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MODELING OUTBREAKS OF ANTIBIOTIC
RESISTANCE IN HOSPITALS
Erika D’Agata,
Beth Israel Deaconess Medical Center
Harvard University
Boston, MA, USA
Pierre Magal
Department of Mathematics
Université du Havre
76058 Le Havre, FRANCE
Shigui Ruan
Department of Mathematics
University of Miami
Coral Gables, FL, 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
Glenn Webb
Department of Mathematics
Vanderbilt University
Nashville, TN USA
WHAT IS A NOSOCOMIAL INFECTION?
nos-o-co-mi-al
adj
originating or occurring in a hospital
“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.
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
Percent Resistance
70
60
50
40
30
20
10
04
20
03
20
02
20
01
20
00
20
99
19
98
19
97
19
96
19
19
95
0
Year
Source: National Nosocomial Infections Surveillance (NNIS) System
Vancomycin-resistant Enteroccoci (VRE)
Among ICU Patients,1995-2004
Percent Resistance
35
30
25
20
15
10
5
04
20
03
20
02
20
01
20
00
20
99
19
98
19
97
19
96
19
19
95
0
Year
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
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
•
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 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

dVF (a)
VF (a) 
 VF (a)F 
,
da
F 

If F >0, then limaVF(a)=F; if F<0, then limaVF(a)=0.

F=12.0log(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) 
V (a)
V  (a) V (a)  

  
  

V (a) V (a),

F
 da
 V (a) V (a)


 
V  (a) V (a)
dV (a)  V  (a)






 
V (a),


 da
F
V (a) V (a)


Equilibria of the model: E0 = (0,0), EF = (F 0), and

  
  F 
 
F
E  
   
,
   


















*
(i) If             0, then E0 is unstable and EF is stable
a V  (a )   F   ,
i.e., lim
lim a  V  (a)  0.
(ii) If             0, then, E0 (uns table), EF is (uns table), and E * (stable), i.e.,
lim a V  (a ) 
 F
 

 
     
,
  

lim a V  (a) 
 F 
 
     
.
  
  
MODEL OF ANTIBIOTIC TREATMENT IN A SINGLE
INFECTED HOST - TRACKING THE BACTERIAL LOAD
Treatment starts at day 3 and lasts 21 days
Treatment starts at day 5 and lasts 14 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 PATIENTS
Patient 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
Number of patients
400*
Number of healthcare workers
100*
Average length of stay for a patient not
infected with either strain
5 days*
Average length of stay for a patient
infected with the nonresistant strain
14 days*
Average length of stay for a patient
infected with the resistant strain
28 days*
Average time between visits of HCW
90 min
Probability of contamination by a HCW
0.4**
Probability of infection by a patient
0.06**
Average time of contamination of HCW
60 min**
*Beth Israel Deaconess Medical Center, Harvard, Boston
** Cook County Hospital, Chicago
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
ratio of HCW to
Interpretation
prob ability for a patient to be visited by a healthcare worker
patients
1/AV
rate at which a healthcare wor ker exits a visit
(AV = average length of a visit)
1/AC
rate at which a healthcare wor ker 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)
 ( N P N (t)   R P R (t))   V V PI H N (t)  H NR (t)  H R (t)PU (t),

 dt
 N 
  N
N
p

p
(
t,a)






P
H
(t)

H
(t)
p
(t,a),




N
V V I
R
NR
t

a
 N
U
p (t,0)   V V PI H N t P (t),


  p RS   p RS ( t,a)   R p RS (t,a),

a
t
 RS
p (t,0)   V V PI H R (t)  H NR (t)P N (t),

 RR
 RR 

p

p ( t,a)   R p RR (t,a),

t

a
 RR
U
p
(t,0)



P
H
(t)P
(t),
V V I
R

 
 NR 
NR
 p 
p ( t,a)   R p NR (t,a),

a
t
p NR (t,0)    P H (t)PU (t),
V V I
NR




EQUATIONS FOR THE HEALTHCARE WORKERS
0   P P I (t) P I (t) P I (t)H (t)  H (t) H (t) H (t),
V C
N
NR
R
U
C
N
NR
R

I
I
I

0  V PC PN (t)HU (t) V PC PNR (t) PR (t)H N (t)  C H N (t),

I
I
I
I
I
0  V PC PNR (t) PR (t)H N (t) V PC PNR (t)HU (t) V PC PN (t) PNR (t)H R (t)  C H NR (t),

I
I
I

0



P
P
(t)
P
(t)
H
(t)

P
P


V C
N
NR
R
V C R (t)H U (t)  C H R (t),

HU (t) H N (t) H NR (t) H R (t)  1.
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

 I
N
N
RS
RS
RR
RR
NR
NR
PN (t)    N (a) p (t,a)   N (a) p (t,a)   N (a) p (t,a)   N (a) p (t,a)da,
0


 I
PR (t)    RN (a) p N (t,a)   RRS (a) p RS (t,a)   RRR (a) p RR (t,a)   RNR (a) p NR (t,a)da,

0


I
N
RS
RR
NR
PNR (t)    NR
(a) p N (t,a)   NR
(a) p RS (t,a)   NR
(a) p RR (t,a)   NR
(a) p NR (t,a)da.

0
1 if a patient of class L is infectious with bacteria of type

 KL (a)   K at age of infection a
0 otherwise.


The infectious functions  NN (a),  NRS (a),  NRR (a), and  NNR (a) are defined
by the solutions of the bacterial load levels obtained from the bacteria
population model for patients undergoing antibiotic therapy.
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
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
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
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
•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), 766773.
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).