Journal of Theoretical Biology 334 (2013) 187–199

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Journal of Theoretical Biology

0022-51http://d

n Corrmatics aArbor, M

E-m

journal homepage: www.elsevier.com/locate/yjtbi

Modeling bacterial colonization and infection routes in health caresettings: Analytic and numerical approaches

Carl P. Simon a,c,n, Bethany Percha b, Rick Riolo a, Betsy Foxman b

a Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, United Statesb School of Public Health, University of Michigan, Ann Arbor, MI 48109, United Statesc School of Public Policy and Departments of Mathematics and Economics, University of Michigan, Weill Hall 735 S. State St. #4203, Ann Arbor, MI 48109,United States

A U T H O R - H I G H L I G H T S

� Construct and analyze a model for the spread of MRSA in health care settings.

� Includes colonization, infection, isolation and contamination among health care workers.� Includes patient–patient and patient–HCW–patient transmission routes.� Analytical computation of the basic reproduction number.� Transmission pathways compared in hospitals and long-term care settings.

a r t i c l e i n f o

Article history:Received 21 July 2012Received in revised form11 May 2013Accepted 21 May 2013Available online 6 June 2013

Keywords:Basic reproduction numberMethicillin-resistant Staphylococcus aureus(MRSA)Transmission modelLong-term care facilityHospital acquired infections

93/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.jtbi.2013.05.016

esponding author at: School of Public Policynd Economics, University of Michigan, Weill HI 48109-3091, United States. Tel.: +1 734 763ail address: cpsimon@umich.edu (C. Simon).

a b s t r a c t

Health-care associated infections are a major problem in our society, accounting for tens of thousands ofpatient deaths and millions of dollars in wasted health care expenditures each year. Many of these infectionsare caused by bacteria that are transmitted from patient to patient either through direct contact or via thehands or clothing of health care workers. Because of the complexity of bacterial transmission routes in healthcare settings, computational approaches are essential, though often analytically intractable. Here we describethe construction and detailed analysis of a model for bacterial transmission in health care settings. Our modelincludes both colonization and disease stages for patients and health care workers, as well as an isolation wardand both patient–patient and patient–HCW–patient transmission pathways. We explicitly derive the basicreproductive ratio for this complex model, a nine-term expression that contains all nine ways with which anew colonization can occur. Using key parameters found in the medical literature, we use our model to gaininsight into the relative importance of various bacterial transmission pathways within health care facilities,and to identify which forms of interventions are likely to prove most effective in hospitals and long-term caresettings. We show that analytical and numerical approaches can complement each other as we seek tountangle the complex web of interactions that occur within a health care facility.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Health care-associated infections (HCAI) are a leading cause ofmorbidity and mortality in the United States. In 2002, approximately1.7 million HCAIs were reported in U.S. hospitals, leading to nearly99 000 deaths (Klevens et al., 2007). The additional cost of careattributable to health-care associated sepsis and pneumonia, twocommon clinical outcomes, are estimated at $5800–$32 900 perpatient for sepsis and $12 700–$46 400 for pneumonia, and all HCAIscombined cause $17–$20 billion in added health care costs each year

ll rights reserved.

and Departments of Mathe-all 735 S. State St. #4203 Ann3074; fax: +1 734 615 4623.

(Klevens et al., 2007; Eber et al., 2010; Zhan and Miller, 2003).Depending on the severity of their infections, patients who contracta HCAI while hospitalized will also increase their length of stay byabout 10–20 days and their risk of mortality by 30–50% (Eber et al.,2010; Pirson et al., 2005; Kothari et al., 2009). Furthermore, recenthospitalization is the single largest risk factor for infection withantibiotic resistant organisms such as methicillin-resistant Staphylo-coccus aureus (MRSA) and vancomycin-resistant Enterococci (VRE) inthe community, making HCAIs a significant threat to the generalpublic (Warshawsky et al., 2000; Chen et al., 2008). For these reasons,identifying the major transmission routes for infectious organisms inhealth-care settings and taking appropriate steps to minimize patientexposure are of crucial importance.

Because HCAI transmission is a complex process and observationof all individual transmission events is impossible, computational

Table 1Description of the infection states of patients and health care workers. A colonizedpatient or HCW is colonized with MRSA but asymptomatic. A contaminated HCW isuncolonized with MRSA but harbors the bacteria on his or her hands or clothing.An isolated patient is quarantined in an isolation ward and does not interact withother patients.

Variable Interpretation

PF Uncolonized patientsP Colonized patients (but disease free)

C. Simon et al. / Journal of Theoretical Biology 334 (2013) 187–199188

approaches have proven extremely useful in helping investigatorsevaluate the relative importance of different transmission routes, planfuture studies, and determine the efficacy of possible interventions.Several recent models have specifically examined the spread of MRSAand other antibiotic-resistant organisms in hospital and communitysettings (Skov and Jensen, 2009; Webb et al., 2009; D'Agata et al.,2009; Cooper et al., 2004; Austin et al., 1999; Cooper and Lipsitch,2004; Ancel Meyers et al., 2003; Bootsma et al., 2006; McBryde et al.,2007; Smith et al., 2004; D'Agata et al., 2007). However, realisticmodels often incorporate a bewildering array of parameters and provedifficult or impossible to approach analytically. In the context ofhospital-acquired infections, this often means that certain importanttransmission pathways are ignored in the interest of model simplicity.For example, one hugely important reservoir of infection in hospitals ishealth care workers (HCWs); several studies have established that thehands of HCWs often act as vectors for the transmission of facility-acquired infections and that HCWs themselves are often colonized orcontaminated with infectious organisms like MRSA (Aiello and Larson,2002; Curtis and Cairncross, 2003; Pittet, 2001; March et al., 2010).Unfortunately, incorporating more complex transmission routes likeHCW vectors into transmission models can easily cause the equationsto become analytically intractable.

Here we model bacterial transmission in two very differenthealth-care environments: hospitals and long-term care facilities(LTCFs). We chose these two institution types because they exist attwo ends of a spectrum; at one end, patients mostly contact HCWs(hospitals) and at the other end, they mostly contact other patients(LTCFs). We wanted to confirm that our model would indeedcapture these differences and reveal the most important transmis-sion routes in each type of facility. Our most general modelincludes HCW vectors, multiple stages of infection, and an isola-tion ward for diseased patients. It contains four patient variables(susceptible, colonized, infected, and isolated), four HCW variables(clean, contaminated, colonized, both), and 25 parameters tocapture the nuances of different types of bacterial infection anddifferent health care settings and institutions. The model could,in principle, be used to compare transmission pathways across avariety of different health care facilities. Our goal is to carry out ananalysis that applies as broadly as possible for all parameter valuesso that comparisons can be made across institutions.

The main step in this general analytic approach is our explicitcalculation of the algebraic expression of the Basic ReproductiveRatio (R0) (Heffernan et al., 2005) for this model, an expressionthat incorporates all of the model parameters and shows how theywork together to contribute to the infectious process. We useLyapunov functions (Simon and Jacquez, 1992; Simon et al., 1991)to perform this calculation, a general approach that should alsowork for other important classes of model. In combination withnumerical simulations and carefully-chosen parameter valuesfrom the scientific literature, our results demonstrate that optimalintervention strategies do indeed differ between hospitals andlong-term care settings. They also provide explanations for thesedifferences in terms of individual parameter values and theirimpact on R0. By attacking the problem in this way, we can gaininsight into the structure of our model and better understand therole of each parameter within its overall architecture. This enablesus to start untangling the complex web of interactions that occurwithin a health care facility, focusing our attention on the subset ofinterventions that is likely to do the most good.

