## Abstract

The longstanding debate as to whether or not chaotic dynamics occur in natural populations (Hastings et al. 1993) arises because various natural populations exhibit apparently “random” fluctuations in abundance. Furthermore, in several instances these fluctuations will look random not only to the naked eye but also through the “eyes” of some standard time series analysis (Ellner 1989). Several simple nonlinear mathematical ecological models that exhibit chaos for apparently reasonable parameter values have been studied over the last few years. Initially chaos was found in simple discrete time models for single species (May 1974, May 1976; May and Oster 1976). Chaotic dynamics have also been encountered in discrete models with age structure (Ebenman 1987; Guckenheimer et al. 1977; Levin 1981; Levin and Goodyear 1980; in discrete models with two species (Allen 1989a,Allen 1989b, Allen 1990a,Allen 1990b, Allen 1991; Beddington et al. 1975; Bellows and Hassell 1988; May 1974; May and Oster 1976; Neubert and Kot 1992); in simple discrete models of parasites (May 1985); in models for host-parasitoid-pathogen systems (Hochberg et al. 1990); in discrete demographic models with two sexes (Caswell and Weeks 1986); and in models that include frequency-dependent selection (Altenberg 1991; Cressman 1988; May and Anderson 1983a).

## Keywords

Human Immunodeficiency Virus Epidemic Model Transmission Dynamic Endemic Equilibrium Theoretical Biology## Preview

Unable to display preview. Download preview PDF.

## References

- Allen, J. C. 1989a. Are natural enemy populations chaotic?
*Estimation and Analysis of Insect Populations*, Lecture Notes in Statistics**55**:190–205.Google Scholar - Allen, J. C. 1989b. Patch efficient parasitoids, chaos, and natural selection.
*Florida Entomologist***79**:52–64.Google Scholar - Allen, J. C. 1990a. Factors contributing to chaos in population feedback systems.
*Ecological Modelling***51**:281–298.Google Scholar - Allen, J. C. 1990b. Chaos in phase-locking in predator-prey models in relation to the functional response.
*Florida Entomologist***73**:100–110.Google Scholar - Allen, J. C. 1991. Chaos and coevolutionary warfare in a chaotic predator-prey system.
*Florida Entomologist***74**:50–59.Google Scholar - Altenberg, L. 1991. Chaos from linear frequency-dependent selection.
*American Naturalist***138**:51–68.Google Scholar - Anderson, R. M. (ed.). 1982.
*Population Dynamics of Infectious Diseases: Theory and Applications*. Chapman & Hall, London and New York.Google Scholar - Anderson, R. M., S. P. Blythe, S. Gupta, and E. Konings. 1989. The transmission dynamics of the Human Immunodeficiency Virus Type I in the male homosexual community in the United Kingdom: The influence of changes in sexual behavior.
*Philosophical Transactions of the Royal Society of London B***325**:145–198.Google Scholar - Anderson, R. M., and R. M. May. 1987. Transmission dynamics of HIV infection.
*Nature***326**:137–142.PubMedGoogle Scholar - Anderson, R. M., and R. M. May. 1991.
*Infectious Diseases of Humans*. Oxford Science Publications, Great Britain.Google Scholar - Anderson, R. M., R. M. May, G. F. Medley, and A. Johnson. 1986. A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS.
*IMA Journal of Mathematics Applied in Medicine and Biology***3**:229–263.PubMedGoogle Scholar - Bailey, N. T. J. 1975.
*The Mathematical Theory of Infectious Diseases and Its Applications*. Griffin, London.Google Scholar - Beddington, J. R., C. A. Free, and J. H. Lawton. 1975. Dynamic complexity in predator-prey models framed in difference equations.
*Nature***255**:58–60.Google Scholar - Bellows, T. S., and M. P. Hassell. 1988. The dynamics of age-structured host-parasitoid interactions.
*Journal of Animal Ecology***57**:259–268.Google Scholar - Blythe, S. P., and R. M. Anderson. 1988a. Distributed incubation and infectious periods in models of the transmission dynamics of the human immunodeficiency virus (HIV).
*IMA Journal of Mathematics Applied in Medicine and Biology***5**:1–19.PubMedGoogle Scholar - Blythe, S. P., and R. M. Anderson. 1988b. Variable infectiousness in HIV transmission models.
*IMA Journal of Mathematics Applied in Medicine and Biology***5**:181–200.PubMedGoogle Scholar - Blythe, S. P., and C. Castillo-Chavez. 1989. Like-with-like preference and sexual mixing models.
*Mathematical Biosciences***96**:221–238.PubMedGoogle Scholar - Blythe, S. P., and C. Castillo-Chavez. 1990. Scaling law of sexual activity.
*Nature***344**:202.PubMedGoogle Scholar - Blythe, S. P., C. Castillo-Chavez, J. Palmer, and M. Cheng. 1991. Towards a unified theory of mixing and pair formation.
*Mathematical Biosciences***107**:379–405.PubMedGoogle Scholar - Blythe, S. P., C. Castillo-Chavez, and G. Casella. 1992. Empirical models for the estimation of the mixing probabilities for socially-structured populations from a single survey sample.
*Mathematical Population Studies*3(3): 199–225.PubMedGoogle Scholar - Brauer, F. 1990. Models for the spread of universally fatal diseases.
*Journal of Mathematical Biology***28**:451–462.PubMedGoogle Scholar - Brauer, F. 1991. “Models for the Spread of Universally Fatal Diseases II.” In S. Busenberg and M. Martelli (eds.),
*Proceedings of the International Conference on Differential Equations and Applications to Biology and Population Dynamics*. Lecture Notes in Biomathematics 92. Springer-Verlag, New York, pp. 57–69.Google Scholar - Busenberg, S., and C. Castillo-Chavez. 1989. “Interaction, Pair Formation and Force of Infection Terms in Sexually Transmitted Diseases.” In C. Castillo-Chavez (ed.),
*Mathematical and Statistical Approaches to AIDS Epidemiology*. Lecture Notes in Biomathematics 83. Springer-Verlag, New York, pp. 289–300.Google Scholar - Busenberg, S., and C. Castillo-Chavez. 1991. A general solution of the problem of mixing subpopulations, and its application to risk- and age-structured epidemic models for the spread of AIDS.
*IMA Journal of Mathematics Applied in Medicine and Biology***8**:1–29.PubMedGoogle Scholar - Busenberg, S., and K. Cooke. 1993.
*Vertically Transmitted Diseases: Models and Dynamics*. Biomathematics 23, Springer-Verlag, New York.Google Scholar - Capasso, V. 1993.
*Mathematical Structures of Epidemic Systems*. Lecture Notes in Biomathematics 97. Springer-Verlag, New York.Google Scholar - Castillo-Chavez, C. (ed.). 1989.
*Mathematical and Statistical Approaches to AIDS Epidemiology*. Lecture Notes in Biomathematics 83. Springer-Verlag, New York.Google Scholar - Castillo-Chavez, C., and S. Busenberg. 1991. “On the Solution of the Two-Sex Mixing Problem.” In S. Busenberg and M. Martelli (eds.),
*Proceedings of the International Conference on Differential Equations and Applications to Biology and Population Dynamics*. Lecture Notes in Biomathematics 92. Springer-Verlag, New York, pp. 80–98.Google Scholar - Castillo-Chavez, C., S. Busenberg, and K. Gerow. 1991. “Pair Formation in Structured Populations.” In J. Goldstein, F. Kappel, and W. Schappacher (eds.),
*Differential Equations with Applications in Biology, Physics and Engineering*. Marcel Dekker, New York, pp. 47–65.Google Scholar - Castillo-Chavez, C., K. Cooke, W. Huang, and S. A. Levin. 1989a. The role of long incubation periods in the dynamics of HIV/AIDS. Part I: Single population models.
