Vanderbilt University Department of Mathematics Mathematical Modelling and Challenges in the Development of Drug Resistance Mary Ann Horn Vanderbilt University Department of Mathematics Joint work with Erika D’Agata, Harvard Medical School and Glenn Webb, Vanderbilt University

Vanderbilt University Department of Mathematics 2 Enterococci What are enterococci? Enterococci are bacteria found in the faeces of most humans and many animals. Two types of enterococci are associated with normal healthy people, Enterococcus faecalis and Enterococcus faecium. Photo credit: University of Oklahoma Health Sciences Center

Vanderbilt University Department of Mathematics 3 Enterococci Issues Associated with both community and hospital-acquired infections Among the vanguard of antibiotic resistant bacteria Have acquired resistance genes to counter antibiotics that were once effective Photo credit: University of Oklahoma Health Sciences Center

Vanderbilt University Department of Mathematics 4 Nosocomial Infections What does “nosocomial” mean? -- Michael J. Berens, Chicago Tribune, July 22, 2002 “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.”

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Vanderbilt University Department of Mathematics 6 What infections are caused by enterococci? Most common infections are urinary tract infections and wound infections. Infections threatening severely ill patients include infection of the bloodstream (bacteraemia), heart valves (endocarditis) and the brain (meningitis). Enterococci frequently colonize open wounds and skin ulcers.

Vanderbilt University Department of Mathematics 7 Antibiotic Resistance Most enterococci have inherent resistance to various drugs –Cephalosporins –Semi-synthetic penicillinase-resistant penicillins –Clindamycin –Aminoglycosides Relatively resistant to other drugs –Penicillin –Ampicillin Tolerant to cell-wall active agents –Ampicillin –Vancomycin

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Vanderbilt University Department of Mathematics 9 Antibiotic Resistance (con’t) Developed resistance –Plasmid-resistance Definition: A plasmid is an extrachromosomal ring of DNA (particularly of bacteria that replicate autonomously) –Transposon-mediated resistance Tetracycline, minocycline, doxycycline Erythromycin, azithromycin, clarithromycin Etc. Developed within the past decade

Vanderbilt University Department of Mathematics 10 Antibiotic Resistance (con’t) Development of Multi-drug Resistance –Variety of different mechanisms for bacterial mating Pheromone responsive plasmids Broad host-range plasmids –Transfer among species of enterococci Conjugative transposons –Transfer genetic information from cell to cell

Vanderbilt University Department of Mathematics 11 Vancomycin Resistance Resistance to vancomycin unknown until First vancomycin-resistant enterococcus found in France. First strain isolated in 1987 in the United Kingdom. Similar strains now found world-wide.

Vanderbilt University Department of Mathematics 12 How does vancomycin resistance arise? Genetic mechanism gives rise to resistance –Models for phenotype evolution incorporating mutation, selection, and recombination exist –Mutation is typically modeled by diffusion Related antibiotics included in animal feed, resulting in acquisition after ingestion Antibiotic therapy in hospitals

Vanderbilt University Department of Mathematics 13 Who is susceptible to VRE? Patients who have been in hospital for extended periods. Patients who have received certain antibiotics (vancomycin, teicoplanin, cephalosporins). Patients fed by naso-gastric tube. Outbreaks primarily reported from renal dialysis, transplant, haematology and ICUs.

Vanderbilt University Department of Mathematics 14 Vancomycin-Resistant Enterococci (VRE) Treatment challenges Range of antibiotics available for treatment are extremely limited. Choice of antibiotics for treatment dependent upon strain. Treatment delay due to time needed for laboratory results.

Vanderbilt University Department of Mathematics 15 Patient Dynamics UNCOLONIZED PATIENTS OFF ANTIBIOTICS P u0 COLONIZED PATIENTS OFF ANTIBIOTICS P c0 UNCOLONIZED PATIENTS ON ANTIBIOTICS P u1 COLONIZED PATIENTS ON ANTIBIOTICS P c1  u0  p  p (1-  )  (Y h /N h )  u1  u1  u0 new patients admitted new patients admitted  c0  c0  u0 Start antibiotics Stop antibiotics  u1  c1  c1 new patients admitted new patients admitted length of stay Start antibiotics  c0  c1 Stop antibiotics

Vanderbilt University Department of Mathematics 16 Health Care Worker Dynamics UNCONTAMINATED HEALTH CARE WORKERS H u CONTAMINATED HEALTH CARE WORKERS H c  p1  h1  (P c1 /N p )  contamination from colonized patients on antibiotics length of contamination contamination from colonized patients on antibiotics  p0  h0  (P c0 /N p )

Vanderbilt University Department of Mathematics 17 Epidemic Model for VRE in a Hospital

