I. MATHEMATICAL MODELS OF NOSOCOMIAL EPIDEMICS

1. ROCKY MOUNTAIN MATHEMATICS CONSORTIUM SUMMER SCHOOL 2012 MATHEMATICAL MODELING IN ECOLOGY AND EPIDEMIOLOGY UNIVERSITY OF WYOMING JUNE 11-22, 2012 I. MATHEMATICAL MODELS OF NOSOCOMIAL EPIDEMICS

2. 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.

3. 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.

4. 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

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

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

7. 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

8. THE BACTERIA FIGHT BACK Endemicity of resistant strains has been observed throughout the evolutionary history of antibiotic therapy (e.g., the replacement of penicillin-susceptible Staphylococcus aureus and vancomycin-susceptible Enterococcus faecium with resistant strains)

9. 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

10. THE HOSPITAL POPULATION CLASSES (i) uninfected patients susceptible to infection (ii) patients infected with the nonresistant strain (iii) patients infected with the resistant strain (iv) uncontaminated HCW (v) HCW contaminated with the nonresistant strain (vi) HCW contaminated with the resistant strain

11. AN INDIVIDUAL BASED MODEL (IBM) FOR THE HOSPITAL POPULATION 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 can be monitored in order to describe the influence of treatment on the infectiousness of patients.

13. Rules of the IBM Model Each healthcare worker hcw h1, h2, h3, … has a contamination status: hi[contamination status] where the contamination status is u (uncontaminated), n (contaminated with the nonresistant strain), or r (contaminated with the resistant strain). Each patient p1, p2, p3, … has an infection status, pj[infection status] wheretheinfection statusisu (uninfected), n (infected with the nonresistant strain), orr (infected with the resistant strain). The model tracks through successive shifts the evolving contamination status of each hcw and the infection status of each patient through their sequential contacts. The status of a hcw or patient may change due to a contact between them. An uncontaminated hcw may be contaminated by a infected patient and a uninfected patient may be infected by a contaminated hcw. The status resulting from a contact of hcw hiand patient pj (changed or unchanged) is indicated by the pairing (hi[contamination status], pj[infection status])

14. Rule 1: All hcw begin the shift uncontaminated. Rule 2: All patients begin the shift with their last infection status on the preceding shift. Rule 3: A patient pjmay exit the hospital after each contact with probability ExitProbabilityPerShift. Rule 4: A patient pj who exits the hospital during the shift is replaced by a new patient at the beginning of the next shift. The new patient has infection status u. Rule 5: A hcw himay be contaminated by contact with patient pj with probability Fh(hi[contamination status], pj[infection status]).

15. Rule 6: A patient pjmay be infected by contact with a contaminated hcw hi with probability Fp(hi[contamination status], pj[infection status]). Rule 7: An infected patient pj retains the same infection status until exit from the hospital, except that a patient infected with the nonresistant strain can be super-infected with the resistant strain. Monte Carlo simulations of the model are made with specified initial conditions of hcw and patients, a specified number of shifts, random generation of hcw-patient contacts each shift, and specified probabilities parameters. Each shift is divided into a specified number of time intervals in which the probabilistic events are updated. The simulations are different each time, so they must be repeated many times to find an average outcome.

17. (1) h1 is uncontaminated and p1 is infected with the nonresistant strain at the beginning of the shift, but h1 is not contaminated with the nonresistant strain during this contact (2) h3 is uncontaminated and p3 is uninfected before and after this contact (3) h2 is uncontaminated and p6 is uninfected before and after this contact (4) h1 is uncontaminated and p5 is uninfected before and after this contact (5) h2 is uncontaminated and p1 is infected with the nonresistant strain before this contact, and h2 is contaminated with the nonresistant strain during this contact (6) h3 is uncontaminated and p2 is uninfected before and after this contact (7) h1 is uncontaminated and p4 is infected with the resistant strain before this contact, and h1 is contaminated with the resistant strain during this contact (8) h3 is uncontaminated and p3 is uninfected before and after this contact (9) h2 is contaminated with the nonresistant strain and p5 is uninfected before this contact and p5 is infected with the nonresistant strain during this contact

