Optimizing hospital infection control: the role of mathematical modeling.

Optimizing Hospital Infection Control: The Role of Mathematical Modeling Author(s): Tan N. Doan, MSc; David C. M. Kong, PhD; Carl M. J. Kirkpatrick, PhD; Emma S. McBryde, PhD, FRACP Source: Infection Control and Hospital Epidemiology, Vol. 35, No. 12 (December 2014), pp. 1521-1530 Published by: The University of Chicago Press on behalf of The Society for Healthcare Epidemiology of America

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infection control and hospital epidemiology

december 2014, vol. 35, no. 12

review article

Optimizing Hospital Infection Control: The Role of Mathematical Modeling Tan N. Doan, MSc;1 David C. M. Kong, PhD;1 Carl M. J. Kirkpatrick, PhD;1 Emma S. McBryde, PhD, FRACP2

Multidrug-resistant bacteria are major causes of nosocomial infections and are associated with considerable morbidity, mortality, and healthcare costs. Preventive strategies have therefore become increasingly important. Mathematical modeling has been widely used to understand the transmission dynamics of nosocomial infections and the quantitative effects of infection control measures. This review will explore the principles of mathematical modeling used in nosocomial infections and discuss the effectiveness of infection control measures investigated using mathematical modeling. Infect Control Hosp Epidemiol 2014;35(12):1521-1530

Nosocomial infections are the most common complications in hospitals.1 Among transmissible organisms (bacteria, fungi, viruses, parasites, and prions), bacteria are the most prevalent pathogens in this setting and have a remarkable ability to acquire and transmit antibiotic-resistant genes.2 Multidrugresistant (MDR) bacteria, such as methicillin-resistant Staphylococcus aureus (MRSA) and vancomycin-resistant enterococci (VRE), represent the most challenging therapeutic hurdles in the twenty-first century and raise the prospect of a “postantibiotic era,” with few antibiotics remaining effective.3 Consequently, preventing the emergence and spread of MDR bacteria is of paramount importance. To design effective preventive strategies, we must understand the transmission dynamics of the pathogen and the effectiveness of infection control strategies.4 Mathematical models have been increasingly used to understand the transmission dynamics of infectious diseases. They provide important insights into the underlying dynamics of infections and help identify factors that are responsible for the observed patterns. However, mathematical models are often presented in a technically complex fashion and therefore are inaccessible to nonspecialists. This has led to a gap between clinical practice and the information arising from the mathematical sciences.5 Van Kleef et al6 recently presented a review of mathematical models of nosocomial infections. However, this review focused on overall trends in the field rather than on the findings of individual studies.6 The present article provides a nontechnical summary of the principles of mathematical modeling of nosocomial infections and a review of the effectiveness of infection control measures investigated using such an approach.

mathematical modeling of infectious diseases: the basics The core structure of a model for infectious diseases is a series of compartments that divide the natural history of an infection in the target patient population. This allows a compartment for each of the various stages of an infection that a patient may undergo, including the following: • Susceptible (S): The stage in which an individual is not infected but could be infected. • Exposed (E): The individual is infected with the pathogen but is not infectious (latent period). • Infectious (I): The individual is infected and able to transmit the infection. This period can reflect colonization or disease (clinical manifestations of infection). • Recovered or removed (R): The individual is immune to the pathogen (recovered) or deceased (removed). Different combinations and arrangements of these compartments result in different model structures (Figure 1). The choice of which compartments to include in a model is determined by the biology of the infection being studied and the research question of interest.7 The simplest model structure is the S-I model. This model assumes that individuals, once infected, will remain so for the rest of their life (eg, HIV infection). In S-I-S models, individuals can become susceptible to the same infection immediately after recovering from an infection (eg, curable sexually transmitted infections). The element of immunity is introduced into the S-I-R and S-IR-S models. In S-I-R models, individuals can be infected only once in their life and then develop lifelong immunity after