C

PD Diseased but non-isolated patientsPD′ Diseased and isolated patientsHF Uncolonized, uncontaminated HCWsHC Colonized but uncontaminated HCWsHF

T Uncolonized but contaminated HCWsHC

T Colonized and contaminated HCWsHD Diseased HCWs (leave facility immediately)

2. The basic reproductive ratio

A key index in any analysis of disease spread is the basicreproductive ratio, R0: the number of new infections attributableto an infected individual over the course of his or her infection in a

population of susceptible individuals. The tipping point betweendisease take-off and die-out occurs at R0¼1. Alternatively, R0 is theratio:

rate at which susceptible individuals become infectedrate at which infected individuals lose their infection

:

For simple epidemiological models involving homogeneous popu-lations composed of susceptible, infected, and recovered indivi-duals,

R0 ¼ χ � β � L;

where χ is the average number of contacts per unit time, β is theprobability of infection per contact, and L is the average lengthof the infectious period. In such simple models, when R041, R0determines the rate of increase in the number of infected indivi-duals and the endemic level of infection, Yn, since Yn ¼ 1−ð1=R0Þ.

Most interesting epidemiological models, however, are ofsufficient complexity that the exact expression for R0 is neitherobvious nor easily calculated. This is true for any realistic model ofbacterial transmission in health-care settings, as well as a wideclass of other interesting models.

3. A simple subsystem

To build intuition for the techniques and conclusions of ouranalyses, we begin with a simple model that displays featuressimilar to those of the more elaborate one described later in thepaper. Patients can either be MRSA-free (PF) or colonized (PC),while HCWs can either be MRSA-free (HF) or contaminated (HF

T),which means they harbor bacteria on their hands or clothing andhave the potential to infect uncolonized patients. Table 1 containsa full description of these different states.

Each model day, the following events occur:

1.

Patients enter the facility at a constant rate U. We assume thatall incoming patients are uncolonized.

2.

Free and colonized patients leave the facility at a rate μ. Becausewe wish to keep the total number of patients in the facilityconstant (i.e. we assume it is always operating at maximumcapacity), we require that U ¼ μPF þ μPC .

3.

Patients and HCWs interact at a rate χ, and each interactionevent can potentially transmit infection or contamination.For instance, during a single contact a colonized patient con-taminates the gloves, skin or clothing of a HCW with prob-ability βCT . In the other direction, a contaminated HCW infects aMRSA-free patient with probability βTP per contact.

4.

Colonized patients lose their colonization at a natural rate γPper day. Colonized patients are treated with antibiotics and the

Table 2Description of the model parameters.

Parameter Interpretation

uC and uD Fraction of incoming patients who are colonized (uC) or diseased (uD) with MRSAμ and μ′ Rates at which free/colonized patients (μ) and diseased patients (μ′) leave the facility per dayχ Contacts per HCW per patient per dayχ′ Contacts per patient per other patient per dayγP and γH Rates of natural loss of MRSA colonization by colonized patients and HCWsδP , δ′P , and δH Rates of successful decolonization of colonized patients, diseased patients, and HCWs, respectively, using drug treatmentαP and αH Rates of progression from colonization to disease for patients and HCWs, respectivelyε Isolation rate of diseased patientsη Rate of loss of contamination by HCWs due to infection control precautions like handwashingρ Patients' recovery rate from disease

Patients infecting each otherβCF Probability that PC infects PF in a single contactβDF Probability that PD infects PF in a single contact

HCWs infecting patientsβTP Probability that HF

T infects PF in a single contactβCP Probability that HC infects PF in a single contactβCTP Probability that HC

T infects PF in a single contact

Patients infecting HCWsβCH Probability that PC infects HF or HF

T in a single contactβDH Probability that PD infects HF or HF

T in a single contactβD′H Probability that PD′ infects HF or HF

T in a single contact

Patients contaminating HCWsβCT Probability that PC contaminates HF/HC in a single contactβDT Probability that PD contaminates HF/HC in a single contactβD′T Probability that PD′ contaminates HF/HC in a single contact

C. Simon et al. / Journal of Theoretical Biology 334 (2013) 187–199 189

rate of successful decolonization due to drug treatment is δPper day.

5.

Fig. 1. Schematic drawing of the initial simplified model, including only contami-nated HCWs and colonized patients. In this diagram, the squares represent patientsand the circles represent HCWs. A light green color denotes a MRSA-free individual,while a darker green denotes a colonized individual. The orange circle shownaround one group of HCWs denotes contamination.

1 D'Agata et al. (2007) construct an individual based model (IBM), formalizedas a system of stochastically determined events, that allows for high levels ofheterogeneity in patient and HCW behavior and includes resistant and non-resistant bacterial strains. These authors also construct a multi-time-scaleddifferential equation model that corresponds to the average behavior of the IBMover a large number of IBM simulations.

Finally, contaminated HCWs lose their contamination throughinfection control measures like glove changing and handwash-ing. Decontamination occurs at a rate η per worker per day.

Table 2 summarizes the parameters discussed above. A graphicalrepresentation of the model is shown in Fig. 1.

Model equations: In this simpler model, patients are eitherMRSA-free or colonized and HCWs can only be MRSA-free orcontaminated—not colonized. Let PF, PC, HF, and HF

T denote thefour subpopulations in this model (see Table 1). In addition, weassume that only MRSA-free patients are allowed to enter thefacility. These assumptions yield a system of just four equations:

_PF ¼ ðμPF þ μPCÞ−μPF−χ � βTP � HTF � PF þ ðγ þ δÞPC

_PC ¼−μPC þ χ � βTP � HTF � PF−ðγ þ δÞPC

_HF ¼−χ � βCT � HF � PC þ ηHTF

_HTF ¼ þ χ � βCT � HF � PC−ηHT

F : ð1ÞSince the second equation is the negative of the first and the

fourth equation is the negative of the third in (1), we can reduce(1) to a system of only two equations. Noting that the totalnumbers of patients and HCWs are constant, we divide the firsttwo equations in (1) by the constant PF þ PC and the last twoequations by the constant HF þ HT

F . We write χP for χ � ðHF þ HTF Þ,

the rate of HCW contacts by patients, and write χH for χ � ðPF þ PCÞ,the rate of patient contacts by HCWs. System (1) then becomes

_p ¼ ð1−pÞ � χP � βTP � h−ðμþ γ þ δÞp_h ¼ ð1−hÞ � χH � βCT � p−η � h ð2Þwhere we write p for PC=ðPF þ PCÞ, the fraction of patients infected,and h for HT

F=ðHF þ HTF Þ, the fraction of HCWs contaminated. This is

basically the system that Austin et al. (1999) and McBryde et al.(2007) analyzed in their studies of Vancomycin-resistant Entero-cocci and MRSA in intensive-care hospital settings. These are

among very few papers that use ordinary differential equationsto model and analyze the spread of bacterial infections and includethe role of HCWs.1

Input–output ratios: To simplify system (2), let

RTP≡χP � βTPμþ γ þ δ

and RCT≡χH � βCT

η: ð3Þ

We can consider RTP the patients' R0 for infection from contami-nated HCWs and RCT the HCWs' R0 for contamination from infected

Fig. 2. These two figures are illustrations of system (2) for different values of theparameters. In both cases, the solid line represents the curve _p ¼ 0, orh¼ p=RTP ð1−pÞ. The dashed line represents the curve _h ¼ 0, or p¼ h=RCT ð1−hÞ.The slope of the _p ¼ 0 curve at the origin is given by 1=RTP , while the slope of the_h ¼ 0 curve at the origin is given by RCT. (a) This figure illustrates a situation where1=RTPoRCT , and the no-disease equilibrium is unstable. The other equilibriumpoint (circled on the graph) is a stable equilibrium with a non-zero prevalence ofdisease. (b) This figure illustrates a situation where 1=RTP≥RCT , and the no-diseaseequilibrium is stable.

