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June  2017, 22(4): 1575-1585. doi: 10.3934/dcdsb.2017076

## Global dynamics of a coupled epidemic model

 1 Department of Mathematics, Tongji University, Shanghai 200092, China 2 Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70503, USA

* Corresponding author

Received  April 2016 Revised  August 2016 Published  February 2017

Fund Project: The first author is supported by the National Natural Science Foundation of China (No.11601392, 11571257), Pujiang Talent Program of Shanghai (No.16PJ1409100) and the Program for Young Excellent Talents at Tongji University.

In this paper, we propose a novel epidemic model coupling direct and indirect transmission of disease and study the global dynamic of the model system. Despite the nonlinearity and complexity of the system, the basic reproduction number exhibits a nice linear property: it is simply the sum of two basic reproduction numbers for direct and indirect disease transmissions respectively. We further demonstrate that the local and global dynamics of the system are related to the basic reproduction number. The new model has the advantage that it generalizes or connects to various disease models on HIV, Zika virus, avian influenza, H1N1 and so on.

Citation: Hongying Shu, Xiang-Sheng Wang. Global dynamics of a coupled epidemic model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1575-1585. doi: 10.3934/dcdsb.2017076
##### References:
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Metz, On the definition and the computation of the basic reproduction ratio$R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. Google Scholar [6] D. J. Earn, D. He, M. B. Loeb, K. Fonseca, B. E. Lee and J. Dushoff, Effects of school closure on incidence of pandemic influenza in Alberta, Canada, Ann Intern Med., 156 (2012), 173-181. doi: 10.7326/0003-4819-156-3-201202070-00005. Google Scholar [7] B. D. Foy, K. C. Kobylinski, J. L. C. Foy, B. J. Blitvich, A. Travassos da Rosa, A. D. Haddow, R. S. Lanciotti and R. B. Tesh, Probable non-vector-borne transmission of Zika virus, Colorado, USA, Emerg. Infect. Dis., 17 (2011), 880-882. doi: 10.3201/eid1705.101939. Google Scholar [8] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. B, 115 (1927), 700-721. Google Scholar [9] A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9. Google Scholar [10] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y. Google Scholar [11] C. M. Kribs-Zaleta, To switch or taper off: The dynamics of saturation, Math. Biosc., 192 (2004), 137-152. doi: 10.1016/j.mbs.2004.11.001. Google Scholar [12] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics Academic Press, San Diego, 1993. Google Scholar [13] X. Lai and X. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM J. Appl. Math., 74 (2014), 898-917. doi: 10.1137/130930145. Google Scholar [14] X. Lai and X. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, J. Math. Anal. Appl., 426 (2015), 563-584. doi: 10.1016/j.jmaa.2014.10.086. Google Scholar [15] C. C. McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence, Math. Biosci. Eng., 7 (2010), 837-850. doi: 10.3934/mbe.2010.7.837. Google Scholar [16] Y. Muroya, T. Kuniya and Y. Enatsu, Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3057-3091. doi: 10.3934/dcdsb.2015.20.3057. Google Scholar [17] A. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586. doi: 10.1126/science.271.5255.1582. Google Scholar [18] H. Pourbashash, S. S. Pilyugin, P. De Leenheer and C. C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3341-3357. doi: 10.3934/dcdsb.2014.19.3341. Google Scholar [19] Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci., 234 (2011), 118-126. doi: 10.1016/j.mbs.2011.09.003. Google Scholar [20] A. Sigal, J. T. Kim, A. B. Balazs, E. Dekel, A. Mayo, R. Milo and D. Baltimore, Cell-to-cell spread of HIV permits ongoing replication despite antiretroviral therapy, Nature, 477 (2011), 95-98. doi: 10.1038/nature10347. Google Scholar [21] 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. Google Scholar [22] X.-S. Wang, J. Wu and Y. Yang, Richards model revisited: Validation by and application to infection dynamics, J. Theoret. Biol., 313 (2012), 12-19. doi: 10.1016/j.jtbi.2012.07.024. Google Scholar [23] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos Texts in Applied Mathematics, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7. Google Scholar [24] Y. Xiao, F. Brauer and S. M. Moghadas, Can treatment increase the epidemic size?, J. Math. Biol., 72 (2016), 343-361. doi: 10.1007/s00285-015-0887-y. Google Scholar

