MODELING OUTBREAKS OF ANTIBIOTIC RESISTANCE IN HOSPITALS Pierre Magal
Department of Mathematics
Université du Havre
76058 Le Havre, FRANCE Shigui Ruan
Department of Mathematics
University of Miami
Coral Gables, FL, USA Glenn Webb
Department of Mathematics
Vanderbilt University
Nashville, TN USA Erika D’Agata,
Beth Israel Deaconess Medical Center
Harvard University
Boston, MA, USA Mary Ann Horn
Mathematical Sciences Division
National Science Foundation
Washington, DC, USA Damien Olivier
Department of Computer Sciences
Université du Havre
76058 Le Havre, FRANCE

WHAT IS A NOSOCOMIAL INFECTION? :

nos-o-co-mi-al adj 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

WHY ARE NOSOCOMIAL INFECTIONS COMMON? :

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 :

A GROWING PROBLEM Approximately 10% of U.S. hospital patients (about 2 million every year) acquire a clinically significant nosocomial infection.
Nosocomial infections are responsible for about 100,000 deaths per year in hospitals
More than 70 percent of bacteria that cause hospital-acquired infections are resistant to at least one of the drugs most commonly used in treatment

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

Methicillin (oxacillin)-resistant Staphylococcus aureus (MRSA) Among ICU Patients, 1995-2004 Source: National Nosocomial Infections Surveillance (NNIS) System

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

Vancomycin-resistant Enteroccoci (VRE) Among ICU Patients,1995-2004 Source: National Nosocomial Infections Surveillance (NNIS) System

WHAT IS THE CONNECTION OF ANTIBIOTIC USE TO NOSOCOMIAL EPIDEMICS? :

WHAT IS THE CONNECTION OF ANTIBIOTIC USE TO NOSOCOMIAL EPIDEMICS? High prevalence of resistant bacterial strains present in the
hospital
High capacity of bacteria to mutate to resistant strains
Selective advantage of mutant strains during antibiotic therapy
Misuse and overuse of antibiotics
Medical practice focused on individual patients rather than the general hospital patient community

TYPES OF MICROBIAL RESISTANCE TO ANTIBIOTICS :

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 :

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 :

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 :

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 a
V(a) = population of resistant bacteria at infection age a
V(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 :

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

MODEL OF BACTERIA IN A SINGLE INFECTED HOST INFECTED WITH PLASMID FREE AND PLASMID BEARING BACTERIA :

MODEL OF BACTERIA IN A SINGLE INFECTED HOST INFECTED WITH PLASMID FREE AND PLASMID BEARING BACTERIA Equilibria of the model: E0 = (0,0), EF = (F 0), and

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

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 :

AN INDIVIDUAL BASED MODEL (IBM) AT THE HOSPITAL POPULATION LEVEL Three stochastic processes:
the admission and exit of patients
the infection of patients by HCW
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 :

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 PATIENTS :

INFECTIOUSNESS OF INDIVIDUAL PATIENTS 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 :

PARAMETERS OF THE IBM AT THE HOSPITAL LEVEL *Beth Israel Deaconess Medical Center, Harvard, Boston
** Cook County Hospital, Chicago

THE INFECTION AND CONTAMINATION PROCESSES :

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 :

SUMMARY OF THE IBM MODEL ASSUMPTIONS each HCW begins the first visit of the shift uncontaminated and subsequent patient visits are randomly chosen among patients without a HCW;
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;
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;
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);
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 :

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) :

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 :

KEY PARAMETERS OF THE DEM

EQUATIONS OF THE DEM :

EQUATIONS OF THE DEM

EQUATIONS FOR THE HEALTHCARE WORKERS :

EQUATIONS FOR THE HEALTHCARE WORKERS 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.

EQUATIONS FOR THE FRACTIONS OF PATIENTS INFECTIOUS FOR THE BACTERIAL STRAINS :

EQUATIONS FOR THE FRACTIONS OF PATIENTS INFECTIOUS FOR THE BACTERIAL STRAINS

COMPARISON OF THE IBM AND THE DEM :

COMPARISON OF THE IBM AND THE DEM Beginning of treatment = day 3
End of treatment = day 21 Beginning of treatment = day 1
End of treatment = day 8

ANALYSIS OF THE PARAMETRIC INPUT :

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 :

EFFECTS OF CHANGING THE DAY TREATMENT BEGINS AND HOW LONG IT LASTS R0R<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, R0R 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 :

EFFECTS OF CHANGING THE LENGTH OF VISITS AND THE LENGTH OF CONTAMINATION OF HCW R0R<1 or R0R>1 depending on the length of HCW visits and the length of HCW contamination. R0R 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 :

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 :

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).

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