# American Institute of Mathematical Sciences

August 2018, 23(6): 2071-2090. doi: 10.3934/dcdsb.2018226

## Epidemic dynamics on complex networks with general infection rate and immune strategies

 1 College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian, 350116, China 2 Mathematics and Science College, Shanghai Normal University, Shanghai, 200234, China

* Corresponding author: Shouying Huang

Received  March 2016 Revised  April 2018 Published  July 2018

Fund Project: This work is supported by the Natural Science Foundation of China (NSFC) under Grant No.11771295 and 11601336, the Natural Science Foundation of Fujian Province under Grant No.2018J01664, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and Shanghai Gaofeng Project for University Academic Program Development

This paper mainly aims to study the influence of individuals' different heterogeneous contact patterns on the spread of the disease. For this purpose, an SIS epidemic model with a general form of heterogeneous infection rate is investigated on complex heterogeneous networks. A qualitative analysis of this model reveals that, depending on the epidemic threshold $R_0$, either the disease-free equilibrium or the endemic equilibrium is globally asymptotically stable. Interestingly, no matter what functional form the heterogeneous infection rate is, whether the disease will disappear or not is completely determined by the value of $R_0$, but the heterogeneous infection rate has close relation with the epidemic threshold $R_0$. Especially, the heterogeneous infection rate can directly affect the final number of infected nodes when the disease is endemic. The obtained results improve and generalize some known results. Finally, based on the heterogeneity of contact patterns, the effects of different immunization schemes are discussed and compared. Meanwhile, we explore the relation between the immunization rate and the recovery rate, which are the two important parameters that can be improved. To illustrate our theoretical results, the corresponding numerical simulations are also included.

Citation: Shouying Huang, Jifa Jiang. Epidemic dynamics on complex networks with general infection rate and immune strategies. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2071-2090. doi: 10.3934/dcdsb.2018226
##### References:
 [1] E. Alfinito, M. Beccaria, A. Fachechi and G. Macorini, Reactive immunization on complex networks, EPL (Europhysics Letters), 117 (2017), 18002. doi: 10.1209/0295-5075/117/18002. [2] A. L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512. doi: 10.1126/science.286.5439.509. [3] 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. [4] R. Cohen, S. Havlin and D. Ben-Avraham, Efficient immunization strategies for computer networks and populations, Phys. Rev. Lett. , 91 (2003), 247901. doi: 10.1103/PhysRevLett.91.247901. [5] 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. [6] 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. [7] S. Huang, F. Chen and L. Chen, Global dynamics of a network-based SIQRS epidemic model with demographics and vaccination, Commun. Nonlinear Sci. Numer. Simul., 43 (2017), 296-310. doi: 10.1016/j.cnsns.2016.07.014. [8] S. Huang and J. Jiang, Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate, Math. Biosci. Eng., 13 (2016), 723-739. doi: 10.3934/mbe.2016016. [9] J. Jiang, On the global stability of cooperative systems, B. Lond. Math. Soc., 26 (1994), 455-458. doi: 10.1112/blms/26.5.455. [10] 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. [11] 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. [12] P. D. Leenheer and H. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905. [13] C. Li, C. Tsai and S. Yang, Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1042-1054. doi: 10.1016/j.cnsns.2013.08.033. [14] X. Li and L. Cao, Diffusion processes of fragmentary information on scale-free networks, Physica A, 450 (2016), 624-634. doi: 10.1016/j.physa.2016.01.035. [15] M. Liu and J. Ruan, Modelling of epidemics with a generalized nonlinear incidence on complex networks, Complex Sciences, Springer Berlin Heidelberg, (2009), 2118-2126. doi: 10.1007/978-3-642-02469-6_88. [16] N. Madar, T. Kalisky, R. Cohen, D. ben-Avraham and S. Havlin, Immunization and epidemic dynamics in complex networks, The European Physical Journal B, 38 (2004), 269-276. doi: 10.1140/epjb/e2004-00119-8. [17] V. Nagy, Mean-field theory of a recurrent epidemiological model, Phys. Rev. E, 79 (2009), 066105. doi: 10.1103/PhysRevE.79.066105. [18] F. Nian and X. Wang, Efficient immunization strategies on complex networks, J. Theor. Biol., 264 (2010), 77-83. doi: 10.1016/j.jtbi.2010.01.007. [19] 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. [20] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett. , 86 (2001), 3200. [21] 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. [22] R. Pastor-Satorras and A. Vespignani, Epidemics and immunization in scale-free networks, Handbook of Graphs and Networks: From the Genome to the Internet, (2003), 111-130. [23] Y. Qin, X. Zhong, H. Jiang and Y. Ye, An environment aware epidemic spreading model and immune strategy in complex networks, Appl. Math. Comput., 261 (2015), 206-215. doi: 10.1016/j.amc.2015.03.110. [24] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026. [25] 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. [26] Q. Wu and X. Fu, Immunization and epidemic threshold of an SIS model in complex networks, Physica A, 444 (2016), 576-581. doi: 10.1016/j.physa.2015.10.043. [27] Q. Wu, X. Fu and G. Zhu, Global attractiveness of discrete-time epidemic outbreak in networks, Int. J. Biomath. , 5 (2012), 1250004, 12pp. doi: 10.1142/S1793524511001441. [28] 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. [29] 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. [30] X. Zhao and Z. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444. [31] 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. [32] 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.

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##### References:
 [1] E. Alfinito, M. Beccaria, A. Fachechi and G. Macorini, Reactive immunization on complex networks, EPL (Europhysics Letters), 117 (2017), 18002. doi: 10.1209/0295-5075/117/18002. [2] A. L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512. doi: 10.1126/science.286.5439.509. [3] 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. [4] R. Cohen, S. Havlin and D. Ben-Avraham, Efficient immunization strategies for computer networks and populations, Phys. Rev. Lett. , 91 (2003), 247901. doi: 10.1103/PhysRevLett.91.247901. [5] 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. [6] 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. [7] S. Huang, F. Chen and L. Chen, Global dynamics of a network-based SIQRS epidemic model with demographics and vaccination, Commun. Nonlinear Sci. Numer. Simul., 43 (2017), 296-310. doi: 10.1016/j.cnsns.2016.07.014. [8] S. Huang and J. Jiang, Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate, Math. Biosci. Eng., 13 (2016), 723-739. doi: 10.3934/mbe.2016016. [9] J. Jiang, On the global stability of cooperative systems, B. Lond. Math. Soc., 26 (1994), 455-458. doi: 10.1112/blms/26.5.455. [10] 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. [11] 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. [12] P. D. Leenheer and H. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905. [13] C. Li, C. Tsai and S. Yang, Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1042-1054. doi: 10.1016/j.cnsns.2013.08.033. [14] X. Li and L. Cao, Diffusion processes of fragmentary information on scale-free networks, Physica A, 450 (2016), 624-634. doi: 10.1016/j.physa.2016.01.035. [15] M. Liu and J. Ruan, Modelling of epidemics with a generalized nonlinear incidence on complex networks, Complex Sciences, Springer Berlin Heidelberg, (2009), 2118-2126. doi: 10.1007/978-3-642-02469-6_88. [16] N. Madar, T. Kalisky, R. Cohen, D. ben-Avraham and S. Havlin, Immunization and epidemic dynamics in complex networks, The European Physical Journal B, 38 (2004), 269-276. doi: 10.1140/epjb/e2004-00119-8. [17] V. Nagy, Mean-field theory of a recurrent epidemiological model, Phys. Rev. E, 79 (2009), 066105. doi: 10.1103/PhysRevE.79.066105. [18] F. Nian and X. Wang, Efficient immunization strategies on complex networks, J. Theor. Biol., 264 (2010), 77-83. doi: 10.1016/j.jtbi.2010.01.007. [19] 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. [20] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett. , 86 (2001), 3200. [21] 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. [22] R. Pastor-Satorras and A. Vespignani, Epidemics and immunization in scale-free networks, Handbook of Graphs and Networks: From the Genome to the Internet, (2003), 111-130. [23] Y. Qin, X. Zhong, H. Jiang and Y. Ye, An environment aware epidemic spreading model and immune strategy in complex networks, Appl. Math. Comput., 261 (2015), 206-215. doi: 10.1016/j.amc.2015.03.110. [24] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026. [25] 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. [26] Q. Wu and X. Fu, Immunization and epidemic threshold of an SIS model in complex networks, Physica A, 444 (2016), 576-581. doi: 10.1016/j.physa.2015.10.043. [27] Q. Wu, X. Fu and G. Zhu, Global attractiveness of discrete-time epidemic outbreak in networks, Int. J. Biomath. , 5 (2012), 1250004, 12pp. doi: 10.1142/S1793524511001441. [28] 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. [29] 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. [30] X. Zhao and Z. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444. [31] 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. [32] 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.
The time series of $I(t)$ with different forms of infection rate
The densities of infected nodes with different degrees. The lines from bottom to top are $I_1(t), I_{10}(t), I_{20}(t), \cdots, I_{90}(t), I_n(t)$
The influence of initial conditions on the density of $I_{30}(t)$
The time evolutions of $I(t)$ with uniform immunization and targeted immunization for a given recovery rate $\gamma = 0.05$
Comparison of the effectiveness of different immunization schemes: uniform immunization with $\sigma = 0.0026$, targeted immunization with $\bar{\sigma} = 0.0026$ (i.e., $k_c = 25$) and acquaintance immunization with $q_{_0} = 0.35$. Here the parameters and initial value are the same as those of Fig. 1(b)
The time evolutions of $I(t)$ for different recovery rate $\gamma$ under the given acquaintance immunization rate $q_{_0}$ and high-risk immunization rate $v_{_0}$. Here, the other parameters and initial value are the same as those of Fig. 4
Effectiveness of high-risk immunization schemes with different value of $v_{_0}$. Here the parameters and initial value are the same as those of Fig. 1(b), except for $n = 30$
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