Affiliations: 1. Centre for Medicine Use and Safety, Faculty of Pharmacy and Pharmaceutical Sciences, Monash University, Melbourne, Victoria, Australia; 2. Victorian Infectious Diseases Service, Royal Melbourne Hospital, Melbourne, Victoria, Australia. Received May 29, 2014; accepted July 28, 2014; electronically published November 14, 2014. 2014 by The Society for Healthcare Epidemiology of America. All rights reserved. 0899-823X/2014/3512-0012$15.00. DOI: 10.1086/678596


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where Yh is the number of contaminated HCWs, Yp is the number of colonized patients, and b is the transmission rate incorporating both direct and indirect transmission.10 This modification is based on the premise that the number of contaminated HCWs in the Ross-Macdonald model is directly proportional to the number of colonized patients.11 This leads to a simple S-I model with admission and discharge (Figure 3).10,12,13 Once the model structure has been determined, the transmission dynamics of an infection can be quantified and visualized by determining the proportion (or number) of individuals in each compartment over time (Figure 4). Of note, the term “dynamic” refers to the fact that the risk of an individual being infected (ie, “force of infection”) changes with time.14 This contrasts with static models, which assume a constant force of infection over time.14 The Basic Reproduction Number: R 0 figure 1. Common model structures for the description of the transmission dynamics of infections. E, exposed; I, infectious; R, recovered; S, susceptible.

recovering from the infection (eg, measles). S-I-R-S models assume that natural immunity is not lifelong but wanes over time or changes when a significant change in the viral makeup occurs (eg, seasonal influenza). The next progression is the S-E-I-R and S-E-I-R-S models, which are modified S-I-R models that include an exposed period. The fundamental difference between the S-E-I-R and S-E-I-R-S models is that individuals develop lifelong immunity for the former, while only short-term immunity develops for the latter. A common model structure in the literature for the description of nosocomial infections is a modified version of the S-I model called the Ross-Macdonald model (Figure 2). It is a host-vector-host model that assumes that the transmission between hosts is indirect via a vector.8 This model framework has been widely used for nosocomial infections because it is generally believed that the transmission between patients (hosts) in a hospital ward occurs predominantly via contaminated hands of healthcare workers (HCWs), who play the role of a vector.9 Direct transmission between patients is considered negligible and therefore is not taken into account in many models.9 In such models, the effect of infection control interventions can be quantified by changing individual parameters or combinations of parameters (Figure 2). The effect of hand hygiene, for instance, can be modeled by changing the decontamination rate of HCWs, whereas the contact rate between patients and HCWs can be used to predict the impact of staff cohorting and staff-patient ratios. Although the Ross-Macdonald model is useful for describing the transmission dynamics of nosocomial infections, it is complex and has a large number of parameters. A more parsimonious model is the 2-compartment model in which the HCW compartments are replaced by the constant Yh ≈ bYp,

The fundamental parameter that governs the transmission dynamics of an infection is the basic reproduction number,

figure 2. The Ross-Macdonald model and impact of infection control measures. Solid lines depict the movement of patients and healthcare workers (HCWs) to and from the 4 compartments. Dashed lines depict transmission between patients and HCWs. L, admission rate; j, proportion of admission colonization; ph, transmission probability per HCW-patient contact; pp, transmission probability per patientHCW contact; Xp, Yp, Xh, Yh, number of uncolonized patients, colonized patients, uncontaminated HCWs, and contaminated HCWs, respectively (Np p Xp ⫹ Yp; Nh p Xh ⫹ Yh); a, per capita contact rate; m, decontamination rate; g, g , discharge rate of uncolonized and colonized patients, respectively. The transmission process is governed by the following differential equations: dXp/dt p L(1 ⫺ j) ⫺ gXp ⫺ appXpYh, dYp/dt p Lj ⫺ g Yp ⫹ appXpYh, dXh/dt p mYh ⫺ aphXhYp, and dYh/dt p ⫺dXh/dt. The infection control measures include screening for and prevention of admission colonization, staff cohorting and staffing level (reduces mixing of patients with contaminated HCWs), antibiotic restriction (reduces the likelihood of colonization per contact with contaminated HCWs), and hand hygiene (allows the decontamination of HCWs). Adapted from Grundmann and Hellriegel.4


