection control measures for pandemic influenza; Brienen, Timen, Wallinga, Van Steenbergen, and Teunis (2010), who analyzed the effect of mask use on the spread of influenza; and King, Noakes, and Sleigh (2015), who developed a stochastic model to link airborne and contact transmission. It is also worth mentioning the recent work by Carruthers et al. (2018), where a zonal ventilation model similar to the one considered in this article is linked to a dose–response approach to estimate the risk of infection after an accidental release of bacteria Francisella tularensis in a microbiology laboratory.
While these studies enable some understanding of the influence of the environment on transmission, they do not consider relationships between ventilation parameters and the progression and control of an infection outbreak. In an earlier study, it was demonstrated that the Wells–Riley model could be coupled to an SI epidemic model to relate ventilation rate and transmission in a fully mixed environment (Noakes, Beggs, Sleigh, & Kerr, 2006). In later work, a zonal air distribution and a stochastic formulation (Noakes & Sleigh, 2009) was considered, and cost benefits of ventilation from an energy and infection risk perspective were explored (Noakes, Sleigh, & Khan, 2012).
The model presented in this article is constructed on a scenario defined in previous work (Noakes & Sleigh, 2009), where the role played by the airflow during a nosocomial outbreak is assessed by linking a deterministic zonal ventilation model with an SI stochastic epidemic model using a computational approach. While the previous approach enabled exploration of the basic interaction between the ventilation and the outbreak, there are a number of limitations:
The epidemic dynamics are represented through a simple SI epidemic model, not accounting for relevant factors such as the discharge and admission of patients, or the detection and declaration of the outbreak.Results reported by Noakes and Sleigh (2009) have high variability, which is related to the fact that they were obtained by means of stochastic simulations of the epidemic process.The large number of parameters associated with each ventilation scenario makes it difficult to identify, from stochastic simulations, the specific factors of the ventilation air distribution that facilitate or mitigate epidemic spread. We refer the reader to Keeling and Ross (2007) where the limitations of analyzing this type of epidemic processes by simulation are discussed in more detail, and where the benefits of following exact analytical approaches instead are highlighted.
Our aim here is to show how this zonal ventilation model can be linked to more complex stochastic epidemic models for the spread of nosocomial pathogens, while accounting for patients, admission and discharge, and different outbreak detection and declaration hypotheses. We show how to implement exact analytical procedures for computing summary statistics of the outbreak (statistics measuring outbreak infectiousness), which by means of a perturbation analysis enables identification of specific characteristics of the ventilation setting that are crucial for the spread or control of the infection.
Finally, we carry out a comprehensive numerical study of six ventilation strategies for a hypothetical hospital ward in order to identify particular ventilation characteristics that may promote or inhibit spread of airborne nosocomial infections. Our results explore the interplay between ward ventilation, location of patients, ward overoccupancy, and outbreak detection management.
A Zonal Ventilation Model for Linking Airflow Dynamics and Infection Rates
In Noakes and Sleigh (2009), this Wells–Riley process is adapted to investigate ventilation scenarios in a hypothetical hospital ward split in M
ventilation zones. The air is assumed to be uniformly mixed within each zone; however, there is incomplete mixing between the zones and unbalanced zone boundaries allow for the effect of directional flow to be examined. In particular, the per capita infection rate λk for susceptible individuals at zone k is defined as λk=pkCk, where Ck is the concentration of infectious material at zone k and pk is the pulmonary rate of these individuals. We note that this concentration could depend on the number of infected individuals, ij, in every zone 1≤j≤M, due to airflow. In Noakes and Sleigh (2009), the spatial distribution of infectious material is represented through the differential equation:
(2)VkdCkdt=qkik−Qo,kCk−∑jβkjCk+∑jβjkCj,1≤k≤M,where Vk is the volume of zone k, qkik is the generation rate of infectious quanta, Qo,k represents the extract ventilation rate in zone k, and ∑jβkjCk and ∑jβjkCj amount to the volume flow of air between zones k and j. Moreover, each interzonal flow rate βkj represents the sum of two contributions:
βkj=β0+βQkj,where β0 is a global mixing rate and βQkj is an additional contribution representing net flow across the k/j zonal boundary, from zone k to zone j.
