Bulletin of Mathematical Biology

, Volume 75, Issue 9, pp 1434–1449 | Cite as

A Single-Parameter Model of the Immune Response to Bacterial Invasion

  • Lester F. CaudillJr.Email author
Original Article


The human immune response to bacterial pathogens is a remarkably complex process, involving many different cell types, chemical signals, and extensive lines of communication. Mathematical models of this system have become increasingly high-dimensional and complicated, as researchers seek to capture many of the major dynamics. In this paper, we argue that, in some important instances, preference should be given to low-dimensional models of immune response, as opposed to their high-dimensional counterparts. One such model is analyzed and shown to reflect many of the key phenomenological properties of the immune response in humans. Notably, this model includes a single parameter that, when combined with a single set of reference parameter values, may be used to quantify the overall immunocompetence of individual hosts.


Immune response Mathematical model Differential equations model Stability analysis Similarity parameter 



I am grateful to the University of Richmond for providing summer research support, and to Dr. Krista Stenger (Department of Biology, University of Richmond) for sharing her expertise in immunology.


  1. Antia, R., Bergstrom, C. T., Pilyugin, S. S., Kaech, S. M., & Ahmed, R. (2003). Models of CD8+ responses: 1. What is the antigen-independent proliferation program. J. Theor. Biol., 221(4), 585–598. MathSciNetCrossRefGoogle Scholar
  2. Asachenkov, A., Pogozhev, I., & Zuev, S. (1993). Parametrization in mathematical models of immune-physiological processes. Russ. J. Numer. Anal. Math. Model., 8(1), 31–46. CrossRefzbMATHGoogle Scholar
  3. Asachenkov, A., Marchuk, G., Mohler, R., & Zuev, S. (1994). Disease dynamics. Boston: Birkhäuser. Google Scholar
  4. Beck, K. (1981). A mathematical model of t-cell effects in the humoral immune response. J. Theor. Biol., 89, 593–610. CrossRefGoogle Scholar
  5. Bell, G. I. (1973). Predator–prey equations simulating an immune response. Math. Biosci., 16, 291–314. CrossRefzbMATHGoogle Scholar
  6. De Boer, R. J., & Boerlijst, M. C. (1994). Diversity and virulence thresholds in aids. Proc. Natl. Acad. Sci. USA, 94, 544–548. CrossRefGoogle Scholar
  7. Boman, H. G. (2000). Innate immunity and the normal microflora. Immunol. Rev., 173, 5–16. CrossRefGoogle Scholar
  8. Bruni, C., Giovenco, M. A., Koch, G., & Strom, R. (1975). A dynamical model of humoral immune response. Math. Biosci., 27, 191–211. CrossRefzbMATHGoogle Scholar
  9. Caudill, L., & Lawson, B. (2013). A hybrid agent-based and differential equations model for simulating antibiotic resistance in a hospital ward. Technical report TR-13-01, University of Richmond Mathematics and Computer Science. Google Scholar
  10. Chaui-Berlinck, J. G., Barbuto, J. A. M., & Monteiro, L. H. A. (2004). Conditions for pathogen elimination by immune systems. Theory Biosci., 123, 195–208. CrossRefGoogle Scholar
  11. Elgert, K. D. (2009). Immunology (2nd ed.). New York: Wiley-Blackwell. Google Scholar
  12. Fishman, M. A., & Perelson, A. S. (1993). Modeling t cell-antigen presenting cell interactions. J. Theor. Biol., 160, 311–342. CrossRefGoogle Scholar
  13. Fouchet, D., & Regoes, R. (2008). A population dynamics analysis of the interaction between adaptive regulatory t cells and antigen presenting cells. PLoS ONE, 3(5), e2306. CrossRefGoogle Scholar
  14. Goldmann, D. A., Weinstein, R. A., Wenzel, R. P., Tablan, O. C., Duma, R. J., Gaynes, R. P., Schlosser, J., & Martone, W. J. (1996). Strategies to prevent and control the emergence and spread of antimicrobial-resistant microorganisms in hospitals. JAMA, 275(3), 234–240. CrossRefGoogle Scholar