Vanderbilt University Department of Mathematics 18 Parameters of the Model

Vanderbilt University Department of Mathematics 19 SIMULATIONS WITH VARIABLE PATIENT-HCW RATIO Patient-HCW ratios are  = 1 (red), 2 (green), 4 (blue), 6 (yellow) and 8 (purple)

Vanderbilt University Department of Mathematics 20 SIMULATIONS WITH VARIABLE HYGIENE COMPLIANCE Hygiene compliance values are  =.1 (red),.3 (green),.5 (blue),.7 (yellow) and.9 (purple)

Vanderbilt University Department of Mathematics 21 SIMULATIONS WITH VARIABLE ANTIBIOTIC STOPPAGE OF COLONIZED PATIENTS Per day discontinuation of antibiotics (  c1) of VRE colonized patients = 4% (red), 7% (green), 10% (blue), 15% (yellow) and 20% (purple)

Vanderbilt University Department of Mathematics 22 SIMULATIONS WITH VARIABLE ANTIBIOTIC STOPPAGE OF UNCOLONIZED PATIENTS Per day discontinuation of antibiotics (  u1) of VRE uncolonized patients = 4% (red), 7% (green), 10% (blue), 15% (yellow) and 20% (purple)

Vanderbilt University Department of Mathematics 23 SIMULATIONS WITH VARIABLE LENGTH OF STAY OF COLONIZED PATIENTS ON ANTIBIOTICS Length of hospital stay for VRE colonized patients on antibiotics (  c1) = 10 days (red), 14 days (green), 21 days (blue), 28 days (yellow) and 35 days (purple)

Vanderbilt University Department of Mathematics 24 SIMULATION OF THE MODEL WITH NO ADMISSIONS OF COLONIZED PATIENTS (  c0 =  c1 =0) All parameters have baseline values except that the handwashing compliance  = 0.7. The colonized patient compartments extinguish over time.

Vanderbilt University Department of Mathematics 25 Vancomycin-Resistant Enterococci (VRE) Preventing the spread of VRE--Conclusions Restrict use of antibiotics, especially vancomycin, teicoplanin and cephalosporins. Enforce scrupulous handwashing by all hospital staff. Lower the ratio of patients to health care workers. Cohort colonized patients.

Vanderbilt University Department of Mathematics 26 Steady States of the Model (with  c0 =  c1 =0) VRE Free Steady State VRE Endemic Steady State

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Vanderbilt University Department of Mathematics 28 Epidemic Model for VRE in a Hospital Conclusions If  c0 and  c1 are assumed to be 0, then R 0 can be calculated for this model (that is, there is no input of colonized patients either on or off antibiotics). R 0 is the number of secondary infections produced by a single infective in a new population of susceptibles. If R 0 is greater than 1, then VRE becomes endemic in the hospital. If R 0 is less than 1, then VRE extinguishes in the hospital. If either  c0 or  c1 is assumed to be positive, then VRE always becomes endemic.

Vanderbilt University Department of Mathematics 29 Lower patient-healthcare worker ratios limit the prevalence of patients colonized with VRE (the benefit is less significant for higher ratios). Epidemic Model for VRE in a Hospital Conclusions

Vanderbilt University Department of Mathematics 30 Epidemic Model for VRE in a Hospital Conclusions Improved compliance with handwashing limits the prevalence of patients colonized with VRE (the benefit is more significant for higher compliance values).

Vanderbilt University Department of Mathematics 31 Epidemic Model for VRE in a Hospital Conclusions Starting unnecessary antimicrobial therapy has a greater impact when targeted to patients who are not colonized with VRE, compared to patients colonized with VRE

Vanderbilt University Department of Mathematics 32 Epidemic Model for VRE in a Hospital Conclusions Stopping unnecessary antimicrobial therapy has a greater impact when targeted to patients who are not colonized with VRE, as compared to patients colonized with VRE

Vanderbilt University Department of Mathematics 33 Epidemic Model for VRE in a Hospital Conclusions Prolonging the duration of hospitalization of colonized patients increases the prevalence of VRE (but the increase is less significant for longer LOS). 

Vanderbilt University Department of Mathematics 34 Final Thoughts Discouraging News First case of vancomycin-resistant Staphylococcus aureus (VRSA) confirmed in 40-year-old Michigan diabetic with kidney failure in July 2002 In the U.S., methicillin-resistant Staphylococcus aureus (MRSA) rates as high as 60% in some facilities

Vanderbilt University Department of Mathematics 35 Final Thoughts Hope for the Future Pharma companies working on veterinary phages that counter E. coli and salmonella in animals are now moving into human infections such as MRSA and VRE Immunization approaches under development –Vaccine preventing middle ear infections in children on the market –Clinical studies underway for an anti-MRSA vaccine called StaphVac –A TB vaccine is now being tested on animals that could counter the multidrug-resistant strain now endemic in the third world