show all references

##### References:
 [1] L. J. S. Allen and E. J. Schwartz, Free-virus and cell-to-cell transmission in models of equine infectious anemia virus infection, Math. Biosci., 270 (2015), 237-248. doi: 10.1016/j.mbs.2015.04.001. Google Scholar [2] M. Bani-Yaghoub, R. Gautam, Z. Shuai, P. van den Driessche and R. Ivanek, Reproduction numbers for infections with free-living pathogens growing in the environment, J. Biol. Dyn., 6 (2012), 923-940. doi: 10.1080/17513758.2012.693206. Google Scholar [3] F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10 (2013), 1335-1349. doi: 10.3934/mbe.2013.10.1335. Google Scholar [4] O. Diekmann, M. C. M. de Jong, A. A. de Koeijer and P. Reijnders, The force of infection in populations of varying size: A modeling problem, J. Biol. Syst., 3 (1995), 519-529. doi: 10.1142/S0218339095000484. Google Scholar [5] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio$R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. Google Scholar [6] D. J. Earn, D. He, M. B. Loeb, K. Fonseca, B. E. Lee and J. Dushoff, Effects of school closure on incidence of pandemic influenza in Alberta, Canada, Ann Intern Med., 156 (2012), 173-181. doi: 10.7326/0003-4819-156-3-201202070-00005. Google Scholar [7] B. D. Foy, K. C. Kobylinski, J. L. C. Foy, B. J. Blitvich, A. Travassos da Rosa, A. D. Haddow, R. S. Lanciotti and R. B. Tesh, Probable non-vector-borne transmission of Zika virus, Colorado, USA, Emerg. Infect. Dis., 17 (2011), 880-882. doi: 10.3201/eid1705.101939. Google Scholar [8] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. B, 115 (1927), 700-721. Google Scholar [9] A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9. Google Scholar [10] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y. Google Scholar [11] C. M. Kribs-Zaleta, To switch or taper off: The dynamics of saturation, Math. Biosc., 192 (2004), 137-152. doi: 10.1016/j.mbs.2004.11.001. Google Scholar [12] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics Academic Press, San Diego, 1993. Google Scholar [13] X. Lai and X. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM J. Appl. Math., 74 (2014), 898-917. doi: 10.1137/130930145. Google Scholar [14] X. Lai and X. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, J. Math. Anal. Appl., 426 (2015), 563-584. doi: 10.1016/j.jmaa.2014.10.086. Google Scholar [15] C. C. McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence, Math. Biosci. Eng., 7 (2010), 837-850. doi: 10.3934/mbe.2010.7.837. Google Scholar [16] Y. Muroya, T. Kuniya and Y. Enatsu, Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3057-3091. doi: 10.3934/dcdsb.2015.20.3057. Google Scholar [17] A. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586. doi: 10.1126/science.271.5255.1582. Google Scholar [18] H. Pourbashash, S. S. Pilyugin, P. De Leenheer and C. C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3341-3357. doi: 10.3934/dcdsb.2014.19.3341. Google Scholar [19] Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci., 234 (2011), 118-126. doi: 10.1016/j.mbs.2011.09.003. Google Scholar [20] A. Sigal, J. T. Kim, A. B. Balazs, E. Dekel, A. Mayo, R. Milo and D. Baltimore, Cell-to-cell spread of HIV permits ongoing replication despite antiretroviral therapy, Nature, 477 (2011), 95-98. doi: 10.1038/nature10347. Google Scholar [21] 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. Google Scholar [22] X.-S. Wang, J. Wu and Y. Yang, Richards model revisited: Validation by and application to infection dynamics, J. Theoret. Biol., 313 (2012), 12-19. doi: 10.1016/j.jtbi.2012.07.024. Google Scholar [23] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos Texts in Applied Mathematics, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7. Google Scholar [24] Y. Xiao, F. Brauer and S. M. Moghadas, Can treatment increase the epidemic size?, J. Math. Biol., 72 (2016), 343-361. doi: 10.1007/s00285-015-0887-y. Google Scholar
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