2 Note that when p¼0, _p40, and when h¼0, _h40. This ensures that solutionsof (4) that start in the positive quadrant stay in the positive quadrant. This can alsobe seen in the vector fields in Fig. 2a and b. The same observations hold for themore complex model we will study in the next section.

3 Of course, V is continuous everywhere, and is infinitely differentiable excepton the line h¼ RCT � p. The above argument shows that V is decreasing along orbits,except possibly on the line h¼ RCT � p. But by continuity, V is decreasing on orbitseven on this line. This argument holds for all the Lyapunov functions used inthis paper.

4 For example, suppose that RCTRTP ¼ 1þ ϵ, where ϵ40. Suppose also that (p,h)are in the triangle T defined by

poϵ=ð1þ ϵÞ and h≤RCT � p:Then, the analysis above becomes

_V ¼ RCT � p¼ RCT � ðμþ γ þ δÞ½RTP � RCT � p � ð1−pÞ−p�¼ RCT � ðμþ γ þ δÞ½ð1þ ϵÞ � p � ð1−pÞ−p�¼ RCT � ðμþ γ þ δÞ � p � ½ϵ−pð1þ ϵÞ�

which is 40, since poϵ=ð1þ ϵÞ. Thus, _V 40 in triangle T; orbits move away fromthe origin to higher level curves of V; the origin is an unstable equilibrium. This

C. Simon et al. / Journal of Theoretical Biology 334 (2013) 187–199190

patients. We can consider RTP to be the ratio of the rate at whichpatients are colonized via contacts with contaminated HCWs tothe rate at which they lose their colonization ðμþ γ þ δÞ. If RTPo1,patients lose their colonization faster than they regain it via HCWcontacts, and the opposite is also true. Similarly, RCT is the ratio ofthe rate at which HCWs become contaminated via contacts withcolonized patients to the rate at which they lose their contamina-tion ðηÞ.

The basic reproductive ratio: System (2) can be rewritten interms of RTP and RCT, respectively, as

_p ¼ ðμþ γ þ δÞ½RTP � hð1−pÞ−p�_h ¼ η½RCT � pð1−hÞ−h� ð4Þ

Since system (4) is planar, we can easily draw its phase portrait(Simon and Blume, 1994) in the plane (Fig. 2). The _p ¼ 0 and _h ¼ 0isoclines are, respectively, the curves

h¼ pRTPð1−pÞ

and p¼ hRCT ð1−hÞ

: ð5Þ

Both are hyperbolas that pass through ð0;0Þ and asymptote to p¼1and h¼1, respectively. The slopes of these hyperbolas at the originare 1=RTP and RCT. As Fig. 2 suggests, if ð1=RTPÞ≥RCT , the _p ¼ 0 curvelies above _h ¼ 0 curve throughout the positive quadrant andFig. 2b holds: the only intersection (system equilibrium) occursat ð0;0Þ. If ð1=RTPÞoRCT , then Fig. 2a holds: there is a crossing(endemic equilibrium) in the interior of ½0;1� � ½0;1�. Filling in theslope field in Fig. 2 shows that the endemic equilibrium is globally

stable in Fig. 2a and that the no-disease equilibrium is globallystable in Fig. 2b.2

Fig. 2a holds if and only if ð1=RTPÞoRCT , i.e.

RTPRCT 41: ð6ÞRTPRCT is the basic reproductive ratio for system (4). Since HCWcontamination and patient infection occur via contacts betweenHCWs and patients, it is natural that the basic reproductive ratio ofthis system is the product of the basic reproductive ratios of thesubsystems. See Austin et al. (1999) for a heuristic proof anddiscussion of RTPRCT as the basic reproductive ratio for system (4).

This means of devising the basic reproductive ratio, however, isnot generalizable to more complicated systems whose phase space isnot as easily visualized and understood. A more general approachrelies instead on the use of Lyapunov functions as stability criteria.Simon and Jacquez (1992) and Simon et al. (1991) showed thatLyapunov functions can be a computationally efficient way ofcomputing basic reproductive ratios. For the simple model examinedhere, a careful inspection of Fig. 2b suggests that, when RTPRCT ≤1,the orbits of system (4) move downward to the right toward ð0;0Þ onthe left of the “diagonal″and move downward to the left toward ð0;0Þon the right of the “diagonal,” as shown in Fig. 3. The dotted lines inFig. 3 can be considered the level curves of the Lyapunov functionVðp;hÞ ¼maxfRCT � p;hg. The derivative _V of V along the orbits ofsystem (4) is

_V≡ddt

½VðpðtÞ;hðtÞÞ� ¼ ∂V∂p

� dpdt

þ ∂V∂h

� dhdt

for any solution or “orbit” ðpðtÞ;hðtÞÞ of system (4). _V measures how Vchanges along solutions of (4).3

If h≤RCT � p, then Vðp;hÞ ¼ RCT � p and

_V ¼ RCT � _p ¼ RCT � ðμþ γ þ δÞ½RTP � h � ð1−pÞ−p�≤RCT � ðμþ γ þ δÞ½ðRTP � RCT � p � ð1−pÞÞ−p�≤RCT � ðμþ γ þ δÞð−p2Þ≤0;

since RTPRCT ≤1. If RCT � p≤h, then Vðp;hÞ ¼ h and

_V ¼ _h ¼ η½RCT � p � ð1−hÞ−h�

≤η RCT �hRCT

� ð1−hÞ−h� �

¼ ηð−h2Þ≤0:

In any case, _V o0 when ðp;hÞ4 ð0;0Þ. So, orbits of system (4) moveto lower and lower level sets of V, and ð0;0Þ is globally asympto-tically stable when RCTRTP ≤1. On the other hand, if RCTRTP41, thenone can use the above analyses to show that for small enough pand h≈RCT � p, _V ðp;hÞ40; i.e., orbits that start close enough to ð0;0Þflow away from ð0;0Þ; and ð0;0Þ is unstable.4

Fig. 3. The flow pattern from Fig. 2b moves across lower and lower level curves ofthe Lyapunov function V, where Vðp; hÞ ¼max RCT � p;h

� �. Because the arrows

always point toward the origin when crossing decreasing sets of level curves ofV, we can be sure that the epidemic will always die out.

