MODELING OUTBREAKS OF ANTIBIOTIC

RESISTANCE IN HOSPITALS

Pierre MagalPierre MagalDepartment of MathematicsDepartment of Mathematics

Université du HavreUniversité du Havre76058 Le Havre, FRANCE76058 Le Havre, FRANCE

Shigui RuanShigui RuanDepartment of MathematicsDepartment of Mathematics

University of MiamiUniversity of MiamiCoral Gables, FL, USACoral Gables, FL, USA

Glenn WebbGlenn WebbDepartment of MathematicsDepartment of Mathematics

Vanderbilt UniversityVanderbilt UniversityNashville, TN USANashville, TN USA

Erika D’Agata, Erika D’Agata, Beth Israel Deaconess Medical CenterBeth Israel Deaconess Medical Center

Harvard Harvard UniversityUniversityBoston, MA, USABoston, MA, USA

Mary Ann HornMary Ann HornMathematical Sciences DivisionMathematical Sciences Division

National Science FoundationNational Science FoundationWashington, DC, USAWashington, DC, USA

Damien OlivierDamien OlivierDepartment of Computer SciencesDepartment of Computer Sciences

Université du HavreUniversité du Havre76058 Le Havre, FRANCE76058 Le Havre, FRANCE

nos-o-co-mi-al nos-o-co-mi-al adj originating or occurring in a hospitaladj originating or occurring in a hospital

WHAT IS A NOSOCOMIAL INFECTION?

“Even a term adopted by the CDC --nosocomial infection obscures the true source of the germs. Nosocomial, derived from Latin, means hospital-acquired. CDC records show that the term was used to shield hospitals from the ‘embarrassment’ of germ-related deaths and injuries.”

--Michael J. Berens, Chicago Tribune, July 21, 2002

• Hospitals house large numbers of people whose immune systems are often in a weakened state.

• Increased use of outpatient treatment means that patients in the hospital are more vunerable.

• Medical staff move from patient to patient, providing a way for pathogens to spread.

• Many medical procedures bypass the body's natural protective barriers.

WHY ARE NOSOCOMIAL INFECTIONS COMMON?

A GROWING PROBLEM

• Approximately 10% of U.S. hospital patients (about Approximately 10% of U.S. hospital patients (about 2 million every year) acquire a clinically significant 2 million every year) acquire a clinically significant nosocomial infectionnosocomial infection. .

• Nosocomial infections are responsible for about Nosocomial infections are responsible for about 100,000 deaths per year in hospitals100,000 deaths per year in hospitals

• More than 70 percent of bacteria that cause More than 70 percent of bacteria that cause hospital-acquired infections are resistant to at least hospital-acquired infections are resistant to at least one of the drugs most commonly used in treatmentone of the drugs most commonly used in treatment

Methicillin (oxacillin)-resistant Staphylococcus aureus (MRSA) Among ICU Patients, 1995-2004

0

10

20

30

40

50

60

70

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Percent Resistance

Source: National Nosocomial Infections Surveillance (NNIS) System

Vancomycin-resistant Enteroccoci (VRE) Among ICU Patients,1995-2004

0

5

10

15

20

25

30

35

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Percent Resistance

Source: National Nosocomial Infections Surveillance (NNIS) System

WHAT IS THE CONNECTION OF ANTIBIOTIC USE TO NOSOCOMIAL EPIDEMICS?

• High prevalence of resistant bacterial strains present in the High prevalence of resistant bacterial strains present in the hospitalhospital

• High capacity of bacteria to mutate to resistant strainsHigh capacity of bacteria to mutate to resistant strains

• Selective advantage of mutant strains during antibiotic therapySelective advantage of mutant strains during antibiotic therapy

• Misuse and overuse of antibioticsMisuse and overuse of antibiotics

• Medical practice focused on individual patients rather than the Medical practice focused on individual patients rather than the general hospital patient community general hospital patient community

TYPES OF MICROBIAL RESISTANCE TO ANTIBIOTICS

• Inherent - microorganisms may be resistant to antibiotics because of physical and biochemical differences.