*Journal of Mathematical Biology***27**:373–398.PubMedGoogle Scholar - Castillo-Chavez, C., K. Cooke, W. Huang, and S. A. Levin. 1989b. “On the Role of Long Incubation Periods in the Dynamics of HIV/AIDS. Part 2: Multiple Group Models.” In C. Castillo-Chavez (ed.),
*Mathematical and Statistical Approaches to AIDS Epidemiology*. Lecture Notes iln Biomathematics 83. Springer-Verlag, New York.Google Scholar - Castillo-Chavez, C., K. Cooke, W. Huang, and S. A. Levin. 1989c. Results on the dynamics for models for the sexual transmission of the human immunodeficiency virus.
*Applied Mathematics Letters***2(4)**:327–331.Google Scholar - Castillo-Chavez, C., S. Fridman, and X. Luo. 1994a. “Stochastic and Deterministic Models in Epidemiology.” In
*Proceedings of the First World Congress of Nonlinear Analysts*, Tampa, FL, August 19–26, 1992. (In press.)Google Scholar - Castillo-Chavez, C., H. Hethcote, V. Andreasen, S. A. Levin, and W.-M. Liu. 1988. “Cross-Immunity in the Dynamics of Homogeneous and Heterogeneous Populations.” In T. G. Hallam, L. G. Gross, and S. A. Levin (eds.),
*Mathematical Ecology*. World Scientific Publishing, Singapore, pp. 303–316.Google Scholar - Castillo-Chavez, C., H. Hethcote, V. Andreasen, S. A. Levin, and W.-M. Liu. 1989d. Epidemiological models with age structure, proportionate mixing, and cross-immunity. Journal of Mathematical Biology
**27(3)**:233–258.PubMedGoogle Scholar - Castillo-Chavez, C., S.-F. Shyu, G. Rubin, and D. Umbauch. 1992. “On the Estimation Problem of Mixing/Pair Formation Matrices with Applications to Models for Sexually-Transmitted Diseases.” In K. Dietz, V. T. Farewell, and N. P. Jewell (eds.),
*AIDS Epidemiology: Methodology Issues*. Birkhäuser, Boston, pp. 384–402.Google Scholar - Castillo-Chavez, C, J. X. Velasco-Hernandez, and S. Fridman. 1994b. “Modeling Contact Structures in Biology.” In S. A. Levin (ed),
*Frontiers of Theoretical Biology*. Lecture Notes in Biomathematics 100. Springer-Verlag, New York. (In press).Google Scholar - Caswell, H., and D. E. Weeks. 1986. Two-sex models: Chaos, extinction, and other dynamic consequences of sex.
*American Naturalist***128**:707–735.Google Scholar - Cressman, R. 1988. Complex dynamical behaviour of frequency dependent variability selection: An example.
*Journal of Theoretical Biology***130**:167–173.PubMedGoogle Scholar - Diekmann, O., J. A. P. Heesterbeek, and J. A. J. Metz. 1990. On the definition of Ro in models for infectious diseases in heterogeneous populations.
*Journal of Mathematical Biology***28**:365–382.PubMedGoogle Scholar - Dietz, K. 1988. On the transmission dynamics of HIV.
*Mathematical Biosciences***90**:397–414.Google Scholar - Dietz, K., and K. P. Hadeler. 1988. Epidemiological models for sexually transmitted diseases.
*Journal of Mathematical Biology***26**:1–25.PubMedGoogle Scholar - Ebenman, B. 1987. Niche differences between age classes and intraspecific competition in age structured populations.
*Journal of Theoretical Biology***124**:25–33.Google Scholar - Ellner, S. 1989. “Inferring the Causes of Population Fluctuations.” In C. Castillo-Chavez, S. A. Levin, and C. A. Shoemaker (eds.),
*Mathematical Approaches to Problems in Resource Management and Epidemiology*. Lecture Notes in Biomathematics 81, Springer-Verlag, New York.Google Scholar - Evans, A. S. 1982.
*Viral Infections of Humans*. Second edition. Plenum Medical Book Company, New York.Google Scholar - Fredrickson, A. G. 1971. A mathematical theory of age structure in sexual populations: Random mating and monogamous marriage models.
*Mathematical Biosciences***20**:117–143.Google Scholar - Gabriel, J. P., C. Lefèvre, and P. Picard (eds.). 1990.
