Bulletin of Mathematical Biology

, Volume 71, Issue 4, pp 845–862

Dynamics of Indirectly Transmitted Infectious Diseases with Immunological Threshold

  • Richard I. Joh
  • Hao Wang
  • Howard Weiss
  • Joshua S. Weitz
Original Article


There are numerous examples of human pathogens which persist in environmental reservoirs while infectious outbreaks remain rare. In this manuscript, we consider the dynamics of infectious diseases for which the primary mode of transmission is indirect and mediated by contact with a contaminated reservoir. We evaluate the realistic scenario in which the number of ingested pathogens must be above a critical threshold to cause infection in susceptible individuals. This minimal infectious dose is a consequence of the clearance effect of the innate immune system. Infected individuals shed pathogens back into the aquatic reservoir, indirectly increasing the transmittability of the pathogen to the susceptible. Building upon prior works in the study of cholera dynamics, we introduce and analyze a family of reservoir mediated SIR models with a threshold pathogen density for infection. Analyzing this family of models, we show that an outbreak can result from noninfinitesimal introductions of either infected individuals or additional pathogens in the reservoir. We devise two new measures of how likely it is that an environmentally persistent pathogen will cause an outbreak: (i) the minimum fraction of infected individuals; and (ii) the minimum fluctuation size of in-reservoir pathogens. We find an additional control parameter involving the shedding rate of infected individuals, which we term the pathogen enhancement ratio, which determines whether outbreaks lead to epidemics or endemic disease states. Thus, the ultimate outcome of disease is controlled by the strength of fluctuations and the global stability of a nonlinear dynamical system, as opposed to conventional analysis in which disease reflects the linear destabilization of a disease free equilibrium. Our model predicts that in the case of waterborne diseases, suppressing the pathogen density in aquatic reservoirs may be more effective than minimizing the number of infected individuals.


