Previous |  Up |  Next


epidemic model; time delay; Hopf bifurcation; equilibrium analysis; differential equations
We study a mathematical model which was originally suggested by Greenhalgh and Das and takes into account the delay in the recruitment of infected persons. The stability of the equilibria are also discussed. In addition, we show that the introduction of a time delay in the transmission term can destabilize the system and periodic solutions can arise by Hopf bifurcation.
[1] R. M. Anderson, R. M. May: Vaccination against rubella and measles; quantitative investigations of different policies. J.  Hyg. 90 (1983), 259–325. DOI 10.1017/S002217240002893X
[2] R. Bellman, K. L.  Cooke: Differential Difference Equations. Academic Press, New York, 1963. MR 0147745
[3] M. C. Boily, R. M.  Anderson: Sexual contact patterns between men and women and the spread of HIV-1 in urban societies in Africa. IMA J. Math. Appl. Med. Biol. 8 (1991), 221–247. DOI 10.1093/imammb/8.4.221
[4] S. N. Busenburg, P. van den Driessche: Analysis of a disease transmission model in a population with varying size. J.  Math. Biol. 29 (1990), 257–270. MR 1047163
[5] S. N. Chow, J. K.  Hale: Methods of Bifurcation Theory. Springer-Verlag, New York, 1982. MR 0660633
[6] R. D. Driver: Ordinary and Delay Differential Equations. Applied Math. Sciences, Springer-Verlag, New York, 1977. MR 0477368 | Zbl 0374.34001
[7] G. Gandolfo: Mathematical Methods and Models in Economic Dynamics. North Holland, London, 1971. Zbl 0227.90010
[8] K. Gopalsamy: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Academic Publisher, Dordrecht-Boston-London, 1992. MR 1163190 | Zbl 0752.34039
[9] D. Greenhalgh, R.  Das: Modelling epidemics with variable contact rates. Theor. Popul. Biol. 47 (1995), 129–179. DOI 10.1006/tpbi.1995.1006
[10] J. K. Hale, S. M. V.  Lunel: Introduction to Functional Differential Equations. Applied Math. Sciences  99. Springer-Verlag, New York, 1993. MR 1243878
[11] H. W. Hethcote,J. A.  Yorke: Gonorrhoea transmission dynamics and control. Lecture Notes in Biomathematics, Vol. 56, Springer-Verlag, New York, 1974. MR 0766910
[12] H. W. Hethcote, D. W. Tudor: Integral equation models for endemic infections diseases. J.  Math. Biol. 9 (1980), 37–47. DOI 10.1007/BF00276034 | MR 0648844
[13] H. W. Hethcote, H. W. Stech, and P. van den Driessche: Nonlinear oscillations in epidemic models. SIAM J.  Appl. Math. 40 (1981), 1–9. DOI 10.1137/0140001 | MR 0602496
[14] H. W. Hethcote, H. W. Stech, and P. van den Driessche: Stability analysis for models of diseases without immunity. J.  Math. Biol. 13 (1981), 185–198. DOI 10.1007/BF00275213 | MR 0661676
[15] H. W. Hethcote, M.  Lewis, and P. van den Driessche: An epidemiological model with a delay and a non-linear incidence rate. J.  Math. Biol. 27 (1989), 49–64. DOI 10.1007/BF00276080 | MR 0984225
[16] J. A. Jacquez, C. P. Simon, J.  Koopman, L. Sattenspiel, and T. Perry: Modelling and analyzing HIV transmissions: The effect of contact patterns. Math. Biosci. 92 (1988), 119–199. DOI 10.1016/0025-5564(88)90031-4 | MR 0975856
[17] M. Kalecki: A macrodynamic theory of business cycles. Econometrica 3 (1935), 327–344. DOI 10.2307/1905325
[18] W. W. Leontief: Lags and stability of dynamic systems. Econometrica 29 (1961), 659–669. DOI 10.2307/1911811 | MR 0145984
[19] N. MacDonald: Time Lags in Biological Models. Lecture Notes in Biomathematics Vol. 28. Springer-Verlag, Berlin-Heidelberg-New York, 1979. MR 0521439
[20] R. M. May, R. M.  Anderson, and A. R. McLean: Possible demographic consequences of HIV/AIDS epidemics I.  Assuming HIV infection always lead to AIDS. Math. Biosci. 90 (1988), 475–505. DOI 10.1016/0025-5564(88)90079-X | MR 0958153
[21] J. E. Marsden, M.  McCracken: The Hopf Bifurcation and Its Applications. Springer-Verlag, New York, 1976. MR 0494309
[22] A. Nold: Heterogeneity in disease transmission modelling. Math. Biosci. 52 (1980), 227–240. DOI 10.1016/0025-5564(80)90069-3 | MR 0596176
[23] H. R. Thieme: Global asymptotic stability for epidemic models. Equadiff 82. In: Lecture Notes in Mathematics Vol.  1017, H.  W.  Knobloch, K.  Schmitt (eds.), Springer-Verlag, Berlin, 1983, pp. 608–615. MR 0726617
[24] H. R. Thieme: Local stability in epidemic models for heterogeneous populations. In: Mathematics in Biology and Medicine. Lecture Notes in Biomathematics Vol. 57, V. Capasso, E.  Grosso and S.  L.  Paveri-Fontana (eds.), Springer-Verlag, Berlin, 1985, pp. 185–189. MR 0812889 | Zbl 0584.92020
Partner of
EuDML logo