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April 2017, 14(2): 421-435. doi: 10.3934/mbe.2017026

## Global stability of a multistrain SIS model with superinfection

 1 Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., Szeged, H-6720, Hungary 2 Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan 3 Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., Szeged, H-6720, Hungary

*Corresponding author: Attila Dénes

Dedicated to the memory of Professor Yoshiaki Muroya who passed away in 2015 after the submission of this paper

Received  October 07, 2015 Revised  August 10, 2016 Published  October 2016

In this paper, we study the global stability of a multistrain SIS model with superinfection. We present an iterative procedure to calculate a sequence of reproduction numbers, and we prove that it completely determines the global dynamics of the system. We show that for any number of strains with different infectivities, the stable coexistence of any subset of the strains is possible, and we completely characterize all scenarios. As an example, we apply our method to a three-strain model.

Citation: Attila Dénes, Yoshiaki Muroya, Gergely Röst. Global stability of a multistrain SIS model with superinfection. Mathematical Biosciences & Engineering, 2017, 14 (2) : 421-435. doi: 10.3934/mbe.2017026
##### References:
 [1] S. Bianco, L. B. Shaw and I. B. Schwartz, Epidemics with multistrain interactions: The interplay between cross immunity and antibody-dependent enhancement, Chaos, 19 (2009), 043123. doi: 10.1063/1.3270261. [2] D. Bichara, A. Iggidr and G. Sallet, Global analysis of multi-strains SIS, SIR and MSIR epidemic models, J. Appl. Math. Comput., 44 (2014), 273-292. doi: 10.1007/s12190-013-0693-x. [3] A. Dénes and G. Röst, Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1101-1117. doi: 10.3934/dcdsb.2016.21.1101. [4] A. Dénes and G. Röst, Structure of the global attractors in a model for ectoparasite-borne diseases, BIOMATH, 1(2012), 1209256, 5 pp. doi: 10.11145/j.biomath.2012.09.256. [5] A. Dénes and G. Röst, Global dynamics for the spread of ectoparasite-borne diseases, Nonlinear Anal. Real World Appl., 18 (2014), 100-107. doi: 10.1016/j.nonrwa.2014.01.003. [6] A. Dénes and G. Röst, Impact of excess mortality on the dynamics of diseases spread by ectoparasites, in: M. Cojocaru, I. S. Kotsireas, R. N. Makarov, R. Melnik, H. Shodiev, (Eds. ), Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science, Springer Proceedings in Mathematics & Statistics, 117 (2015), 177-182. doi: 10.1007/978-3-319-12307-3_25. [7] T. Faria and Y. Muroya, Global attractivity and extinction for Lotka-Volterra systems with infinite delay and feedback controls, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 301-330. doi: 10.1017/S0308210513001194. [8] S. Kryazhimskiy, U. Dieckmann, S. A. Levin and J. Dushoff, On state-space reduction in multi-strain pathogen models, with an application to antigenic drift in influenza A, PLoS Comput. Biol., 3 (2007), 1513-1525. doi: 10.1371/journal.pcbi.0030159. [9] M. Martcheva, A non-autonomous multi-strain SIS epidemic model, J. Biol. Dyn., 3 (2009), 235-251. doi: 10.1080/17513750802638712. [10] M. A. Nowak, Evolutionary Dynamics, The Belknap Press of Harvard University Press, Cambridge, MA, 2006. [11] H. R. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mountain J. Math., 24 (1994), 351-380. doi: 10.1216/rmjm/1181072470.

show all references

##### References:
 [1] S. Bianco, L. B. Shaw and I. B. Schwartz, Epidemics with multistrain interactions: The interplay between cross immunity and antibody-dependent enhancement, Chaos, 19 (2009), 043123. doi: 10.1063/1.3270261. [2] D. Bichara, A. Iggidr and G. Sallet, Global analysis of multi-strains SIS, SIR and MSIR epidemic models, J. Appl. Math. Comput., 44 (2014), 273-292. doi: 10.1007/s12190-013-0693-x. [3] A. Dénes and G. Röst, Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1101-1117. doi: 10.3934/dcdsb.2016.21.1101. [4] A. Dénes and G. Röst, Structure of the global attractors in a model for ectoparasite-borne diseases, BIOMATH, 1(2012), 1209256, 5 pp. doi: 10.11145/j.biomath.2012.09.256. [5] A. Dénes and G. Röst, Global dynamics for the spread of ectoparasite-borne diseases, Nonlinear Anal. Real World Appl., 18 (2014), 100-107. doi: 10.1016/j.nonrwa.2014.01.003. [6] A. Dénes and G. Röst, Impact of excess mortality on the dynamics of diseases spread by ectoparasites, in: M. Cojocaru, I. S. Kotsireas, R. N. Makarov, R. Melnik, H. Shodiev, (Eds. ), Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science, Springer Proceedings in Mathematics & Statistics, 117 (2015), 177-182. doi: 10.1007/978-3-319-12307-3_25. [7] T. Faria and Y. Muroya, Global attractivity and extinction for Lotka-Volterra systems with infinite delay and feedback controls, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 301-330. doi: 10.1017/S0308210513001194. [8] S. Kryazhimskiy, U. Dieckmann, S. A. Levin and J. Dushoff, On state-space reduction in multi-strain pathogen models, with an application to antigenic drift in influenza A, PLoS Comput. Biol., 3 (2007), 1513-1525. doi: 10.1371/journal.pcbi.0030159. [9] M. Martcheva, A non-autonomous multi-strain SIS epidemic model, J. Biol. Dyn., 3 (2009), 235-251. doi: 10.1080/17513750802638712. [10] M. A. Nowak, Evolutionary Dynamics, The Belknap Press of Harvard University Press, Cambridge, MA, 2006. [11] H. R. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mountain J. Math., 24 (1994), 351-380. doi: 10.1216/rmjm/1181072470.
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