  15. Grossman, Z., Asofsky, R., & DeLisi, C. (1980). The dynamics of antibody-secreting cell production: regulation of growth and oscillations in the response to t-independent antigen. J. Theor. Biol., 84(1), 49–92. CrossRefGoogle Scholar
  16. Huang, X.-C. (1990). Uniqueness of limit cycles in a predator–prey model simulating an immune response. In R. Mohler & A. Asachenkov (Eds.), Selected topics on mathematical models in immunology and medicine, Laxenburg, Austria (pp. 147–153). IIASA. Google Scholar
  17. Klein, P., & Dolezal, J. (1980). A mathematical model of antibody response dynamics. Probl. Control Inf. Theory, 9, 407–419. zbMATHGoogle Scholar
  18. Lee, H. Y., Topham, D. J., Park, S. Y., Hollenbaugh, J., Treanor, J., Mosmann, T. R., Jin, X., Ward, B. M., Miao, H., Holden-Wiltse, J., Perelson, A. S., Zand, M., & Wu, H. (2009). Simulation and prediction of the adaptive immune response to influenza a virus infection. J. Virol., 83(14), 7151–7165. CrossRefGoogle Scholar
  19. Mackay, I., & Rosen, F. S. (2000). Advances in immunology. N. Engl. J. Med., 343, 338–344. CrossRefGoogle Scholar
  20. Marchuk, G. I. (1997). Mathematical modeling of immune response in infectious diseases. Boston: Kluwer. CrossRefGoogle Scholar
  21. McLean, A. R. (1994). Modeling t cell memory. J. Theor. Biol., 170, 63–74. CrossRefGoogle Scholar
  22. Moellering, R., & Blumgart, H. (2002). Understanding antibiotic resistance development in the immunocompromised host. Int. J. Infect. Dis., 6, S3–S4. CrossRefGoogle Scholar
  23. Mohler, R. R., Barton, C. F., & Hsu, C.-S. (1978). T and b cells in the immune system. In G. I. Bell, A. S. Perelson, & G. H. Pimbley (Eds.), Theoretical immunology (pp. 415–435). New York: Marcel-Dekker. Google Scholar
  24. Nowak, M. A., & Bangham, C. R. M. (1996). Population dynamics of immune responses to persistent viruses. Science (NS), 272(5258), 74–79. CrossRefGoogle Scholar
  25. Nowak, M. A., May, R. M., & Sigmund, K. (1995). Immune responses against multiple epitopes. J. Theor. Biol., 175, 325–353. CrossRefGoogle Scholar
  26. Pogozhev, I., Usmanov, R., & Zuev, S. (1993). Models of processes in organism and population characteristics. Russ. J. Numer. Anal. Math. Model., 8(5), 441–452. CrossRefzbMATHGoogle Scholar
  27. Prikrylova, D., Jilek, M., & Waniewski, J. (1992). Mathematical modeling of the immune response. Boca Raton: CRC Press. zbMATHGoogle Scholar
  28. Rundell, A., DeCarlo, R., HogenEsch, H., & Doerschuk, P. (1998). The humoral immune response hemophilus influenzae type: a mathematical model based on t-zone ad germinal center b-cell dynamics. J. Theor. Biol., 194, 341–381. CrossRefGoogle Scholar
  29. Spellberg, B., Guidos, R., Gilbert, D., Bradley, J., Boucher, H. W., Scheld, W. M., Bartlett, J. G., & Edwards, J. (2008). The epidemic of antibiotic-resistant infections: a call to action for the medical community from the Infectious Diseases Society of America. Clin. Infect. Dis., 46, 155–164. CrossRefGoogle Scholar
  30. Usmanov, R., & Zuev, S. (1993). Parametrization in mathematical models of diseases. Russ. J. Numer. Anal. Math. Model., 8(3), 275–284. MathSciNetCrossRefzbMATHGoogle Scholar
  31. Waltman, P. (1978). A threshold model of antigen-stimulated antibody production. In G. I. Bell, A. S. Perelson, & G. H. Pimbley (Eds.), Theoretical immunology (pp. 437–453). New York: Marcel-Dekker. Google Scholar
  32. Weinand, R. G., & Conrad, M. (1988). Maturation of the immune response: a computational model. J. Theor. Biol., 133, 409–428. CrossRefGoogle Scholar
  33. Wodarz, D., & Nowak, M. A. (2000). Correlates of cytotoxic T-lymphocytemediated virus control: implications for immuno-suppressive infections and their treatment. Philos. Trans. R. Soc. Lond. B, 355, 1059–1070. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of RichmondRichmondUSA

Personalised recommendations