• Previous Article
    Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy
  • MBE Home
  • This Issue
  • Next Article
    Cilium height difference between strokes is more effective in driving fluid transport in mucociliary clearance: A numerical study
2015, 12(5): 1083-1106. doi: 10.3934/mbe.2015.12.1083

Global stability of a multi-group model with vaccination age, distributed delay and random perturbation

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China, China

Received  September 2014 Revised  January 2015 Published  June 2015

A multi-group epidemic model with distributed delay and vaccination age has been formulated and studied. Mathematical analysis shows that the global dynamics of the model is determined by the basic reproduction number $\mathcal{R}_0$: the disease-free equilibrium is globally asymptotically stable if $\mathcal{R}_0\leq1$, and the endemic equilibrium is globally asymptotically stable if $\mathcal{R}_0>1$. Lyapunov functionals are constructed by the non-negative matrix theory and a novel grouping technique to establish the global stability. The stochastic perturbation of the model is studied and it is proved that the endemic equilibrium of the stochastic model is stochastically asymptotically stable in the large under certain conditions.
Citation: Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083
References:
[1]

K. L. Cooke, Stability analysis for a vector disease model,, Rocky Mount. J. Math., 9 (1979), 31. doi: 10.1216/RMJ-1979-9-1-31. Google Scholar

[2]

E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays,, J. Math. Biol., 33 (1995), 250. doi: 10.1007/BF00169563. Google Scholar

[3]

H. Y. Shu, D. J. Fan and J. J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission,, Nonlinear Anal.: Real World Appl., 13 (2012), 1581. doi: 10.1016/j.nonrwa.2011.11.016. Google Scholar

[4]

A. Lajmanovich and J. A. York, A deterministic model for gonorrhea in a nonhomogeneous population,, Math. Biosci., 28 (1976), 221. doi: 10.1016/0025-5564(76)90125-5. Google Scholar

[5]

D. Q. Ding and X. H. Ding, Global stability of multi-group vaccination epidemic models with delays,, Nonlinear Anal.: Real World Appl., 12 (2011), 1991. doi: 10.1016/j.nonrwa.2010.12.015. Google Scholar

[6]

R. Y. Sun and J. P. Shi, Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates,, Appl. Math. Comput., 218 (2011), 280. doi: 10.1016/j.amc.2011.05.056. Google Scholar

[7]

H. Chen and J. T. Sun, Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates,, Appl. Math. Comput., 218 (2011), 4391. doi: 10.1016/j.amc.2011.10.015. Google Scholar

[8]

T. Kuniya, Global stability of a multi-group SVIR epidemic model,, Nonlinear Anal.: Real World Appl., 14 (2013), 1135. doi: 10.1016/j.nonrwa.2012.09.004. Google Scholar

[9]

H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models,, Canad. Appl. Math. Quart., 14 (2006), 259. Google Scholar

[10]

H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions,, Proc. Amer. Math. Soc., 136 (2008), 2793. doi: 10.1090/S0002-9939-08-09341-6. Google Scholar

[11]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Differential Equations, 248 (2010), 1. doi: 10.1016/j.jde.2009.09.003. Google Scholar

[12]

Z. Shuai and P. van den Driessche, Impact of heterogeneity on the dynamics of an SEIR epidemic model,, Math. Biosci. Eng., 9 (2012), 393. doi: 10.3934/mbe.2012.9.393. Google Scholar

[13]

J. Q. Li, Y. L. Yang and Y. C. Zhou, Global stability of an epidemic model with latent stage and vaccination,, Nonlinear Anal.: Real World Appl., 12 (2011), 2163. doi: 10.1016/j.nonrwa.2010.12.030. Google Scholar

[14]

S. M. Blower and A. R. McLean., Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco,, Science, 265 (1994), 1451. doi: 10.1126/science.8073289. Google Scholar

[15]

Y. Xiao and S. Tang, Dynamics of infection with nonlinear incidence in a simple vaccination model,, Nonlinear Anal.: Real World Appl., 11 (2010), 4154. doi: 10.1016/j.nonrwa.2010.05.002. Google Scholar

[16]

X. Y. Song, Y. Jiang and H. M. Wei, Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays,, Appl. Math. Comput., 214 (2009), 381. doi: 10.1016/j.amc.2009.04.005. Google Scholar

[17]

G. P. Sahu and J. Dhar, Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate,, Appl. Math. Model., 36 (2012), 908. doi: 10.1016/j.apm.2011.07.044. Google Scholar

[18]

