zbMATH — the first resource for mathematics

Epidemiological models with age structure, proportionate mixing, and cross-immunity. (English) Zbl 0715.92028
Summary: Infection by one strain of influenza type A provides some protection (cross-immunity) against infection by a related strain. It is important to determine how this influences the observed co-circulation of comparatively minor variants of the H1N1 and H3N2 subtypes. To this end, we formulate discrete and continuous time models with two viral strains, cross-immunity, age structure, and infectious disease dynamics.
Simulation and analysis of models with cross-immunity indicate that sustained oscillations cannot be maintained by age-specific infection activity level rates when the mortality rate is constant; but are possible if mortalities are age-specific, even if activity levels are independent of age. Sustained oscillations do not seem possible for a single-strain model, even in the presence of age-specific mortalities; and thus it is suggested that the interplay between cross-immunity and age-specific mortalities may underlie observed oscillations.

92D30 Epidemiology
35B35 Stability in context of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
PDF BibTeX Cite
Full Text: DOI
[1] Anderson, R. M.: The epidemiology of HIV infection: variable incubation plus infectious period and heterogeneity in sexual activity. J. R. Stat. Soc. A 151, 66-93 (1988) · Zbl 1159.92313
[2] Anderson, R. M., May, R. M.: Transmission dynamics of HIV infection. Nature 326, 137-142 (1987)
[3] Anderson, R. M., May, R. M., Medley, G. F., Johnson, A.: A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS. IMA J. Math. Med. Biol. 3, 229-263 (1986) · Zbl 0609.92025
[4] Barré-Sinoussi, F., Chermann, J.-C., Rey, F., Nugeyre, M. T., Chamaret, S., Gruest, J., Dauguet, C., Axler-Blin, C., Brun-Vézinet, F., Rouzioux, C., Rozenbaum, W., Montagnier, L.: Isolation of a T-lymphotropic retrovirus from a patient at risk for acquired immune deficiency syndrome (AIDS). Science 220, 868-870 (1983)
[5] Blythe, S. P., Anderson, R. M.: Distributed incubation and infectious periods in models of the transmission dynamics of the human immunodeficiency virus (HIV). IMA J. Math. Med. Bio. 5, 1-19 (1988) · Zbl 0686.92015
[6] Castillo-Chavez, C., Cooke, K., Huang, W., Levin, S. A.: The role of long periods of infectiousness in the dynamics of acquired immunodeficiency syndrome. In: Castillo-Chavez, C., Levin, S. A., Shoemaker, C. (eds.) Mathematical approaches to resource management and epidemiology. (Lect. Notes Biomath., in press) Berlin Heidelberg New York: Springer 1989a · Zbl 0682.92013
[7] Castillo-Chavez, C., Cooke, K., Huang, W., Levin, S. A.: Results on the dynamics for models for the sexual transmission of the human immunodeficiency virus. Appl. Math. Lett., in press (1989b) · Zbl 0703.92022
[8] Dietz, K., Hadeler, K. P.: Epidemiological models for sexually transmitted diseases. J. Math. Biol. 26, 1-26 (1988) · Zbl 0643.92015
[9] Francis, D. F., Feorino, P. M., Broderson, J. R., McClure, H. M., Getchell, J. P., McGrath, C. R., Swenson, B., McDougal, J. S., Palmer, E. L., Harrison, A. K., Barré-Sinoussi, F., Chermann, J.-C., Montagnier, L., Curran, J. W., Cabradilla, C. D., Kalyanaraman, V. S.: Infection of chimpanzees with lymphadenopathy-associated virus. Lancet 2, 1276-1277 (1984)