Epidemic Endemic Cholera SIR Minimum infectious dose 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bove, F.J., Fulcomer, M.C., Klotz, J.B., Esmart, J., Dufficy, E.M., Savrin, J.E., 1995. Public drinking water contamination and birth outcomes. Am. J. Epidemiol. 141, 850–62. Google Scholar
  2. Brayton, P.R., Tamplin, M.L., Huq, A., Colwell, R.R., 1987. Enumeration of vibrio cholerae O1 in Bangladesh waters by fluorescent-antibody direct viable count. Appl. Environ. Microbiol. 53, 2862–865. Google Scholar
  3. Capasso, V., Paveri-Fontana, S.L., 1979. A mathematical model for the 1973 cholera epidemic in the European Mediterranean region. Rev. Epidém. Santé Pub. 27, 121–32. Google Scholar
  4. Chtsulo, L., Engles, D., Montresor, A., Savioli, L., 2000. The global status of schistosomiasis and its control. Acta Trop. 77, 41–1. CrossRefGoogle Scholar
  5. Codeço, C.T., 2001. Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir. BMC Infect. Dis. 1, 1. CrossRefGoogle Scholar
  6. Colwell, R.R., Bryaton, P., Herrington, D., Tall, B., Huq, A., Levine, M.M., 1996. Viable but nonculturable vibrio cholerae revert to a cultivable state in the human intestine. World J. Microbiol. Biotechnol. 12, 28–1. CrossRefGoogle Scholar
  7. Craig, M.H., Snow, R.W., le Sueur, D., 1999. A climate-based distribution model of malaria transmission in sub-Saharan Africa. Parasitol. Today 15, 105–11. CrossRefGoogle Scholar
  8. Dietz, K., 1993. The estimation of the basic reproduction number for infectious diseases. Stat. Meth. Med. Res. 2, 23–1. CrossRefGoogle Scholar
  9. DuPont, H.L., Chappell, C.L., Sterling, C.R., Okhuysen, P.C., Rose, J.B., Jakubowski, W., 1995. The infectivity of cryptosporidium parvum in healthy volunteers. N. Eng. J. Med. 332, 855–59. CrossRefGoogle Scholar
  10. Estes, M.K., Palmer, E.L., Obijeski, J.F., 1983. Rotavirus: a review. Curr. Top. Microbiol. Immunol. 105, 123–84. Google Scholar
  11. Fields, B.S., Benson, R.F., Besser, R.E., 2002. Legionella and legionnaires’ disease: 25 years of investigation. Clin. Microbiol. Rev. 15, 506–26. CrossRefGoogle Scholar
  12. Häder, D.P., Kumar, H.D., Smith, R.C., Worrest, R.C., 1998. Effects on aquatic ecosystems. J. Photochem. Photobiol. B: Biol. 46, 53–8. CrossRefGoogle Scholar
  13. Hartley, D.M., Morris, J.G., Smith, D.L., 2006. Hyperinfectivity: A critical element in the ability of v. cholerae to cause epidemics? PLoS Med. 3, e7. CrossRefGoogle Scholar
  14. Holling, C.S., 1959. The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Can. Entomol. 91, 293–20. CrossRefGoogle Scholar
  15. Jensen, P.K., Ensink, J.H.J., Jayasinghe, G., van de Hoek, W., Cairncross, S., Dalsgaard, A., 2002. Domestic transmission routes of pathogens: the problem of in-house contamination of drinking water during storage in developing countries. Trop. Med. Int. Health 7, 604–09. CrossRefGoogle Scholar
  16. Jensen, M.A., Faruque, S.M., Mekalanos, J.J., Levin, B.R., 2006. Modeling the role of bacteriophage in the control of cholera outbreaks. Proc. Natl. Acad. Sci. USA 103, 4652–657. CrossRefGoogle Scholar
  17. Kaper, J.B., Morris Jr., J.G., Levine, M.M., 1995. Cholera. Clin. Microbiol. Rev. 8, 48–6. Google Scholar
  18. Kermack, W.O., McKendrik, A.G., 1927. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A 115, 700–21. CrossRefGoogle Scholar
  19. LeChevallier, M.W., Norton, W.D., Lee, R.G., 1991. Giardia and cryptosporidium spp. in filtered drinking water supplies. Appl. Environ. Microbiol. 57, 2617–621. Google Scholar
  20. Levine, M.M., Black, R.E., Clements, M.L., Nalin, D.R., Cisneros, L., Finkelstein, R.A., 1981. Volunteer studies in development of vaccines against cholera and enterotoxigenic escherichia coli: a review. In: Holme, T., Holmgren, J., Merson, M.H., Mollby, R. (Eds.), Acute Enteric Infections in Children. New Prospects for Treatment and Prevention, pp. 443–59. Elsevier/North-Holland Biomedical Press, Amsterdam. Google Scholar
  21. Macdonald, G., 1952. The analysis of equilibrium in malaria epidemiology. Trop. Dis. Bull. 49, 813–29. Google Scholar
  22. Mourino-Pérez, R.R., Worden, A.Z., Azam, F., 2003. Growth of vibrio cholerae O1 in red tide waters off California. Appl. Environ. Microbiol. 69, 6923–931. CrossRefGoogle Scholar
  23. Murphy, K.M., Travers, P., Walport, M., 2007. Janeway’s Immunobiology, 7th edn. Garland Science. Google Scholar
  24. Pascual, M., Rodó, X., Ellner, S.P., Colwell, R., Bouma, M.J., 2000. Cholera dynamics and El Niño-Southern oscillation. Science 289, 1766–769. CrossRefGoogle Scholar
  25. Real, L., 1977. The kinetics of functional response. Am. Nat. 111, 289–00. CrossRefGoogle Scholar
  26. Rendtorff, R.C., 1954. The experimental transmission of human intestinal protozoan parasites: Giardia lamblia cysts given in capsules. Am. J. Hyg. 59, 209–20. Google Scholar
  27. Rose, J.B., 1997. Environmental ecology of crptosporidiumm and public health implications. Annu. Rev. Public Health 18, 135–61. CrossRefGoogle Scholar
  28. Ross, R., 1908. Report on the Prevention of Marlaria in Mauritius. Waterlow and Sons Ltd. Google Scholar
  29. Strogatz, S.H., 1994. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Perseus Books. Google Scholar
  30. Ward, R.L., Bernstein, D.I., Young, E.C., Sherwood, J.R., Knowlton, D.R., Shiff, G.M., 1986. Human rotavirus studies in volunteers: determination of infectious dose and serological response to infection. J. Infectious Dis. 154, 871–80. Google Scholar
  31. Webster, A.D., 1980. Giardiasis and immunodeficiency diseases. Trans. R. Soc. Trop. Med. Hyg. 74, 440–43. CrossRefGoogle Scholar
  32. White, P.O., Fenner, F.J., 1994. Medical Virology, 4th edn. Academic, San Diego. Google Scholar
  33. Wolfe, M.S., 1992. Giardiasis. Clin. Microbiol. Rev. 5, 93–00. Google Scholar
  34. Wolfe, N.D., Dunovan, C.P., Diamond, J., 2007. Origins of major human infectious diseases. Nature 447, 279–83. CrossRefGoogle Scholar
  35. Yoganathan, D., Rom, W.N., 2001. Medical aspects of global warming. Am. J. Ind. Med. 40, 199–10. CrossRefGoogle Scholar
  36. Zaki, S.R., et al., 1995. Hantavirus pulmonary syndrome—pathogenesis of an emerging infectious disease. Am. J. Pathol. 146, 552–79. Google Scholar

Copyright information

© Society for Mathematical Biology 2008

Authors and Affiliations

  • Richard I. Joh
    • 1
  • Hao Wang
    • 2
  • Howard Weiss
    • 2
  • Joshua S. Weitz
    • 1
    • 3
  1. 1.School of PhysicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  3. 3.School of BiologyGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations