# American Institute of Mathematical Sciences

2016, 13(4): 723-739. doi: 10.3934/mbe.2016016

## Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate

 1 Mathematics and Science College, Shanghai Normal University, Shanghai, 200234, China 2 Mathematics and Science College, Shanghai Normal University, Shanghai 200234

Received  October 2015 Revised  February 2016 Published  May 2016

In this paper, we develop and analyze an SIS epidemic model with a general nonlinear incidence rate, as well as degree-dependent birth and natural death, on heterogeneous networks. We analytically derive the epidemic threshold $R_0$ which completely governs the disease dynamics: when $R_0<1$, the disease-free equilibrium is globally asymptotically stable, i.e., the disease will die out; when $R_0>1$, the disease is permanent. It is interesting that the threshold value $R_0$ bears no relation to the functional form of the nonlinear incidence rate and degree-dependent birth. Furthermore, by applying an iteration scheme and the theory of cooperative system respectively, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. Our results improve and generalize some known results. To illustrate the theoretical results, the corresponding numerical simulations are also given.
Citation: Shouying Huang, Jifa Jiang. Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate. Mathematical Biosciences & Engineering, 2016, 13 (4) : 723-739. doi: 10.3934/mbe.2016016
##### References:
 [1] L.-M. Cai and X.-Z. Li, Analysis of a SEIV epidemic model with a nonlinear incidence rate, Appl. Math. Modelling, 33 (2009), 2919-2926. doi: 10.1016/j.apm.2008.01.005. [2] X. Chu, Z. Zhang, J. Guan and S. Zhou, Epidemic spreading with nonlinear infectivity in weighted scale-free networks, Physica A, 390 (2011), 471-481. doi: 10.1016/j.physa.2010.09.038. [3] X. Fu, M. Small, D. M. Walker and H. Zhang, Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization, Phys. Rev. E, 77 (2008), 036113, 8pp. doi: 10.1103/PhysRevE.77.036113. [4] H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [5] S. Huang, Dynamic analysis of an SEIRS model with nonlinear infectivity on complex networks, Int. J. Biomath., 9 (2016), 1650009, 25pp. doi: 10.1142/S1793524516500091. [6] J. Jiang, On the global stability of cooperative systems, B. Lond. Math. Soc., 26 (1994), 455-458. doi: 10.1112/blms/26.5.455. [7] Z. Jin, G. Sun and H. Zhu, Epidemic models for complex networks with demographics, Math. Biosci. Eng., 11 (2014), 1295-1317. doi: 10.3934/mbe.2014.11.1295. [8] H. Kang and X. Fu, Epidemic spreading and global stability of an SIS model with an infective vector on complex networks, Commun. Nonlinear Sci. Numer. Simul., 27 (2015), 30-39. doi: 10.1016/j.cnsns.2015.02.018. [9] A. Lahrouz, L. Omari, D. Kiouach and A. Belmaâtic, Complete global stability for an SIRS epidemic model with generalized nonlinear incidence and vaccination, Appl. Math. Comput., 218 (2012), 6519-6525. doi: 10.1016/j.amc.2011.12.024. [10] A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5. [11] C.