C. Simon et al. / Journal of Theoretical Biology 334 (2013) 187–199 191

For the case of RCTRTP41, it is straightforward to set the righthand sides of system (4) equal to zero to compute its endemicequilibrium:

pn ¼ RCTRTP−1RCT ðRTP þ 1Þ and hn ¼ RCTRTP−1

RTPðRCT þ 1Þ : ð7Þ

As expected, both pn and hn are increasing functions of RCT and RTP,respectively. We can write the expressions in (7) as

pn ¼ RTP

RTP þ 11−

1R0

� �and hn ¼ RCT

RCT þ 11−

1R0

� �:

which have a similar structure to the equilibrium prevalenceYn ¼ 1−ð1=R0Þ in the very simplest (“SIS”) epidemic models.

The assumption that everyone enters the health care facilitybacteria free is certainly unrealistic. To handle the case in whicha positive fraction uC of the patients enter the facility colonizedwith MRSA, one simply adds a μuC term to the _p equation insystem (4). Since infected patients are always entering the system,there is no disease-free equilibrium in this case.

4. A more complex model

Having illustrated the use of the Lyapunov function techni-que for calculating R0 for a simple subsystem, we now move to asystem of more practical interest: one incorporating coloniza-tion and disease among both patients and HCWs, as well aspatient–patient transmission and the isolation of diseasedpatients. Our model population consists of a group of patientsin a health care facility. We assume that the staff of the facilityconsists of identical HCWs, all of whom are in contact withpatients for equal amounts of time each day. Patients can be inone of the three states: free, colonized, or diseased (see Table 1).HCWs can be in one of the five states: free or colonized (butuncontaminated), diseased (in which case they leave the facil-ity), and free/colonized and contaminated which means that, inaddition to their colonization status, they harbor bacteria ontheir hands or clothing.

Each model day, the following events occur:

1.

(fooargwil

Patients enter the facility at a constant rate U. The fractions ofincoming patients who are colonized and diseased are uC anduD, respectively. The rest of the incoming patients areuncolonized.

tnote continued)ument holds for the rest of the Lyapunov function arguments in this paper; wel, however, focus on showing when the origin is globally stable.

2.

Free and colonized patients leave the facility at a rate μ, anddiseased patients leave at a rate μ′. Because we wish to keep thetotal number of patients in the facility constant (i.e. we assumeit is always operating at maximum capacity), we require thatU ¼ μPF þ μPC þ μ′PD þ μ′PD′.

3.

Patients and HCWs interact at a rate χ, and patients contacteach other at a rate χ′. Each interaction event can potentiallytransmit infection or contamination according to the β para-meters described in Table 2.

4.

Colonized patients and HCWs lose their colonization at naturalrates γP and γH per day. Colonized patients and colonized HCWsare also treated with antibiotics and the rate of successfuldecolonization due to drug treatment is δP per day for patientsand δH for HCWs.

5.

Both colonized patients and HCWs can also progress to adiseased state. The probability of progression to disease is αPper day for patients and αH per day for HCWs.

6.

Diseased patients are also treated with drugs (δ′P) and canrecover naturally from their disease ðρÞ. Natural disease recov-ery returns them to the colonized state, where they thenrecover/respond to drug treatment at the same rates ascolonized patients. If a HCW becomes diseased, we assumethat he/she is immediately replaced by a MRSA-free,uncontaminated HCW.

7.

Diseased patients are moved to an isolation ward at rate ε.Patients leave this ward and return to the main part of thefacility only when they are completely MRSA-free. Whileisolated, these patients do not directly infect other patients;however, HCWs who interact with them may or may notcontact the rest of the patients.

8.

Finally, contaminated HCWs lose their contamination throughinfection control measures like glove changing and handwashing.Decontamination occurs at a rate η per worker per day.

Table 2 summarizes the parameters discussed above. A graphicalrepresentation of the model is shown in Fig. 4.

Model equations: This more complex model can be representedusing a system of eight differential equations. We write _PF fordPF=dt, and use similar abbreviations for the time derivatives ofthe other variables. If we use uF to denote the fraction of patientsentering free of MRSA (uF ¼ 1−uC−uD), the differential equationsare

_PF ¼ uFU−μPF−GPPF þ ðγP þ δPÞPC þ δ′PðPD þ PD′Þ_PC ¼ uCU−μPC þ GPPF−ðγP þ δPÞPC−αPPC þ ρPD

_PD ¼ uDU−μ′PD þ αPPC−ðρþ δ′PÞPD−εPD

_PD′ ¼−μ′PD′−δ′PPD′ þ εPD

_HF ¼ −HF � ðGH1 þ GH2 þ GH3Þ þ ðγH þ δHÞHC þ αHðHC þ HTCÞ þ ηHT

F

_HTF ¼ þ HF � GH2−HT

F � GH5 þ ðγH þ δHÞHTC−ηH

TF

_HC ¼ þ HF � GH1−HC � GH4−ðγH þ δHÞHC−αHHC þ ηHTC

_HTC ¼ þ HF � GH3 þ HC � GH4 þ HT

F � GH5−ðγH þ δHÞHTC−αHH

TC−ηH

TC

ð8Þwhere

U ¼ μPF þ μPC þ μ′PD þ μ′PD′

is the facility's total entry and exit rate,

GP ¼ χ′ � βCF � PC þ χ′ � βDF � PD þ χ � βTP � HTF þ χ � βCP � HC þ χ � βCTP � HT

C

is the force of infection for patients,

GH1 ¼ χ � βCHð1−βCT Þ � PC þ χ � βDHð1−βDT Þ � PD þ χ � βD′Hð1−βD′T Þ � PD′

Fig. 4. Diagram of the basic model. The squares represent patients and the circles represent HCWs. The light green color signifies an individual who is free of MRSAcolonization; darker green signifies colonization; blue signifies disease. The orange circles represent contamination of hands or clothing. The red square represents isolation.The parameters shown on the arrows correspond to those described in Table 2. Colonization of HF

Ts and contamination of HCs are also possible, but those lines are omitted inthe interest of clarity.

C. Simon et al. / Journal of Theoretical Biology 334 (2013) 187–199192

is the force of colonization (without contamination) on HF's by PC's,PD's, and PD′ 's,

GH2 ¼ χ � βCT � ð1−βCHÞ � PC þ χ � βDT � ð1−βDHÞ � PD þ χ � βD′T � ð1−βD′HÞ � PD′

is the force of contamination (without colonization) on HF's,

GH3 ¼ χ � βCT � βCH � PC þ χ � βDT � βDH � PD þ χ � βD′T � βD′H � PD′

is the force of [simultaneous] colonization and contamination onHF's,

GH4 ¼ χ � βCT � PC þ χ � βDT � PD þ χ � βD′T � PD′

is the force of contamination on HCs, and

GH5 ¼ χ � βCH � PC þ χ � βDH � PD þ χ � βD′H � PD′

is the force of colonization on HFT's.

Input–output ratios: As we did for the simple subsystem, webegin by defining separate input–output ratios for different sub-systems within our model. The first two are the same ratiosintroduced for the simple subsystem. The first, RTP, is for con-taminated HCWs infecting patients:

RTP≡χP � βTP

μþ γP þ δP þ αP

where now χP ¼ χ � ðHF þ HTF þ HC þ HT

CÞ, the number of HCWs asingle patient contacts per day. The second ratio, RCT, is forcolonized patients contaminating HCWs:

RCT≡χH � βCT

η;

where χH ¼ χ � ðPF þ PC þ PD þ PD′Þ, the number of patients a singleHCW contacts per day. Once again, the first ratio is R0 for newpatient infections from contaminated HCWs, and the second is R0for new HCW contaminations from colonized patients.