mathematical modeling of infection control

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from an infectious individual is called the effective reproduction number, R E.18 The R E of a particular infection control measure is given by the following equation: figure 3. The S-I model with admission and discharge. Assuming a hospital ward with fixed size N (N p S ⫹ I ) and homogenous mixing between patients, colonization due to cross transmission is described by the mass-action term bSI, where b is the transmission rate. Colonization can also be acquired via the discharge of an uncolonized patient and immediate replacement with a colonized patient, which occurs at a rate jSg, where j is the proportion of admission colonization and g is the discharge rate of uncolonized patients. Decreases in the number of colonized patients occur only via their discharge, occurring at a rate (1 ⫺ j)Ig , where g is the discharge rate of colonized patients. The differential equations governing this model are dS/dt p ⫺bSI ⫺ jSg ⫹ (1 ⫺ j)Ig and dI/dt p ⫺dS/dt.

R 0, defined as “the average number of secondary cases produced by a single infectious individual in a totally susceptible population.”15(p35) In the hospital setting, when R 0 is less than or equal to 1, the infection will die out if there is no admission of colonized patients but may continue at a low level if colonized patients are admitted (Figure 5).16 When R 0 is more than 1, the pathogen can spread rapidly, and epidemics can occur (Figure 5). Once introduced, the pathogen is likely to persist, irrespective of whether colonized patients are admitted.16 The larger the R 0, the more rapidly the epidemic grows from one generation to the next (Figure 5). For the RossMacdonald model, R 0 consists of two components: (i) the number of colonized patients resulting from a single contaminated HCW (R p) and (ii) the number of contaminated HCWs generated by a single infectious patient (R h).17 R p and R h are calculated as follows (see Figure 2 for parameter definitions):17 R p p patient to HCW contact rate (a)

R E p R 0 # p,

(4)

where p is the relative infectivity after the intervention takes place.18 Assuming 100% efficacy, 1 ⫺ p represents the rate of compliance to the intervention. Types of Modeling Methods and How Models Are Formulated A model depicts the movement of individuals from one compartment to another. For example, if we consider the S-I model with admission and discharge in a hospital ward of fixed size in which individuals progress from S to I and from I to S (Figure 3), such transitions are determined by 5 distinct factors: the number of colonized/infected individuals, the transmission rate b, the proportion of patients who are already colonized on admission j, and the discharge rates of uncolonized and colonized patients g and g , respectively. Numerical examples of how these parameters can be derived from a hospital infection data set can be found elsewhere.7 Once these input parameters have been specified, a set of differential equations (for deterministic models) or stochastic equations (for stochastic models) are formulated to describe the transmission dynamics of the infection. Deterministic models describe the average outcome of an infection’s cycle/ outbreak.7 These models do not take into account the role of random (stochastic) events in the transmission dynamics. In deterministic models, a given set of parameters and initial conditions will always lead to the same outcome (Figure 6).7 In contrast, stochastic models allow for the fact that random events can affect the transmission outcome of an infection (Figure 6).7 In these models, the change of compartments is

# no. of HCWs (N h) # probability of colonization (pp ) # duration of colonization (1/discharge rate g ),

(1)

R h p HCW to patient contact rate (a) # no. of patients (Np ) # probability of contamination (p h) # duration of contamination (1/decontamination rate m).

(2)

R 0 is the given product R 0 p Rp # Rh p

a 2pp p h Np N h . g m

(3)

For the S-I model with a fixed population size N, R 0 is given by bN/g , where g is the rate at which colonized patients are discharged (Figure 3).7 The formulas of R 0 for various model frameworks are discussed elsewhere.7 The main purpose of many infection control interventions, such as hand hygiene, is to reduce infectiousness. In such circumstances, the average number of new infections resulting

figure 4. Example of the time course of an infection, using the structural model from Figure 3. The figure is plotted assuming b p 0.002, g p 0.1; g p 0.08; and j p 0.02, N p 40.