• Acquired - bacteria can develop resistance to antibiotics driven by two genetic processes:

• (a) mutation and selection (vertical evolution)

• (b) exchange of genes (plasmids) between strains and species (horizontal evolution).

OBJECTIVES OF THE MODELING PROJECT

• Construct a model based on observable hospital

parameters, focusing on healthcare worker (HCW)

contamination by patients, patient infection by healthcare

workers, and infectiousness of patients undergoing

antibiotic therapy.

• Analyze the elements in the model and determine strategies

to mitigate nosocomial epidemics

THE TWO LEVELS OF A NOSOCOMIAL EPIDEMIC

• Bacteria population level in a single infected host:

(i) host infected with the nonresistant strain

(ii) host infected with the resistant strain

• Patient and healthcare worker level in the hospital:

(i) uninfected patients susceptible to infection

(ii) patients infected with the nonresistant strain

(iii) patients infected with the resistant strain

(iv) uncontaminated HCW

(v) contaminated HCW

AN ORDINARY DIFFERENTIAL EQUATIONS MODEL AT THE BACTERIA POPULATION LEVEL

A. Bacteria in a host infected only with the nonresistant strain

VF(a) = population of nonresistant bacteria at infection age a

F(a) = proliferation rate

F = carrying capacity parameter of the host

B. Bacteria in a host infected with both nonresistant and resistant strains

V(a) = population of nonresistant bacteria at infection age aV(a) = population of resistant bacteria at infection age aV(a) = V(a)+ V(a)

_(a)(a) = proliferation rates

= recombination rate, = reversion rate

MODEL OF PLASMID FREE BACTERIA IN A SINGLE INFECTED HOST INFECTED WITH ONLY PLASMID FREE BACTERIA

If F >0, then limaVF(a)=F; if F<0, then limaVF(a)=0.

€

dVF(a)

da= V

F(a) β

F−

VF(a)

κF

⎛

⎝ ⎜

⎞

⎠ ⎟,

F=12.0log(2) before treatment (doubling time = 2 hr), F=-2.0 after treatment, F=1010

MODEL OF BACTERIA IN A SINGLE INFECTED HOST INFECTED

WITH PLASMID FREE AND PLASMID BEARING BACTERIA

€

dV − (a)

da= −

τV+(a)

V − (a)+V+(a)+ β

−−

V − (a)+V+(a)

κF

⎛

⎝ ⎜

⎞

⎠ ⎟V − (a)+γV+(a),

dV+(a)

da=

τV − (a)

V − (a)+V+(a)+ β

+−

V − (a)+V+(a)

κF

− γ ⎛

⎝ ⎜

⎞

⎠ ⎟V+(a),

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

Equilibria of the model: E0 = (0,0), EF = (F 0), and

€

E* =γκ

F

γ +σβ

−+σ −

τσ

γ +σ

⎛

⎝ ⎜

⎞

⎠ ⎟,σκ

F

γ +σβ

−+σ −

τσ

γ +σ

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

(i) If ,0<−+−=−+σ then 0Ei sunstabl eandFEi s stable

i.e., ,)(lim−−∞→=FaaV .0)(lim=+∞→aVa(ii)I f ,0>−+−=−+σ the ,n 0E(unstable),FEi (s unstable),and *E(stable),i.e.,,)(lim ⎟⎟⎠⎞⎜⎜⎝⎛+−++=−−∞→ σσσσFaaV .)(lim ⎟⎟⎠⎞⎜⎜⎝⎛+−++=−+∞→ σσσσσFaaV

MODEL OF ANTIBIOTIC TREATMENT IN A SINGLE INFECTED HOST - TRACKING THE BACTERIAL LOAD

Treatment starts at day 5 and lasts 14 days

Treatment starts at day 3 and lasts 21 days

AN INDIVIDUAL BASED MODEL (IBM) AT THE HOSPITAL POPULATION LEVEL

Three stochastic processes:

1) the admission and exit of patients

2) the infection of patients by HCW

3) the contamination of HCW by patients

These processes occur in the hospital over a period of months or years

as the epidemic evolves day by day. Each day is decomposed into 3 shifts

of 8 hours for the HCW. Each HCW begins a shift uncontaminated, but

may become contaminated during a shift. During the shift a time step t

delimits the stochastic processes. The bacterial load of infected patients

during antibiotic treatment is monitored in order to describe the

influence of treatment on the infectiousness of patients.