We now define nine more R0s in addition to the first two.The third and fourth R0s, RCF and RDF, refer to colonized anddiseased patients colonizing other patients. For these ratios,we define a new quantity χ′P ¼ χ′ � ðPF þ PC þ PD þ PD′Þ, which is

the number of other-patient contacts a given patient has in a singleday. The ratios are

RCF≡χ′P � βCF

μþ γP þ δP þ αPand RDF≡

χ′P � βDFμþ γP þ δP þ αP

:

The fifth and sixth R0s, RDT and RD′T , are similar to the RCT ratiodefined above, but are for diseased and diseased-isolated patientscontaminating HCWs:

RDT≡χH � βDT

ηand RD′T≡

χH � βD′Tη

:

The seventh, eighth, and ninth R0s are for patients infecting HCWs:

RCH≡χH � βCH

γH þ δH þ αHRDH≡

χH � βDHγH þ δH þ αH

RD′H≡χH � βD′H

γH þ δH þ αH:

Finally, the tenth and eleventh R0s are similar to the RTP ratio definedabove, but are for colonized or colonized-and-contaminated HCWsinfecting patients. They are

RCP≡χP � βCP

μþ γP þ δP þ αPand RCTP≡

χP � βCTPμþ γP þ δP þ αP

:

Besides contamination and colonization via patient–HCW inter-actions, there are other ways for patients to move from one classto another: disease progression and drug treatment. We use theletter A (not R) to denote these input–output ratios, of which thereare three. The first ratio, ACD, describes the ways in which patientsnaturally progress from colonization to disease, and the secondratio, ADC, is for the reverse process, PD-PC . The third ratio, ADD′,refers to the process PD-PD′ as patients are isolated. The ratios are

ACD≡αP

μ′þ δ′P þ ρþ εADC≡

ρ

μþ δP þ γP þ αPADD′≡

ε

μ′þ δ′P:

For example, the numerator in ADC is the rate by which a patientmoves from PD to PC by recovering from his disease. The denomi-nator of ADC then includes the four ways the patient can leave thePC class: by exiting the facility (μ), naturally losing colonization

Table 3Estimated values of the model parameters, both for a large, acute-care hospital and for a long-term care facility. Descriptions of the meanings of these parameters can befound in Table 2 and in Appendix B. See Appendix B for explanations of those parameter values for which no reference is provided.

Parameter Value for hospital Reference Value for LTCF Reference

uF 0.927–0.960 Hidron et al. (2005), Chaix et al. (1999),Jarvis et al. (2006)

0.750–0.884 Bradley et al. (1991),Minary-Dohen et al. (2003)

uC 0.040–0.073 Hidron et al. (2005), Chaix et al. (1999),Jarvis et al. (2006)

0.116–0.250 Bradley et al. (1991),Minary-Dohen et al. (2003)

uD 0.000 0.000μ 0.217 Wier et al. (2010) 0.00058 Drinka et al. (2005)μ′ 0.020–0.060 Wier et al. (2010), Eber et al. (2010),

Filice et al. (2010)0.00071 Strausbaugh and Joseph (2000)

χH 22.1 Needleman et al. (2002), Austin et al. (1999),Slaughter et al. (1996)

6.45 Decker et al. (2001), Harrington et al. (1998),Cardona et al. (1997)

χP 13.8 Needleman et al. (2002), Austin et al. (1999),Slaughter et al. (1996)

1.29 Decker et al. (2001), Harrington et al. (1998),Cardona et al. (1997)

χ′P 0.0 5.0γP 0.0033–0.0060 Robicsek et al. (2009), Robicsek et al. (2009),

Cooper et al. (2004), Scanvic et al. (2001), Lucet et al. (2009)0.0005–0.0146 O'Sullivan and Keane (2000)

γH 0.423–0.923 Cookson et al. (1989) 0.423–0.923 Cookson et al. (1989)δP 0.0036–0.0177 Boyce et al. (1981), Cooper et al. (2004),

Scanvic et al. (2001), Lucet et al. (2009),Robicsek et al. (2009), Robicsek et al. (2009)

0.0036–0.0177 Boyce et al. (1981), Cooper et al. (2004),Scanvic et al. (2001), Lucet et al. (2009),Robicsek et al. (2009), Robicsek et al. (2009)

δ′P 0.1025 Fitzpatrick et al. (2000) 0.1025 Fitzpatrick et al. (2000)δH 0.00 Cookson et al. (1989) 0.00 Cookson et al. (1989)αP 0.0003 Robicsek et al. (2009) 0.00072–0.0016 Muder et al. (1991), Datta and Huang (2008)αH 0.0001 0.0001ε 0.2 0.0 Manzur and Gudiol (2009), Nicolle (2001)ρ 0.0 0.0βCT 0.17 McBryde et al. (2004) 0.17 McBryde et al. (2004)βDT 0.17 McBryde et al. (2004) 0.17 McBryde et al. (2004)βD′T 0.17 McBryde et al. (2004) 0.17 McBryde et al. (2004)βCH 0.026 Austin et al. (1999) 0.026 Austin et al. (1999)βDH 0.026 Austin et al. (1999) 0.026 Austin et al. (1999)βD′H 0.026 Austin et al. (1999) 0.026 Austin et al. (1999)βTP 0.026 0.026βCP 0.000 0.000βCTP 0.026 0.026βCF NA 0.0070 Bradley et al. (1991)βDF NA 0.0070 Bradley et al. (1991)η 4.3 Smith et al. (2008) 4.3 Smith et al. (2008)Staffing ratio 1.60 Austin et al. (1999), Slaughter et al. (1996) 5.00 Decker et al. (2001), Harrington et al. (1998)(patients/HCWs)

C. Simon et al. / Journal of Theoretical Biology 334 (2013) 187–199 193

(γP), losing colonization due to prophylactic drug treatment (δP), orreturning to the diseased state via natural disease progression (αP).

The basic reproductive ratio: We now use the same Lyapunovfunction technique employed in our analysis of the simple sub-systems (1) and (2) to compute the basic reproductive ratio of theentire system of equations in (8), above. In order to ensure thepossibility of a disease-free equilibrium – which is necessary as wesearch for a threshold for disease take-off – we assume that nocolonized or diseased patients enter the facility (uC ¼ uD ¼ 0).We write the full system (8) in terms of proportions of patientsand HCWs in the various classes; that is, we write pC ; pD;h

TF ;hC , and

hCT in place of the corresponding capital letters. We also write the

system in terms of the input–output ratios described above. Aftermaking these modifications and eliminating the redundant equa-tions for pF and hF, we are left with the system

_pC ¼ pF � ðμþ γP þ δP þ αPÞ�½pCRCF þ pDRDF þ hTFRTP þ hCRCP þ hTCRCTP �þðμþ γP þ δP þ αPÞ � ½pDADC−pC �

_pD ¼ ðμ′þ ρþ δ′P þ εÞ � ½pCACD−pD�_pD′ ¼ ðμ′þ δ′PÞ � ½pDADD′−pD′�_hTF ¼ hF � η � ½pCRCT ð1−βCHÞ þ pDRDT ð1−βDHÞ þ pD′RD′T ð1−βD′HÞ�

