March  2018, 23(2): 957-974. doi: 10.3934/dcdsb.2018050

Modeling the transmission of dengue fever with limited medical resources and self-protection

1. 

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

2. 

School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, China

* Corresponding author

Received  January 2017 Revised  July 2017 Published  December 2017

Fund Project: The work is supported by the NSFC of China (Grant No. 11371311) and Graduate Research and Innovation Projects of Jiangsu Province (Grant No. KYZZ16−0489).

To capture the impacts of limited medical resources and self-protection on the transmission of dengue fever, we formulate an SIS v.s. SI dengue model with the nonlinear recovery rate and contact transmission rate. The spatial heterogeneity of environment is also taken into consideration. With the aid of the relevant eigenvalue problem, we explore some properties of the basic reproduction number, and show that it still plays its "traditional" role in determining the stability of equilibria, that is, the extinction and persistence of dengue fever. Moreover, we consider a special diffusive pattern in which there is only human diffusion, but no mosquitoes diffusion, then present the explicit expression of the basic reproduction number and exhibit the corresponding transmission dynamics. This paper ends up with some numerical simulations and epidemiological explanations, which confirm our analytical findings.

Citation: Min Zhu, Zhigui Lin. Modeling the transmission of dengue fever with limited medical resources and self-protection. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 957-974. doi: 10.3934/dcdsb.2018050
References:
[1]

A. AbdelrazecJ. BelairC. H. Shan and H. P. Zhu, Modeling the spread and control of dengue with limited public health resources, Math. Biosci., 271 (2016), 136-145.  doi: 10.1016/j.mbs.2015.11.004.  Google Scholar

[2]

I. AhnS. Baek and Z. G. Lin, The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Modelling, 40 (2016), 7082-7101.  doi: 10.1016/j.apm.2016.02.038.  Google Scholar

[3]

L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

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R. H. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.  Google Scholar

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X. M. FengS. G. RuanZ. D. Teng and K Wang, Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China, Math. Biosci., 266 (2015), 52-64.  doi: 10.1016/j.mbs.2015.05.005.  Google Scholar

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J. GeK. I. KimZ. G. Lin and H. P. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.  Google Scholar

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J. GeZ. G. Lin and H. P. Zhu, Environmental risks in a diffusive SIS model incorporating use efficiency of the medical resource, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1469-1481.  doi: 10.3934/dcdsb.2016007.  Google Scholar

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[20]

Z. G. Lin and M. Pedersen, Stability in a diffusive food-chain model with Michaelis-Menten functional response, Nonlinear Anal., 57 (2004), 421-433.  doi: 10.1016/j.na.2004.02.022.  Google Scholar

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Y. J. 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.  Google Scholar

[22]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar

[23]

M. Safan and A. Ghazi, Demographic impact and controllability of malaria in an SIS model with proportional fatality, Bull. Malays. Math. Sci. Soc., 39 (2016), 65-86.  doi: 10.1007/s40840-015-0181-6.  Google Scholar

[24]

C. H. Shan and H. P. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Differential Equations, 257 (2014), 1662-1688.  doi: 10.1016/j.jde.2014.05.030.  Google Scholar

[25]

A. K. TarboushZ. G. Lin and M. Y. Zhang, Spreading and vanishing in a West Nile virus model with expanding fronts, Sci. China Math., 60 (2017), 841-860.  doi: 10.1007/s11425-016-0367-4.  Google Scholar

[26]

J. J. TewaJ. L. Dimi and S. Bowong, Lyapunov functions for a dengue disease transmission model, Chaos Solitons Fractals, 39 (2009), 936-941.  doi: 10.1016/j.chaos.2007.01.069.  Google Scholar

[27]

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

[28]

W. D. 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.  Google Scholar

[29]

Z. Wang and X. Q. Zhao, Global dynamics of a time-delayed dengue transmission model, Can. Appl. Math. Q., 20 (2012), 89-113.   Google Scholar

[30]

WHO, Fact Sheet 117: Dengue and dengue hemorrhagic fever, (2009). Google Scholar

[31]

N. Z. Xu and H. M. Yang, Mosquitoes and prevent(Chinese), Disease Monitor and Control, 4 (2010), 635. Google Scholar

[32]

H. M. YangM. L. G. MacorisK. C. Galvani and M. T. Andrighetti, Follow up estimation of Aedes aegypti entomological parameters and mathematical modellings, Biosystems, 103 (2011), 360-371.  doi: 10.1016/j.biosystems.2010.11.002.  Google Scholar

[33]

X. H. ZhangS. Y. Tang and R. A. Cheke, Models to assess how best to replace dengue virus vectors with $Wolbachia$-infected mosquito populations, Math. Biosci., 269 (2015), 164-177.  doi: 10.1016/j.mbs.2015.09.004.  Google Scholar

show all references

References:
[1]

A. AbdelrazecJ. BelairC. H. Shan and H. P. Zhu, Modeling the spread and control of dengue with limited public health resources, Math. Biosci., 271 (2016), 136-145.  doi: 10.1016/j.mbs.2015.11.004.  Google Scholar

[2]

I. AhnS. Baek and Z. G. Lin, The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Modelling, 40 (2016), 7082-7101.  doi: 10.1016/j.apm.2016.02.038.  Google Scholar

[3]

L. J. S. AllenB. M. BolkerY. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1-20.  doi: 10.3934/dcds.2008.21.1.  Google Scholar

[4]

P. Álvarez-CaudevillaY. H. Du and R. Peng, Qualitative analysis of a cooperative reaction-diffusion system in a spatiotemporally degenerate environment, SIAM J. Math. Anal., 46 (2014), 499-531.  doi: 10.1137/13091628X.  Google Scholar

