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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 and Continuous Dynamical Systems - B, 2017, 22 (4) : 1575-1585. doi: 10.3934/dcdsb.2017076
##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics Academic Press, San Diego, 1993. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics Academic Press, San Diego, 1993. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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