Equation (2) leads to a ventilation matrix that characterizes the ventilation air distribution under study,
V=Qo,1+∑kβ1k−β21⋯−βM−1,1−βM1−β12Qo,2+∑kβ2k⋯−βM−1,2−βM2−β13−β23⋯−βM−1,3−βM3⋮⋮⋱⋮⋮−β1M−β2M⋯−βM−1,MQo,M+∑kβMkrepresenting ventilation in a hospital ward divided into M ventilation zones.
By assuming steady‐state conditions for airflow, and taking into account Equation (2), Noakes and Sleigh (2009) propose to link infection rates λj, j∈{1,⋯,M} with the ventilation matrix V as follows:
(3)λ1p1λ2p2⋮λMpM=V−1q1i1q2i2⋮qMiM.This means that per capita infection rates {λ1,⋯,λM} for susceptible individuals at zones {1,⋯,M} depend on how many infective individuals (i1,⋯,iM) there are in any zone at any given time, computed in a specialized manner (Equation (3)) that takes into account the ventilation distribution through matrix V. Once this procedure for computing infection rates is proposed, stochastic simulations for the SI epidemic dynamics are carried out in Noakes and Sleigh (2009) by following steps (i)–(iv) (Noakes & Sleigh, 2009), which assume exponentially distributed interevent times and make use of the property depicted in Equation (1).
We note here that Equation (3) means considering per capita infection rates at each zone 1≤j≤M as functions λj(i1,⋯,iM) of the number of infectives (i1,⋯,iM) within each ventilation zone at the hospital ward. In Section 2.2, we now go on to exploit this, to link the zonal ventilation model with a multicompartment SIS model with detection, to evaluate the infection spread dynamics within the hospital ward until detection of the outbreak. Instead of carrying out stochastic simulations, we present an exact approach for analyzing a summary statistic of the outbreak: the total number of infections occurring until the outbreak ends or is detected and declared. This exact approach does not only allow us to compute this quantity of interest, but it also allows one to carry out a sensitivity analysis on the model parameters, so that the impact that different characteristics of the ventilation setting has on this summary statistic can be evaluated.
A Multicompartment SIS Stochastic Model for the Infection Spread Dynamics
At the epidemic level, we assume that patients in each zone i are discharged at rate γi, so that γi−1 amounts to the average length of stay (LOS) of patients in zone i. Discharges are immediately replaced by new admitted patients, a reasonable approximation for hospital wards under high demand (Pelupessy, Bonten, & Diekmann, 2002; Wolkewitz, Dettenkofer, Bertz, Schumacher, & Huebner, 2008). Moreover, we consider that the nosocomial outbreak will go undetected by health‐care workers for some time, and incorporate this fact into our model by considering that each infected individual in zone i can be discovered/detected at some rate δi. The reciprocal δi−1 represents the average time until some symptoms arise that alert health‐care workers to a patient's infection, or the average time until the infected individual is detected through screening policies put in place at this hospital ward. Fig. 1 represents the epidemic dynamics for an individual in zone i.
This leads to a multicompartment SIS epidemic model that can be described as a CTMC X={(I1(t),⋯,IM(t)):t≥0}, where Ij(t) represents the number of infective individuals in zone j at time t≥0, defined over the space of states S={(i1,i2,⋯,iM):ij∈{0,⋯,Nj},j∈{1,⋯,M}}∪{Δ}=C∪{Δ}. Nj is the total number of patients in zone j, leading to N=∑j=1MNj patients in the hospital ward. State Δ represents that the nosocomial outbreak has been detected and declared by health‐care workers by the first detection of an infected patient in the hospital ward. We note that absorbing state (0, …, 0) represents the end of the outbreak (lack of infective individuals), due to patients' discharge (i.e., if all the patients infected by the pathogen are discharged before the outbreak is actually detected). We consider Δ also as an absorbing state in this process, since we are only interested in the dynamics of the process until the end or declaration of the outbreak, and the transitions (obtained from diagram in Fig. 1) described in Table I. We note that, according to our comments in Section 2.1, λj(i1,⋯,iM) is a function of the state (i1,⋯,iM), representing the per capita infection rate of susceptible individuals in zone j when we have (i1,⋯,iM) infective individuals within the ward, computed from Equation (3) for each (i1,⋯,iM)∈C.