PATIENT AND HCW POPULATION LEVEL

Top: Healthare workers are divided into four classes: uncontaminated (HU), contaminated only with non-resistant bacteria (HN), contaminated with both non-resistant and resistant bacteria (HNR), and contaminated only with resistant bacteria (HR)

Bottom: Patients are divided into five classes: uninfected patients (PU), patients infected only by the non-resistant strain (PN), and three classes of patients infected by resistant bacteria (PRS), (PNR), and (PRR). PRS consists of super-infected patients, that is, patients that were in class PN and later become infected with resistant bacteria. PRR consists of patients that were uninfected and then became infected by resistant bacteria. PNR consists of patients that were uninfected, and then become infected with both non-resistant and resistant bacteria.

INFECTIOUSNESS OF INDIVIDUAL PATIENTSPatient PN Patient PRS

Patient PRR Patient PNR

Infectiousness periods when the antibiotic treatment starts on day 3 and stops on day 21 (inoculation occurs on day 0). The blue and red curves represent, respectively, the bacterial load of resistant and non-resistant bacteria during the period of infection. The green horizontal lines represent the threshold of infectiousness TH=1011. The green bars represent the treatment period. The yellow, red, and orange bars represent the periods of infectiousness for the non-resistant, resistant, and both non-resistant and resistant classes, respectively.

PARAMETERS OF THE IBM AT THE HOSPITAL LEVEL

*Beth Israel Deaconess Medical Center, *Beth Israel Deaconess Medical Center, Harvard, BostonHarvard, Boston** Cook ** Cook County Hospital,County Hospital, Chicago Chicago

Number of patientsNumber of patients 400400**

Number of healthcare workersNumber of healthcare workers 100100**

Average length of stay for a patient not Average length of stay for a patient not infected with either straininfected with either strain

55 daysdays**

Average length of stay for a patient Average length of stay for a patient infected with the nonresistant straininfected with the nonresistant strain

14 days14 days**

Average length of stay for a patient Average length of stay for a patient infected with the resistant straininfected with the resistant strain

28 days28 days**

Average time between visits of HCW Average time between visits of HCW 90 min

Probability of contamination by a HCWProbability of contamination by a HCW 0.40.4****

Probability of infection by a patientProbability of infection by a patient 0.060.06****

Average time of contamination of HCWAverage time of contamination of HCW 60 min60 min****

THE INFECTION AND CONTAMINATION PROCESSES

Patient-HCW contact diagram for 4 patients and 1 HCW during one shift. Patient status: uninfected (green), infected with the non-resistant strain (yellow), infected with the resistant strain (red). HCW status: uncontaminated (______ ), contaminated with the non-resistant strain (………), contaminated with the resistant strain (- - - - - ).

SUMMARY OF THE IBM MODEL ASSUMPTIONS

(i) each HCW begins the first visit of the shift uncontaminated and subsequent patient visits are randomly chosen among patients without a HCW;

(ii) at the end of a visit a HCW becomes contaminated from an infectious patient with probability PC and a patient becomes infected from a contaminated HCW with probability PI;

(iii) the bacterial load of an infected patient is dependent on treatment scheduling and infected patients are infectious to a HCW when their bacterial load is above a threshold TH;

(iv) each time step t a contaminated HCW exits contamination with probability 1 - exp(-t/AC) (AC = average period of contamination) and exits a visit with probability 1 - exp(-t/AV) (AV = average length of visit);

(v) each time step t a patient of type L exits the hospital with probability 1 - exp(-t/AL), where AL = average length of stay and L = U,N,R.

