Download infected with the non-resistant strain

MODELING ANTIBIOTIC RESISTANCE
EPIDEMICS 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
Nosocomial infections are infections which are a result
of treatment in a hospital or a healthcare service unit,
but secondary to the patient's original condition.
Infections are considered nosocomial if they first appear
48 hours or more after hospital admission or within 30
days after discharge.
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
people who are in the hospital are sicker on average.
•
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 hospitalacquired 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 Enterococi (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
 _(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 
 
     
.
  
  
-=8.0log(2), +=4.0log(2), =10 -5, = 10-3 , F=1010, 2.770.
-=8.0log(2), +=9.0log(2), =10 -5, = 10-3 , F=1010, 0.690.
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
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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 MODEL 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) and exits a visit with probability 1 - exp(-t/AV);
(v)
(v) each time step t a patient of type L exits the hospital with probability 1
- exp(-t/AL), where 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).
SIMULATIONS OF THE IBM
Beginning of treatment day 3
End of treatment day 21
Beginning of treatment day 1
End of treatment day 8
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 DEM corresponds to the average behavior of the IBM over a
large number of simulations. Denote by PU(t), PN(t), PR(t) the fraction
of patients in the class (U), (N), (R) respectively.
To describe the infectiousness status of patients, we use the age of
infection a, which represents the time already spent in the class of
infected patients (N), (RS), (RR), or (NR). For K=N, RS, RR, NR, we
denote by pK(t,a) the density of the fraction of patients with infection
status (K) and infection-age a at time t. For K=N, RS, RR, NR
P (t) 
K

p
0

K
(t,a)da.
STATE VARIABLES OF THE (DEM)
Symbol
Interpretation
U
H (t) fraction of HCW wo rkers uncontaminated
H N (t)
H NR (t)











fraction of HCW contaminated only by non-resistant strain
fraction of HCW contaminated by both resistant and nonresistant strains
R
H (t) fraction of HCW contaminated only by resistant strain
PU (t) fraction of patients uninfected
P N (t) fraction of patients infected only by non-resistant strain
P RS (t) fraction of patients infected first by non-resistant strain and the n
by resistant strain
P NR (t) fraction of patients infected first by both resistant and nonresistant strain
RR
P (t) fraction of patients infected only by resistant strain
p K (t,a) infection age density of the fraction of infected patients of class
K=N,RS,RR,NR
I
PK (t) fraction of patients infectious wi th the strain K=N,NR,R
Note: Infected patients may or may not be infectious, depending on
stage of infection and use of antibiotics.
PARAMETERS OF THE (DEM)
Symbol
Interpretation
V  NBH /NBP probabi lity for a patie nt to be visited by a healthcare
worker





 V  1/ AV
rate at which a healthcare worker exits a visit
 C  1/ AC
rate at which a healthcare worker becomes
unconta minated
 N 1/ AN
rate at which a class N patient exits the hospital
 R  1/ AR
rate at which a class R patient exits the hospital
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.
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 nonresistant, resistant, and both non-resistant and resistant classes, respectively.
EQUATIONS FOR 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




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 - R0
A major advantage of the DEM is that the parametric input can be analyzed
through the basic reproductive numbers R0 , which predict 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.
R0N for patients infected only by the non-resistant strain is
R 
N
0
2
V  V PI PC   N (a)exp(
C

N
N
a)da.
0
If R0N<1, then R0R for patients infected only by the resistant strain is

R 
R
0
2
V  V PI PC
C
r(A),
where r(A) is the largest eigenvalue of the matrix
 RR
   (a)exp( a)da
 R
R
0
A  
 RR
   NR (a)exp( R a)da
 0

  (a)exp( R a)da 
0
.


NR

(a)exp(

a)da
 NR
R


0

NR
R
EFFECTS OF CHANGING THE DAY TREATMENT
BEGINS AND HOW LONG IT LASTS
R0N<1 always, but R0R<1 or R0R>1 depending on the starting day and the duration of
treatment. Both R0N and R0R are 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, R0N decreases and 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
Both R0N and R0R decrease 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
•Start treatment as soon as possible after infection is diagnosed and
minimize its duration. Recent clinical trials suggest that the duration of
antimicrobial therapy can be decreased substantially in the treatment
of certain community- and hospital-acquired infections with equivocal
patient outcomes.
•Because R0N and R0N depend linearly on AC and quadratically on 1/AV,
extending the average length of visits (which is correlated to the
allocation of HCW resources) may have less benefit than reducing the
average length of the contamination (which is correlated to
improvement in hygienic measures).
•The care of individual patients and the general patient population
welfare must be balanced.
•Mathematical models provide a framework to analyze the dynamic
elements in antibiotic resistance epidemics and quantify their impact in
specific hospital environments.
REFERENCES
D.J. Austin, K.G. Kristinsson, and R.M. Anderson, The relationship between the volume
of antimicrobial consumption in human communities and the frequency of resistance,
Proc. Natl., Acad. Sci., Vol. 96 (1999), 1152-1156.
C.R. Bergstrom, M. Lipsitch, and B.R. Levin, Natural selection, infectious transfer and
the existence conditions for bacterial plasmids, Genetics, Vol. 155 (2000), 1505-1519.
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 vancomycinresistant 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).
I. Pelupessy, M. C. M. Bonten and O. Diekmann, How to assess the relative importance of
different colonization routes of pathogens within hospital settings, Proc. Natl., Acad. Sci.
99 (2002), 5601-5605.