Analyzing R
For an initial state (i1,⋯,iM)∈C, our aim is to compute probabilities
p(i1,i2,⋯,iM)(n)=P(R=n|(I1(0),I2(0),⋯,IM(0))=(i1,i2,⋯,iM)),for n≥0; that is, the probability distribution of R for some initial state (i1,⋯,iM). We can compute these probabilities from a system of linear equations, which is obtained by a first‐step argument. In particular, by proposing notation
i=(i1,⋯,iM),i+(s)=(i1,⋯,is+1,⋯,iM),i−(s)=(i1,⋯,is−1,⋯,iM),we get
(4)pi(0)∑j=1M(λj(i)(Nj−ij)+(γj+δj)ij)=∑k=1Mγkikpi−(k)(0)+δkik,
(5)pi(n)∑j=1M(λj(i)(Nj−ij)+(γj+δj)ij)=∑k=1Mγkikpi−(k)(n)+λk(i)(Nk−ik)pi+(k)(n−1),for n≥1 and any (i1,⋯,iM)∈C, and with boundary conditions p(0,0,⋯,0)(0)=1, p(0,0,⋯,0)(n)=0 for all n≥1; a detailed explanation on how Equations (4) and (5) are obtained is in the Appendix. This means that probabilities for n=0 (Equation (4)) can be computed by solving a system of
(6)#C=∏i=1M(Ni+1)linear equations.1 Once these are in hand, probabilities for n≥1 can be computed by solving the system of linear equations given by Equation (5), which also consists of #C equations.
Algorithm 1 in the Appendix computes probabilities p(i1,⋯,iM)(n) for any (i1,⋯,iM)∈C, n≥0. It works sequentially, computing probabilities p(i1,⋯,iM)(n) for i1+i2+⋯+iM=I for increasing values of n=0,1,2,⋯ and I=1,2,3,⋯,N.
Local Sensitivity Analysis
Our analysis allows one to identify the most important characteristics of the ventilation scenario, regarding the infectious potential of the outbreak until detection, by means of computing partial derivatives of the form ∂E[R]/∂θ with respect to ventilation parameters θ∈{β0}∪{βQij:i,j∈{1,⋯,M}}∪{Qo,i:i∈{1,⋯,M}}. We note that, for an initial state (i1,⋯,iM)∈C, E[R]=∑n=0+∞np(i1,⋯,iM)(n), so that
(7)∂E[R]∂θ=∑n=0+∞n∂p(i1,⋯,iM)(n)∂θ.Partial derivatives ∂p(i1,⋯,iM)(n)∂θ can be computed from direct differentiation of Equations (4) and (5). In Equations (4) and (5), the only quantities that depend on parameter θ∈{β0}∪{βQij:i,j∈{1,⋯,M}}∪{Qo,i:i∈{1,⋯,M}} are infection rates λj(i1,⋯,iM) and probabilities p(i1,⋯,iM)(n). Thus, we get
∑j=1M∂λj(i)∂θ(Nj−ij)pi(0)+∑j=1Mλj(i)(Nj−ij)+(γj+δj)ij∂pi(0)∂θ=∑k=1Mγkik∂pi−(k)(0)∂θ,∑j=1M∂λj(i)∂θ(Nj−ij)pi(n)+∑j=1Mλj(i)(Nj−ij)+(γj+δj)ij∂pi(n)∂θ=∑k=1Mγkik∂pi−(k)(n)∂θ+∂λk(i)∂θ(Nk−ik)pi+(k)(n−1)+λk(i)(Nk−ik)∂pi+(k)(n−1)∂θ,for n≥1, and any (i1,⋯,iM)∈C. Partial derivatives ∂p(i1,⋯,iM)(n)∂θ can then be computed from the equations above by following arguments similar to those in Algorithm 1 in the Appendix. In order to solve these equations, one needs to have in hand values of p(i1,⋯,iM)(n) (previously computed from Algorithm 1), as well as derivatives ∂λj(i1,⋯,iM)∂θ. These derivatives can be straightforwardly obtained from Equation (3) as:
1p1·∂λ1(i1,⋯,iM)∂θ1p2·∂λ2(i1,⋯,iM)∂θ⋮1pM·∂λM(i1,⋯,iM)∂θ=−V−1V(θ)V−1q1i1q2i2⋮qMiM,
where V(θ) represents the element‐by‐element partial derivative of matrix V with respect to parameter θ (Gómez‐Corral & López‐García, 2018).
