Abstract
Three behavioral-epidemic models (i.e., epidemic systems including feedbacks (FB) that the information about an infectious disease has on its spreading) are introduced. Two relevant FB are explicitly considered: the pseudo-rational exemption to vaccination and the information-related changes in contact patterns by healthy subjects. The global stability analysis of the endemic states is performed by means of the geometric approach to stability, with particular focus on a model of vaccination of adult susceptible subjects. Biological implications of the results are discussed.
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Notes
- 1.
That is, for the generic matrix A = (a ij ), \(\vert A\vert =\max _{1\leq k\leq n}\sum\limits _{j=1}^{n}\vert a_{jk}\vert \) and \(\mathcal{L}(A) =\max _{1\leq k\leq n}(a_{kk} + \sum\limits _{j=1(j\neq k)}^{n}\vert a_{jk}\vert )\).
- 2.
A function \(g \in {C}^{1}(D \rightarrow {\text{R}}^{n})\) is called a C 1 local ε-perturbation of f at x 0 ∈ D if there exists an open neighborhood U of x 0 in D such that the support supp(f − g)\(\subset U\) and \(f - g_{{C}^{1}} < \epsilon \), where \(f - g_{{C}^{1}} =\sup \left \{f(x) - g(x) + f_{x}(x) - g_{x}(x) : x \in D\right \}\).
- 3.
A point x 0 ∈ D is said to be non-wandering for Eq. (33) if for any neighborhood U of x 0 in D and there exists arbitrarily large t such that \(U \cap x(t,U)\neq \varnothing \). For example, any equilibrium, alpha limit point or omega limit point, is non-wandering.
References
Auld, C.: J. Health Econ. 22, 361–377 (2003)
Bauch, C.T.: Proc. Royal Soc. London B 272, 1669–1675 (2005)
Bauch, C.T., Earn, D.: PNAS 101, 13391–13394 (2004)
Beretta, E., Kon, R., Takeuchi, Y.: Nonlinear Anal. RWA 3, 107–129 (2002)
Beretta, E., Solimano, F., Takeuchi, Y.: Nonlinear Anal. TMA 50, 941–966 (2002)
Brito, D.L., Sheshinski, E., Intriligator, M.D.: J. Public Econ. 45, 69–90 (1991)
Buonomo, B., d’Onofrio, A., Lacitignola, D.: Math. Biosci. 216, 9–16 (2008)
Buonomo, B., d’Onofrio, A., Lacitignola, D.: Math. Biosci. Eng. 7, 561–578 (2010)
Buonomo, B., d’Onofrio, A., Lacitignola, D.: Appl. Math. Lett. 25, 1056–1060 (2012)
Buonomo, B., Lacitignola, D.: Nonlinear Anal. RWA 5, 749–762 (2004)
Buonomo, B., Lacitignola, D.: Proc. Dyn. Syst. Appl. 4, 53–57 (2004)
Buonomo, B., Lacitignola, D.: J. Math. Anal. Appl. 348, 255–266 (2008)
Buonomo, B., Lacitignola, D.: In: Manganaro, N., et al. (eds.) Proceedings of Waves and Stability in Continuous Media, Scicli, Italy, June 2007, pp 78–83. World Scientific, Hackensack (2008)
Buonomo, B., Vargas De-León, C.: J. Math. Anal. Appl. 385, 709–720 (2012)
Capasso, V.: Mathematical structures of epidemic systems. In: Lecture Notes in Biomathematics. Springer, Berlin (2008)
d’Onofrio, A., Manfredi, P.: J. Theor. Biol. 256, 473–478 (2009)
d’Onofrio, A., Manfredi, P., Salinelli, E.: Theor. Pop. Biol. 71, 301–317 (2007)
d’Onofrio, A., Manfredi, P., Salinelli, E.: Math. Med. Biol. 25, 337–357 (2008)
Efimov, D.V., Fradkov, A.L.: Math. Biosc. 216, 187–191 (2008)
Efimov, D.V., Fradkov, A.L.: Siam J. Contr. Optim. 48, 618–640 (2009)
Fine, P., Clarkson, J.: Am. J. Epidemiol. 124, 1012–1020 (1986)
Freedman, H.I., Ruan, S., Tang, M.: J. Diff. Eq. 6, 583–600 (1994)
Geoffard, P.Y., Philipson, T.: Am. Econ. Rev. 87, 222–230 (1997)
Hethcote, H.W.: Math. Biosci. 28, 335–356 (1976)
Hutson, V., Schmitt, K.: Math. Biosci. 111, 1–71 (1992)
Iwami, S., Takeuchi, Y., Liu, X.: Math. Biosci. 207, 1–25 (2007)
Li, M.Y., Graef, J.R., Wang, L., Karsai, J.: Math. Biosci. 160, 191–213 (1999)
Li, M.Y., Muldowney, J.S.: J. Diff. Eq. 106, 27–39 (1993)
Li, M.Y., Muldowney, J.S.: Math. Biosci. 125, 155–164 (1995)
Li, M.Y., Muldowney, J.S.: SIAM J. Math. Anal. 27, 1070–1083 (1996)
Li, G., Wang, W., Jin, Z.: Chaos Soliton. Fract. 30, 1012–1019 (2006)
MacDonald, N.: Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge (1989)
Martin Jr, R.H.: J. Math. Anal. Appl. 45, 432–454 (1974)
Muldowney, J.S.: Rocky Mount. J. Math. 20, 857–872 (1990)
Murray, J.D.: Mathematical Biology. Interdisciplinary Applied Mathematics, vol. 17. Springer, Heidelberg (2002)
Reluga, T.C., Bauch, C.T., Galvani, A.P.: Math. Biosci. 204, 185–198 (2006)
Salmon, D.A., Teret, S.P., Raina MacIntyre, C., et al.: Lancet 367, 436–442 (2006)
Vardavas, R., Breban, R., Blower, S.: PLoS Comp. Biol. 3, e85 (2007)
Wang, L., Li, M.Y.: Math. Biosci. 200, 44–57 (2006)
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Buonomo, B., d’Onofrio, A., Lacitignola, D. (2013). The Geometric Approach to Global Stability in Behavioral Epidemiology. In: Manfredi, P., D'Onofrio, A. (eds) Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5474-8_18
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