# American Institute of Mathematical Sciences

January  2013, 18(1): 147-161. doi: 10.3934/dcdsb.2013.18.147

## Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation

 1 Department of Mathematics, National Chung Cheng University, Min-Hsiung, Chia-Yi 621 2 Department of Natural Science in the Center for General Education, Chang Gung University, Kwei-Shan, Taoyuan 333, Taiwan

Received  September 2011 Revised  April 2012 Published  September 2012

Dengue fever is a virus-caused disease in the world. Since the high infection rate of dengue fever and high death rate of its severe form dengue hemorrhagic fever, the control of the spread of the disease is an important issue in the public health. In an effort to understand the dynamics of the spread of the disease, Esteva and Vargas [2] proposed a SIR v.s. SI epidemiological model without crowding effect and spatial heterogeneity. They found a threshold parameter $R_0,$ if $R_0<1,$ then the disease will die out; if $R_0>1,$ then the disease will always exist.
To investigate how the spatial heterogeneity and crowding effect influence the dynamics of the spread of the disease, we modify the autonomous system provided in [2] to obtain a reaction-diffusion system. We first define the basic reproduction number in an abstract way and then employ the comparison theorem and the theory of uniform persistence to study the global dynamics of the modified system. Basically, we show that the basic reproduction number is a threshold parameter that predicts whether the disease will die out or persist. Further, we demonstrate the basic reproduction number in an explicit way and construct suitable Lyapunov functionals to determine the global stability for the special case where coefficients are all constant.
Citation: Tzy-Wei Hwang, Feng-Bin Wang. Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 147-161. doi: 10.3934/dcdsb.2013.18.147
##### References:
 [1] 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 the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [2] L. Esteva and C. Vargas, Analysis of a dengue disease transmission model, Mathematical Biosciences, 150 (1998), 131-151. doi: 10.1016/S0025-5564(98)10003-2. [3] D. J. Gubler, Dengue, in "The arbovirus: Epidemiology and Ecology" (ed. T. P. Monath), CRC Press, Florida, USA, II (1986), 213-261. [4] J. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society Providence, RI, 1988. [5] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin, New York, 1981. [6] S. B. Hsu, A survey of constructing lyapunov function for mathematical models in population biology, Taiwanese Journal of Mathematics, 9 (2005), 151-173. [7] J. S. B. Hsu, J. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Diff. Eqns., 248 (2010), 2470-2496. doi: 10.1016/j.jde.2009.12.014. [8] T.-W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model, Can. Appl. Math. Q., 10 (2002), 473-499. Mathematical Biosciences and Engineering, 2 (2005), 743-751. [9] S. B. Hsu, F. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, Journal of Dynamics and Differential Equations, 23 (2011), 817-842. doi: 10.1007/s10884-011-9224-3. [10] F. X. Jousset, Geographic A. aegypti strains and dengue-2 virus: susceptibility, ability to transmit to vertebrate and transovarial transmission, Ann. Virol, E, 132 (1981), pp. 357. [11] A. Korobeinikov, Global properties of basic virus dynamics models, Bulletin of Mathematical Biology, 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001. [12] A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Mathematical Biosciences and Engineering, 1 (2004), 57-60. [13] Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568. doi: 10.1007/s00285-010-0346-8. [14] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. of A. M. S., 321 (1990), 1-44. [15] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173. [16] T. Ouyang, On the positive solutions of semilinear equations $\Delta u+ \lambda u-hu^p=0$ on the compact manifolds, Trans. of A. M. S., 331 (1992), 503-527. doi: 10.2307/2154124. [17] A. Pazy, "Semigroups of Linear Operators and Applicationto Partial Differential Equations," Springer-Verlag, 1983. [18] L. Rosen, D. A. Shroyer, R. B. Tesh, J. E. Freirer and J. Ch. Lien, Transovarial transmission of dengue viruses by mosquitoes A. Alhopictus and a. Aegypti, Am. J. Trop. Med. Hyg. 32, (1983), pp. 1108. [19] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, 1984. [20] H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995. [21] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2. [22] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. [23] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM, J. Appl. Math., 70 (2009), 188-211. [24] 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. [25] , Dengue haemorrhagic fever: Diagnosis, treatment and control,, World Health Organization, (1986). [26] F. B. Wang, A system of partial differential equations modeling the competition for two complementary resources in flowing habitats, J. Diff. Eqns., 249 (2010), 2866-2888. doi: 10.1016/j.jde.2010.07.031. [27] W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890. [28] X.-Q. Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003.

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##### References:
 [1] 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 the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [2] L. Esteva and C. Vargas, Analysis of a dengue disease transmission model, Mathematical Biosciences, 150 (1998), 131-151. doi: 10.1016/S0025-5564(98)10003-2. [3] D. J. Gubler, Dengue, in "The arbovirus: Epidemiology and Ecology" (ed. T. P. Monath), CRC Press, Florida, USA, II (1986), 213-261. [4] J. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society Providence, RI, 1988. [5] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin, New York, 1981. [6] S. B. Hsu, A survey of constructing lyapunov function for mathematical models in population biology, Taiwanese Journal of Mathematics, 9 (2005), 151-173. [7] J. S. B. Hsu, J. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Diff. Eqns., 248 (2010), 2470-2496. doi: 10.1016/j.jde.2009.12.014. [8] T.-W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model, Can. Appl. Math. Q., 10 (2002), 473-499. Mathematical Biosciences and Engineering, 2 (2005), 743-751. [9] S. B. Hsu, F. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, Journal of Dynamics and Differential Equations, 23 (2011), 817-842. doi: 10.1007/s10884-011-9224-3. [10] F. X. Jousset, Geographic A. aegypti strains and dengue-2 virus: susceptibility, ability to transmit to vertebrate and transovarial transmission, Ann. Virol, E, 132 (1981), pp. 357. [11] A. Korobeinikov, Global properties of basic virus dynamics models, Bulletin of Mathematical Biology, 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001. [12] A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Mathematical Biosciences and Engineering, 1 (2004), 57-60. [13] Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568. doi: 10.1007/s00285-010-0346-8. [14] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. of A. M. S., 321 (1990), 1-44. [15] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173. [16] T. Ouyang, On the positive solutions of semilinear equations $\Delta u+ \lambda u-hu^p=0$ on the compact manifolds, Trans. of A. M. S., 331 (1992), 503-527. doi: 10.2307/2154124. [17] A. Pazy, "Semigroups of Linear Operators and Applicationto Partial Differential Equations," Springer-Verlag, 1983. [18] L. Rosen, D. A. Shroyer, R. B. Tesh, J. E. Freirer and J. Ch. Lien, Transovarial transmission of dengue viruses by mosquitoes A. Alhopictus and a. Aegypti, Am. J. Trop. Med. Hyg. 32, (1983), pp. 1108. [19] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, 1984. [20] H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995. [21] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2. [22] H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. [23] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM, J. Appl. Math., 70 (2009), 188-211. [24] 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. [25] , Dengue haemorrhagic fever: Diagnosis, treatment and control,, World Health Organization, (1986). [26] F. B. Wang, A system of partial differential equations modeling the competition for two complementary resources in flowing habitats, J. Diff. Eqns., 249 (2010), 2866-2888. doi: 10.1016/j.jde.2010.07.031. [27] W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890. [28] X.-Q. Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003.
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