*Stochastic Processes in Epidemic Theory*. Lecture Notes in Biomathematics 86. Springer-Verlag, New York.Google Scholar - Gardini, L., R. Lupini, C. Mammana, and M. G. Messia. 1987. Bifurcation and transition to chaos in the three dimensional Lotka Volterra map.
*SIAM Journal of Applied Mathematics***47**:455–482.Google Scholar - Gilpin, M. E. 1992. Spiral chaos in a predator prey model.
*American Naturalist***113**:306–308.Google Scholar - Guckenheimer, J., G. Oster, and A. Ipaktchi. 1977. The dynamics of density-dependent population models.
*Journal of Mathematical Biology***4**:101–147.Google Scholar - Gupta, S., R. M. Anderson, and R. M. May. 1989. Network of sexual contacts: Implications for the pattern of spread of HIV.
*AIDS***3**:1–11.Google Scholar - Hadeler, K. P. 1989a. Pair formation in age-structured populations.
*Acta Applicandae Mathematicae***14**:91–102.PubMedGoogle Scholar - Hadeler, K. P. 1989b. “Modeling AIDS in Structured Populations.” In
*Proceedings of the 47th Session of the International Statistical Institute*, Paris, August/September, pp. 83–99.Google Scholar - Hadeler, K. P., and K. Nagoma. 1990. Homogeneous models for sexually transmitted diseases.
*Rocky Mountain Journal of Mathematics***20**:967–986.Google Scholar - Hastings, A., C. Horn, S. Ellner, P. Turchin, and H. C. J. Godfray. 1993. Chaos in ecology: Is Mother Nature a strange attractor?
*Annual Review of Ecological Systems***24**:1–33.Google Scholar - Hastings, A., and T. Powell. 1991. Chaos in a three species food chain.
*Ecology***72**:896–903.Google Scholar - Hethcote, H. W. 1976. Qualitative analysis for communicable disease models.
*Mathematical Biosciences***28**:335–356.Google Scholar - Hethcote, H. W. 1978. An immunization model for a heterogeneous population.
*Theoretical Population Biology***14**:338–349.PubMedGoogle Scholar - Hethcote, H. W. 1989. “Three Basic Epidemiological Models.” In S. A. Levin, T. G. Hallam, and J. Gross (eds.),
*Applied Mathematical Ecology*. Biomathematics 18, Springer-Verlag, New York, pp. 119–144.Google Scholar - Hethcote, H. W., and S. A. Levin. 1989. “Periodicity in Epidemiological Models.” In: S.A. Levin, T. G. Hallam, and L. J. Gross (eds.),
*Applied Mathematical Ecology*. Biomathematics 18. Springer-Verlag, New York.Google Scholar - Hethcote, H. W., and J. W. van Ark. 1992.
*Modeling HIV Transmission and AIDS in the United States*. Lecture Notes in Biomathematics 95, Springer-Verlag, New York.Google Scholar - Hethcote, H. W., and J. A. Yorke. 1984.
*Gonorrhea Transmission Dynamics and Control*. Lecture Notes in Biomathematics 56, Springer-Verlag, New York.Google Scholar - Hochberg, M. E., M. P. Hassell, and R. M. May. 1990. The dynamics of host-parasitoid-pathogen interactions.
*American Naturalist***135**:74–94.Google Scholar - Hsu Schmitz, S.-F. 1994. Some theories, estimation methods, and applications of marriage and mixing functions to demography and epidemiology. Ph.D. dissertation, Cornell University, Ithaca, NY.Google Scholar
- Hsu Schmitz, S.-F., S. Busenberg, and C. Castillo-Chavez. 1993. On the evolution of marriage functions: It Takes Two to Tango. Biometrics Unit Technical Report BU-1210-M, Cornell University, Ithaca, NY.Google Scholar
- Hsu Schmitz, S.-F., and C. Castillo-Chavez. 1993. “Completion of Mixing Matrices for Non-Closed Social Networks.” In
*Proceedings of First World Congress of Nonlinear Analysts*, Tampa, FL, August 19–26, 1992. (In press.)Google Scholar - Hsu Schmitz, S.-F., and C. Castillo-Chavez. 1994. “Parameter Estimation in Non-Closed Social Networks Related to the Dynamics of Sexually-Transmitted Diseases.” In E. Kaplan and M. Brandeau (eds.),
*Modeling the AIDS Epidemic*. Raven, New York. (In press.)Google Scholar - Huang, W. 1989. Studies in differential equations and applications. Ph.D. dissertation, The Claremont Graduate School, Claremont, CA.Google Scholar
- Huang, W., K. Cook, and C. Castillo-Chavez. 1992. Stability and bifurcation for a multiple group model for the dynamics of HIV/AIDS transmission.