M. Iannelli, M. Martcheva and X. Z. Li, Strain replacement in an epidemic model with super-infection and perfect vaccination,, Math. Biosci., 195 (2005), 23. doi: 10.1016/j.mbs.2005.01.004. Google Scholar

[19]

X. Z. Li, J. Wang and M. Ghosh, Stability and bifurcation of an SIVS epidemic model with treatment and age of vaccination,, Appl. Math. Model., 34 (2010), 437. doi: 10.1016/j.apm.2009.06.002. Google Scholar

[20]

X. C. Duan, S. L. Yuan and X. Z. Li, Global stability of an SVIR model with age of vaccination,, Appl. Math. Comput., 226 (2014), 528. doi: 10.1016/j.amc.2013.10.073. Google Scholar

[21]

F. Hoppensteadt, An age-dependent epidemic model,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4. Google Scholar

[22]

F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epiemics,, Philadelphia: Society for industrial and applied mathematics, (1975). Google Scholar

[23]

R. K. Miller, Nolinear Volterra Integral Equations,, W. A. Benjamin, (1971). Google Scholar

[24]

F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331. Google Scholar

[25]

J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, in: Applied Mathematical Sciences,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar

[26]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324. Google Scholar

[27]

J. R. Haddock and J. Terjeki, Liapunov-Razumikhin functions and an invariance principle for functional-differential equations,, J. Differential Equations, 48 (1983), 95. doi: 10.1016/0022-0396(83)90061-X. Google Scholar

[28]

J. R. Haddock, T. Krisztin and J. Terjeki, Invariance principles for autonomous functional-differential equations,, J. Integral Equations, 10 (1985), 123. Google Scholar

[29]

S. Spencer, Stochastic Epidemic Models for Emerging Diseases,, Ph.D. thesis, (2008). Google Scholar

[30]

J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment,, Science, 197 (1977), 463. doi: 10.1126/science.197.4302.463. Google Scholar

[31]

X. R. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics,, Stoch Process Appl., 97 (2002), 95. doi: 10.1016/S0304-4149(01)00126-0. Google Scholar

[32]

N. Dalal, D. Greenhalgh and X. R. Mao, A stochastic model for internal HIV dynamics,, J Math Anal Appl., 341 (2008), 1084. doi: 10.1016/j.jmaa.2007.11.005. Google Scholar

[33]

C. Ji, D. Jiang and N. Shi, Multigroup SIR epidemic model with stochastic perturbation,, Phys A: Stat Mech Appl., 390 (2011), 1747. doi: 10.1016/j.physa.2010.12.042. Google Scholar

[34]

P. S. Mandal, S. Abbas and M. Banerjee, A comparative study of deterministic and stochastic dynamics for a non-autonomous allelopathic phytoplankton model,, Appl. Math. Comput., 238 (2014), 300. doi: 10.1016/j.amc.2014.04.009. Google Scholar

[35]

M. Liu, C. Bai and K. Wang, Asymptotic stability of a two-group stochastic SEIR model with infinite delays,, Commun Nonlinear Sci Numer Simulat., 19 (2014), 3444. doi: 10.1016/j.cnsns.2014.02.025. Google Scholar

[36]

Q. S. Yang and X. R. Mao, Stochastic dynamic of SIRS epidemic models with random perturbation,, Math. Biosci. Eng., 11 (2014), 1003. doi: 10.3934/mbe.2014.11.1003. Google Scholar

[37]

X. R. Mao, Stochastic Differential Equations and Their Applications,, Chichester: Horwood publishing, (1997). Google Scholar

[38]

D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Rev., 43 (2001), 525. doi: 10.1137/S0036144500378302. Google Scholar

show all references

References:
[1]

K. L. Cooke, Stability analysis for a vector disease model,, Rocky Mount. J. Math., 9 (1979), 31. doi: 10.1216/RMJ-1979-9-1-31. Google Scholar

[2]

E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays,, J. Math. Biol., 33 (1995), 250. doi: 10.1007/BF00169563. Google Scholar

[3]

H. Y. Shu, D. J. Fan and J. J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission,, Nonlinear Anal.: Real World Appl., 13 (2012), 1581. doi: 10.1016/j.nonrwa.2011.11.016. Google Scholar

[4]

A. Lajmanovich and J. A. York, A deterministic model for gonorrhea in a nonhomogeneous population,, Math. Biosci., 28 (1976), 221. doi: 10.1016/0025-5564(76)90125-5. Google Scholar

[5]

D. Q. Ding and X. H. Ding, Global stability of multi-group vaccination epidemic models with delays,, Nonlinear Anal.: Real World Appl., 12 (2011), 1991. doi: 10.1016/j.nonrwa.2010.12.015. Google Scholar

