2017, 14(4): 1019-1033. doi: 10.3934/mbe.2017053

Global stability of infectious disease models with contact rate as a function of prevalence index

1. 

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

2. 

Maestría en Ciencias de la Complejidad, Universidad Autónoma de la Ciudad de México, San Lorenzo 290, Col. Del Valle Sur Del.Benito Juárez, 03100, Ciudad de México, Mexico

3. 

International Prevention Research Institute, 96 Cours Lafayette, 69006 Lyon, France

* Corresponding author: leoncruz82@yahoo.com.mx

Received  August 1 2015 Revised  January 2017 Accepted  January 26, 2017 Published  March 2017

In this paper, we consider a SEIR epidemiological model with information-related changes in contact patterns. One of the main features of the model is that it includes an information variable, a negative feedback on the behavior of susceptible subjects, and a function that describes the role played by the infectious size in the information dynamics. Here we focus in the case of delayed information. By using suitable assumptions, we analyze the global stability of the endemic equilibrium point and disease-free equilibrium point. Our approach is applicable to global stability of the endemic equilibrium of the previously defined SIR and SIS models with feedback on behavior of susceptible subjects.
Citation: Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053
References:
[1]

C. Auld, Choices, beliefs, and infectious disease dynamics, J. Health. Econ., 22 (2003), 361-377. doi: 10.1016/S0167-6296(02)00103-0.

[2]

C. T. Bauch, D. J. D. Earn, Vaccination and the theory of games, Proc. Natl. Acad. Sci. U S A., 101 (2004), 13391-13394. doi: 10.1073/pnas.0403823101.

[3]

C. T. Bauch, Imitation dynamics predict vaccinating behavior, Proc. R. Soc. London B, 272 (2005), 1669-1675.

[4]

E. Beretta, V. Capasso, On the general structure of epidemic systems. Global asymptotic stability, Comput. Math. Appl., Part A, 12 (1986), 677-694. doi: 10.1016/0898-1221(86)90054-4.

[5]

S. Bhattacharyya, C. T. Bauch, ''Wait and see'' vaccinating behaviour during a pandemic: A game theoretic analysis, Vaccine, 29 (2011), 5519-5525. doi: 10.1016/j.vaccine.2011.05.028.

[6]

D. L. Brito, E. Sheshinski, M. D. Intriligator, Externalities and compulsory vaccinations, J. Public Econ., 45 (1991), 69-90.

[7]

B. Buonomo, A. d'Onofrio, D. Lacitignola, Global stability of an SIR epidemic model with information dependent vaccination, Math. Biosci., 216 (2008), 9-16. doi: 10.1016/j.mbs.2008.07.011.

[8]

B. Buonomo, A. d'Onofrio, D. Lacitignola, Rational exemption to vaccination for non-fatal SIS diseases: globally stable and oscillatory endemicity, Math. Biosci. Eng., 7 (2010), 561-578. doi: 10.3934/mbe.2010.7.561.

[9]

B. Buonomo, A. d'Onofrio, D. Lacitignola, Globally stable endemicity for infectious diseases with information-related changes in contact patterns, Appl. Math. Lett., 25 (2012), 1056-1060. doi: 10.1016/j.aml.2012.03.016.

[10]

B. Buonomo, 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), 255-266. doi: 10.1016/j.jmaa.2008.07.021.

[11]

B. Buonomo, C. Vargas-De-León, Global stability for an HIV-1 infection model including an eclipse stage of infected cells, J. Math. Anal. Appl., 385 (2012), 709-720. doi: 10.1016/j.jmaa.2011.07.006.

[12]

B. Buonomo, C. Vargas-De-León, Stability and bifurcation analysis of a vector-bias model of malaria transmission, Math. Biosci., 242 (2013), 59-67. doi: 10.1016/j.mbs.2012.12.001.

[13]

V. Capasso, G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math.Biosci., 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8.

[14] V. Capasso, Mathematical Structures of Epidemic Systems, 2 printing, Springer-Verlag, Berlin, 2008.
[15]

A. d'Onofrio, P. Manfredi, E. Salinelli, Vaccinating behaviour, information, and the dynamics of $SIR$ vaccine preventable diseases, Theor. Popul. Biol., 71 (2007), 301-317. doi: 10.1016/j.tpb.2007.01.001.

[16]

A. d'Onofrio, P. Manfredi, E. Salinelli, Bifurcation threshold in an SIR model with information-dependent vaccination, Math. Model. Nat. Phenom., 2 (2007), 23-38. doi: 10.1051/mmnp:2008009.