-H. Li, Dynamics of a network-based SIS epidemic model with nonmonotone incidence rate, Physica A, 427 (2015), 234-243. doi: 10.1016/j.physa.2015.02.023. [12] J. Liu, Y. Tang and Z. R. Yang, The spread of disease with birth and death on networks, J. Stat. Mech., 2004 (2004), p08008. doi: 10.1088/1742-5468/2004/08/P08008. [13] M. Liu and Y. Chen, An SIRS model with differential susceptibility and infectivity on uncorrelated networks, Math. Biosci. Eng., 12 (2015), 415-429. doi: 10.3934/mbe.2015.12.415. [14] M. Liu and J. Ruan, Modelling of epidemics with a generalized nonlinear incidence on complex networks, Complex Sciences, Springer Berlin Heidelberg, 5 (2009), 2118-2126. doi: 10.1007/978-3-642-02469-6_88. [15] Z. Ma, Y. Zhou, W. Wang and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases, China sci. press, Beijing, 2004. [16] R. Olinky and L. Stone, Unexpected epidemic thresholds in heterogeneous networks: The role of disease transmission, Phys. Rev. E, 70 (2004), 030902. doi: 10.1103/PhysRevE.70.030902. [17] R. Pastor-Satorras and A. Vespignani, Epidemic dynamics and endemic states in complex networks, Phys. Rev. E, 63 (2001), 066117. doi: 10.1103/PhysRevE.63.066117. [18] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X. [19] J. Sanz, L. Floría and Y. Moreno, Spreading of persistent infections in heterogeneous populations, Phys. Rev. E, 81 (2010), 056108, 9pp. doi: 10.1103/PhysRevE.81.056108. [20] L. Wang and G.-Z. Dai, Global stability of virus spreading in complex heterogeneous networks, SIAM J. Appl. Math., 68 (2008), 1495-1502. doi: 10.1137/070694582. [21] R. Yang, B. Wang, J. Ren, W. Bai, Z. Shi, W. Wang and T. Zhou, Epidemic spreading on heterogeneous networks with identical infectivity, Phys. Lett. A, 364 (2007), 189-193. doi: 10.1016/j.physleta.2006.12.021. [22] H. Zhang and X. Fu, Spreading of epidemics on scale-free networks with nonlinear infectivity, Nonlinear Anal. Theory Methods Appl., 70 (2009), 3273-3278. doi: 10.1016/j.na.2008.04.031. [23] J. Zhang and J. Sun, Stability analysis of an SIS epidemic model with feedback mechanism on networks, Physica A, 394 (2014), 24-32. doi: 10.1016/j.physa.2013.09.058. [24] J. Zhang and J. Sun, Analysis of epidemic spreading with feedback mechanism in weighted networks, Int. J. Biomath., 8 (2015), 1550007, 11pp. doi: 10.1142/S1793524515500072. [25] J. Zhang and Z. Jin, The analysis of an epidemic model on networks, Appl. Math. Comput., 217 (2011), 7053-7064. doi: 10.1016/j.amc.2010.09.063. [26] G. Zhu, X. Fu and G. Chen, Global attractivity of a network-based epidemic SIS model with nonlinear infectivity, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2588-2594. doi: 10.1016/j.cnsns.2011.08.039. [27] G. Zhu, X. Fu and G. Chen, Spreading dynamics and global stability of a generalized epidemic model on complex heterogeneous networks, Appl. Math. Modell., 36 (2012), 5808-5817. doi: 10.1016/j.apm.2012.01.023.