[6]

R. Y. Sun and J. P. Shi, Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates,, Appl. Math. Comput., 218 (2011), 280. doi: 10.1016/j.amc.2011.05.056. Google Scholar

[7]

H. Chen and J. T. Sun, Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates,, Appl. Math. Comput., 218 (2011), 4391. doi: 10.1016/j.amc.2011.10.015. Google Scholar

[8]

T. Kuniya, Global stability of a multi-group SVIR epidemic model,, Nonlinear Anal.: Real World Appl., 14 (2013), 1135. doi: 10.1016/j.nonrwa.2012.09.004. Google Scholar

[9]

H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models,, Canad. Appl. Math. Quart., 14 (2006), 259. Google Scholar

[10]

H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions,, Proc. Amer. Math. Soc., 136 (2008), 2793. doi: 10.1090/S0002-9939-08-09341-6. Google Scholar

[11]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Differential Equations, 248 (2010), 1. doi: 10.1016/j.jde.2009.09.003. Google Scholar

[12]

Z. Shuai and P. van den Driessche, Impact of heterogeneity on the dynamics of an SEIR epidemic model,, Math. Biosci. Eng., 9 (2012), 393. doi: 10.3934/mbe.2012.9.393. Google Scholar

[13]

J. Q. Li, Y. L. Yang and Y. C. Zhou, Global stability of an epidemic model with latent stage and vaccination,, Nonlinear Anal.: Real World Appl., 12 (2011), 2163. doi: 10.1016/j.nonrwa.2010.12.030. Google Scholar

[14]

S. M. Blower and A. R. McLean., Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco,, Science, 265 (1994), 1451. doi: 10.1126/science.8073289. Google Scholar

[15]

Y. Xiao and S. Tang, Dynamics of infection with nonlinear incidence in a simple vaccination model,, Nonlinear Anal.: Real World Appl., 11 (2010), 4154. doi: 10.1016/j.nonrwa.2010.05.002. Google Scholar

[16]

X. Y. Song, Y. Jiang and H. M. Wei, Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays,, Appl. Math. Comput., 214 (2009), 381. doi: 10.1016/j.amc.2009.04.005. Google Scholar

[17]

G. P. Sahu and J. Dhar, Analysis of an SVEIS epidemic model with partial temporary immunity and saturation incidence rate,, Appl. Math. Model., 36 (2012), 908. doi: 10.1016/j.apm.2011.07.044. Google Scholar

[18]

M. Iannelli, M. Martcheva and X. Z. Li, Strain replacement in an epidemic model with super-infection and perfect vaccination,, Math. Biosci., 195 (2005), 23. doi: 10.1016/j.mbs.2005.01.004. Google Scholar

[19]

X. Z. Li, J. Wang and M. Ghosh, Stability and bifurcation of an SIVS epidemic model with treatment and age of vaccination,, Appl. Math. Model., 34 (2010), 437. doi: 10.1016/j.apm.2009.06.002. Google Scholar

[20]

X. C. Duan, S. L. Yuan and X. Z. Li, Global stability of an SVIR model with age of vaccination,, Appl. Math. Comput., 226 (2014), 528. doi: 10.1016/j.amc.2013.10.073. Google Scholar

[21]

F. Hoppensteadt, An age-dependent epidemic model,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4. Google Scholar

[22]

F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epiemics,, Philadelphia: Society for industrial and applied mathematics, (1975). Google Scholar

[23]

R. K. Miller, Nolinear Volterra Integral Equations,, W. A. Benjamin, (1971). Google Scholar

[24]

F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331. Google Scholar

[25]

J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, in: Applied Mathematical Sciences,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar

[26]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324. Google Scholar

[27]

J. R. Haddock and J. Terjeki, Liapunov-Razumikhin functions and an invariance principle for functional-differential equations,, J. Differential Equations, 48 (1983), 95. doi: 10.1016/0022-0396(83)90061-X. Google Scholar

[28]

J. R. Haddock, T. Krisztin and J. Terjeki, Invariance principles for autonomous functional-differential equations,, J. Integral Equations, 10 (1985), 123. Google Scholar

[29]

S. Spencer, Stochastic Epidemic Models for Emerging Diseases,, Ph.D. thesis, (2008). Google Scholar

[30]

J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment,, Science, 197 (1977), 463. doi: 10.1126/science.197.4302.463. Google Scholar

[31]

X. R. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics,, Stoch Process Appl., 97 (2002), 95. doi: 10.1016/S0304-4149(01)00126-0. Google Scholar