−hTF � ðγH þ δH þ αHÞ � ½pCRCH þ pDRDH þ pD′RD′H�−hTF � ηþ hTC � ðγH þ δHÞ

_hC ¼ hF � ðγH þ δH þ αHÞ � ½pCð1−βCT ÞRCH

þpDð1−βDT ÞRDH þ pD′ð1−βD′T ÞRD′H �−hC � η � ½pCRCT þ pDRDT þ pD′RD′T �−hC � ðγH þ δH þ αHÞ þ hT

C � η_hTC ¼ hF � η � ½pCβCHRCT þ pDβDHRDT þ pD′βD′HRD′T �

þhC � η � ½pCRCT þ pDRDT þ pD′RD′T �þhTF � ðγH þ δH þ αHÞ � ½pCRCH þ pDRDH þ pD′RD′H�−hTC � ðγH þ δH þ αH þ ηÞ ð9Þ

where hF ¼ 1−hC−hTF−h

TC and pF ¼ 1−pC−pD−pD′. Note that we can

also write the last equation as

_hTC ¼ hF � ðγH þ δH þ αHÞ � ½pCβCTRCH þ pDβDTRDH þ pD′βD′TRD′H �

þhC � η � ½pCRCT þ pDRDT þ pD′RD′T �þhTF � ðγH þ δH þ αHÞ � ½pCRCH þ pDRDH þ pD′RD′H�−hTC � ðγH þ δH þ αH þ ηÞ:

We will use this trick in future calculations.We must now choose a Lyapunov function that will enable us

to calculate the basic reproductive ratio of this system of sixequations. As it turns out, the natural choice,

VðpC ; pD; pD′;hTF ;hC ;h

TCÞ ¼max ApC ;BpD;CpD′;Dh

TF ; EhC ; Fh

TC

n o

Table 4Comparison of different R0 terms for hospitals and LTCFs.

Term Components Hospitals LTCFs

1 RCF 0.0 (1.02, 6.48)2 ACDRDF 0.0 (0.02, 0.05)3 RCHRCP 0.0 0.04 ACDRDHRCP 0.0 0.05 ACDADD′RD′HRCP 0.0 0.06 RCTRTP (1.30, 1.40) (0.25, 1.58)7 ACDRDTRTP (0.0011, 0.0013) (0.0040, 0.0110)8 ACDADD′RD′TRTP (0.0013, 0.0021) 0.09 ACDADC 0.0 0.0

C. Simon et al. / Journal of Theoretical Biology 334 (2013) 187–199194

does not work. However, a variant of it,

VðpC ; pD; pD′;hTF ;hC ;hTCÞ

¼max ApC ;BpD;CpD′;DðhC þ hTCÞ; EðhTF þ hTCÞn o

does work, as we show in Appendix A. Our goal is to find values ofA;B;C;D; E for which the derivative _V of V along the orbits of thesystem of Eq. (9) is always negative.

We show in Appendix A that the following nine-term expres-sion:

RCF PC directly infects PF

þACDRDF PC progresses to PD before infecting PF

þRCHRCP PC infects a HCW who then infects PF

þACDRDHRCP PC progresses to PD;

then infects HCW who infects PF

þACDADD′RD′HRCP PC progresses to

PD′; then infects HCW who infects PF

þRCTRTP PC contaminates a HCW who then infects PF

þACDRDTRTP PC progresses to PD;

then contaminates HCW who infects PF

þACDADD′RD′TRTP PC progresses toPD′;

then contaminates HCW who infects PF

þACDADC PC progresses to PD;

drugs=natural recovery return him to PC ð10Þis a basic reproduction ratio for system (9) in which (1) when it is≤1, _V is globally negative and so the no-disease equilibrium isglobally asymptotically stable, and (2) it contains all nine ways inwhich a colonized patient spawns another colonized patient in thefacility.

It is important to note that our expression for R0 is a combina-tion solely of the parameters of the model. It does not depend, forexample, on compartment sizes that change over time. Diekmannet al. (1990) and Hill and Longini (2003), among others, point outthat R0 is closely related (often identical) to the dominanteigenvalue of the next generation matrix. We suggest that ourLyapunov function approach is much less daunting than calculat-ing a general formula for the dominant eigenvalue of an 8�8matrix. Heffernan et al. (2005) present a survey of variousinterpretations and calculation methods for R0, but do not includethe current Lyapunov function approach.

5 All of our numerical simulations were performed using custom Pythonscripts, and all subsequent regression analyses were done using R.

5. Applications

This expression for the basic reproductive ratio provides uswith a tool to evaluate the probable effectiveness of variousintervention strategies at reducing the prevalence of MRSA in ahealth care facility, since the endemic prevalence will correlatehighly with R0 if we assume random mixing. We can use ourexpression for R0 to gauge the relative impacts of different modelparameters on the endemic equilibrium, and from there toascertain which interventions are most likely to reduce it. Lookingat expression (10), we see that new cases of patient colonization inhealth care settings are due to three main factors: contact withcolonized patients, contact with contaminated HCWs, and contactwith colonized HCWs. Health care workers, in turn, attain theircolonization/contamination through patient contact, as illustrated

in Fig. 4. The question then becomes: which of these pathways islikely to lead to most new colonization events? And: does thispathway differ substantially between hospitals and long-term carefacilities (LTCFs)?

Table 3 contains estimated values of the model parameters forboth a hospital and a LTCF, gleaned from a review of the literatureexamining MRSA and VRE epidemics in these settings. If we plugthese values into expression (10), we can begin to see which termsin R0 contribute most to the epidemic process in each type offacility, as well as how these differences might lead to variations inthe type of intervention strategies that will be most effective ineach setting.

Estimates of the ranges for each of the nine R0 terms, forhospitals and LTCFs, are shown in Table 4. We see that the majorcontributors to R0 differ between the two settings. For example,transmission from colonized or diseased patients through transi-ently contaminated HCWs (terms 6 and 7) plays an important rolein both settings; in hospitals, it is enough to push R0 above one forall ranges of the other terms, and it is the most important source ofnew colonization. In LTCFs, however, transmission via contami-nated HCWs is not as important as transmission via direct patient–patient contact (terms 1 and 2). Most patient–patient transmissionappears to take place through uncolonized patients’ contact withcolonized patients who do not display symptoms of MRSA infec-tion and may not be aware of their own colonization status.

To confirm this result numerically, we performed numericalintegration on the equations in system (8) using parameterschosen5 uniformly from the ranges in Table 3,5. For those para-meters for which only a single point estimate was available, weused a range of 710% around the estimated values. Initially, allHCWs were bacteria-free. The fractions of colonized and diseasedpatients started off as random values chosen within the ranges ofuC and uD values from Table 3, but once the simulations began, noadditional colonized or diseased patients were allowed to enter.We repeated the simulations 100 000 times for each type of healthcare facility, using different parameter values chosen uniformlyfrom these ranges. We then fit a linear regression model for eachtype of facility, using the steady-state level of colonization+diseasein the patient population as our response variable and theparameter values as our predictor variables. The resultant coeffi-cients and p-values can be found in Table 5.

The parameters that attained statistical significance in theregression model were contributors to the dominant R0 terms inTable 4. For example, all of the parameters included in ourexpressions for RCF and RDF ðχ′P ; βCF ; βDF ; μ; γP ; δP ;αPÞ, the dominantcontributors to R0 for LTCFs, were significant in the LTCF linearregression model. By merging both analytical and numericalanalyses, therefore, we are able to say not only that certain modelparameters play a greater or lesser role in the spread of infection

Table 5Significant (po0:05) coefficients and p-values from the linear regression models.