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transitions may occur within a single fixed time step, whereas in continuous-time models one event occurs at a time and each time step is variable.7 Figure 7 provides a basic summary of the steps for setting up deterministic and stochastic compartmental models, using the S-I model with admission and discharge as an example.

effectiveness of infection c ontrol measures investigated using mathematical modeling A number of models have been developed to investigate the impact of infection control strategies on the transmission dynamics of nosocomial infections (Table 1). These measures targeted HCWs (hand hygiene, staff cohorting, staffing level),11,17,19-23 patients (admission colonization, active screening),9,19,24,25 antibiotic restriction,24,26-28 or the hospital environment.21,29 Hand Hygiene, Staff Cohorting, and Staffing Level figure 5. Basic reproduction number R 0 and characteristics of epidemic, using the structural model from Figure 3. When R 0 is less than or equal to 1, the infection will die out if there is no admission of colonized patients (solid line) but may continue at a low level if colonized patients are admitted (dashed line). When R 0 is more than 1, the pathogen can spread rapidly and epidemics can occur, irrespective of whether colonized patients are admitted (dashed line). The larger the R 0, the more rapidly the pathogen spreads.

dependent on the model parameters and the number of individuals in each compartment; however, the change represents a probability rather than a predetermined flow.7 To allow for stochastic fluctuations, each individual has a unique probability calculated of an event occurring at a given time point.7 As an example, while a deterministic model may—at a particular model configuration—have a flow of 1 patient per day moving from S to I, a corresponding stochastic model may allow all patients in S a probability (1/S) of moving into I. The exact number of individuals who actually move from S to I is determined using random number generators, available in mathematical and statistical software packages. The number of individuals in each compartment is then updated, and the process is repeated for the next time step until the transmission ceases. The model is then run a large number of times to determine the range of outcomes possible for a given set of parameters and initial conditions. As such, stochastic models allow for a more realistic presentation of the transmission dynamics of an infection in small populations such as hospital wards, where chance events are likely to be dominant in the transmission process.7 Stochastic models can be broadly divided into discrete-time compartmental models, continuous-time compartmental models, and agent (individual)-based models.7 Agent-based models keep track of the state of each individual in the population, whereas compartmental models keep track of the total number of individuals at each time step.7 In discrete-time models, many

It is well recognized that contaminated hands of HCWs are the most important source of nosocomial infections.30 As such, hand hygiene of HCWs remains the cornerstone of hospital infection control. This strategy has been repeatedly demonstrated to be the most effective infection control measure in a number of studies.11,17,19,20,22 McBryde et al11 used the Ross-Macdonald model to evaluate the transmission dynamics of MRSA in an intensive care unit. The outcome measure used was attack rate, defined as the number of MRSA transmissions per uncolonized patient-day. The model found that hand hygiene was the most effective infection control measure; specifically, it improved hand hygiene from 0% to

figure 6. Illustration of the difference in predictions between deterministic (dashed line) and stochastic (solid line) methods, using the structural model from Figure 3.



