Mathematical Tools for Understanding Infectious Disease Dynamics by Diekmann Odo; Heesterbeek Hans; Britton Tom

Author:Diekmann, Odo; Heesterbeek, Hans; Britton, Tom
Language: eng
Format: epub
Publisher: Princeton University Press

These two equations can be simplified to one equation for one unknown by defining a new variable y = αsπsxs + αcπcxc.

iii) Compute and the overall fraction infected for the three cases treated above.

1The function sign is defined in the usual way: sign (y) = y/|y|if y ≠ 0, and sign (0) = 0.

2See e.g., H. Inaba: Threshold and stability results for an age-structured epidemic model. J. Math. Biol, 28 (1990), 411–434; H.J.A.M. Heijmans: The dynamical behaviour of the age-size distribution of a cell population. In: J.A.J. Metz and O. Diekmann (1986), pp. 185–202. H.R. Thieme: Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math., 70 (2009), 188–211.

3O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel and H.-O. Walther: Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Springer-Verlag, Berlin, 1995; G. Gripenberg, S-O. Londen and O. Staffans: Volterra Integral and Functional Equations. Cambridge University Press, Cambridge, 1990; O. Diekmann, P. Getto and M. Gyllenberg: Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. SIAM J. Math. Anal., 39 (2007), 1023–1069.

4The proof is based on ideas in C.K. Li and H. Schneider: Applications of Perron-Probenius theory to population dynamics. J. Math. Biol., 44 (2002), 450–462, who addressed a similar problem in population dynamics in a discrete-time setting — building, in turn, on ideas in J.M. Cushing and Zhou Yicang: The net reproductive value and stability in matrix population models. Nat. Res. Mod., 8 (1994), 297–333. In P. Van den Driessche and J. Watmough: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 180 (2002), 29–48, a proof is presented in terms of M-matrices, and we refer to H.R. Thieme: Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math., 70 (2009), 188–211, for the analogous result for the infinite dimensional case.

5 A. Nold: Heterogeneity in disease-transmission modeling. Math. Biosci., 52 (1980), 227–240.

6 See e.g., R.D. Nussbaum: The Fixed Point Index and Some Applications. Séminaire de Mathématiques Supérieures. les Presses de l’Université de Montreal, Canada, 1985.

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