(vi) The number of patients in the hospital is assumed constant, so that a patient leaving the hospital is immediately replace by a new patient in class (U).

TWO SIMULATIONS OF THE IBM WITH DIFFERENT TREATMENT SCHEDULES

From the two IBM simulations we see that when treatment starts earlier and has a shorter period, both non-resistant and resistant strains are eliminated. Earlier initiation of treatment reduces the non-resistant bacterial load and shorter treatment intervals reduce the time that patients infected by the resistant strain are infectious for this strain.

A COMPLEMENTARY DIFFERENTIAL EQUATIONS MODEL (DEM)

The Individual Based Model (IBM) provides a stochastic simulation of the epidemic based on probabilistic assumptions on events occurring in the hospital. But the IBM is different every time it is simulated, and it is difficult to analyse the effects of various elements in the model. For example, what is the effect of modify the length of HCW contamination periods or the length of treatment periods?

We develop a Differential Equations Model (DEM) that corresponds to the average behavior of the IBM over a large number of simulations. The DEM is based on dynamic rates of change of the processes occurring over the course of the epidemic.

KEY PARAMETERS OF THE DEM

Parameter Interpretation

ratio of HCW to

patients

probability for a patient to be visited by a healthcare worker

1/AV

rate at which a healthcare worker exits a visit

(AV = average length of a visit)

1/AC

rate at which a healthcare worker becomes uncontaminated

(AC = the average period of contamination)

1/ AN

rate at which a patient infected with the nonresistant strain

exits the hospital (AN = the average LOS of a patient infected

with the nonresistant strain)

1/ AR

rate at which a patient infected with the resistant strain exits

the hospital (AR = the average LOS of a patient infected with

the resistant strain)

EQUATIONS OF THE DEM

€

dPU (t)

dt= (ν N PN (t) + ν R PR (t)) −ν V βV PI HN (t) + HNR (t) + HR (t)( )P

U (t),

∂

∂tpN +

∂

∂apN ⎛

⎝ ⎜

⎞

⎠ ⎟(t,a) = − ν N + ν V βV PI HR (t) + HNR (t)( )( )pN (t,a),

pN (t,0) = ν V βV PI HN t( )PU (t),

∂

∂tpRS +

∂

∂apRS ⎛

⎝ ⎜

⎞

⎠ ⎟(t,a) = −ν R pRS (t,a),

pRS (t,0) = ν V βV PI HR (t) + HNR (t)( )PN (t),

∂

∂tpRR +

∂

∂apRR ⎛

⎝ ⎜

⎞

⎠ ⎟(t,a) = −ν R pRR (t,a),

pRR (t,0) = ν V βV PI HR (t)PU (t),

∂

∂tpNR +

∂

∂apNR ⎛

⎝ ⎜

⎞

⎠ ⎟(t,a) = −ν R pNR (t,a),

pNR (t,0) = ν V βV PI HNR (t)PU (t),

⎧

⎨

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

EQUATIONS FOR THE HEALTHCARE WORKERS

€

0 = −νV PC PNI (t)+ PNR

I ( t)+ PRI (t)[ ]HU( t)+ν C H N ( t)+ H NR( t)+ H R(t)[ ],

0 = νV PC PNI ( t)HU( t)− νV PC PNR

I ( t)+ PRI (t)[ ]H N ( t)− ν C H N ( t),

0 = νV PC PNRI ( t)+ PR

I (t)[ ]H N ( t)+νV PC PNRI (t)HU(t)+νV PC PN

I (t)+ PNRI ( t)[ ]H R(t)− ν C H NR( t),

0 = −νV PC PNI (t)+ PNR

I ( t)[ ]H R(t)+νV PC PRI (t)HU(t)− ν C H R(t),

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

The equations for the HCW are motivated by a singular perturbation technique. The idea is that the time scale of the HCW is much smaller than the time scale for the evolution of the epidemic at the patient level. These equations are solved for the HCW fractions.

€

HU( t)+ H N (t)+ H NR( t)+ H R(t) = 1.