[10] Gallo, R. C.: The first human retrovirus. Scientific American 255, 88-98 (1986)
[11] Gallo, R. C.: The AIDS virus. Scientific American 256, 47-56 (1987)
[12] Gallo, R. C., Salahuddin, S. Z., Popovic, M., Shearer, G. M., Kaplan, M., Haynes, B. F., Palker, T. J., Redfield, R., Oleske, J., Safai, B., White, G., Foster, P., Markhamet, P. D.: Frequent detection and isolation of sytopathic retroviruses (HTLV-III) from patients with AIDS and at risk for AIDS. Science 224, 500-503 (1984)
[13] Hethcote, H. W., Stech, H. W., van den Driessche, P.: Nonlinear oscillations in epidemic models. SIAM J. Appl. Math. 40, 1-9 (1981) · Zbl 0469.92012
[14] Hethcote, H. W., van Ark, J. W.: Epidemiological methods for heterogeneous populations: proportional mixing, parameter estimation, and immunization programs. Math. Biosci. 84, 85-118 (1987) · Zbl 0619.92006
[15] Huang, W., Castillo-Chavez, C., Cooke, K., Levin, S. A.: On the role of long incubation periods of in the dynamics of acquired immunodeficiency syndrome. Part 2: Multiple group models. In: Castillo-Chavez, C. (ed.) Mathematical and statistical approaches to AIDS transmission and epidemiology. (Lect. Notes Biomath., in preparation) Berlin Heidelberg New York: Springer 1989a · Zbl 0715.92029
[16] Huang, W., Cooke, K., Castillo-Chavez, C.: Multiple group models for the dynamics of HIV/AIDS transmission with proportionate and preferred mixing, and multiple endemic equilibria (in preparation, 1989b)
[17] Hyman, J.M., Stanley, E. A.: A risk base model for the spread of the AIDS virus. Math. Biosci. 90, 415-473 (1988) · Zbl 0727.92025
[18] Kingsley, R. A., Kaslow, R., Rinaldo, C. R., Jr., Detre, K., Odaka, N., Van Raden, M., Detels, R., Polk, B. F., Chimel, J., Kersey, S. F., Ostrow, D., Visscher, B.: Risk factors for seroconversion to human immunodeficiency virus among male homosexuals, Lancet 1, 345-348 (1987)
[19] Lange, J. M. A., Paul, D. A., Huisman, H. G., De Wolf, F., Van den Berg, H., Roel, C. A., Danner, S. A., Van der Noordaa, J., Goudsmit, J.: Persistent HIV antigenaemia and decline of HIV core antibodies associated with transition to AIDS. Brit. Med. J. 293, 1459-62 (1986)
[20] Medley, G. F., Anderson, R. M., Cox, D. R., Billiard, L.: Incubation period of AIDS in patients infected via blood transfusions. Nature 328, 719-721 (1987)
[21] Miller, R. K.: On the linearization of Volterra integral equations. J. Math. Anal. Appl. 23, 198-208 (1968) · Zbl 0167.40902
[22] Miller, R. K.: Nonlinear Volterra integral equations. Menlo Park: Benjamin 1971 · Zbl 0448.45004
[23] Pickering J., Wiley, J. A., Padian, N. S., et al.: Modeling the incidence of acquired immunodeficiency syndrome (AIDS) in San Francisco, Los Angeles, and New York. Math. Modelling 7, 661-688 (1986)
[24] Salahuddin, S. Z., Groopman, J. E., Markham, P. D., Sarngaharan, M. G., Redfield, R. R., McLane, M. F., Essex, M., Sliski, A., Gallo, R. C.: HTLV-III in symptom-free seronegative persons. Lancet 2, 1418-1420 (1984)
[25] Thieme, H., Castillo-Chavez, C., Cooke, K.: On the effects of variable infectivity in the dynamics of HIV/AIDS. In: Castillo-Chavez, C., Levin, S. A., Shoemaker, C. (eds.) Mathematical approaches to resource management and epidemiology. (Lect. Notes Biomath., in press) Berlin Heidelberg New York: Springer 1989
[26] Wong-Staal, F., Gallo, R. C.: Human T-lymphotropic retroviruses. Nature 317, 395-403 (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.