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48% and reduced the attack rate from 0.55 to 0. Using the S-I model with admission and discharge, Pressley et al22 found that a similar increase (from 0% to 50%) in hand hygiene compliance rate reduced the prevalence of MRSA from 50% to 8%. Although it is a simple, inexpensive, and effective measure, hand hygiene compliance often remains obstinately low, with estimated compliance rates below 50%.31,32 The ideal targets for hand hygiene compliance rates have been estimated in a number of studies. McBryde et al11 estimated that a compliance rate of at least 48% was required to eradicate MRSA colonization, whereas the figure estimated by Pressley et al22 was 70% for the same pathogen. Poor adherence to hand hygiene among HCWs has been attributable to high workload, which is associated with understaffing.33 As such, maintaining adequate staffing levels is essential. Increasing the staff-to-patient ratio has been associated with decreased colonization prevalence.11,19,23 A staffto-patient ratio of 1 : 1 was needed to prevent the transmission of MRSA11,19 or VRE.23 However, such a staffing level seems unrealistic in hospital settings, where staff deficit has been a persistent problem.34 To overcome staff deficit, trained volunteers have been employed in Chinese hospitals. Their work involves taking care of patients, transferring them from one unit to another, and reporting irregular results of patients to doctors.21 Using the Ross-Macdonald model with an additional volunteers’ compartment, Wang et al21 found that employing trained volunteers reduced transmission. Admission Colonization and Active Screening The effect of admission colonization on the prevalence of MDR organisms was investigated by Cooper et al.9 To study the effect, the authors assumed that all patients and HCWs were uncolonized at the start of the study period. Transmissions occurred when there was admission of infected or colonized cases. The model found that the prevalence of colonization increased with the proportion of admission colonization, from 0% to 70% when the proportion of patients admitted with colonization increased from 0% to 10%. This finding was confirmed by D’Agata et al,24 who found that preventing the admission of colonized patients would eradicate VRE infections. Therefore, reducing admission colonization appears to be an attractive infection control measure. Active screening is essential to identify admission colonization. It allows early identification and isolation of colonized patients, who are important reservoirs for transmission.35 Perencevich et al25 developed a model to investigate the potential benefits of active screening. The screening test was assumed to have 100% sensitivity and specificity and a compliance rate of 90%. Two active screening strategies were investigated: strategy 1, screening of all patients at admission and subsequent isolation of patients with positive culture results; and strategy 2, isolation of all patients at admission until culture results were available, with subsequent discontinuation of iso-

figure 7. Summary of the steps for setting up deterministic and stochastic models.

lation of patients with negative culture results. Over a 1-year period, strategy 1 reduced the VRE incidence by 39%, compared with no active screening, whereas strategy 2 resulted in a 65% reduction. The benefits of active screening were further verified by Raboud et al,19 who found that such practice reduced MRSA incidence by 35% under an assumption of 80% compliance rate with the screening. Despite these positive results, active screening programs have not been widely implemented in hospitals. This may be due in part to concerns about their high costs and lack of evidence of their effects on patient outcomes.35 Furthermore, the implementation of such programs may be resisted by HCWs, who are already working under time and resource constraints. Antibiotic Restriction Antibiotic exposure plays an important role in the transmission dynamics of MDR bacteria.36 This is because antibiotic therapy can disrupt the normal human gastrointestinal microbiota ecosystem and eradicate susceptible strains, facilitating the overgrowth of resistant strains and the spread of their resistant genes.36 As such, reducing unnecessary anti-


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Setting

Stochastic

Raboud et al19 (2005)

Hospital

Deterministic Hospital

ICU

Hemodialysis unit

D’Agata et al24 (2005)

Perencevich et al25 Stochastic (2004)

Stochastic

Cooper et al9 (1999)


Stochastic

Austin et al17 (1999)

Medical-surgical ward

Model type

Deterministic, ICU stochastic

Study (year)

Interventions tested Admission colonization, hand hygiene, staff cohorting

MRSA

VRE

VRE

VRE

Active screening, hand hygiene, staff-to-patient ratio

Admission colonization, antibiotic exposure, hand hygiene, staff-to-patient ratio

Active screening

Admission colonization, staffto-patient ratio

Nonspecific Admission colonization, LOS, hand hygiene, transmissibility

VRE

Organism

Model assumptions Transmission among patients is indirect, via contaminated hands of HCWs Precaution procedures are 100% effective Once colonized, patients remain so for their entire LOS Homogenous mixing of patients and HCWs 100% bed occupancy rate Transmission among patients is indirect, via contaminated hands of HCWs 100% bed occupancy rate Fixed patient and HCW population size HCW-patient transmissibility p patient-HCW transmissibility Homogenous mixing of patients and HCWs Once detected, colonized patients are removed from the ward Hand washing is 100% effective Transmission among patients is indirect, via contaminated hands of HCWs LOS of uncolonized patients p life expectancy of patients receiving chronic hemodialysis (6.5 years) LOS of colonized patients p duration of colonization (3 months) Risk of transmission to an uncolonized patient is proportional to the number of colonized patients Population is at steady state (no. of admission p no. of discharge) Patients can be colonized with only 1 VRE strain Transmission among patients is indirect, via contaminated hands of HCWs Patients receiving antibiotics can move only from an uncolonized state to a colonized state Patients not receiving antibiotics can move only from a colonized state to an uncolonized state

