Abstract
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.
Similar content being viewed by others
References
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):216–222
Asano E, Gross L, Lenhart S, Real L (2008) Optimal control of vaccine distribution in a rabies metapopulation model. Math Biosci Eng 5(2):219–238
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):561–567
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
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):628–645
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):4–20
Fister R, Lenhart S, McNally J (1998) Optimizing chemotherapy in an hiv model. Electron J Differ Equ 32:1–12
Gaff H, Schaefer E (2009) Optimal control applied vaccination and treatment strategies for various epidemiological models. Math Biosci Eng 6:469–492
Ghosh-Dasitar U, Lenhart S (2015) Modeling the effect of vaccines on cholera. J Biol Syst 23(02):323–338
Johnson S, Gerding D (1998) Clostridium difficile-associated diarrhea. Clin Infect Dis 26(5):1027–1034
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 10:222–249
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):390–397
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):390–397
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):553–561
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):5245–5253
Leffler D, Lamont T (2015) Clostridium difficile infection. N Engl J Med 372(16):1539–1548
Lenhart S, Workman J (2007) Optimal control applied to biological models. CRC Press, New York
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):825–834
Leuzzi R, Adamo R, Scarselli M (2014) Vaccines against Clostridium difficile. Hum Vaccines Immunother 10(6):1466–1477
Lowden J, Beilan R, Yahdi M (2014) Optimal control of vancomycin-resistant enterococci using preventive care and treatment of infections. Math Biosci 249:8–17
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):2433–2441
McFarland L (2008) Update on the changing epidemiology of Clostridium difficile-associated disease. Nat Clin Pract Gastroenterol Hepatol 5(1):40–48
McFarland L, Mulligan M, Kwok R, Stamm W (1989) Nosocomial acquisition of Clostridium difficile infection. N Engl J Med 320(4):204–210
Miller-Neilan R, Schaefer E, Gaff H, Fister K, Lenhart S (2010) Modeling optimal intervention strategies for cholera. Bull Math Biol 1(4):379–393
Pontryagin L, Boltyanskii V, Gamkrelize R, Mishchenko E (1962) The mathematical theory of optimal processes. Wiley, New York
Rafii F, Sutherland J, Cerniglia C (2008) Effects of treatment with antimicrobial agents on the human colonic microflora. Ther Clin Risk Manag 4(6):1343–1358
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):32–40
Siewe N, Yakubu A, Satoskar A, Friedman A (2016) Immune response to infection by Leishmania: a mathematical model. Math Biosci 276:28–43
Smith H (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. American Mathematical Society, Rhode Island
Starr J, Campbell A, Renshaw E, Poxton I, Gibson G (2009) Spatiotemporal stochastic modeling of Clostridium difficile. J Hosp Infect 71(1):49–56
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):93–100
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the joint DMS/NIGMS Mathematical Biology Program through NIH award #R01GM113239.
Rights and permissions
About this article
Cite this article
Stephenson, B., Lanzas, C., Lenhart, S. et al. Optimal control of vaccination rate in an epidemiological model of Clostridium difficile transmission. J. Math. Biol. 75, 1693–1713 (2017). https://doi.org/10.1007/s00285-017-1133-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-017-1133-6