*SIAM Journal of Applied Mathematics*.**52(3)**:835–854.Google Scholar - Hyman, J. M., and E. A. Stanley. 1988. Using mathematical models to understand the AIDS epidemic.
*Mathematical Biosciences***90**:415–473.Google Scholar - Hyman, J. M., and E. A. Stanley. 1989. “The Effect of Social Mixing Patterns on the Spread of AIDS.” In C. Castillo-Chavez, S. A. Levin, and C. Shoemaker (eds.),
*Mathematical Approaches to Problems in Resource Management and Epidemiology*. Lecture Notes in Biomathematics 81. Springer-Verlag, New York, pp. 190–219.Google Scholar - Jacquez, J. A., C. P. Simon, and J. Koopman. 1989. “Structured Mixing: Heterogeneous Mixing by the Definition of Mixing Groups.” In C. Castillo-Chavez (ed.),
*Mathematical and Statistical Approaches to AIDS Epidemiology*. Lecture Notes in Biomathematics. Springer-Verlag, New York, pp. 301–315.Google Scholar - Jacquez, J. A., C. P. Simon, J. Koopman, L. Sattenspiel, and T. Perry. 1988. Modelling and analyzing HIV transmission: The effect of contact patterns.
*Mathematical Biosciences***92**:119–199.Google Scholar - Jewell, N. P., K. Dietz, and V. T. Farewell. 1991.
*AIDS Epidemiology: Methodology Issues*. Birkhäuser. Boston.Google Scholar - Kaplan, E., and M. Brandeau. (eds.). 1994.
*Modeling AIDS and the AIDS Epidemic*. Raven, New York. (In press.)Google Scholar - Kendall, D. G. 1949. Stochastic processes and population growth.
*Royal Statistical Society, Series B***2**:230–264.Google Scholar - Kermack, W. O., and A. G. McKendrick. 1927. A contribution to the mathematical theory of epidemics.
*Proceedings of the Royal Society of London, Series A***115**:700–721.Google Scholar - Kermack, W. O., and A. G. McKendrick. 1932. A contribution to the mathematical theory of epidemics.
*Proceedings of the Royal Society of London, Series A***138**:55–83.Google Scholar - Keyfitz, N. 1949. The mathematics of sex and marriage.
*Proceedings of the Sixth Berkeley Symposium on Mathematical and Statistical Problems***4**:89–108.Google Scholar - Kot, M., W. M. Schaffer, G. L. Truty, D. J. Graser, and L. F. Olsen. 1988. Changing criteria for imposing order.
*Ecological Modelling***43**:75–110.Google Scholar - Leslie, P. H. 1945. On the use of matrices in certain population mathematics.
*Biometrika***33**:183–212.PubMedGoogle Scholar - Levin, S. A. 1981. Age structure and stability in multiple-age populations.
*Renewable Resources Management***40**:21–45.Google Scholar - Levin, S. A. 1983a. “Coevolution.” In H. I. Freedman and C. Strobeck (eds.),
*Population Biology*. Lecture Notes in Biomathematics 52. Springer-Verlag, New York, pp. 328–334.Google Scholar - Levin, S. A. 1983b. “Some Approaches in the Modeling of Coevolutionary Interactions.” In M. Nitecki (ed.),
*Coevolution*. University of Chicago Press, Chicago, pp. 21–65.Google Scholar - Levin, S. A., and C. Castillo-Chavez. 1990. “Topics in Evolutionary Biology.” In S. Lessard (ed),
*Mathematical and Statistical Developments of Evolutionary Theory*. NATO ASI Series. Kluwer, Boston, pp. 327–358.Google Scholar - Levin, S. A., and C. Goodyear. 1980. Analysis of an age-structured fishery model.