Reducing antibiotic exposure was more effective in uncolonized patients than in colonized patients Eliminating admission colonization would eradicate infection Increasing staff-to-patient ratio and hand hygiene reduced colonization Hand hygiene and screening were Transmission among patients is indirect, via contaminated the most effective measures hands of HCWs Screening at admission reduced Hand hygiene compliance is higher when patients are in transmission rate by 41% isolation Increasing staff-to-patient ratio reduced colonization

Active screening reduced VRE incidence by 39% compared with no screening

Constant influx of newly colonized patients would sustain endemicity Increasing staff-to-patient ratio led to reduction of colonization

Reducing admission colonization was the most effective measure LOS had little impact on transmission Hand washing frequency 140% would eradicate infection

R0 reduced from 3.8 (without interventions) to 0.7 (with interventions)

Main findings

table 1. Overview of the Reviewed Studies of the Transmission Dynamics of Nosocomial Bacterial Infections

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Deterministic, Hospital stochastic


Stochastic


McBryde et al11 (2007)

Pressley et al22 (2010)

Admission colonization, antibiotic exposure, environmental contamination, hand hygiene, staff-to-patient ratio



Deterministic, Emergency ward MRSA stochastic and respiratory ICU

Chamchod and Ruan27 (2012)

Grima et al26 (2012)

Wang et al21 (2012)

Hand hygiene, environmental contamination, use of volunteers

Impact of nonantibiotic treatment for Clostridium difficile on prevalence of VRE

Admission colonization, antibiotic exposure

Hand hygiene, use of volunteers

Competitive exclusion of CAMRSA and HA-MRSA, hand hygiene

Hand hygiene, staff cohorting, staff-to-patient ratio

Environmental contamination reTransmission among patients is via contaminated hands of duced the effectiveness of infection HCWs and environment control measures Patients receiving antibiotics can move only from an uncolonized state to a colonized state Patients not receiving antibiotics can move only from a colonized state to an uncolonized state Probability of HCW hand contamination is greater from contact with colonized patients receiving antibiotics than from colonized patients not receiving antibiotics Reducing duration of antibiotic treat- Without treatment, nonresistant strain has a selective adment reduced colonization vantage over resistant train; with treatment, nonresistant strain is reduced to a very low level Each HCW begins his or her first visit of the shift uncontaminated Hand hygiene was the most effective Transmission among patients is indirect, via contaminated measure hands of HCWs Hand hygiene compliance rate 148% Decolonization rate is zero in the absence of interventions was needed to eradicate infection Once colonized, patients remain so for their entire LOS Staff-to-patient ratio of 1 : 1 was Homogenous mixing of patients and HCWs needed to eradicate infection HCW-to-patient ratio p 1 : 1 No competitive exclusion Transmission among patients is indirect, via contaminated Hand hygiene compliance rate ≥70% hands of HCWs was required to eradicate infection 100% bed occupancy rate The main distinguishing characteristic among strains is the LOS of colonized patients Hand hygiene was the most effective Transmission among patients is indirect, via contaminated measure hands of HCWs Using volunteers reduced Transmission rate for volunteers is lower than that for transmission HCWs With admission of colonized paTransmission among patients is indirect, via contaminated tients, MRSA always persisted hands of HCWs Colonization increased with increas- Homogenous mixing of patients and HCWs ing duration of antibiotic Hand hygiene is 100% effective treatment Nonantibiotic treatment in 50% of Transmission among patients is indirect, via contaminated C. difficile patients reduced VRE hands of HCWs colonization by 18% compared Patients receiving antibiotics can move only from an uncowith antibiotic use lonized state to a colonized state Patients not receiving antibiotics can move only from a colonized state to an uncolonized state Patients receiving antibiotics are more likely to contaminate HCWs Using volunteers reduced Transmission among patients is indirect, via contaminated transmission hands of HCWs and free-living bacteria in the Improving hand hygiene of volunenvironment teers was more effective than im- Once colonized, patients remain so for their entire LOS proving that of HCWs Hand hygiene is 100% effective