Parameter Hospital LTCF

Coefficient p-value Coefficient p-value

μ −3.50�102 o0:001 −6.19�103 o0:001μ′

χH 3.26 o0:001 1.72 o0:001χP 5.91 o0:001 9.80 o0:001χ′P 12.17 o0:001γP −3.47�102 o0:001 −6.48�103 o0:001γHδP −3.48�102 o0:001 −6.49�103 o0:001δ′P −1.76 o0:001 −13.49 o0:001δHαP −5.22�103 o0:001αHε

βCT 4.24�102 o0:001 63.92 o0:001βDTβD′T 6.53�10−1 0.026 2.27 0.030βCHβDHβD′HβTP 2.72�103 o0:001 4.07�102 o0:001βCPβCTP 4.18�102 o0:001 82.40 o0:001βCF 1.72�104 o0:001βDF 1.78�102 o0:001η −16.75 o0:001 −2.52 o0:001

Fig. 5. Illustration of the effects of changes in parameter values on the prevalence of MREach simulation was run 100,000 times with parameter values taken by treating the po

C. Simon et al. / Journal of Theoretical Biology 334 (2013) 187–199 195

within health care facilities, but also that they play such arole because of their involvement with certain pathways ofinfection. Numerical analyses tell us, for instance, that the rate atwhich patients leave a facility has a greater effect on patientcolonization rates in LTCFs than in hospitals (coefficients: −6:19�103 and −3:50� 102) but our analytical formula for R0 reveals akey to the interpretation of this result: μ impacts two importanttransmission pathways in LTCFs (patient–patient and patient–HCW–patient) while in hospitals the patient–patient pathway isirrelevant.

We examined the time course of the epidemic process byplotting the fractions of free, colonized, and diseased patients andfree, contaminated and colonized HCWs as functions of time for3000 simulations with parameters chosen uniformly from theranges in Table 3. At the beginning of each simulation, 1% of thepatients were colonized with MRSA, and the infectious processwas then allowed to run unchecked, with no further colonizedpatients or HCWs introduced into the facility at any point.The results are shown in Fig. 5. We see that, in accordance withthe results for R0 in Table 4 (R0 is always ≥1 for both facility types),the epidemic never dies out in either cases. Recall that to produceexpression (10) for R0, we assumed that no colonized patientsentered the facility. This means that according to our best para-meter estimates, patient–patient and patient–HCW contacts areenough to produce nonzero endemic levels of MRSA colonizationin either settings. Although real-world behavior should exhibitmore stochasticity, and therefore some random die-out would beexpected, we should also point out that a sizeable fraction of

SA colonization and infection in hospitals (left) and long-term care facilities (right).ssible ranges in Table 3 as uniform distributions.

C. Simon et al. / Journal of Theoretical Biology 334 (2013) 187–199196

patients enter LTCFs (11.6–25.0%) and hospitals (4.0–7.3%) alreadycolonized. These results help explain why total eradication ofMRSA in both hospitals and LTCFs has proven to be so challenging(Dulon et al., 2011). Finally, Fig. 5 and the R0 calculation in Table 4show us that we should expect a higher degree of colonizationamong patients in LTCFs than in hospitals, although the level ofcolonization among HCWs is expected to be similar in both places.One recent study showed that the average MRSA carrier rateamong HCWs was 4.6% (Albrich and Harbarth, 2008), which is inthe range of values predicted by our model.

An explicit calculation of R0 for this relatively complex model,therefore, helps us in three ways. First, it reveals that differentpathways serve as the major drivers of infection in hospitals andLTCFs. In hospitals, patient–HCW contact is the main route for newcolonization events, so interventions aimed at improving HCWhand hygiene and other care practices should be most efficient forreducing the spread of MRSA. In LTCFs, in contrast, patient–patientcontact is the major driver, so efforts should focus on decolonizingpatients and limiting colonized patients’ contact with otherpatients and fomites in common areas.

Second, our calculation of R0 shows us how different modelparameters work together. Numerical analyses help illuminate theeffect of each parameter on patient prevalence, but they do notexpose the interactions inherent in the model. Focusing on path-ways in addition to single parameters can be revealing. Forexample, the expressions for RCF and RDF in LTCFs tell us thatreducing patient–patient contacts ðχ′PÞ has the same effect asincreasing prophylactic drug treatment of colonized patients ðδPÞor removing colonized patients temporarily from the facility ðμÞ.Certain of these options may prove easier to implement in somecases than others, and it is useful to be able to compare the likelyoutcomes quantitatively.

Finally, pathway analysis helps us notice certain unexpectedeffects. For instance, an increased rate of progression from colo-nization to disease among patients ðαPÞ is actually predicted to beprotective of LTCF patients on a population level because drugtreatment happens much more aggressively once patients aresymptomatic. This seems to indicate that HCWs in LTCFs shouldbe on guard for symptoms of MRSA infection and start aggressiveantibiotic therapy early on, even if the patient is experiencing onlyminor distress.

6. Conclusions

Explicit calculation of the basic reproductive ratio is useful for awide class of interesting epidemiologic models. Here we haveshown how Lyapunov functions can be used to calculate R0 for acomplex model of bacterial transmission in health care facilities,illuminating nine specific routes by which one colonized patientdirectly or indirectly causes the colonization of another. Thecalculations involved remain tractable even when multiple stagesof infection and explicit modeling of disease dynamics amongHCWs are incorporated. We hope other epidemiologists will findsimilar approaches useful for their own complex, multicompart-mental models.

Appendix A

This appendix proves that expression (10) is the basic repro-duction number for system (9). In our treatment of the simpletwo-dimensional system (2), we showed that Vðp;hÞ ¼maxfRCT � p;hg is a Lyapunov function for system (2) by examiningthe two cases: RCT � p≥h and RCT � p≤h. We concluded that RTP � RCT

is the basic reproduction ratio for system (2). Here we carry out asimilar analysis for the six-dimensional system (9) by showingthat

VðpC ; pD; pD′;hTF ;hC ;h

TCÞ

¼max ApC ;BpD;CpD′;DðhC þ hTCÞ; EðhTF þ hTCÞn o

works as a Lyapunov function for the more complicated system(9). Our goal is to find values of A;B;C;D; E for which the derivativeof V along the orbits of the system of equations in (9) is alwaysnegative.

If one quintuplet of values works, so will any multiple of thatquintuplet since V is linear in A;B;C;D; E. In particular, we can (andwill) set one of A;B;C;D; E equal to 1. We now examine situationsin which each of the five arguments of V is the maximizer.

1.