[32]

N. Dalal, D. Greenhalgh and X. R. Mao, A stochastic model for internal HIV dynamics,, J Math Anal Appl., 341 (2008), 1084. doi: 10.1016/j.jmaa.2007.11.005. Google Scholar

[33]

C. Ji, D. Jiang and N. Shi, Multigroup SIR epidemic model with stochastic perturbation,, Phys A: Stat Mech Appl., 390 (2011), 1747. doi: 10.1016/j.physa.2010.12.042. Google Scholar

[34]

P. S. Mandal, S. Abbas and M. Banerjee, A comparative study of deterministic and stochastic dynamics for a non-autonomous allelopathic phytoplankton model,, Appl. Math. Comput., 238 (2014), 300. doi: 10.1016/j.amc.2014.04.009. Google Scholar

[35]

M. Liu, C. Bai and K. Wang, Asymptotic stability of a two-group stochastic SEIR model with infinite delays,, Commun Nonlinear Sci Numer Simulat., 19 (2014), 3444. doi: 10.1016/j.cnsns.2014.02.025. Google Scholar

[36]

Q. S. Yang and X. R. Mao, Stochastic dynamic of SIRS epidemic models with random perturbation,, Math. Biosci. Eng., 11 (2014), 1003. doi: 10.3934/mbe.2014.11.1003. Google Scholar

[37]

X. R. Mao, Stochastic Differential Equations and Their Applications,, Chichester: Horwood publishing, (1997). Google Scholar

[38]

D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Rev., 43 (2001), 525. doi: 10.1137/S0036144500378302. Google Scholar

[1]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with generalized nonlinear incidence and vaccination age. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 977-996. doi: 10.3934/dcdsb.2016.21.977

[2]

Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109

[3]

Rui Wang, Xiaoyue Li, Denis S. Mukama. On stochastic multi-group Lotka-Volterra ecosystems with regime switching. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3499-3528. doi: 10.3934/dcdsb.2017177

[4]

Jinliang Wang, Xianning Liu, Toshikazu Kuniya, Jingmei Pang. Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2795-2812. doi: 10.3934/dcdsb.2017151

[5]

Gunduz Caginalp, Mark DeSantis. Multi-group asset flow equations and stability. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 109-150. doi: 10.3934/dcdsb.2011.16.109

[6]

Chunmei Zhang, Wenxue Li, Ke Wang. Graph-theoretic approach to stability of multi-group models with dispersal. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 259-280. doi: 10.3934/dcdsb.2015.20.259

[7]

Jinliang Wang, Hongying Shu. Global analysis on a class of multi-group SEIR model with latency and relapse. Mathematical Biosciences & Engineering, 2016, 13 (1) : 209-225. doi: 10.3934/mbe.2016.13.209

[8]

Toshikazu Kuniya, Yoshiaki Muroya. Global stability of a multi-group SIS epidemic model for population migration. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1105-1118. doi: 10.3934/dcdsb.2014.19.1105

[9]

Yoshiaki Muroya. A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model). Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 999-1008. doi: 10.3934/dcdss.2015.8.999

[10]

Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057

[11]

Lili Liu, Xianning Liu, Jinliang Wang. Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2615-2630. doi: 10.3934/dcdsb.2016064

[12]

Xiaomei Feng, Zhidong Teng, Fengqin Zhang. Global dynamics of a general class of multi-group epidemic models with latency and relapse. Mathematical Biosciences & Engineering, 2015, 12 (1) : 99-115. doi: 10.3934/mbe.2015.12.99

[13]

Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080

[14]

Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241

[15]

Qun Liu, Daqing Jiang, Ningzhong Shi, Tasawar Hayat, Ahmed Alsaedi. Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2479-2500. doi: 10.3934/dcdsb.2017127

[16]

Fuke Wu, Yangzi Hu. Stochastic Lotka-Volterra system with unbounded distributed delay. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 275-288. doi: 10.3934/dcdsb.2010.14.275

[17]

Fuke Wu, Shigeng Hu. The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1065-1094. doi: 10.3934/dcds.2012.32.1065

[18]

Ya Wang, Fuke Wu, Xuerong Mao, Enwen Zhu. Advances in the LaSalle-type theorems for stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-14. doi: 10.3934/dcdsb.2019182

[19]

Mohamed Baouch, Juan Antonio López-Ramos, Blas Torrecillas, Reto Schnyder. An active attack on a distributed Group Key Exchange system. Advances in Mathematics of Communications, 2017, 11 (4) : 715-717. doi: 10.3934/amc.2017052

[20]

Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]