Journal of Mathematical Biology

, Volume 75, Issue 6–7, pp 1693–1713 | Cite as

Optimal control of vaccination rate in an epidemiological model of Clostridium difficile transmission

  • Brittany Stephenson
  • Cristina Lanzas
  • Suzanne Lenhart
  • Judy Day


The spore-forming, gram-negative bacteria Clostridium difficile can cause severe intestinal illness. A striking increase in the number of cases of C. difficile infection (CDI) among hospitals has highlighted the need to better understand how to prevent the spread of CDI. In our paper, we modify and update a compartmental model of nosocomial C. difficile transmission to include vaccination. We then apply optimal control theory to determine the time-varying optimal vaccination rate that minimizes a combination of disease prevalence and spread in the hospital population as well as cost, in terms of time and money, associated with vaccination. Various hospital scenarios are considered, such as times of increased antibiotic prescription rate and times of outbreak, to see how such scenarios modify the optimal vaccination rate. By comparing the values of the objective functional with constant vaccination rates to those with time-varying optimal vaccination rates, we illustrate the benefits of time-varying controls.


Optimal control Clostridium difficile Vaccination Ordinary differential equations Healthcare setting 

Mathematics Subject Classification

49J15 92B05 92C60 


  1. Alasmari F, Seiler S, Hink T, Burnham C, Dubberke E (2014) Prevalence and risk factors for asymptomatic Clostridium difficile carriage. Clin Infect Dis 59(2):216–222CrossRefGoogle Scholar
  2. Asano E, Gross L, Lenhart S, Real L (2008) Optimal control of vaccine distribution in a rabies metapopulation model. Math Biosci Eng 5(2):219–238MathSciNetCrossRefMATHGoogle Scholar
  3. Clabots C, Johnson S, Olson M, Peterson L, Gerding D (1992) Acquisition of Clostridium difficile by hospitalized patients: evidence for colonized new admissions as a source of infection. J Infect Dis 166(3):561–567CrossRefGoogle Scholar
  4. Ding W, Webb G (2016) Optimal control applied to community-acquired methicillin-resistant Staphylococcus aureus in hospitals. J Biol Dyn. doi: 10.1080/17513758.2016.1151564
  5. Dubberke E, Carling P, Carrico R, Donskey C, Loo V, McDonald L, Maragakis L, Sandora T, Weber D, Yokoe D et al (2014) Strategies to prevent Clostridium difficile infections in acute care hospitals: 2014 update. Infect Control Hosp Epidemiol 35(06):628–645CrossRefGoogle Scholar
  6. Dubberke E, Haslam D, Lanzas C, Bobo L, Burnham CA, Grohn Y (2011) The ecology and pathobiology of Clostridium difficile infections: an interdisciplinary challenge. Zoonoses Public Health 58(1):4–20CrossRefGoogle Scholar
  7. Fister R, Lenhart S, McNally J (1998) Optimizing chemotherapy in an hiv model. Electron J Differ Equ 32:1–12MathSciNetMATHGoogle Scholar
  8. Gaff H, Schaefer E (2009) Optimal control applied vaccination and treatment strategies for various epidemiological models. Math Biosci Eng 6:469–492MathSciNetCrossRefMATHGoogle Scholar
  9. Ghosh-Dasitar U, Lenhart S (2015) Modeling the effect of vaccines on cholera. J Biol Syst 23(02):323–338CrossRefMATHGoogle Scholar
  10. Johnson S, Gerding D (1998) Clostridium difficile-associated diarrhea. Clin Infect Dis 26(5):1027–1034CrossRefGoogle Scholar
  11. Kelly M Jr, Tien J, Eisenberg M, Lenhart S (2016) The impact of spatial arrangements on epidemic disease dynamics and intervention strategies. J Biol Dyn 10:222–249MathSciNetCrossRefGoogle Scholar
  12. Kyne L, Warny M, Qamar A, Kelly C (2000) Asymptomatic carriage of Clostridium difficile and serum levels of igg antibody against toxin a. N Engl J Med 342(6):390–397CrossRefGoogle Scholar