[17]

A. d'Onofrio, P. Manfredi, E. Salinelli, Fatal SIR diseases and rational exemption to vaccination, Math. Med. Biol., 25 (2008), 337-357. doi: 10.1093/imammb/dqn019.

[18]

A. d'Onofrio, P. Manfredi, Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases, J. Theor. Biol., 256 (2009), 473-478. doi: 10.1016/j.jtbi.2008.10.005.

[19]

P. E. M. Fine, J. A. Clarkson, Individual versus public priorities in the determination of optimal vaccination policies, Am. J. Epidemiol., 124 (1986), 1012-1020. doi: 10.1093/oxfordjournals.aje.a114471.

[20]

S. Funk, M. Salathe, V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, J. Royal Soc. Interface, 7 (2010), 1247-1256. doi: 10.1098/rsif.2010.0142.

[21]

P. Y. Geoffard, T. Philipson, Disease eradication: Private versus public vaccination, Am. Econ. Rev., 87 (1997), 222-230.

[22]

B. S. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318. doi: 10.1007/BF00275063.

[23]

V. Hatzopoulos, M. Taylor, P. L. Simon, I. Z. Kiss, Multiple sources and routes of information transmission: Implications for epidemic dynamics, Math. Biosci., 231 (2011), 197-209. doi: 10.1016/j.mbs.2011.03.006.

[24]

I. Z. Kiss, J. Cassell, M. Recker, P. L. Simon, The impact of information transmission on epidemic outbreaks, Math. Biosci., 225 (2010), 1-10. doi: 10.1016/j.mbs.2009.11.009.

[25]

A. Korobeinikov, G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960. doi: 10.1016/S0893-9659(02)00069-1.

[26]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1093/imammb/21.2.75.

[27]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001.

[28]

A. Korobeinikov, P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128. doi: 10.1093/imammb/dqi001.

[29]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9.

[30]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y.

[31]

A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence, and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225-239.

[32]

A. Korobeinikov, Stability of ecosystem: Global properties of a general prey-predator model, Math. Med. Biol., 26 (2009), 309-321. doi: 10.1093/imammb/dqp009.

[33] J. La Salle, Stability by Liapunov's Direct Method with Applications, 1 printing, Academic Press, New York-London, 1961.
[34]

M. Y. Li, J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155-164. doi: 10.1016/0025-5564(95)92756-5.

[35]

M. Y. Li, J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083. doi: 10.1137/S0036141094266449.

[36]

M. Y. Li, L. Wang, Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonlinear Anal. Real World Appl., 17 (2014), 147-160. doi: 10.1016/j.nonrwa.2013.11.002.

[37]

G. Lu, Z. Lu, Geometric approach for global asymptotic stability of three-dimensional Lotka-Volterra systems, J. Math. Anal. Appl., 389 (2012), 591-596. doi: 10.1016/j.jmaa.2011.11.075.

[38] A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor and Francis, London, 1992.
[39] P. Manfredi, A. d'Onofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4614-5474-8.
[40]

L. Pei, J. Zhang, Losing weight and elimination of weight cycling by the geometric approach to global-stability problem, Nonlinear Anal. RWA, 14 (2013), 1865-1870. doi: 10.1016/j.nonrwa.2012.12.003.

[41]

A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan, A. V. Pokrovskii, D. Rachinskii, Memory effects in population dynamics: Spread of infectious disease as a case study, Math. Model. Nat. Phenom., 7 (2012), 204-226. doi: 10.1051/mmnp/20127313.

[42]

A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan, D. Rachinskii, Adaptive behaviour and multiple equilibrium states in a predator-prey model, Theor. Popul. Biol., 101 (2015), 24-30. doi: 10.1016/j.tpb.2015.02.004.

[43]

T. C. Reluga, C. T. Bauch, A. P. Galvani, Evolving public perceptions and stability in vaccine uptake, Math. Biosci., 204 (2006), 185-198. doi: 10.1016/j.mbs.2006.08.015.

[44]

P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[45]

R. Vardavas, R. Breban, S. Blower, Can influenza epidemics be prevented by voluntary vaccination?, PLoS Comp. Biol., 3 (2007), e85. doi: 10.1371/journal.pcbi.0030085.

[46]

C. Vargas-De-León, A. Korobeinikov, Global stability of a population dynamics model with inhibition and negative feedback, Math. Med. Biol., 30 (2013), 65-72. doi: 10.1093/imammb/dqr027.

