Article Contents
Article Contents

# Stability analysis of a two-strain epidemic model on complex networks with latency

• In this paper, a two-strain epidemic model on a complex network is proposed. The two strains are the drug-sensitive strain and the drug-resistant strain. The related basic reproduction numbers $R_s$ and $R_r$ are obtained. If $R_0=\max\{R_s,R_r\}<1$, then the disease-free equilibrium is globally asymptotically stable. If $R_r>1$, then there is a unique drug-resistant strain dominated equilibrium $E_r$, which is locally asymptotically stable if the invasion reproduction number $R_r^s<1$. If $R_s>1$ and $R_s>R_r$, then there is a unique coexistence equilibrium $E^*$. The persistence of the model is also proved. The theoretical results are supported with numerical simulations.
Mathematics Subject Classification: 92D30, 92D25, 90B10.

 Citation:

•  [1] A. L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512.doi: 10.1126/science.286.5439.509. [2] T. Cohen and M. Murray, Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness, Nat Med., 10 (2004), 1117-1121.doi: 10.1038/nm1110. [3] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.doi: 10.1016/S0025-5564(02)00108-6. [4] Z. Feng, C. Castillo-Chavez and A. Capurro, A model for Tuberculosis with exogenous reinfection, Theoretical Population Biology, 57 (2000), 235-247.doi: 10.1006/tpbi.2000.1451. [5] Z. Feng, M. Iannelli and F. A. Milner, A two-strain tuberculosis model with age of infection, SIAM J. Appl. Math., 62 (2002), 1634-1656.doi: 10.1137/S003613990038205X. [6] X. C. Fu, S. Michael and G. R. Chen, Propagation Dynamics on Complex Networks, Models, Methods and Stability Analysis, Higher Education Press. Beijing, 2014.doi: 10.1002/9781118762783. [7] J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.doi: 10.1137/0520025. [8] C. H. Li, C. C. Tsai and S. Y. 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. [9] M. E. J. Newman, Threshold effects for two pathogens spreading on a network, Phys. Rev. Lett., 95 (2005), 108701.doi: 10.1103/PhysRevLett.95.108701. [10] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett., 86 (2001), 3200-3203.doi: 10.1103/PhysRevLett.86.3200. [11] 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. [12] R. Pastor-Satorras and A. Vespignani, Immunization of complex networks, Phys. Rev. E., 65 (2002), 036104.doi: 10.1103/PhysRevE.65.036104. [13] L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1991.doi: 10.1007/978-1-4684-0392-3. [14] R. J. Smith, J. T. Okano, J. S. Kahn, E. N. Bodine and S. Blower, Evolutionary dynamics of complex networks of HIV drug-resistant strains: The case of San Francisco, Science, 327 (2010), 697-701.doi: 10.1126/science.1180556. [15] 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. [16] Y. Wang, J. Cao, Z. Jin, H. Zhang and G. Sun, Impact of media coverage on epidemic spreading in complex networks, Phys. A., 392 (2013), 5824-5835.doi: 10.1016/j.physa.2013.07.067. [17] 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. [18] L. Wang and G. Z. Dai, Scale-free Phenomanas of Complex Network, Science Press. Beijing, 2009. [19] Y. Wang and Z. Jin, Global analysis of multiple routes of disease transmission on heterogeneous networks, Physica A, 392 (2013), 3869-3880.doi: 10.1016/j.physa.2013.03.042. [20] Y. Wang, Z. Jin, Z. Yang, Z. Zhang, T. Zhou and G. Sun, Global analysis of an SIS model with an infective vector on complex networks, Nonlinear Anal. Real World Appl., 13 (2012), 543-557.doi: 10.1016/j.nonrwa.2011.07.033. [21] Y. B. Wang, G. X. Xiao and J. Liu, Dynamics of competing ideas in complex social systems, New J. Phys., 14 (2012), 013015.doi: 10.1088/1367-2630/14/1/013015. [22] Q. C. Wu, X. C. Fu and M. Yang, Epidemic thresholds in a heterogenous population with competing strains, Chin. Phys. B., 20 (2011), 046401.doi: 10.1088/1674-1056/20/4/046401. [23] Q. Wu, M. Small and H. Liu, Superinfection behaviors on scale-free networks with competing strains, J. Nonlinear Sci., 23 (2013), 113-127.doi: 10.1007/s00332-012-9146-1. [24] H. Zhang and X. Fu, Spreading of epidemics on scale-free networks with nonlinear infectivity, Nonlinear Anal., 70 (2009), 3273-3278.doi: 10.1016/j.na.2008.04.031. [25] J. Zhang, Z. Jin and Y. Chen, Analysis of sexually transmitted disease spreading in heterosexual and homosexual populations, Math. Biosci., 242 (2013), 143-152.doi: 10.1016/j.mbs.2013.01.005.