  13. Kyne L, Warny M, Qamar A, Kelly C (2000) Asymptomatic carriage of Clostridium difficile and serum levels of IgG antibody against toxin A. N Engl J Med 342(6):390–397CrossRefGoogle Scholar
  14. Lanzas C, Dubberke E, Lu Z, Reske K, Grohn Y (2011) Epidemiological model for Clostridium difficile transmission in healthcare settings. Infect Control Hosp Epidemiol 32(6):553–561CrossRefGoogle Scholar
  15. Lee B, Popovich M, Tian Y, Bailey R, Ufberg P, Wiringa A, Muder R (2010) The potential value of Clostridium difficile vaccine: an economic computer simulation model. Vaccine 28(32):5245–5253CrossRefGoogle Scholar
  16. Leffler D, Lamont T (2015) Clostridium difficile infection. N Engl J Med 372(16):1539–1548CrossRefGoogle Scholar
  17. Lenhart S, Workman J (2007) Optimal control applied to biological models. CRC Press, New YorkMATHGoogle Scholar
  18. Lessa F, Mu Y, Bamberg W, Beldavs Z, Dumyati G, Dunn J, Farley M, Holzbauer S, Meek J, Phipps E, Wilson L, Winston L, Cohen J, Limbago B, Fridkin S, Gerding D, McDonald L (2015) Burden of Clostridium difficile infection in the united states. N Engl J Med 372(9):825–834CrossRefGoogle Scholar
  19. Leuzzi R, Adamo R, Scarselli M (2014) Vaccines against Clostridium difficile. Hum Vaccines Immunother 10(6):1466–1477CrossRefGoogle Scholar
  20. Lowden J, Beilan R, Yahdi M (2014) Optimal control of vancomycin-resistant enterococci using preventive care and treatment of infections. Math Biosci 249:8–17MathSciNetCrossRefMATHGoogle Scholar
  21. McDonald L, Killgore G, Thompson A, Owens RJ, Kazakova S, Sambol S, Johnson S, Gerding D (2005) An epidemic, toxin gene-variant strain of Clostridium difficile. N Engl J Med 353(23):2433–2441CrossRefGoogle Scholar
  22. McFarland L (2008) Update on the changing epidemiology of Clostridium difficile-associated disease. Nat Clin Pract Gastroenterol Hepatol 5(1):40–48MathSciNetCrossRefGoogle Scholar
  23. McFarland L, Mulligan M, Kwok R, Stamm W (1989) Nosocomial acquisition of Clostridium difficile infection. N Engl J Med 320(4):204–210CrossRefGoogle Scholar
  24. Miller-Neilan R, Schaefer E, Gaff H, Fister K, Lenhart S (2010) Modeling optimal intervention strategies for cholera. Bull Math Biol 1(4):379–393MathSciNetMATHGoogle Scholar
  25. Pontryagin L, Boltyanskii V, Gamkrelize R, Mishchenko E (1962) The mathematical theory of optimal processes. Wiley, New YorkGoogle Scholar
  26. Rafii F, Sutherland J, Cerniglia C (2008) Effects of treatment with antimicrobial agents on the human colonic microflora. Ther Clin Risk Manag 4(6):1343–1358CrossRefGoogle Scholar
  27. Samore M, Venkataraman L, DeGirolami P, Arbeit R, Karchmer A (1996) Clinical and molecular epidemiology of sporadic and clustered cases of nosocomial Clostridium difficile infection. N Engl J Med 100(1):32–40Google Scholar
  28. Siewe N, Yakubu A, Satoskar A, Friedman A (2016) Immune response to infection by Leishmania: a mathematical model. Math Biosci 276:28–43MathSciNetCrossRefMATHGoogle Scholar
  29. Smith H (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. American Mathematical Society, Rhode IslandMATHGoogle Scholar
  30. Starr J, Campbell A, Renshaw E, Poxton I, Gibson G (2009) Spatiotemporal stochastic modeling of Clostridium difficile. J Hosp Infect 71(1):49–56CrossRefGoogle Scholar
  31. Viscidi R, Laughon B, Yolken R, Bo-Linn P, Moench T, Ryder R, Bartlett J (1983) Serum antibody response to toxins a and b of Clostridium difficile. J Infect Dis 148(1):93–100CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TennesseeKnoxvilleUSA
  2. 2.Department of Population Health and PathobiologyNorth Carolina State UniversityRaleighUSA
  3. 3.Departments of Mathematics and Electrical Engineering and Computer ScienceUniversity of TennesseeKnoxvilleUSA

Personalised recommendations