[47]

C. Vargas-De-León, Global properties for virus dynamics model with mitotic transmission and intracellular delay, J. Math. Anal. Appl., 381 (2011), 884-890. doi: 10.1016/j.jmaa.2011.04.012.

[48]

C. Vargas-De-León, Global properties for a virus dynamics model with lytic and nonlytic immune responses and nonlinear immune attack rates, J. Biol. Syst., 22 (2014), 449-462. doi: 10.1142/S021833901450017X.

show all references

References:
[1]

C. Auld, Choices, beliefs, and infectious disease dynamics, J. Health. Econ., 22 (2003), 361-377. doi: 10.1016/S0167-6296(02)00103-0.

[2]

C. T. Bauch, D. J. D. Earn, Vaccination and the theory of games, Proc. Natl. Acad. Sci. U S A., 101 (2004), 13391-13394. doi: 10.1073/pnas.0403823101.

[3]

C. T. Bauch, Imitation dynamics predict vaccinating behavior, Proc. R. Soc. London B, 272 (2005), 1669-1675.

[4]

E. Beretta, V. Capasso, On the general structure of epidemic systems. Global asymptotic stability, Comput. Math. Appl., Part A, 12 (1986), 677-694. doi: 10.1016/0898-1221(86)90054-4.

[5]

S. Bhattacharyya, C. T. Bauch, ''Wait and see'' vaccinating behaviour during a pandemic: A game theoretic analysis, Vaccine, 29 (2011), 5519-5525. doi: 10.1016/j.vaccine.2011.05.028.

[6]

D. L. Brito, E. Sheshinski, M. D. Intriligator, Externalities and compulsory vaccinations, J. Public Econ., 45 (1991), 69-90.

[7]

B. Buonomo, A. d'Onofrio, D. Lacitignola, Global stability of an SIR epidemic model with information dependent vaccination, Math. Biosci., 216 (2008), 9-16. doi: 10.1016/j.mbs.2008.07.011.

[8]

B. Buonomo, A. d'Onofrio, D. Lacitignola, Rational exemption to vaccination for non-fatal SIS diseases: globally stable and oscillatory endemicity, Math. Biosci. Eng., 7 (2010), 561-578. doi: 10.3934/mbe.2010.7.561.

[9]

B. Buonomo, A. d'Onofrio, D. Lacitignola, Globally stable endemicity for infectious diseases with information-related changes in contact patterns, Appl. Math. Lett., 25 (2012), 1056-1060. doi: 10.1016/j.aml.2012.03.016.

[10]

B. Buonomo, 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), 255-266. doi: 10.1016/j.jmaa.2008.07.021.

[11]

B. Buonomo, C. Vargas-De-León, Global stability for an HIV-1 infection model including an eclipse stage of infected cells, J. Math. Anal. Appl., 385 (2012), 709-720. doi: 10.1016/j.jmaa.2011.07.006.

[12]

B. Buonomo, C. Vargas-De-León, Stability and bifurcation analysis of a vector-bias model of malaria transmission, Math. Biosci., 242 (2013), 59-67. doi: 10.1016/j.mbs.2012.12.001.

[13]

V. Capasso, G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math.Biosci., 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8.

[14] V. Capasso, Mathematical Structures of Epidemic Systems, 2 printing, Springer-Verlag, Berlin, 2008.
[15]

A. d'Onofrio, P. Manfredi, E. Salinelli, Vaccinating behaviour, information, and the dynamics of $SIR$ vaccine preventable diseases, Theor. Popul. Biol., 71 (2007), 301-317. doi: 10.1016/j.tpb.2007.01.001.

[16]

A. d'Onofrio, P. Manfredi, E. Salinelli, Bifurcation threshold in an SIR model with information-dependent vaccination, Math. Model. Nat. Phenom., 2 (2007), 23-38. doi: 10.1051/mmnp:2008009.

[17]

A. d'Onofrio, P. Manfredi, E. Salinelli, Fatal SIR diseases and rational exemption to vaccination, Math. Med. Biol., 25 (2008), 337-357. doi: 10.1093/imammb/dqn019.

[18]

A. d'Onofrio, P. Manfredi, Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases, J. Theor. Biol., 256 (2009), 473-478. doi: 10.1016/j.jtbi.2008.10.005.