18. (10) h1 is contaminated with the resistant strain and p1 is infected with the nonresistant strain before this contact, and p1 is super-infected with the resistant strain during this contact (11) h2 is contaminated with the nonresistant strain and p2 is uninfected before this contact, but h2 is not contaminated and p2 is not infected during this contact (12) h3 is uncontaminated and p5 is infected with the nonresistant strain before this contact, and h3 is contaminated with the nonresistant strain during this contact (13) h1 is contaminated with the resistant strain and p6 is uninfected before this contact, and p6 is not infected with the resistant strain during this contact (14) h2 is uncontaminated and p1 is infected with the resistant strain before this contact, but h2 is not contaminated with the resistant strain during this contact (15) h3 is contaminated with the nonresistant strain and p3 is uninfected before this contact, and h3 is not contaminated with the nonresistant strain and p3 is not infected with the nonresistant strain during this contact (16) h3 is uncontaminated and p4 is infected with the resistant strain before and after this contact (17) h2 is uncontaminated and p5 is infected with the nonresistant strain before and after this contact (18) h1 is uncontaminated and p2 is uninfected before and after this contact

19. A Mathematica code for the IBM Model

26. The status of hcw and patients at the end of the first shift is

27. The status of hcw and patients at the end of the last shift is

28. The epidemic over 3 days

29. The epidemic over a total 60 days

31. The epidemic over a total 60 days – the non-resistant strain extinguishes and the resistant strain dominates

32. The epidemic over a total 60 days – the non-resistant strain extinguishes and the resistant strain dominates

33. Parameters yielding the nonresistant strain as dominant (the other parameters and initial conditions remain the same)

34. The epidemic with new parameters over 60 days – the resistant strain extinguishes and the non-resistant strain becomes endemic

35. The epidemic with new parameters over 60 days – the resistant strain extinguishes and the non-resistant strain becomes endemic

36. Incorporation of the age of infection into the modelTwo levels of the 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

37. An ordinary differential equations model of the bacterial load at the individual patient 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

38. Model of nonresistant bacterial load V(a) at age a days of infection in a patient infected with the nonresistant strain If F >0, then limaVF(a)=F; if F<0, then limaVF(a)=0. bF=12.0log(2) before treatment (doubling time = 2 hr), bF=-2.0 after treatment, kF=1010.

39. Model of nonresistant bacterial load V-(a) and resistant bacterial load V+(a) at age a days of infection in a patient infected with both strains Equilibria of the model:E0 = (0,0), EF =(F0), and

40. Incorporation of antibiotic use into the model – contamination depends on the bacterial load and the number of day on antibioticsPatients infected with the non-resistant strain

41. Incorporation of antibiotic use into the model – contamination depends on the bacterial load and the number of day on antibioticsPatients infected with the resistant strain

42. Incorporation of antibiotic use into the model – contamination depends on the bacterial load and the number of day on antibioticsPatients infected with both the resistant and non-resistant strains

43. Application to an Intensive Care Unit (ICU) • ICUs have a critical problem with nosocomial epidemics and IBM models are very applicable because of the small number of patients and hcw involved. Objective: Specialize the basic model to an ICU and parameterize the models from recent scientific literature. • Add antibiotic use to the model: let the patient status depend on infection status • (s = u, n, r) and also the age of infection a on antibiotics pj[s,a]. • Allow the patient exit probability to depend on patient status and allow patient admission to depend on infection status. • Implement a control strategy of rotating antibiotics (1,2, or more) over time intervals (1, 2, or more months) for all patients. • Evaluate the efficacy of rotation strategies using the IBM model.

44. REFERENCES 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, D. Olivier, S. Ruan, and G.F. Webb, Modeling antibiotic resistance in hospitals: The impact of minimizing treatment duration, J. Theoret. Biol., Vol. 249 (2007), 487-499. D. Gruson, et. al, Rotation and restricted use of antibiotics in a medical intensive care unit, Am. J. Respir. Crit. Care Med., Vol. 162 (2000). D. Raymond, et. al, Impact of a rotating empiric antibiotic schedule on infectious mortality in an intensive care unit, Crit. Care Med., Vol. 29 (2001). K. Prubaker and R Weinstein, Trends in antimicrobial resistance in intensive care units in the United States, Current Opinion in Critical Care, Vol. 17, 2011.