Spread until the Dth Individual Detection
As outlined above, declaration of the outbreak is identified with the first detection of an infective patient, where each patient is detected in zone j at rate δj. If detection of an infective patient occurs because this patient shows symptoms, outbreak declaration might require several (D>1) patients showing some common symptoms, since for some nosocomial pathogens, associated symptoms are quite common and pass unnoticed (Ekkert, 2015). For example, norovirus causes gastrointestinal symptoms such as nausea, vomiting, or diarrhea that are common to many diseases and conditions. The National Guidelines on the Management of Outbreaks of Norovirus Infection in Health‐Care Settings (National Disease Surveillance Centre, 2003), issued by the National Disease Surveillance Centre in Ireland, requires for D=2 patients to show these symptoms in a hospital ward for a potential norovirus outbreak declaration. Once the outbreak has been declared, control strategies such as immediate cleaning and decontamination, frequent handwashing, or cohorting of affected patients are recommended.
Thus, our interest in this subsection is to analyze the summary statistic R when the detection of the outbreak requires D patients to show symptoms, for some value D≥1, and results in the subsections above can be seen as the particular case D=1. We define the augmented process Xaug={(I1(t),⋯,IM(t),D(t)):t≥0}, where the increasing variable D(t) amounts to the number of detected patients up to time t≥0. We consider that the outbreak is declared once D(t)=D, and the space of states of this CTMC is given by:
Saug={(i1,i2,⋯,iM,d):ij∈{0,⋯,Nj},j∈{1,⋯,M},0≤d≤D−1}∪{Δ}.Thus, Saug=Caug∪Δ, with state Δ representing outbreak declaration (i.e., the detection of the Dth infected patient). Events occurring in this process, at different rates, are described in Table II.
Our arguments in the subsections above can be adapted for process Xaug. For example, Equations (4) and (5) become
p(i1,⋯,iM,d)(0)∑j=1M(λj(i1,⋯,iM)(Nj−ij)+(γj+δj)ij)=∑k=1Mγkikp(i1,⋯,ik−1,⋯,iM,d)(0)+δkik(1d=D−1+1d
Impact of Ventilation Setting on Spread Dynamics
In Fig. 3, we plot the probability mass function of R versus different values of the global mixing rate β0, the average time δ−1 at which each infective patient shows symptoms, and for ventilation scenarios A–F. For these results, it is assumed that an infective patient in zone 1a starts the outbreak, and we report in Table III the mean values E[R] computed for these distributions.
Ventilation setting C can be identified in Fig. 3 and Table III as the best one, while settings A, D, and E are identified as the worst ones depending on the detection parameter δ and the global mixing rate β0. We note that ventilation setting C has significant extract ventilation at the initially infected zone 1a, so that the airflow is directed from 1a outward to the hospital ward. On the other hand, ventilation setting D represents a well‐mixed ward (βQik=0 for all i and k) with no extract ventilation at zone 1b (Qo,1b=0), which might favor the spread of pathogens from 1a toward other zones within the ward, leading to more infections occurring until outbreak detection.
