×

Some epidemiological models with nonlinear incidence. (English) Zbl 0722.92015

Summary: Epidemiological models with nonlinear incidence rates can have very different dynamic behaviors than those with the usual bilinear incidence rate. The first model considered here includes vital dynamics and a disease process where susceptibles become exposed, then infectious, then removed with temporary immunity and then susceptible again. When the equilibria and stability are investigated, it is found that multiple equilibria exist for some parameter values and periodic solutions can arise by Hopf bifurcation from the larger endemic equilibrium. Many results analogous to those in the first model are obtained for the second model which has a delay in the removed class but no exposed class.

MSC:

92D30 Epidemiology
34C23 Bifurcation theory for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Busenberg, S., Cooke, K. L.: The population dynamics of two vertically transmitted infections. Theor. Popul. Biol. 33, 181-198 (1988) · Zbl 0638.92009
[2] Capasso, V., Serio, G.: A generalization of the Kermack-McKendrick deterministic epidemic model. Math. Biosci. 42, 41-61 (1978) · Zbl 0398.92026
[3] Hale, J. K.: Ordinary differential equations. New York: Wiley-Interscience 1969 · Zbl 0186.40901
[4] Hao, D.-Y., Brauer, F.: Analysis of a characteristic equation. J. Integral Equations Appl. 3, (1990). In press.
[5] Hethcote, H. W.: An immunization model for a heterogeneous population. Theor. Popul. Biol. 14, 338-349 (1978) · Zbl 0392.92009
[6] Hethcote, H. W., Levin, S. A.: Periodicity in epidemiological models. In: Gross, L., Hallam, T. G., Levin, S. A. (eds.) Applied mathematical ecology, pp. 193-211. Berlin Heidelberg New York: Springer 1989
[7] Hethcote, H. W., Lewis, M. A., van den Driessche, P.: An epidemiological model with a delay and a nonlinear incidence rate. J. Math. Biol. 27, 49-64 (1989) · Zbl 0714.92021
[8] Hethcote, H. W., Stech, H. W., van den Driessche, P.: Nonlinear oscillations in epidemic models. SIAM J. Appl. Math. 40, 1-9 (1981a) · Zbl 0469.92012
[9] Hethcote, H. W., Stech, H. W., van den Driessche, P.: Stability analysis for models of diseases without immunity. J. Math. Biol. 13, 185-198 (1981b) · Zbl 0475.92014
[10] Hethcote, H. W., Stech, H. W., van den Driessche, P.: Periodicity and stability in epidemic models: A survey. In: Busenberg, S. N., Cooke, K. L. (eds.) Differential equations and applications in ecology, epidemics and population problems, pp. 65-82. New York: Academic Press 1981c · Zbl 0477.92014
[11] Hethcote, H. W., Tudor, D. W.: Integral equation models for endemic infectious diseases. J. Math. Biol. 9, 37-47 (1980) · Zbl 0433.92026
[12] Hethcote, H. W., Van Ark, J. W.: Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation and immunization programs. Math. Biosci. 84, 85-118 (1987) · Zbl 0619.92006
[13] Holling, C. S.: Some characteristics of simple types of predation and parasitism. Can. Ent. 91, 385-395 (1959)
[14] Liu, W. M., Hethcote, H. W., Levin, S. A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25, 359-380 (1987) · Zbl 0621.92014
[15] Liu, W. M., Levin, S. A., Iwasa, Y.: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23, 187-204 (1986) · Zbl 0582.92023
[16] Miller, R. K., Michel, A. N.: Ordinary differential equations. New York: Academic Press 1982 · Zbl 0552.34001
[17] van den Driessche, P.: A cyclic epidemic model with temporary immunity and vital dynamics. In: Freedman, H. I., Strobeck, C. (eds.) Population biology, (Lect. Notes Biomath., vol. 52, pp. 433-440) Berlin Heidelberg New York: Springer 1983 · Zbl 0519.92025
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.