*Journal of Mathematical Biology***9**:245–274.Google Scholar - Levin, S. A., and D. Pimentel. 1981. Selection of intermediate rates of increase in parasite-host systems.
*American Naturalist***117**:308–315.Google Scholar - Lotka, A. J. 1922. The stability of the normal age distribution.
*Proceedings of the National Academy of Sciences***8**:339–345.Google Scholar - Lotka, A. J. 1923. Contributions to the analysis of malaria epidemiology.
*American Journal of Hygiene*, 3 January Supplement.Google Scholar - Lubkin, S., and C. Castillo-Chavez. 1994. “A Pair-Formation Approach to Modeling Inheritance of Social Traits.” In
*Proceedings of First World Congress of Nonlinear Analysts*, Tampa, FL, August 19–26, 1992. (In press.)Google Scholar - Luo, X., and C. Castillo-Chavez. 1991. Limit behavior of pair formation for a large dissolution rate.
*Journal of Mathematical Systems Estimation and Control***3**:247–264.Google Scholar - May, R. M. 1974. Biological populations with non-overlapping generations: Stable points, stable cycles, and chaos.
*Journal of Theoretical Biology***5**:511–524.Google Scholar - May, R. M. 1976. Simple mathematical models with very complicated dynamics.
*Nature***261**:459–467.PubMedGoogle Scholar - May, R. M. 1985. Regulation of populations with nonoverlapping generations by microparasites: A purely chaotic system.
*American Naturalist***125**:573–584.Google Scholar - May, R. M. 1987. Chaos and the dynamics of biological populations.
*Proceedings of the Royal Society of London, Series A***413**:27–44.Google Scholar - May, R. M., and R. M. Anderson. 1983a. Epidemiology and genetics in the coevolution of parasites and hosts.
*Proceedings of the Royal Society of London, Series B***219**:281–313.Google Scholar - May, R. M. 1983b. “Parasite-Host Coevolution.” In D. Futuyama and M. Slatkin (eds.),
*Coevolution*. Sinauer, Sunderland, MA.Google Scholar - May, R. M. 1989. The transmission dynamics of human immunodeficiency virus (HIV).
*Philosophical Transactions of the Royal Society, Series B***321**:565–607.Google Scholar - May, R. M., and G. F. Oster. 1976. Bifurcations and dynamics complexity in simple ecological models.
*American Naturalist***110**:573–599.Google Scholar - McFarland, D. D. 1972. “Comparison of Alternative Marriage Models.” In T. N. E. Greville (ed.),
*Population Dynamics*. Academic Press, New York, pp. 89–106.Google Scholar - McKendrick, A. G. 1926. Applications of mathematics to medical problems.
*Proceedings of the Edinburgh Mathematics Society***44**:98–130.Google Scholar - Neubert, M. G., and M. Kot. 1992. The subcriticai collapse of predator-prey models.
*Mathematical Biosciences***110**:45–66.PubMedGoogle Scholar - Nold, A. 1980. Heterogeneity in disease-transmission modeling.
*Mathematical Biosciences***52**:227–240.Google Scholar - Olsen, L. F., and W. M. Schaffer. 1990. Chaos versus noisy periodicity: Alternative hypotheses for childhood epidemics.
*Science***249**:499–504.PubMedGoogle Scholar - Parlett, B. 1972. “Can There Be a Marriage Function?” In T. N. E. Greville (ed.),
*Population Dynamics*. Academic Press, New York, pp. 107–135.Google Scholar - Pickering, J. J., A. Wiley, N. S. Padian, L. E. Lieb, D. F. Echenberg, and J. Walker. 1986. Modelling the incidence of acquired immunodeficiency syndrome (AIDS) in San Francisco, Los Angeles and New York.
*Mathematical Modelling***7**:661–698.Google Scholar - Pollard, J. H. 1973.
