MODELING ANTIBIOTIC RESISTANCE EPIDEMICS IN HOSPITALS Pierre Magal Department of Mathematics Université du Havre 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 Le Havre, FRANCE

nos-o-co-mi-al adj originating or occurring in a hospital 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. WHAT IS A NOSOCOMIAL INFECTION?

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. WHY ARE NOSOCOMIAL INFECTIONS COMMON?

A GROWING PROBLEM Approximately 10% of U.S. hospital patients (about 2 million every year) acquire a clinically significant nosocomial infection.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 hospitalsNosocomial 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 treatmentMore 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, Source: National Nosocomial Infections Surveillance (NNIS) System

Vancomycin-resistant Enterococi (VRE) Among ICU Patients, 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 theHigh prevalence of resistant bacterial strains present in the hospital hospital 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 general hospital patient communityMedical 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 V F (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 If  F >0, then lim a  V F (a)=  F ; if  F <0, then lim a  V F (a)=0.  F =12.0  log(2) before treatment (doubling time = 2 hr),  F =-2.0 after treatment,  F =10 10 

MODEL OF BACTERIA IN A SINGLE INFECTED HOST INFECTED WITH PLASMID FREE AND PLASMID BEARING BACTERIA Equilibria of the model: E 0 = (0,0), E F = (  F    0), and

 - =8.0  log(2),  + =4.0  log(2),  =10 -5,  = 10 -3,  F =10 10,   -=8.0  log(2),  + =9.0  log(2),  =10 -5,  = 10 -3,  F =10 10, 

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 (H U ), contaminated only with non-resistant bacteria (H N ), contaminated with both non-resistant and resistant bacteria (H NR ), and contaminated only with resistant bacteria (H R ) Bottom: Patients are divided into five classes: uninfected patients (P U ), patients infected only by the non-resistant strain (P N ), and three classes of patients infected by resistant bacteria (P RS ), (P NR ), and (P RR ). P RS consists of super-infected patients, that is, patients that were in class P N and later become infected with resistant bacteria. P RR consists of patients that were uninfected and then became infected by resistant bacteria. P NR consists of patients that were uninfected, and then become infected with both non-resistant and resistant bacteria.

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 T H = 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 *Beth Israel Deaconess Medical Center, Harvard, Boston ** Cook County Hospital, Chicago 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 **

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 P C and a patient becomes infected from a contaminated HCW with probability P I ; (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 T H ; (iv)each time step  t a contaminated HCW exits contamination with probability 1 - exp(-  t/A C ) and exits a visit with probability 1 - exp(-  t/A V ); (v)(v) each time step  t a patient of type L exits the hospital with probability 1 - exp(-  t/A L ), 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 P U (t), P N (t), P R (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 p K (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

STATE VARIABLES OF THE (DEM) Note: Infected patients may or may not be infectious, depending on stage of infection and use of antibiotics.

PARAMETERS OF THE (DEM)

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

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 T H = 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.

EQUATIONS FOR 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 - R 0 A major advantage of the DEM is that the parametric input can be analyzed through the basic reproductive numbers R 0, which predict the expected number of secondary cases per primary case. When R 0 1, then the epidemic becomes endemic. R 0 N for patients infected only by the non-resistant strain is If R 0 N <1, then R 0 R for patients infected only by the resistant strain is where r(A) is the largest eigenvalue of the matrix

EFFECTS OF CHANGING THE DAY TREATMENT BEGINS AND HOW LONG IT LASTS R 0 N 1 depending on the starting day and the duration of treatment. Both R 0 N and R 0 R 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, R 0 N decreases and R 0 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 Both R 0 N and R 0 R decrease as the length of visits A C increases and increase as the length of contamination A V increases, but the dependence is linear in A C and quadratic in 1/A V. The reason is A C is specific to HCW, but A V 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 R 0 N and R 0 N depend linearly on A C and quadratically on 1/A V, 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.

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