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Global stability for a multi-group SVIR model with age of vaccination. (English) Zbl 1395.35184

Summary: In this paper, a multi-group SVIR epidemic model with age of vaccination is considered. The model allows the vaccinated individuals to become susceptible after the vaccine loses its protective properties, and the vaccination classes satisfy first-order the partial differential equations structured by vaccination age. Combining the Lyapunov functional method with a graph-theoretic approach, we show that the global stability of endemic equilibrium for the strongly connected system is determined by the basic reproduction number. In addition, the dynamics for non-strongly connected model are also investigated, depending on the basic reproduction numbers corresponding to each strongly connected component. Numerical simulations are carried out to support the theoretical conclusions.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D30 Epidemiology
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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[1] Alexander, M. E.; Bowman, C.; Moghadas, S. M.; Summers, R.; Gumel, A. B.; Sahai, B. M., A vaccination model for transmission dynamics of influenza, SIAM J. Appl. Dynam. Syst., 3, 503-524, (2004) · Zbl 1067.92051
[2] Andersson, H.; Britton, T., Heterogeneity in epidemic models and its effect on the spread of infection, J. Appl. Prob., 35, 651-661, (1998) · Zbl 0913.92021
[3] Andreasen, V.; Arino, O.; Axelrod, D.; Kimmel, M.; Langlais, M., Mathematical Population Dynamics: Analysis of Heterogeneity, Instability in an SIR-model with age-dependent susceptibility, 3-14, (1995), Wuerz Publications, Winnipeg
[4] Busenberg, S. N.; Cooke, K., Vertically Transmitted Diseases: Models and Dynamics, (1993), Springer-Verlag, New York · Zbl 0837.92021
[5] Busenberg, S. N.; Iannelli, M.; Thieme, H. R., Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22, 1065-1080, (1991) · Zbl 0741.92015
[6] Cha, Y.; Iannelli, M.; Milner, F. A., Stability change of an epidemic model, Dynam. Syst. Appl., 9, 361-376, (2000) · Zbl 0984.92029
[7] D’Agata, E.; Magal, P.; Ruan, S.; Webb, G. F., Asymptotic behavior in nosocomial epidemic models with antibiotic resistance, Differ. Integral Eqs., 19, 573-600, (2006) · Zbl 1212.35229
[8] Du, P.; Li, M. Y., Impact of network connectivity on the synchronization and global dynamics of coupled systems of differential equations, Phys. D, 286, 32-42, (2014) · Zbl 1349.34197
[9] Duan, X.; Yuan, S.; Li, X., Global stability of an SVIR model with age of vaccination, Appl. Math. Comput., 226, 528-540, (2014) · Zbl 1354.92083
[10] Hagenaars, T. J.; Donnelly, C. A.; Ferguson, N. M., Spatial heterogeneity and the persistence of infectious diseases, J. Theor. Biol., 229, 349-359, (2004) · Zbl 1440.92064
[11] Hale, J. K., Asymptotic Behavior of Dissipative Systems, (1988), American Mathematical Society, Providence · Zbl 0642.58013
[12] Hattaf, K.; Lashari, A. A.; Louartassi, Y., A delayed SIR epidemic model with general incidence rate, Electron. J. Qual. Theory Differ. Eqs., 3, 1-9, (2014)
[13] Hoppensteadt, F., An age-dependent epidemic model, J. Franklin Inst., 297, 325-338, (1974) · Zbl 0305.92010
[14] Hoppensteadt, F., Mathematical Theories of Populations: Demographics, Genetics and Epidemics, (1975), SIAM Publications, Philadelphia · Zbl 0304.92012
[15] Huang, G.; Liu, X.; Takeuchi, Y., Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72, 25-38, (2012) · Zbl 1238.92032
[16] Iannelli, M., Applied Mathematics Monographs CNR, 7, Mathematical theory of age-structured population dynamics, (1994), Applied Mathematics Monographs-C.N.R., Pisa
[17] Inaba, H., Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28, 411-434, (1990) · Zbl 0742.92019
[18] Kuniya, T., Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Anal. Real World Appl., 12, 2640-2655, (2011) · Zbl 1219.35330
[19] Li, M. Y.; Shuai, Z., Global-stability problem for coupled systems of differential equations on networks, J. Differ. Eqs., 248, 1-20, (2010) · Zbl 1190.34063
[20] Li, X. Z.; Wang, J.; Ghosh, M., Stability and bifurcation of an SIVS epidemic model with treatment and age of vaccination, Appl. Math. Model., 34, 437-450, (2010) · Zbl 1185.34100
[21] Li, J. Q.; Ma, Z.; Zhou, Y., Global analysis of SIS epidemic model with a simple vaccination and multiple endemic equilibria, Acta Math. Sci., 26, 83-93, (2006) · Zbl 1092.92041
[22] Li, M. Y.; Shuai, Z.; Wang, C., Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361, 38-47, (2010) · Zbl 1175.92046
[23] Liu, X.; Takeuchi, Y.; Iwami, S., SVIR epidemic models with vaccination strategies, J. Theor. Biol., 253, 1-11, (2008) · Zbl 1398.92243
[24] MaCluskey, C., Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6, 603-610, (2009) · Zbl 1190.34108
[25] Magal, P.; Ruan, S., Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202, 1-76, (2009) · Zbl 1191.35045
[26] Magal, P., Compact attractors for time periodic age-structured population models, Electron. J. Differ. Eqs., 65, 1-35, (2001) · Zbl 0992.35019
[27] Magal, P.; McCluskey, C. C.; Webb, G. F., Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89, 1109-1140, (2010) · Zbl 1208.34126
[28] Melnik, A. V.; Korobeinikov, A., Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, Math. Biosci. Engrg., 10, 369-378, (2013) · Zbl 1260.92106
[29] Roy, P. K.; Chatterjee, A. N.; Greenhalgh, D., Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model, Nonlinear Anal. Real World Appl., 14, 1621-1633, (2013) · Zbl 1317.92073
[30] Smith, H. L.; Thieme, H. R., Dynamical Systems and Population Persistence, (2011), American Mathematical Society, Providence · Zbl 1214.37002
[31] Sun, R., Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60, 2286-2291, (2010) · Zbl 1205.34066
[32] Thieme, H. R., “intergrated semigroups” and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152, 416-447, (1990) · Zbl 0738.47037
[33] Thieme, H. R., Semiflows generated by Lipschitz perturbations of non-densely defined operators, Diff. Integral Eqs., 3, 1035-1066, (1990) · Zbl 0734.34059
[34] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29-48, (2002) · Zbl 1015.92036
[35] Wang, J.; Zu, J.; Liu, X.; Huang, G.; Zhang, J., Global dynamics of a multi-group epidemic model with general relapse distribution and nonlinear incidence rate, J. Biol. Syst., 20, 235-258, (2012) · Zbl 1279.92083
[36] Webb, G. F., Theory of Nonlinear Age-dependent Population Dynamics, (1985), Marcel Dekker, New York · Zbl 0555.92014
[37] Xu, J.; Zhou, Y., Global stability of a multi-group model with generalized nonlinear incidence and vaccination age, Discrete Contin. Dynam. Syst. Ser. B, 21, 977-996, (2016) · Zbl 1342.92290
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