If ApC is the maximizer, V ¼ ApC and

_V ¼ A _pC

¼ A � pF � ðμþ γP þ δP þ αPÞ�½pCRCF þ pDRDF þ hT

FRTP þ hCRCP þ hTCRCTP �þA � ðμþ γP þ δP þ αPÞ � ½pDADC−pC �:

We now make one small assumption, which is that βCTP ≤βTP þ βCP , and consequently that RCTP ≤RTP þ RCP . This is logicalbecause βCTP represents infection of PFs by HC

Ts, and it ishighly unlikely that this probability is greater than the sum ofthe probabilities of infection from HF

Ts and HCs (or rather,that the joint effect of colonization and contamination isgreater than the sum of the individual effects). Assumingthis is true, and using the fact that pF is always less than 1,we have

_V ≤A � ðμþ γP þ δP þ αPÞ�½pCRCF þ pDRDF þ ðhTF þ hTCÞRTP þ ðhC þ hTCÞRCP þ pDADC−pC �:

We then factor out pC, which yields

_V ≤A � ðμþ γP þ δP þ αPÞ � pC

� RCF þpDpC

RDF þðhTF þ hTCÞ

pCRTP þ

ðhC þ hTCÞpC

RCP þpDpC

ADC−1

" #

≤A � ðμþ γP þ δP þ αPÞ � pC� RCF þ

ABRDF þ

AERTP þ

ADRCP þ

ABADC−1

� �

since our initial assumption that ApC≥BpD means that pD=pC ≤A=B, and similarly for the other terms. This last expression willalways be negative if and only if

RCF þABRDF þ

AERTP þ

ADRCP þ

ABADC ≤1: ð11Þ

2.

If BpD is the maximizer, V ¼ BpD and

_V ¼ B _pD

¼ B � ðμ′þ ρþ δ′P þ εÞ � ½pCACD−pD�

¼ B � ðμ′þ ρþ δ′P þ εÞ � pD � pCpD

ACD−1� �

≤B � ðμ′þ ρþ δ′P þ εÞ � pD � BAACD−1

� �;

which is always ≤0 if and only if

BAACD≤1: ð12Þ

C. Simon et al. / Journal of Theoretical Biology 334 (2013) 187–199 197

3.

If CpD′ is the maximizer, V ¼ CpD′ and

_V ¼ C _pD′¼ C � ðμ′þ δ′PÞ � ½pDADD′−pD′�

¼ C � ðμ′þ δ′PÞ � pD′ �pDpD′

ADD′−1� �

≤C � ðμ′þ δ′PÞ � pD′ �CBADD′−1

� �;

which is always ≤0 if and only if

CBADD′≤1: ð13Þ

4.

If DðhC þ hTCÞ is the maximizer, V ¼DðhC þ hTCÞ and_V ¼Dð _hC þ _h

TCÞ

¼D � hF � ðγH þ δH þ αHÞ � ½pCð1−βCT ÞRCH

þpDð1−βDT ÞRDH þ pD′ð1−βD′T ÞRD′H �−D � hC � η � ½pCRCT þ pDRDT þ pD′RD′T �−D � hC � ðγH þ δH þ αHÞ þ D � hT

C � ηþD � hF � ðγH þ δH þ αHÞ�½pCβCTRCH þ pDβDTRDH þ pD′βD′TRD′H �þD � hC � η � ½pCRCT þ pDRDT þ pD′RD′T �þD � hTF � ðγH þ δH þ αHÞ�½pCRCH þ pDRDH þ pD′RD′H �−D � hTC � ðγH þ δH þ αH þ ηÞ

¼D � ðhF þ hTF Þ � ðγH þ δH þ αHÞ�½pCRCH þ pDRDH þ pD′RD′H �−D � ðhC þ hTCÞ � ðγH þ δH þ αHÞ

¼D � ð1−hC−hTCÞ � ðγH þ δH þ αHÞ

�½pCRCH þ pDRDH þ pD′RD′H �−D � ðhC þ hTCÞ � ðγH þ δH þ αHÞ

¼D � ðγH þ δH þ αHÞ�½pCRCH þ pDRDH þ pD′RD′H−ðhC þ hTCÞ�−D � ðhC þ hTCÞ � ðγH þ δH þ αHÞ�½pCRCH þ pDRDH þ pD′RD′H �

¼D � ðγH þ δH þ αHÞ � ðhC þ hTCÞ

� pCðhC þ hTCÞ

RCH þ pDðhC þ hTCÞ

RDH þ pD′ðhC þ hT

CÞRD′H−1

" #

≤D � ðγH þ δH þ αHÞ � ðhC þ hTCÞ

� DARCH þ D

BRDH þ D

CRD′H−1

� �;

which is always negative if and only if

DARCH þ D

BRDH þ D

CRD′H ≤1: ð14Þ

5.

Finally, if EðhTF þ hTCÞ is the maximizer, V ¼ EðhTF þ hTCÞ and_V ¼ Eð _hT

F þ _hTCÞ

¼ E � hF � η � ½pCRCT ð1−βCHÞþpDRDT ð1−βDHÞ þ pD′RD′T ð1−βD′HÞ�−E � hTF � ðγH þ δH þ αHÞ�½pCRCH þ pDRDH þ pD′RD′H �−E � hTF � ηþ E � hT

C � ðγH þ δHÞþE � hF � η � ½pCβCHRCT

þpDβDHRDT þ pD′βD′HRD′T �þE � hC � η � ½pCRCT þ pDRDT þ pD′RD′T �þE � hTF � ðγH þ δH þ αHÞ�½pCRCH þ pDRDH þ pD′RD′H�

−E � hTC � ðγH þ δH þ αH þ ηÞ¼ E � ðhF þ hCÞ � η � ½pCRCT þ pDRDT þ pD′RD′T �−E � ðhTF þ hTCÞ � η−E � hT

C � αH¼ E � ð1−hTF−hTCÞ � η�½pCRCT þ pDRDT þ pD′RD′T �−E � ðhTF þ hTCÞ � η−E � hT

C � αH≤E � η � ½pCRCT þ pDRDT þ pD′RD′T−ðhTF þ hTCÞ�¼ E � η � ðhTF þ hTCÞ

� pCðhTF þ hTCÞ

RCT þpD

ðhTF þ hTCÞRDT þ

pD′ðhTF þ hTCÞ

RD′T−1

" #

≤E � η � ðhTF þ hTCÞ

� EARCT þ

EBRDT þ

ECRD′T−1

� �

which is always negative if and only if

EARCT þ

EBRDT þ

ECRD′T ≤1: ð15Þ

As suggested above, we can set B¼1 without loss of generalityand find A;C;D; E that satisfy all of these conditions simulta-neously. We begin by solving inequalities (12) and (13) as equal-ities, which lead to

A¼ ACD and C ¼ 1ADD′

:

Substituting these values into inequalities (14) and (15) whiletreating them as equalities results in

D¼ RCH

ACDþ RDH þ RD′HADD′

� �−1

E¼ RCT

ACDþ RDT þ RD′TADD′

� �−1:

Finally, putting all of these values for A;B;C;D; E into inequality(11), we obtain the threshold condition

RCF PC directly infects PF

þACDRDF PC progresses to

PD before infecting PF

þRCHRCP PC infects a HCW who then infects PF

þACDRDHRCP PC progresses to PD;

then infects HCW who infects PF

þACDADD′RD′HRCP PC progresses to

PD′; then infects HCW who infects PF

þRCTRTP PC contaminates a HCW who then infects PF

þACDRDTRTP PC progresses to

PD; then contaminates HCW who infects PF

þACDADD′RD′TRTP PC progresses to

PD′; then contaminates HCW who infects PF

þACDADC PC progresses to

PD; drugs=natural recovery return him to PC

is ≤1. This expression is indeed a basic reproductive ratio since itcontains all nine ways in which a colonized patient “spawns”another colonized patient in the facility.

C. Simon et al. / Journal of Theoretical Biology 334 (2013) 187–199198

Appendix B. Supplementary data

Supplementary data associated with this article can be found inthe online version at http://dx.doi.org.10.1016/j.jtbi.2013.05.016.

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