*Models for the Growth of Human Populations*. Cambridge University Press, London.Google Scholar - Pugliese, A. 1990a. Population models for disease with no recovery.
*Journal of Mathematical Biology***28**:65–82.PubMedGoogle Scholar - Pugliese, A. 1990b. “An S→E→I Epidemic Model with Varying Population Size.” In S. Busenberg and M. Martelli (eds.),
*Proceedings of the International Conference on Differential Equations and Applications to Biology and Population Dynamics*. Lecture Notes in Biomathematics 93. Springer-Verlag, New York, pp. 121–138.Google Scholar - Ross, R. 1911.
*The Prevention of Malaria*. Second edition, with addendeum. John Murray, London.Google Scholar - Rubin, G., D. Umbauch, S.-F. Shyu, and C. Castillo-Chavez. 1992. Application of capture-recapture methodology to estimation of size of population at risk of AIDS and/ or other sexually transmitted diseases.
*Statistics in Medicine***11**:1533–1549.PubMedGoogle Scholar - Sattenspiel, L. 1987a. Population structure and the spread of disease.
*Human Biology***59**:411–438.PubMedGoogle Scholar - Sattenspiel, L. 1987b. Epidemics in nonrandomly mixing populations: A simulation.
*American Journal of Physical Anthropology***73**:251–265.PubMedGoogle Scholar - Sattenspiel, L., and C. Castillo-Chavez. 1990. Environmental context, social interactions, and the spread of HIV.
*American Journal of Human Biology***2**:397–417.Google Scholar - Sattenspiel, L., and C. P. Simon. 1988. The spread and persistence of infectious diseases in structured populations.
*Mathematical Biosciences***90**:341–366.Google Scholar - Schaffer, W. M. 1985a. Can nonlinear dynamics elucidate mechanisms in ecology and epidemiology?
*IMA Journal of Mathematics Applied in Medicine and Biology***2**:221–252.PubMedGoogle Scholar - Schaffer, W. M., and M. Kot. 1985. Nearly one dimensional dynamics in an epidemic.
*Journal of Theoretical Biology***112**:403–427.PubMedGoogle Scholar - Shaffer, W. M., L. F. Olsen, G. L. Truty, and S. L. Fulmer. 1990. “The Case of Chaos in Childhood Epidemics.” In S. Krasner (ed.),
*The Ubiquity of Chaos*. American Association for the Advancement of Science, Washington, D.C., pp. 138–166.Google Scholar - Soper, H. E. 1929. Interpretation of periodicity in disease prevalence.
*Journal of the Royal Statistical Society B***92**:34–73.Google Scholar - Takeuchi, Y., and N. Adachi. 1983. Existence and bifurcation of stable equilibrium in two-prey, one-predator communities.
*Bulletin of Mathematical Biology***45**:877–900.Google Scholar - Thieme, H. R., and C. Castillo-Chavez. 1989. “On the Role of Variable Infectivity in the Dynamics of the Human Immunodeficiency Virus Epidemic.” In C. Castillo-Chavez (ed.),
*Mathematical and Statistical Approaches to AIDS Epidemiology*. Lecture Notes in Biomathematics 83. Springer-Verlag, New York, pp. 157–176.Google Scholar - Thieme, H. R., and C. Castillo-Chavez. 1994. How may infection-age dependent infectivity affect the dynamics of HIV/AIDS?
*SIAM Journal of Applied Mathematics*. (In press.)Google Scholar - Velasco-Hernandez, J. X., and C. Castillo-Chavez. 1994. “Modeling Vector-Host Disease Transmission and Food Web Dynamics Through the Mixing/Pair-Formation Approach.” In
*Proceedings in the First World Congress of Nonlinear Analysts*, Tampa, FL, August 19–26, 1992. (In press.)Google Scholar - Waldstätter, R. 1989. “Pair Formation in Sexually Transmitted Diseases.” In C. Castillo-Chavez (ed.),
*Mathematical and Statistical Approaches to AIDS Epidemiology*. Lecture Notes in Biomathematics 83, Springer Verlag, New York, pp. 260–274.Google Scholar