Global stability of infectious disease models with
contact rate as a function of prevalence index
Pages: 1019  1033,
Issue 4,
August
2017
doi:10.3934/mbe.2017053 Abstract
References
Full text (410.1K)
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Cruz VargasDeLeón  Maestría en Ciencias de la Salud, Escuela Superior de Medicina, Instituto Politécnico Nacional, Plan de San Luis y Díaz Mirón s/n, Col. Casco de Santo Tomas, Del. Miguel Hidalgo, 11340, Ciudad de México, Mexico (email)
Alberto d'Onofrio  International Prevention Research Institute, 96 Cours Lafayette, 69006 Lyon, France (email)
1 
C. Auld, Choices, beliefs, and infectious disease dynamics, J. Health. Econ., 22 (2003), 361377. 

2 
C. T. Bauch and D. J. D. Earn, Vaccination and the theory of games, Proc. Natl. Acad. Sci. U S A., 101 (2004), 1339113394. 

3 
C. T. Bauch, Imitation dynamics predict vaccinating behavior, Proc. R. Soc. London B, 272 (2005), 16691675. 

4 
E. Beretta and V. Capasso, On the general structure of epidemic systems. Global asymptotic stability, Comput. Math. Appl., Part A, 12 (1986), 677694. 

5 
S. Bhattacharyya and C. T. Bauch, "Wait and see'' vaccinating behaviour during a pandemic: A game theoretic analysis, Vaccine, 29 (2011), 55195525. 

6 
D. L. Brito, E. Sheshinski and M. D. Intriligator, Externalities and compulsory vaccinations, J. Public Econ., 45 (1991), 6990. 

7 
B. Buonomo, A. d'Onofrio and D. Lacitignola, Global stability of an $SIR$ epidemic model with information dependent vaccination, Math. Biosci., 216 (2008), 916. 

8 
B. Buonomo, A. d'Onofrio and D. Lacitignola, Rational exemption to vaccination for nonfatal $SIS$ diseases: globally stable and oscillatory endemicity, Math. Biosci. Eng., 7 (2010), 561578. 

9 
B. Buonomo, A. d'Onofrio and D. Lacitignola, Globally stable endemicity for infectious diseases with informationrelated changes in contact patterns, Appl. Math. Lett., 25 (2012), 10561060. 

10 
B. Buonomo and D. Lacitignola, On the use of the geometric approach to global stability for three dimensional ODE systems: a bilinear case, J. Math. Anal. Appl., 348 (2008), 255266. 

11 
B. Buonomo and C. VargasDeLeón, Global stability for an HIV1 infection model including an eclipse stage of infected cells, J. Math. Anal. Appl., 385 (2012), 709720. 

12 
B. Buonomo and C. VargasDeLeón, Stability and bifurcation analysis of a vectorbias model of malaria transmission, Math. Biosci., 242 (2013), 5967. 

13 
V. Capasso and G. Serio, A generalization of the KermackMcKendrick deterministic epidemic model, Math.Biosci., 42 (1978), 4361. 

14 
V. Capasso, Mathematical Structures of Epidemic Systems, $2^{nd}$ printing, SpringerVerlag, Berlin, 2008. 

15 
A. d'Onofrio, P. Manfredi and E. Salinelli, Vaccinating behaviour, information, and the dynamics of $SIR$ vaccine preventable diseases, Theor. Popul. Biol., 71 (2007), 301317. 

16 
A. d'Onofrio, P. Manfredi and E. Salinelli, Bifurcation threshold in an $SIR$ model with informationdependent vaccination, Math. Model. Nat. Phenom., 2 (2007), 2338. 

17 
A. d'Onofrio, P. Manfredi and E. Salinelli, Fatal $SIR$ diseases and rational exemption to vaccination, Math. Med. Biol., 25 (2008), 337357. 

18 
A. d'Onofrio and P. Manfredi, Informationrelated changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases, J. Theor. Biol., 256 (2009), 473478. 

19 
P. E. M. Fine and J. A. Clarkson, Individual versus public priorities in the determination of optimal vaccination policies, Am. J. Epidemiol., 124 (1986), 10121020. 

