Previous |  Up |  Next

Article

Keywords:
epidemic model; time delay; Hopf bifurcation; equilibrium analysis; differential equations
Summary:
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.
References:
[1] R. M. Anderson, R. M. May: Vaccination against rubella and measles; quantitative investigations of different policies. J.  Hyg. 90 (1983), 259–325.
[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.
[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.
[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. 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. 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. 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. 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. MR 0975856
[17] M. Kalecki: A macrodynamic theory of business cycles. Econometrica 3 (1935), 327–344.
[18] W. W. Leontief: Lags and stability of dynamic systems. Econometrica 29 (1961), 659–669. 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. 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. 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