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 & 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.  doi: 10.1007/BF00178324.  Google Scholar

[2]

L. Esteva and C. Vargas, Analysis of a dengue disease transmission model,, Mathematical Biosciences, 150 (1998), 131.  doi: 10.1016/S0025-5564(98)10003-2.  Google Scholar

[3]

D. J. Gubler, Dengue,, in, II (1986), 213.   Google Scholar

[4]

J. Hale, "Asymptotic Behavior of Dissipative Systems,", American Mathematical Society Providence, (1988).   Google Scholar

[5]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Math., (1981).   Google Scholar

[6]

S. B. Hsu, A survey of constructing lyapunov function for mathematical models in population biology,, Taiwanese Journal of Mathematics, 9 (2005), 151.   Google Scholar

[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.  doi: 10.1016/j.jde.2009.12.014.  Google Scholar

[8]

T.-W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model,, Can. Appl. Math. Q., 10 (2002), 473.   Google Scholar

[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.  doi: 10.1007/s10884-011-9224-3.  Google Scholar

[10]

F. X. Jousset, Geographic A. aegypti strains and dengue-2 virus: susceptibility, ability to transmit to vertebrate and transovarial transmission,, Ann. Virol, 132 (1981).   Google Scholar

[11]

A. Korobeinikov, Global properties of basic virus dynamics models,, Bulletin of Mathematical Biology, 66 (2004), 879.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s00285-010-0346-8.  Google Scholar

[14]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. of A. M. S., 321 (1990), 1.   Google Scholar

[15]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM. J. Math. Anal., 37 (2005), 251.  doi: 10.1137/S0036141003439173.  Google Scholar

[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.  doi: 10.2307/2154124.  Google Scholar

[17]

A. Pazy, "Semigroups of Linear Operators and Applicationto Partial Differential Equations,", Springer-Verlag, (1983).   Google Scholar

[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, 32 (1983).   Google Scholar

[19]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (1984).   Google Scholar

[20]

H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,", Math. Surveys Monogr 41, 41 (1995).   Google Scholar

[21]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems,, Nonlinear Anal., 47 (2001), 6169.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[22]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  doi: 10.1007/BF00173267.  Google Scholar

[23]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM, 70 (2009), 188.   Google Scholar

[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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[25]

, Dengue haemorrhagic fever: Diagnosis, treatment and control,, World Health Organization, (1986).   Google Scholar

[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.  doi: 10.1016/j.jde.2010.07.031.  Google Scholar

[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.  doi: 10.1137/090775890.  Google Scholar

[28]

X.-Q. Zhao, "Dynamical Systems in Population Biology,", CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16 (2003).   Google Scholar

show all references

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.  doi: 10.1007/BF00178324.  Google Scholar

[2]

L. Esteva and C. Vargas, Analysis of a dengue disease transmission model,, Mathematical Biosciences, 150 (1998), 131.  doi: 10.1016/S0025-5564(98)10003-2.  Google Scholar

[3]

D. J. Gubler, Dengue,, in, II (1986), 213.   Google Scholar

[4]

J. Hale, "Asymptotic Behavior of Dissipative Systems,", American Mathematical Society Providence, (1988).   Google Scholar

[5]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Math., (1981).   Google Scholar

[6]

S. B. Hsu, A survey of constructing lyapunov function for mathematical models in population biology,, Taiwanese Journal of Mathematics, 9 (2005), 151.   Google Scholar

[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.  doi: 10.1016/j.jde.2009.12.014.  Google Scholar

[8]

T.-W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model,, Can. Appl. Math. Q., 10 (2002), 473.   Google Scholar

[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.  doi: 10.1007/s10884-011-9224-3.  Google Scholar

[10]

F. X. Jousset, Geographic A. aegypti strains and dengue-2 virus: susceptibility, ability to transmit to vertebrate and transovarial transmission,, Ann. Virol, 132 (1981).   Google Scholar

[11]

A. Korobeinikov, Global properties of basic virus dynamics models,, Bulletin of Mathematical Biology, 66 (2004), 879.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s00285-010-0346-8.  Google Scholar

[14]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. of A. M. S., 321 (1990), 1.   Google Scholar

[15]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM. J. Math. Anal., 37 (2005), 251.  doi: 10.1137/S0036141003439173.  Google Scholar

[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.  doi: 10.2307/2154124.  Google Scholar

[17]

A. Pazy, "Semigroups of Linear Operators and Applicationto Partial Differential Equations,", Springer-Verlag, (1983).   Google Scholar

[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, 32 (1983).   Google Scholar

[19]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (1984).   Google Scholar

[20]

H. L. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,", Math. Surveys Monogr 41, 41 (1995).   Google Scholar

[21]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems,, Nonlinear Anal., 47 (2001), 6169.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[22]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  doi: 10.1007/BF00173267.  Google Scholar

[23]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM, 70 (2009), 188.   Google Scholar

[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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[25]

, Dengue haemorrhagic fever: Diagnosis, treatment and control,, World Health Organization, (1986).   Google Scholar

[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.  doi: 10.1016/j.jde.2010.07.031.  Google Scholar

[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.  doi: 10.1137/090775890.  Google Scholar

[28]

X.-Q. Zhao, "Dynamical Systems in Population Biology,", CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16 (2003).   Google Scholar

[1]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[2]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268

[3]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

[4]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

[5]

Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020288

[6]

Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2020  doi: 10.3934/jcd.2021006

[7]

Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020378

[8]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331

[9]

Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020371

[10]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004

[11]

Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021016

[12]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[13]

Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028

[14]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[15]

Hua Zhong, Xiaolin Fan, Shuyu Sun. The effect of surface pattern property on the advancing motion of three-dimensional droplets. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020366

[16]

Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028

[17]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[18]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304

[19]

Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021027

[20]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (72)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]