August  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 11, 2015 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
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B. BuonomoA. d'Onofrio and 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.  Google Scholar

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B. Buonomo and 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.  Google Scholar

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

[22]

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

[23]

V. HatzopoulosM. TaylorP. L. Simon and 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.  Google Scholar

[24]

I. Z. KissJ. CassellM. Recker and 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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

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

M. Y. Li and 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.  Google Scholar

[35]

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

[36]

M. Y. Li and 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.  Google Scholar

[37]

G. Lu and 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.  Google Scholar

[38] A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor and Francis, London, 1992.   Google Scholar
[39] P. Manfredi and 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.  Google Scholar
[40]

L. Pei and 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.  Google Scholar

[41]

A. PimenovT. C. KellyA. KorobeinikovM. J. A. O'CallaghanA. V. Pokrovskii and 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.  Google Scholar

[42]

A. PimenovT. C. KellyA. KorobeinikovM. J. A. O'Callaghan and 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.  Google Scholar

[43]

T. C. RelugaC. T. Bauch and 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.  Google Scholar

[44]

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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[45]

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

[46]

C. Vargas-De-León and 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

C. T. Bauch and 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.  Google Scholar

[3]

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

[4]

E. Beretta and 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.  Google Scholar

[5]

S. Bhattacharyya and 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.  Google Scholar

[6]

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

[7]

B. BuonomoA. d'Onofrio and 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.  Google Scholar

[8]

B. BuonomoA. d'Onofrio and 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.  Google Scholar

[9]

B. BuonomoA. d'Onofrio and 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.  Google Scholar

[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), 255-266.  doi: 10.1016/j.jmaa.2008.07.021.  Google Scholar

[11]

B. Buonomo and 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.  Google Scholar

[12]

B. Buonomo and 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.  Google Scholar

[13]

V. Capasso and 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.  Google Scholar

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

A. d'OnofrioP. Manfredi and 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.  Google Scholar

[16]

A. d'OnofrioP. Manfredi and 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.  Google Scholar

[17]

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

[18]

A. d'Onofrio and 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.  Google Scholar

[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), 1012-1020.  doi: 10.1093/oxfordjournals.aje.a114471.  Google Scholar

[20]

S. FunkM. 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), 1247-1256.  doi: 10.1098/rsif.2010.0142.  Google Scholar

[21]

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

[22]

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

[23]

V. HatzopoulosM. TaylorP. L. Simon and 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.  Google Scholar

[24]

I. Z. KissJ. CassellM. Recker and 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.  Google Scholar

[25]

A. Korobeinikov and 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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

M. Y. Li and 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.  Google Scholar

[35]

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

[36]

M. Y. Li and 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.  Google Scholar

[37]

G. Lu and 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.  Google Scholar

[38] A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor and Francis, London, 1992.   Google Scholar
[39] P. Manfredi and 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.  Google Scholar
[40]

L. Pei and 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.  Google Scholar

[41]

A. PimenovT. C. KellyA. KorobeinikovM. J. A. O'CallaghanA. V. Pokrovskii and 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.  Google Scholar

[42]

A. PimenovT. C. KellyA. KorobeinikovM. J. A. O'Callaghan and 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.  Google Scholar

[43]

T. C. RelugaC. T. Bauch and 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.  Google Scholar

[44]

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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[45]

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

[46]

C. Vargas-De-León and 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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