[5]

S. BhattP. W. Gething and O. J. Brady, The global distribution and burden of dengue, Nature, 496 (2013), 504-507.  doi: 10.1038/nature12060.  Google Scholar

[6]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd, 2003.  Google Scholar

[7]

R. H. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343.  doi: 10.1016/j.jde.2016.05.025.  Google Scholar

[8]

Dengue Fever, Available from: https://en.wikipedia.org/wiki/Dengue_fever. Google Scholar

[9]

Dengue Fever: The Fastest-spreading Vector-borne Infectious Disease Worldwide, Available from: http://www.jianke.com/crbpd/1681049.html. Google Scholar

[10]

L. Esteva and C. Vargas, Analysis of a dengue disease transmission model, Math. Biosci., 150 (1998), 131-151.  doi: 10.1016/S0025-5564(98)10003-2.  Google Scholar

[11]

X. M. FengS. G. RuanZ. D. Teng and K Wang, Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China, Math. Biosci., 266 (2015), 52-64.  doi: 10.1016/j.mbs.2015.05.005.  Google Scholar

[12]

Z. L. Feng and J. X. Velasco-Hernández, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (1997), 523-544.  doi: 10.1007/s002850050064.  Google Scholar

[13]

D. Fischer and S. Halstead, Observations related to pathogenesis of dengue hemorrhagic fever. V. Examination of age specific sequential infection rates using a mathemactical model, J. Biol. Med., 42 (1970), 329-349.   Google Scholar

[14]

D. Z. Gao, Y. J. Lou and D. H. He, et al., Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: A mathematical modeling ananlysis Scientific Reports 6, Article number: 28070 (2016), 28070 doi: doi:10.1038/srep28070.  Google Scholar

[15]

S. M. GarbaA. B. Gumel and M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics, Math. Biosci., 215 (2008), 11-25.  doi: 10.1016/j.mbs.2008.05.002.  Google Scholar

[16]

J. GeK. I. KimZ. G. Lin and H. P. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035.  Google Scholar

[17]

J. GeZ. G. Lin and H. P. Zhu, Environmental risks in a diffusive SIS model incorporating use efficiency of the medical resource, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1469-1481.  doi: 10.3934/dcdsb.2016007.  Google Scholar

[18]

H. Heesterbeek and R. M. Anderson, et al. Modeling infectious disease dynamics in the complex landscape of global health Science, 347 (2015), aaa4339. doi: 10.1126/science.aaa4339.  Google Scholar

[19]

T. W. Hwang and F. B. Wang, Dynamics of a Dengue fever transmission model with crowding effect in human population and spatial variation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 147-161.   Google Scholar

[20]

Z. G. Lin and M. Pedersen, Stability in a diffusive food-chain model with Michaelis-Menten functional response, Nonlinear Anal., 57 (2004), 421-433.  doi: 10.1016/j.na.2004.02.022.  Google Scholar

[21]

Y. J. 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.  Google Scholar

[22]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  Google Scholar

[23]

M. Safan and A. Ghazi, Demographic impact and controllability of malaria in an SIS model with proportional fatality, Bull. Malays. Math. Sci. Soc., 39 (2016), 65-86.  doi: 10.1007/s40840-015-0181-6.  Google Scholar

[24]

C. H. Shan and H. P. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Differential Equations, 257 (2014), 1662-1688.  doi: 10.1016/j.jde.2014.05.030.  Google Scholar

[25]

A. K. TarboushZ. G. Lin and M. Y. Zhang, Spreading and vanishing in a West Nile virus model with expanding fronts, Sci. China Math., 60 (2017), 841-860.  doi: 10.1007/s11425-016-0367-4.  Google Scholar

[26]

J. J. TewaJ. L. Dimi and S. Bowong, Lyapunov functions for a dengue disease transmission model, Chaos Solitons Fractals, 39 (2009), 936-941.  doi: 10.1016/j.chaos.2007.01.069.  Google Scholar

[27]

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

[28]

W. D. 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.  Google Scholar

[29]

Z. Wang and X. Q. Zhao, Global dynamics of a time-delayed dengue transmission model, Can. Appl. Math. Q., 20 (2012), 89-113.   Google Scholar

[30]

WHO, Fact Sheet 117: Dengue and dengue hemorrhagic fever, (2009). Google Scholar

[31]

N. Z. Xu and H. M. Yang, Mosquitoes and prevent(Chinese), Disease Monitor and Control, 4 (2010), 635. Google Scholar

[32]

H. M. YangM. L. G. MacorisK. C. Galvani and M. T. Andrighetti, Follow up estimation of Aedes aegypti entomological parameters and mathematical modellings, Biosystems, 103 (2011), 360-371.  doi: 10.1016/j.biosystems.2010.11.002.  Google Scholar

[33]

X. H. ZhangS. Y. Tang and R. A. Cheke, Models to assess how best to replace dengue virus vectors with $Wolbachia$-infected mosquito populations, Math. Biosci., 269 (2015), 164-177.  doi: 10.1016/j.mbs.2015.09.004.  Google Scholar

Figure 1.  $p = 10$ and $h = 1.65$. Graphs (a) and (b) show that the solution $(I_H, I_V)$ decays to zero, which means the dengue virus is vanishing.
Figure 2.  $p = 5$ and $h = 1.65$. From graphs (a) and (b), we can see that the solution $(I_H, I_V)$ keeps positive and stabilizes to an equilibrium, which is globally asymptotically stable, that is to say the dengue virus is spreading.
Figure 3.  $p = 10$ and $h = 0.7$. One can observe the long-time behavior of the solution $(I_H, I_V)$, where the dengue virus $I_H$ and $I_V$ don't vanish, and spread gradually.
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