In general terms, worse scenarios can be identified for values δ−1=48 h and β0=27m3/ min , where the long‐tailed distribution of R for these scenarios in Fig. 3 indicates that large outbreaks occur with significant probability. Larger differences among ventilation settings are also found for value δ−1=48 h. Thus, our results suggest that ventilation of the ward should be of special concern for pathogens that have longer infectious asymptomatic periods, or in hospital wards with more limited surveillance policies. It is also clear that the average individual detection time δ−1 has a higher impact on the infection spread than the specific ventilation setting in the ward, so that outbreak detection seems to dominate ventilation regarding infection spread.
Dependence on Location of Initial Infective
In Fig. 4, we plot analogous results to those in Fig. 3 when the infective patient starting the outbreak is located in zones {1a,1b,2a,2b}, for δ−1=12 h and β0=9m3/ min . Corresponding mean values E[R] are reported in Table IV. We note that zones 3a and 3b are equivalent to zones 1a and 1b, for all ventilation settings in Fig. 2, and thus we do not test them. For zones near the corridor (i.e., 1b and 2b), ventilation setting B is identified as the best one, while D is identified as the worst one. We note that ventilation setting B has no extract ventilation in zones 1b and 2b (Qo,1b=Qo,2b=0), but it directs the airflow instead toward corridor areas. In this setting B, corridor areas have no patients and significant ventilation, with airflow unbalance from bays to corridor areas acting in practice as an infection control measure. Thus, our results suggest that the spread control ability of a given ventilation setting depends on the location of the patient starting the outbreak as well as the airflow direction. However, from results in Fig. 4 and Table IV, ventilation setting D seems to perform poorly regardless of the initial infective location, suggesting that some ventilation settings might be inadvisable regardless of this location (i.e., if this location is unknown).
When focusing on a particular location for the initial infective, comments above are supported by the sensitivity analysis on the ventilation parameters. For example, in Tables V and VI we report, for ventilation parameters θ∈{β0,Qo,1a,⋯,Qo,c3,βQ1a,1b,⋯,βQc3,3b}, partial derivatives ∂E[R]/∂θ, and elasticities (∂E[R]/∂θ)·(θ/E[R]) for ventilation settings B and D, δ−1=12 h, β0=9m3/ min and an infective patient starting the outbreak in zone 1b. We note that while dimensionless elasticities are useful for comparison purposes, they equal zero if parameter θ is zero.
Regime B requires airflow toward the corridor in order to expel pathogens from zone 1b, since Qo,1b=0. Thus, rates βQ1b,c1, Qo,c1, Qo,c2, and Qo,c3 correspond to significantly large negative elasticities reported in Table V (i.e., increasing the values of these rates would lead to decreasing values of E[R]). Global mixing rate β0 has a significant impact (large positive elasticity) favoring disease spread, since increasing the value of β0 represents increasing the rate at which pathogens flow among all zones, instead of flowing specifically toward the extract ventilation areas (corridors in this setting).
According to results in Table VI, ventilation setting D could be significantly improved by increasing extract ventilation (especially in zones 1a, 1b, and c1), as well as increasing airflow from 1b to 1a and to c1. This is directly related to the fact that, since there is no extract ventilation in zone 1b, infectious material in this zone can only be expelled by directing it toward adjacent zones 1a and c1.
Decreasing Hospital Ward Infection Spread Risk Might Increase Risk at Specific Bays
It is clear that the number, R, of infections occurring until the end or detection of the outbreak can be split according to where these infections actually occur as
R=R(1)+R(2)+R(3),where R(j) is the number of infections occurring at bay j. Although probabilities P(R(j)=n) can be analytically computed by adapting arguments in Section 3, details are omitted here for the sake of brevity, and results in Table VII are obtained from 106 stochastic simulations of the process.
