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Stability of the endemic equilibrium in epidemic models with subpopulations. (English) Zbl 0582.92024
The authors study some epidemic models in the case when the population is divided into subpopulations. The main problems that are considered are to identify thresholds and to prove that beyond these there is a unique endemic equilibrium and finally to find conditions for when the equilibrium is locally stable. For the existence of equilibria the authors use fixed-point theorems and the problem of stability is solved with the aid of linearization and Fourier analysis. One of the models considered is a system of ordinary differential equations and includes immunization and the second one is a system of integrodifferential equations and includes class-age infectivity. The authors also review earlier work on these problems.
Reviewer: G.Gripenberg

MSC:
92D25Population dynamics (general)
45M10Stability theory of integral equations
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References:
[1] Aronsson, G.; Mellander, I.: A deterministic model in biomathematics: asymptotic behavior and threshold conditions. Math. biosci. 49, 207-222 (1980) · Zbl 0433.92025
[2] Berman, A.; Plemmons, R. J.: Nonnegative matrices in the mathematical sciences. (1979) · Zbl 0484.15016
[3] Diekmann, O.: Integral equations and populations dynamics. Math. centrum syllabus 41, 117-149 (1979) · Zbl 0432.92020
[4] Diekmann, O.; Montijn, R.: Prelude to Hopf bifurcation in an epidemic model: analysis of the characteristic equation associated with a nonlinear Volterra integral equation. J. math. Biol. 14, 117-127 (1982) · Zbl 0487.92023
[5] Diekmann, O.; Van Gils, S. A.: Invariant manifolds for Volterra integral equations of convolution type. J. differential equations 54, 139-180 (1984) · Zbl 0543.45004
[6] K. Dietz and D. Schenzle, Mathematical models for infectious disease statistics, in A Celebration of Statistics (A.C. Atkinson and S.E. Fienberg, Eds.), to appear. · Zbl 0586.92017
[7] Gripenberg, G.: Periodic solutions to an epidemic model. J. math. Biol. 10, 271-280 (1980) · Zbl 0446.92022
[8] Gripenberg, G.: On some epidemic models. Quart. appl. Math. 39, 317-327 (1981) · Zbl 0476.92017
[9] Gripenberg, G.: On a nonlinear integral equation modelling an epidemic in an age-structured population. J. reine angew. Math. 341, 54-67 (1983) · Zbl 0514.92027
[10] Gripenberg, G.: Stability of periodic solutions of some integral equations. J. reine angew. Math. 331, 16-31 (1983) · Zbl 0468.45009
[11] Hethcote, H. W.: Qualitative analysis for communicable disease models. Math. biosci. 28, 335-356 (1976) · Zbl 0326.92017
[12] Hethcote, H. W.: An immunization model for a heterogeneous population. Theoret. population biol. 14, 338-349 (1978) · Zbl 0392.92009
[13] Hethcote, H. W.; Stech, H. W.; Van Der Driessche, P.: Nonlinear oscillations in epidemic models. SIAM J. Appl. math. 40, 1-9 (1981) · Zbl 0469.92012
[14] Hethcote, H. W.; Stech, H. W.; Den Driessche, P. Van: Stability analysis for models of diseases without immunity. J. math. Biol. 13, 185-198 (1981) · Zbl 0475.92014
[15] Hethcote, H. W.; Stech, H. W.; Den Driessche, P. Van: Periodicity and stability in epidemic models: A survey. Differential equations and applications in ecology, epidemics and population problems, 65-82 (1981)
[16] Hethcote, H. W.; Tudor, D. W.: Integral equation models for endemic infectious diseases. J. math. Biol. 9, 37-47 (1980) · Zbl 0433.92026
[17] Hethcote, H. W.; Yorke, J. A.: Gonorrhea transmission dynamics and control. Springer-verlag lecture notes in biomathematics 56 (1984) · Zbl 0542.92026
[18] Hirsch, M. W.: The differential equations approach to dynamical systems. Bull. amer. Math. soc. 11, 1-64 (1984) · Zbl 0541.34026
[19] Hoppensteadt, G.: Mathematical theories of populations: demographics, genetics and epidemics. (1975) · Zbl 0304.92012
[20] Kermack, W. O.; Mckendrick, A. G.: A contribution to the mathematical theory of epidemics. Proc. roy. Soc. London ser. A 155, 700-721 (1927) · Zbl 53.0517.01
[21] Krasnoselskii, M. A.: Positive solutions of operator equations. (1964)
[22] Lajmanovich, A.; Yorke, J. A.: A deterministic model for gonorrhea in a nonhomogeneous population. Math. biosci. 28, 221-236 (1976) · Zbl 0344.92016
[23] Londen, S. O.: Integral equations of Volterra type. Mathematics of biology (1981)
[24] Miller, R. K.: On the linearization of Volterra integral equations. J. math. Anal. appl. 23, 198-208 (1968) · Zbl 0167.40902
[25] Nallaswamy, R.; Shukla, J. B.: Effects of dispersal on the stability of a gonorrhea endemic model. Math. biosci. 61, 63-72 (1982) · Zbl 0523.92021
[26] Nold, A.: Heterogeneity in disease-transmission modeling. Math. biosci. 52, 227-240 (1980) · Zbl 0454.92020
[27] Post, W. M.; Deangelis, D. L.; Travis, C. C.: Endemic disease in environments with spatially heterogeneous host populations. Math. biosci. 63, 289-302 (1983) · Zbl 0528.92018
[28] Rushton, S.; Mautner, A. J.: The deterministic model of a simple epidemic for more than one community. Biometrika 42, 126-132 (1955) · Zbl 0064.39102
[29] D. Schenzle, An age structured model of pre- and post-vaccination measles transmission, IMA J. Math. Appl. Biol. Med., to appear. · Zbl 0611.92021
[30] Thieme, H. R.: Global asymptotic stability in epidemic models. Lecture notes in mathematics 1017, 608-615 (1983)
[31] H.R. Thiema, Local stability in epidemic models for heterogeneous populations, in Conference on Mathematics in Biology and Medicine, Bari, 1983 (V. Capasso, Ed.), Springer Lecture in Biomathematics, to appear.
[32] H.R. Thieme, Renewal theorems for some mathematical models in epidemiology, J. Integral Equations, to appear. · Zbl 0565.92020
[33] Tudor, D. W.: An age dependent epidemic model with application to measles. Math. biosci. 73, 131-147 (1985) · Zbl 0572.92023
[34] Varga, R. S.: Matrix iterative analysis. (1962)