note. CA-MRSA, community-acquired MRSA; HA-MRSA, hospital-acquired MRSA; HCW, healthcare worker; ICU, intensive care unit; LOS, length of stay; MRSA, methicillin-resistant Staphylococcus aureus; R0, basic reproduction number; VRE, vancomycin-resistant enterococci.

VRE

MRSA


MRSA

MRSA

MRSA

Nonspecific Duration of antibiotic treatment

VRE

Wang et al20 (2011)

Hospital


McBryde and McElwain29 (2006)

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biotic use and treatment duration may be an effective strategy to reduce the spread of MDR bacteria. Such strategies have been investigated in a number of models, with consistent results.24,26-28 Using a Ross-Macdonald model with two patient groups (those receiving antibiotics and those not receiving antibiotics), D’Agata et al24 found that increasing antibiotic exposure increased the prevalence of VRE colonization. Discontinuing antibiotics had a greater impact on VRE prevalence in uncolonized than in colonized patients. Increasing the proportion of uncolonized patients who received antibiotics from 15% to 30% increased the prevalence of VRE from 12% to 22%. The same increase in antibiotics applied to colonized patients only resulted in a modest increase in VRE prevalence, from 12% to 13.5%. These findings could be explained by the higher proportion of uncolonized patients compared with those colonized. In another study, Grima et al26 found that discontinuing antibiotic treatment in half of the patients with Clostridium difficile reduced the prevalence of VRE from 15% to 12.3%. Unnecessarily prolonged antimicrobial treatment has been associated with increased risk of toxicity and resistance.37 The concept of minimizing antibiotic treatment duration has been investigated in an agent-based model by D’Agata et al.28 By varying the duration of antibiotic treatment while keeping all other variables constant, the authors predicted that shorter duration was associated with lower MRSA prevalence. Such an association was also reported by Chamchod and Ruan.27 Nevertheless, further studies are needed to determine the comparative clinical outcomes of a short versus a prolonged course of antibiotic therapies. Contaminated Environment Contaminated environments play an important role in the transmission of nosocomial infections. Pathogens such as MRSA and VRE can survive on dry surfaces for many weeks or months.38 When living freely in the environment, they contribute to the spread of infections by transmitting to susceptible patients or by causing hand contamination among HCWs.38 The role of environmental contamination in the transmission of nosocomial pathogens has been investigated by McBryde and McElwain29 and by Wang et al.21 These models included an environmental component to the Ross-Macdonald model, thus allowing for both direct HCW-patient transmission and indirect transmission via a contaminated environment. Both models suggested that even when sources of direct transmission had been removed, there were still colonized cases initiated by free-living infective bacteria in the environment. This finding highlights the importance of environmental cleaning in preventing the spread of nosocomial infections.

discussion and future d irections MDR bacteria continue to emerge and spread in hospitals. The spread of these organisms is a dynamic and complex