20 
S. Funk, M. Salathe and V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, J. Royal Soc. Interface, 7 (2010), 12471256. 

21 
P. Y. Geoffard and T. Philipson, Disease eradication: Private versus public vaccination, Am. Econ. Rev., 87 (1997), 222230. 

22 
B. S. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313318. 

23 
V. Hatzopoulos, M. Taylor, P. L. Simon and I. Z. Kiss, Multiple sources and routes of information transmission: Implications for epidemic dynamics, Math. Biosci., 231 (2011), 197209. 

24 
I. Z. Kiss, J. Cassell, M. Recker and P. L. Simon, The impact of information transmission on epidemic outbreaks, Math. Biosci., 225 (2010), 110. 

25 
A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for $SIR$, $SIRS$, and $SIS$ epidemiological models, Appl. Math. Lett., 15 (2002), 955960. 

26 
A. Korobeinikov, Lyapunov functions and global properties for $SEIR$ and $SEIS$ epidemic models, Math. Med. Biol., 21 (2004), 7583. 

27 
A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879883. 

28 
A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113128. 

29 
A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with nonlinear transmission, Bull. Math. Biol., 68 (2006), 615626. 

30 
A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 18711886. 

31 
A. Korobeinikov, Global asymptotic properties of virus dynamics models with dosedependent parasite reproduction and virulence, and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225239. 

32 
A. Korobeinikov, Stability of ecosystem: Global properties of a general preypredator model, Math. Med. Biol., 26 (2009), 309321. 

33 
J. La Salle, Stability by Liapunov's Direct Method with Applications, $1^{st}$ printing, Academic Press, New YorkLondon, 1961. 

34 
M. Y. Li and J. S. Muldowney, Global stability for the $SEIR$ model in epidemiology, Math. Biosci., 125 (1995), 155164. 

35 
M. Y. Li and J. S. Muldowney, A geometric approach to globalstability problems, SIAM J. Math. Anal., 27 (1996), 10701083. 

36 
M. Y. Li and L. Wang, Backward bifurcation in a mathematical model for HIV infection in vivo with antiretroviral treatment, Nonlinear Anal. Real World Appl., 17 (2014), 147160. 

37 
G. Lu and Z. Lu, Geometric approach for global asymptotic stability of threedimensional LotkaVolterra systems, J. Math. Anal. Appl., 389 (2012), 591596. 

38 
A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor and Francis, London, 1992. 

39 
P. Manfredi and A. d'Onofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, SpringerVerlag, New York, 2013. 

40 
L. Pei and J. Zhang, Losing weight and elimination of weight cycling by the geometric approach to globalstability problem, Nonlinear Anal. RWA, 14 (2013), 18651870. 

41 
A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan, A. V. Pokrovskii and D. Rachinskii, Memory effects in population dynamics: Spread of infectious disease as a case study, Math. Model. Nat. Phenom., 7 (2012), 204226. 

42 
A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan and D. Rachinskii, Adaptive behaviour and multiple equilibrium states in a predatorprey model, Theor. Popul. Biol., 101 (2015), 2430. 

43 
T. C. Reluga, C. T. Bauch and A. P. Galvani, Evolving public perceptions and stability in vaccine uptake, Math. Biosci., 204 (2006), 185198. 

44 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 2948. 

45 
R. Vardavas, R. Breban and S. Blower, Can influenza epidemics be prevented by voluntary vaccination?, PLoS Comp. Biol., 3 (2007), e85. 

46 
C. VargasDeLeón and A. Korobeinikov, Global stability of a population dynamics model with inhibition and negative feedback, Math. Med. Biol., 30 (2013), 6572. 

47 
C. VargasDeLeón, Global properties for virus dynamics model with mitotic transmission and intracellular delay, J. Math. Anal. Appl., 381 (2011), 884890. 

48 
C. VargasDeLeón, Global properties for a virus dynamics model with lytic and nonlytic immune responses and nonlinear immune attack rates, J. Biol. Syst., 22 (2014), 449462. 

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