In Table VII, we report values of E[R]=E[R(1)]+E[R(2)]+E[R(3)] for β0=9m3/ min , δ−1=48 h, ventilation settings A, D, and E, and an infective patient starting the outbreak in zone 1a. Results suggest that epidemic spread can be limited by switching ward ventilation from setting D to A, and further containment is obtained by switching to ventilation setting E. However, infection risk in bay 1 (in terms of E[R(1)]) behaves contrarily; although the global hospital ward infection risk (in terms of E[R]) is lower for setting E, this is at the expense of expelling pathogens from the infected zone 1a toward zones 1b and c1, and thus posing a greater risk to patients in bay 1.
Interplay with Detection Management
As explained in Section 3.3, detection and declaration of an outbreak in the hospital ward may require several patients showing symptoms, and not only one. In Fig. 5, we plot analogous results to those in Fig. 3 when declaration of the outbreak occurs after D=2 patients show symptoms (each after average time δ−1). The corresponding mean values of E[R] are reported in Table VIII.
We note that values in Table VIII are significantly larger than those in Table III, since outbreak declaration takes longer to occur, allowing for more infections to take place. This increase is significantly larger than the differences that can be observed, in Table III, between different ventilation settings, suggesting again that detection policy is likely to dominate ventilation as an infection control strategy. Under slow detection scenarios (δ−1=48 h), we observe in Fig. 5 a clear bimodality for the distribution of R. Thus, our model predicts that under slow detection, a two‐output situation can be expected: either the initially infective patient is discharged before infecting any other patient (so that R=0), or this patient infects a second patient, leading to a large outbreak (represented by the second mode in Fig. 5).
Screening at Admission
In Sections 4.1–4.4, we analyze infection spread under the assumption that each individual in zone j is detected (by showing symptoms) at rate δj, with δj=δ for all j, and where the outbreak is detected and declared after one (or several) infective patients are detected. This leads to the contribution ∑j=1Mδjij in Equations (4) and (5). However, if the detection of the outbreak is due instead to the screening of the newly admitted patient who starts the outbreak, and results of this screening arrive after an average time δ−1, then the outbreak is detected at rate δ, and one needs to replace ∑j=1Mδjij by δ in Equations (4) and (5).
Under this hypothesis, we plot in Fig. 6 the probability mass function of the number R of infections until the end or detection of the outbreak, when the results of this screening (and thus, the declaration of the outbreak) arrive after an average time δ−1∈{4h,8h,12h,24h}. Corresponding mean values of E[R] are reported in Table IX. If results arrive after δ−1=4 h, ventilation has a less significant impact on the nosocomial spread, and low values of E[R] are reported in Table IX. The number of infections until the end or detection of the outbreak proportionally increases with the delay δ−1 in obtaining the screening results. In particular, for δ−1=24 h significant differences in E[R] can be noticed among the different ventilation settings, and a marked bimodality can be observed in Fig. 6.
Ventilation and Overoccupancy
In this subsection, our aim is to shed some light on the interplay between ventilation, nosocomial spread, and overoccupancy of the hospital ward. We represent hospital ward overoccupancy by locating three additional patients at the corridor areas; in particular, we set N=21 and locate one additional patient in each of the {c1,c2,c3} zones. This practice is common in U.K. hospitals during times of high demand. We assume that the outbreak is detected and declared after the first patient shows symptoms, each patient showing symptoms after an average time δ−1=12 h. For interzonal mixing β0=9m3/ min , we report in Table X the mean number R of infections until the end or detection of the outbreak, for ventilation settings A–F, and for the initial infective patient being located in zones {1a,c1,c2,c3}.
We first note that results in Table X are exactly the same for an initially infective individual being located in zones c1 and c3. This is explained by noting that bays 1 and 3 are completely symmetric for all ventilation settings as noted in Section 4.2; see diagrams and matrices in Fig. 2. On the other hand, when the initial infective patient is located in zone 1a, results in Table X can be compared to those in Table III for (δ−1,β0)=(12h,9m3/ min ). For zone 1a, values of E[R] are larger in Table X than in Table III; if the nosocomial outbreak is initiated by an infective patient in zone 1a, more infections during this outbreak should be expected under ward overoccupancy. This might not be only related to having more patients in the ward under overoccupancy (21 instead of 18), but also to the potential of patients in the corridor to act as infection links between bays. For example, under overoccupancy, an infective individual in zone 1a might infect individuals in bay 2 by, as a first step, infecting individuals in the corridor areas. These people then might more easily infect individuals in bay 2, before being discharged, due to being in closer proximity and depending on the particular ventilation setting in place in the ward.