process involving numerous factors, such as patients, HCWs, and the environment. Epidemiological studies have conventionally been used to investigate the spread of nosocomial infections and to identify risk factors that contribute to the spread. These studies, however, provide only individual patient-level data and thus are unable to fully capture the spread of infections, which is a complex and dynamic process involving numerous interrelated factors.39 Mathematical modeling provides a theoretical framework to conceptualize the dynamic interactions between interdependent variables. It is a powerful tool to advance our understanding of the transmission dynamics of infections and the role that different factors play in the transmission process. It also helps us design epidemiological studies and enables us to quantify the potential impact of infection control measures without conducting those interventions. This is particularly useful in the healthcare setting, where resources are scarce. Furthermore, mathematical models make it possible to consider measures that would not be feasible in epidemiological studies due to operational, financial, and/or ethical constraints.4 They also enable system parameters to be freely modified and the impact of such modifications on outcome variables quantified.4 As such, mathematical modeling allows us to formulate new hypotheses and test “what-if ” scenarios for the design of optimal infection control measures.40 The predictive value of mathematical models relies heavily on the quality and detail of input data. Consequently, reliable epidemiological data are essential for models to make realistic and meaningful predictions. To ensure the validity of a model, results must be quantitatively tested using empirical data.4 This implies that mathematical modeling and epidemiological studies should be complementary—rather than alternative— to each other.4 Recent advances in electronic data management, such as microbiology databases, antibiotic prescribing databases, and electronic surveillance systems, have made the collection of extensive epidemiological data possible. The availability of such data will allow for more complex models to be developed. The choice of model type and complexity are often the subject of debate. The degree of model complexity should be determined on the basis of the principle of parsimony.14 An ideal model should be the simplest one that is able to answer the research question of interest. Models must also be designed by taking into consideration the available data, researchers’ skills, and the level of computational difficulties. Unnecessarily complex models will require more assumptions and input data, introducing greater uncertainties into the model. Complex models also increase the number of parameters to be estimated and make interpretation of results difficult. Mathematical models are inherently based on assumptions.4 Most studies described in this review assumed that the only source of patient-to-patient transmission was via the contaminated hands of HCWs.9,11,17,19,20,22,24,26,27 This assumption is unlikely to be true, given the well-known importance



of environmental contamination and endogenous flora systems in colonization.36,38 Understanding the importance of different sources of transmission is essential to design effective infection control measures. When cross-transmission is dominant, strategies such as hand hygiene and contact precautions should be enforced. When endogenous colonization is important, antibiotic policy should be a priority. Unfortunately, little is known about the relative importance of various transmission sources of nosocomial bacteria. Many bacteria can remain viable freely in the environment for prolonged periods and serve as important reservoirs for nosocomial infections.38 Despite this, the role of environmental contamination in the transmission of bacteria has received little attention. Other factors, such as host defense mechanisms and patient comorbidities, that facilitate colonization and transmission have also been excluded from the studies reviewed. Incorporating these elements is important to provide a more realistic and complete picture of the dynamic process of an infection. Mathematical modeling provides valuable insights into the spread of infections. With reliable input data, models will provide useful outputs for the design of optimal infection control strategies. Future models will need to take into account the role of free-living bacteria in the environment, the relative importance of transmission routes, and other transmission determinants. Studies that investigate the transmission dynamics of gram-negative bacteria are warranted. Mathematical modeling should also be used to guide the collection of epidemiological data. The availability of such model-driven data will in turn strengthen the utility and validity of models developed and will enable the models to address more complex research questions.

acknowledgments Financial support. T.N.D. was supported by a Monash Graduate Scholarship and a Monash International Postgraduate Research Scholarship. E.S.M. has received financial support from a National Health and Medical Research Council Career Development Fellowship. Potential conflicts of interest. D.C.M.K. reports having sat on advisory boards for Pfizer and receiving financial/travel support (unrelated to the current work) from Pfizer, Roche, Merck, Novartis, and Gilead Sciences. C.M.J.K. reports having undertaken collaborative research projects unrelated to the current work with Roche, Pfizer, CSL, and d3 Medicine. All other authors report no conflicts of interest relevant to this article. All authors submitted the ICMJE Form for Disclosure of Potential Conflicts of Interest, and the conflicts that the editors consider relevant to this article are disclosed here. Address correspondence to Emma S. McBryde, PhD, FRACP, Victorian Infectious Diseases Service, Royal Melbourne Hospital, Peter Doherty Institute for Infection and Immunity, Level 4, 792 Elizabeth Street, Melbourne, Victoria 3000, Australia ([email protected]); or, David C. M. Kong, PhD, Centre for Medicine Use and Safety, Faculty of Pharmacy and Pharmaceutical Sciences, Monash University, 381 Royal Parade, Melbourne, Victoria 3052, Australia ([email protected]).

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