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

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October 2016, 21(8): 2851-2866. doi: 10.3934/dcdsb.2016076

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

Junyuan Yang ^1, , Yuming Chen ^2, and Jiming Liu ^3,

1.	Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi
2.	Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5
3.	Department of Computer Science, Hongkong Baptist University, Kowloon Tong, Hongkong, China

Received September 2015 Revised March 2016 Published September 2016

Abstract
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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.

Keywords: complex network, drug-resistant strain, stability., Drug-sensitive strain.

Mathematics Subject Classification: 92D30, 92D25, 90B1.

Citation: Junyuan Yang, Yuming Chen, Jiming Liu. Stability analysis of a two-strain epidemic model on complex networks with latency. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2851-2866. doi: 10.3934/dcdsb.2016076

References:

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[2]	T. Cohen and M. Murray, Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness,, Nat Med., 10 (2004), 1117. 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. 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. 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. 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. 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. 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). 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. 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). doi: 10.1103/PhysRevE.63.066117.
[12]	R. Pastor-Satorras and A. Vespignani, Immunization of complex networks,, Phys. Rev. E., 65 (2002). doi: 10.1103/PhysRevE.65.036104.
[13]	L. Perko, Differential Equations and Dynamical Systems,, Springer-Verlag, (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. 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. 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. 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. 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. 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. 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). 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). 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. 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. 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. doi: 10.1016/j.mbs.2013.01.005.*

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References:

[1]	A. L. Barabási and R. Albert, Emergence of scaling in random networks,, Science, 286 (1999), 509. 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. 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. 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. 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. 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. 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. 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). 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. 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). doi: 10.1103/PhysRevE.63.066117.
[12]	R. Pastor-Satorras and A. Vespignani, Immunization of complex networks,, Phys. Rev. E., 65 (2002). doi: 10.1103/PhysRevE.65.036104.
[13]	L. Perko, Differential Equations and Dynamical Systems,, Springer-Verlag, (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. 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. 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. 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. 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. 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. 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). 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). 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. 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. 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. doi: 10.1016/j.mbs.2013.01.005.*

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