Infection dynamics related to the scenario above highly depend on the particular ventilation setting under study, which can be noticed by inspecting rows corresponding to zones {c1,c2,c3} in Table X. While an individual in zone 1a has a larger infectious potential (in terms of E[R]) than individuals located in the corridor when ventilation settings B and E are in place, this is not the case for ventilation settings C, D, and F, and these infectious potentials are comparable under ventilation setting A, which represents a well‐mixed scenario. Our results then indicate that overoccupancy leads in general to higher airborne spread risks, and that this increase can be especially significant depending on the specific ventilation in place.
The Unit of Infection
We note that parameter q is highly pathogen dependent, ranging from q∼0.01 quanta/min for rhinovirus to q∼10 quanta/min for measles (Noakes & Sleigh, 2009). We perform a parametric analysis by varying q to assess the sensitivity of our conclusions, and report expected infections E[R] until the end of the outbreak in Table XI. In particular, we are interested in the mean number E[R] of infections if the outbreak is detected on the first patient showing symptoms, each patient showing symptoms after an average time δ−1=12 h, and where we consider β0=9 m3/min. We note that, as expected, increasing values of q lead to increasing mean number E[R] of infections. However, this does not seem to affect the relative infectiousness of ventilation setting C, which is identified as the best scenario regardless of the value of q. On the other hand, less advantageous ventilation schemes are dependent on the value of q. For example, ventilation setting A and E can be identified as the worst for q=0.1 quanta/min, while setting D can be seen as the worst one for q=50.0 quanta/min.
DISCUSSION AND CONCLUSIONS
In this work, we link a zonal ventilation model for the generation and airborne spread of infectious material within a hospital ward with a multicompartment SIS Markovian model for the infection of patients within this ward. Our model incorporates the possibility of considering a wide range of ventilation settings, the discharge and arrival of patients within the ward, as well as different hypotheses regarding how outbreak detection and declaration occurs. Moreover, it allows us to explore the interplay between ventilation, outbreak management, ward overoccupancy, and the location of the infective patient starting the outbreak.
Our results suggest that detection time dominates ventilation when the variable of interest is the number of infections occurring before the declaration or end of the outbreak, with longer detection times leading to significantly more infections happening. Longer detection times can arise when analyzing pathogens with long infectious asymptomatic periods, when declaration of an outbreak requires for several patients to show symptoms, or when this declaration depends on screening events for which results take longer to arrive. The interplay between ventilation of the hospital ward and location of the initially infective patient starting the outbreak implies that recommendations on where to locate potentially infected (e.g., newly admitted) patients in a given hospital ward could be issued depending on the ventilation in place in the ward. Our model also predicts that decreasing the infection spread risk in the hospital ward can sometimes come at the expense of increasing the risk in particular areas of the ward.
Similar models have already been considered in the literature for linking zonal ventilation scenarios with epidemic spread models (Carruthers et al., 2018; Ko et al., 2004; Noakes & Sleigh, 2009), where epidemic dynamics are usually analyzed by means of stochastic simulations. To the best of our knowledge, this is the first time that this link is carried out by defining in detail the CTMC for the infection spread, where infection rates at each ventilation zone are in fact functions λj(i1,⋯,iM) of the number of infectives in each zone at any given time, and where (i1,⋯,iM) represents in fact a state of the CTMC under study. This detailed mathematical construction allows for the analytical computation of summary statistics (such as R in this work), and for carrying out a local sensitivity analysis that allows one to identify the particular factors of each ventilation setting having the most significant impact on the infection spread.
