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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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [21] F. Hoppensteadt, An age-dependent epidemic model,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4. [22] F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epiemics,, Philadelphia: Society for industrial and applied mathematics, (1975). [23] R. K. Miller, Nolinear Volterra Integral Equations,, W. A. Benjamin, (1971). [24] F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331. [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. [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. [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. [28] J. R. Haddock, T. Krisztin and J. Terjeki, Invariance principles for autonomous functional-differential equations,, J. Integral Equations, 10 (1985), 123. [29] S. Spencer, Stochastic Epidemic Models for Emerging Diseases,, Ph.D. thesis, (2008). [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. [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. [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. [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. [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. [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. [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. [37] X. R. Mao, Stochastic Differential Equations and Their Applications,, Chichester: Horwood publishing, (1997). [38] D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Rev., 43 (2001), 525. doi: 10.1137/S0036144500378302.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [21] F. Hoppensteadt, An age-dependent epidemic model,, J. Franklin Inst., 297 (1974), 325. doi: 10.1016/0016-0032(74)90037-4. [22] F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epiemics,, Philadelphia: Society for industrial and applied mathematics, (1975). [23] R. K. Miller, Nolinear Volterra Integral Equations,, W. A. Benjamin, (1971). [24] F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331. [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. [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. [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. [28] J. R. Haddock, T. Krisztin and J. Terjeki, Invariance principles for autonomous functional-differential equations,, J. Integral Equations, 10 (1985), 123. [29] S. Spencer, Stochastic Epidemic Models for Emerging Diseases,, Ph.D. thesis, (2008). [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. [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. [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. [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. [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. [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. [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. [37] X. R. Mao, Stochastic Differential Equations and Their Applications,, Chichester: Horwood publishing, (1997). [38] D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Rev., 43 (2001), 525. doi: 10.1137/S0036144500378302.
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