[19]

P. E. M. Fine, J. A. Clarkson, Individual versus public priorities in the determination of optimal vaccination policies, Am. J. Epidemiol., 124 (1986), 1012-1020. doi: 10.1093/oxfordjournals.aje.a114471.

[20]

S. Funk, M. Salathe, V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, J. Royal Soc. Interface, 7 (2010), 1247-1256. doi: 10.1098/rsif.2010.0142.

[21]

P. Y. Geoffard, T. Philipson, Disease eradication: Private versus public vaccination, Am. Econ. Rev., 87 (1997), 222-230.

[22]

B. S. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318. doi: 10.1007/BF00275063.

[23]

V. Hatzopoulos, M. Taylor, P. L. Simon, I. Z. Kiss, Multiple sources and routes of information transmission: Implications for epidemic dynamics, Math. Biosci., 231 (2011), 197-209. doi: 10.1016/j.mbs.2011.03.006.

[24]

I. Z. Kiss, J. Cassell, M. Recker, P. L. Simon, The impact of information transmission on epidemic outbreaks, Math. Biosci., 225 (2010), 1-10. doi: 10.1016/j.mbs.2009.11.009.

[25]

A. Korobeinikov, G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960. doi: 10.1016/S0893-9659(02)00069-1.

[26]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1093/imammb/21.2.75.

[27]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001.

[28]

A. Korobeinikov, P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128. doi: 10.1093/imammb/dqi001.

[29]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9.

[30]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y.

[31]

A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence, and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225-239.

[32]

A. Korobeinikov, Stability of ecosystem: Global properties of a general prey-predator model, Math. Med. Biol., 26 (2009), 309-321. doi: 10.1093/imammb/dqp009.

[33] J. La Salle, Stability by Liapunov's Direct Method with Applications, 1 printing, Academic Press, New York-London, 1961.
[34]

M. Y. Li, J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155-164. doi: 10.1016/0025-5564(95)92756-5.

[35]

M. Y. Li, J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083. doi: 10.1137/S0036141094266449.

[36]

M. Y. Li, L. Wang, Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonlinear Anal. Real World Appl., 17 (2014), 147-160. doi: 10.1016/j.nonrwa.2013.11.002.

[37]

G. Lu, Z. Lu, Geometric approach for global asymptotic stability of three-dimensional Lotka-Volterra systems, J. Math. Anal. Appl., 389 (2012), 591-596. doi: 10.1016/j.jmaa.2011.11.075.

[38] A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor and Francis, London, 1992.
[39] P. Manfredi, A. d'Onofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4614-5474-8.
[40]

L. Pei, J. Zhang, Losing weight and elimination of weight cycling by the geometric approach to global-stability problem, Nonlinear Anal. RWA, 14 (2013), 1865-1870. doi: 10.1016/j.nonrwa.2012.12.003.

[41]

A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan, A. V. Pokrovskii, D. Rachinskii, Memory effects in population dynamics: Spread of infectious disease as a case study, Math. Model. Nat. Phenom., 7 (2012), 204-226. doi: 10.1051/mmnp/20127313.

[42]

A. Pimenov, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan, D. Rachinskii, Adaptive behaviour and multiple equilibrium states in a predator-prey model, Theor. Popul. Biol., 101 (2015), 24-30. doi: 10.1016/j.tpb.2015.02.004.

[43]

T. C. Reluga, C. T. Bauch, A. P. Galvani, Evolving public perceptions and stability in vaccine uptake, Math. Biosci., 204 (2006), 185-198. doi: 10.1016/j.mbs.2006.08.015.

[44]

P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[45]

R. Vardavas, R. Breban, S. Blower, Can influenza epidemics be prevented by voluntary vaccination?, PLoS Comp. Biol., 3 (2007), e85. doi: 10.1371/journal.pcbi.0030085.

[46]

C. Vargas-De-León, A. Korobeinikov, Global stability of a population dynamics model with inhibition and negative feedback, Math. Med. Biol., 30 (2013), 65-72. doi: 10.1093/imammb/dqr027.

[47]

C. Vargas-De-León, Global properties for virus dynamics model with mitotic transmission and intracellular delay, J. Math. Anal. Appl., 381 (2011), 884-890. doi: 10.1016/j.jmaa.2011.04.012.

[48]

C. Vargas-De-León, Global properties for a virus dynamics model with lytic and nonlytic immune responses and nonlinear immune attack rates, J. Biol. Syst., 22 (2014), 449-462. doi: 10.1142/S021833901450017X.

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