EQUATIONS FOR THE FRACTIONS OF PATIENTS INFECTIOUS FOR THE BACTERIAL STRAINS

€

PNI (t) = γ N

N

0

+∞

∫ (a) pN (t,a) + γ NRS (a) pRS (t,a) + γ N

RR (a) pRR (t,a) + γ NNR (a) pNR (t,a)da,

PRI (t) = γ R

N

0

+∞

∫ (a) pN (t,a) + γ RRS (a) pRS (t,a) + γ R

RR (a) pRR (t,a) + γ RNR (a) pNR (t,a)da,

PNRI (t) = γ NR

N

0

+∞

∫ (a)pN (t,a) + γ NRRS (a)pRS (t,a) + γ NR

RR (a)pRR (t,a) + γ NRNR (a) pNR (t,a)da.

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

€

The infectious functions γ NN(a), γ N

RS(a), γ NRR(a), and γ N

NR(a) are defined

by the solutions of the bacterial load levels obtained from the bacteria

population model for patients undergoing antibiotic therapy.

€

KL (a) =

1 if a patient of class L is infectious with bacteria of type

K at age of infection a

0 otherwise.

⎧

⎨ ⎪

⎩ ⎪

COMPARISON OF THE IBM AND THE DEM

Beginning of treatment = day 3 End of treatment = day 21

Beginning of treatment = day 1End of treatment = day 8

ANALYSIS OF THE PARAMETRIC INPUT

A major advantage of the DEM is that the parametric input can be analyzed in terms of the parametric input. This is accomplished by calculating:

R0 = epidemic basic reproductive number.

R0 predicts the expected number of secondary cases per primary case. When R0 <1, then the epidemic extinguishes and when R0 >1, then the epidemic becomes endemic.

R0 is a function of all the parameters in the model, and a sensitivity analysis of R0 can be carried out by holding some of the parameters fixed and varying some of the other parameters.

EFFECTS OF CHANGING THE DAY TREATMENT BEGINS AND HOW LONG IT LASTS

R0

R<1 or R0R>1 depending on the starting day and the duration of treatment.

R0R is increasing when the starting day of treatment increases, because the

bacterial loads of both strains are higher if treatment is delayed and thus more likely to reach threshold Further, R0

R increases as the length of treatment duration increases, because the resistant strain prevails during treatment.

EFFECTS OF CHANGING THE LENGTH OF VISITS AND THE LENGTH OF CONTAMINATION OF HCW

R0R<1 or R0

R>1 depending on the length of HCW visits and the length of HCW contamination. R0

R decreases as the length of visits AC increases and increase as the length of contamination AV increases, but the dependence is linear in AC

and quadratic in 1/AV. The reason is AC is specific to HCW, but AV is specific to both HCW and patients.

CONCLUSIONS OF THE MODEL

•Antibiotic therapy regimens should balance the care of individual patients and the general patient population welfare.

•Antibiotic treatment should start as soon as possible after infection is diagnosed and its duration should be minimized.

•Mathematical models can create virtual hospitals and analyze measures to control nosocomial epidemics (hospital acquired infection epidemics) in specific hospital environments.

REFERENCES

E. D’Agata, M.A. Horn, and G.F. Webb, The impact of persistent gastrointestinal colonization on the transmission dynamics of vancomycin-resistant enterococci, J. Infect. Dis. ,Vol. 185 (2002), 766-773.

E. D’Agata, M.A. Horn, and G.F. Webb, A mathematical model quantifying the impact of antibiotic exposure and other interventions on the endemic prevalence of vancomycin-resistant enterococci, J. Infect. Dis., Vol. 192 (2005), 2004-2011.

E. D’Agata, P. Magal, S. Ruan, and G.F. Webb, A model of antibiotic resistant bacterial epidemics in hospitals, Proc. Nat. Acad. Sci. Vol. 102, No. 37, (2005), 13343-13348.

E. D'Agata, P. Magal, S. Ruan, and G.F. Webb, Modeling antibiotic resistance in hospitals: The impact of minimizing treatment duration, J. Theoret. Biol., Available online Aug. 27 (2007).