It should be noted that the primary objective in this study is to demonstrate this detailed mathematical analysis and how it can be applied to evaluate the relative influence of different parameters. The model is applied to a hypothetical hospital ward, which, while it is representative of multibed ward environments in many hospitals, is a very simplified model of reality. The results demonstrate that the ventilation flow settings may influence the dispersion of airborne pathogens and hence the risk of transmission; however, these should be interpreted with caution. We assume a steady‐state ventilation scenario with the flow pattern replicated exactly between neighboring bays 1, 2, and 3. In reality, the flows will not be exactly identical for every bay, and other factors such as heat sources and movement of people will alter the mixing with and between zones. In particular, corridor ventilation often has a directional flow due to wider spacing of ventilation supply/extract grilles, which may hinder or improve the control of infection. However, the analysis we have carried out gives some clear insight into why particular directional flows influence risk, and the relative importance of detection strategies, ventilation control, and occupancy.
It is clear that some of our conclusions could be highly dependent on the hospital ward structure, and therefore the flexibility of our methodology comes into play. It can be applied to any hospital ward of interest by appropriately adapting the corresponding ventilation matrix V. Although carrying out a detailed mathematical analysis of a number of potentially different hospital ward structures is out of the scope of this article, we include a short numerical study of an alternative hospital ward in the Supplementary Material. The aim of this is twofold: (i) to show how our methodology can be easily implemented for a different hospital ward to that in Fig. 2 by just adapting the ventilation matrix V, illustrating how this matrix varies with hospital ward structure; and (ii) to show that while some of our conclusions might be hospital ward structure dependent, others seem to be valid for a wide range of hospital ward structures (e.g., detection dominates ventilation as well in this alternative hospital ward).
In this article, we go beyond the SI epidemic model in Noakes and Sleigh (2009), proposing an SIS‐type model, which allows us to incorporate patients' arrival/discharge and outbreak detection and declaration. This is similar to the model recently proposed by López‐García and Kypraios (2018), as a unified framework for modeling the spread of nosocomial infections. We note that this epidemic model structure would be especially relevant for pathogens with no or short (i.e., negligible compared to the average patient's LOS) noninfectious or latent periods, and where the infectious period is long enough so that recovery of patients does not occur before discharge (or detection). Depending on the hospital ward under analysis and the average LOS of patients in this ward, this could be the case for influenza or norovirus. Pathogens with nonnegligible incubation periods (e.g., 7–21 days for measles) might require more complex stochastic epidemic model structures such as the SEIR (susceptible–exposed–infective–recovered). On the other hand, when analyzing hospital wards with longer average patient LOS, so that individuals may become infected and recover during their stay, SIRS‐type epidemic models would be required to represent the recovery of patients (I→R) before discharge (R→S) occurs. We note here that in principle, the methodology outlined in Section 3 can be extended to any of these compartmental‐based epidemic models for the disease spread dynamics, where the link between the deterministic zonal ventilation model for the airflow dynamics and the stochastic epidemic model for disease spread dynamics would still be as in Section 2. In a similar way, more complex epidemic model structures could allow one to study the infection spread dynamics after outbreak detection and declaration occurs. In this article, we have focused instead on the impact of ventilation on disease spread until the end or declaration of the outbreak. Considering these alternative compartmental‐based epidemic model structures could be the aim of future work.
Finally, we note that when carrying out our analysis, the main computational effort lies in solving systems of linear equations, where the number of equations is determined by the number of states of the corresponding CTMC, given by Equation (6) in our model. Limitations of our approach are then of a computational nature, since highly complex epidemic models (here, a multicompartment SIS stochastic model with detection) linked to large hospital wards split in many different ventilation zones (here, M=9 zones with three empty zones and six zones containing three patients each) would lead to an intractable number of equations, and stochastic simulation approaches would prevail.
DATA, SOFTWARE, AND REPRODUCIBILITY
Computer codes (in Python) in order to reproduce our numerical results are available López‐García, King, and Noakes (2019).