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Impact of group mixing on disease dynamics. (English) Zbl 1200.92027
Summary: A general mathematical model is proposed to study the impact of group mixing in a heterogeneous host population on the spread of a disease that confers temporary immunity upon recovery. The model contains general distribution functions that account for the probabilities that individuals remain in the recovered class after recovery. For this model, the basic reproduction number 0 is identified. It is shown that if 0 <1, then the disease dies out in the sense that the disease free equilibrium is globally asymptotically stable; whereas if 0 >1, this equilibrium becomes unstable. In this latter case, depending on the distribution functions and the group mixing strengths, the disease either persists at a constant endemic level or exhibits sustained oscillatory behavior.
MSC:
92C60Medical epidemiology
34D23Global stability of ODE
37N25Dynamical systems in biology
92D30Epidemiology
References:
[1]Brauer, F.; Den Driessche, P. Van; Wang, L.: Oscillations in a patchy environment disease model, Math. biosci. 215, 1 (2008) · Zbl 1176.34098 · doi:10.1016/j.mbs.2008.05.001
[2]Gantmacher, F. R.: Applications of the theory of matrices, (1959) · Zbl 0085.01001
[3]Hethcote, H. W.; Stech, H. W.; Den Driessche, P. Van: Nonlinear oscillations in epidemic models, SIAM J. Appl. math. 40, 1 (1981) · Zbl 0469.92012 · doi:10.1137/0140001
[4]Lajmanovich, A.; Yorke, J. A.: A deterministic model for gonorrhea in a nonhomogeneous population, Math. biosci. 28, 221 (1976) · Zbl 0344.92016 · doi:10.1016/0025-5564(76)90125-5
[5]Lloyd, A. L.; May, R. M.: Spatial heterogeneity in epidemic models, J. theor. Biol. 179, 1 (1996)
[6]Miller, R. K.: Nonlinear Volterra integral equations, (1971)
[7]Smith, H. L.: Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical surveys and monographs 41 (1995) · Zbl 0821.34003
[8]Den Driessche, P. Van; Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci. 180, 29 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6