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##### References:
 [1] L.-M. Cai and X.-Z. Li, Analysis of a SEIV epidemic model with a nonlinear incidence rate, Appl. Math. Modelling, 33 (2009), 2919-2926. doi: 10.1016/j.apm.2008.01.005. [2] X. Chu, Z. Zhang, J. Guan and S. Zhou, Epidemic spreading with nonlinear infectivity in weighted scale-free networks, Physica A, 390 (2011), 471-481. doi: 10.1016/j.physa.2010.09.038. [3] X. Fu, M. Small, D. M. Walker and H. Zhang, Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization, Phys. Rev. E, 77 (2008), 036113, 8pp. doi: 10.1103/PhysRevE.77.036113. [4] H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [5] S. Huang, Dynamic analysis of an SEIRS model with nonlinear infectivity on complex networks, Int. J. Biomath., 9 (2016), 1650009, 25pp. doi: 10.1142/S1793524516500091. [6] J. Jiang, On the global stability of cooperative systems, B. Lond. Math. Soc., 26 (1994), 455-458. doi: 10.1112/blms/26.5.455. [7] Z. Jin, G. Sun and H. Zhu, Epidemic models for complex networks with demographics, Math. Biosci. Eng., 11 (2014), 1295-1317. doi: 10.3934/mbe.2014.11.1295. [8] H. Kang and X. Fu, Epidemic spreading and global stability of an SIS model with an infective vector on complex networks, Commun. Nonlinear Sci. Numer. Simul., 27 (2015), 30-39. doi: 10.1016/j.cnsns.2015.02.018. [9] A. Lahrouz, L. Omari, D. Kiouach and A. Belmaâtic, Complete global stability for an SIRS epidemic model with generalized nonlinear incidence and vaccination, Appl. Math. Comput., 218 (2012), 6519-6525. doi: 10.1016/j.amc.2011.12.024. [10] A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5. [11] C.-H. Li, Dynamics of a network-based SIS epidemic model with nonmonotone incidence rate, Physica A, 427 (2015), 234-243. doi: 10.1016/j.physa.2015.02.023. [12] J. Liu, Y. Tang and Z. R. Yang, The spread of disease with birth and death on networks, J. Stat. Mech., 2004 (2004), p08008. doi: 10.1088/1742-5468/2004/08/P08008. [13] M. Liu and Y. Chen, An SIRS model with differential susceptibility and infectivity on uncorrelated networks, Math. Biosci. Eng., 12 (2015), 415-429. doi: 10.3934/mbe.2015.12.415. [14] M. Liu and J. Ruan, Modelling of epidemics with a generalized nonlinear incidence on complex networks, Complex Sciences, Springer Berlin Heidelberg, 5 (2009), 2118-2126. doi: 10.1007/978-3-642-02469-6_88. [15] Z. Ma, Y. Zhou, W. Wang and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases, China sci. press, Beijing, 2004. [16] R. Olinky and L. Stone, Unexpected epidemic thresholds in heterogeneous networks: The role of disease transmission, Phys. Rev. E, 70 (2004), 030902. doi: 10.1103/PhysRevE.70.030902. [17] R. Pastor-Satorras and A. Vespignani, Epidemic dynamics and endemic states in complex networks, Phys. Rev. E, 63 (2001), 066117. doi: 10.1103/PhysRevE.63.066117. [18] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X. [19] J. Sanz, L. Floría and Y. Moreno, Spreading of persistent infections in heterogeneous populations, Phys. Rev. E, 81 (2010), 056108, 9pp. doi: 10.1103/PhysRevE.81.056108. [20] L. Wang and G.-Z. Dai, Global stability of virus spreading in complex heterogeneous networks, SIAM J. Appl. Math., 68 (2008), 1495-1502. doi: 10.1137/070694582. [21] R. Yang, B. Wang, J. Ren, W. Bai, Z. Shi, W. Wang and T. Zhou, Epidemic spreading on heterogeneous networks with identical infectivity, Phys. Lett. A, 364 (2007), 189-193. doi: 10.1016/j.physleta.2006.12.021. [22] H. Zhang and X. Fu, Spreading of epidemics on scale-free networks with nonlinear infectivity, Nonlinear Anal. Theory Methods Appl., 70 (2009), 3273-3278. doi: 10.1016/j.na.2008.04.031. [23] J. Zhang and J. Sun, Stability analysis of an SIS epidemic model with feedback mechanism on networks, Physica A, 394 (2014), 24-32. doi: 10.1016/j.physa.2013.09.058. [24] J. Zhang and J. Sun, Analysis of epidemic spreading with feedback mechanism in weighted networks, Int. J. Biomath., 8 (2015), 1550007, 11pp. doi: 10.1142/S1793524515500072. [25] J. Zhang and Z. Jin, The analysis of an epidemic model on networks, Appl. Math. Comput., 217 (2011), 7053-7064. doi: 10.1016/j.amc.2010.09.063. [26] G. Zhu, X. Fu and G. Chen, Global attractivity of a network-based epidemic SIS model with nonlinear infectivity, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2588-2594. doi: 10.1016/j.cnsns.2011.08.039. [27] G. Zhu, X. Fu and G. Chen, Spreading dynamics and global stability of a generalized epidemic model on complex heterogeneous networks, Appl. Math. Modell., 36 (2012), 5808-5817. doi: 10.1016/j.apm.2012.01.023.
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