American Institute of Mathematical Sciences

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  March 2018 Early access  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 and Continuous Dynamical Systems - B, 2018, 23 (2) : 957-974. doi: 10.3934/dcdsb.2018050
References:
 [1] A. Abdelrazec, J. Belair, C. 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. [2] I. Ahn, S. 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. [3] L. J. S. Allen, B. M. Bolker, Y. 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. [4] P. Álvarez-Caudevilla, Y. 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. [5] S. Bhatt, P. W. Gething and O. J. Brady, The global distribution and burden of dengue, Nature, 496 (2013), 504-507.  doi: 10.1038/nature12060. [6] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd, 2003. [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. [8] Dengue Fever, Available from: https://en.wikipedia.org/wiki/Dengue_fever. [9] Dengue Fever: The Fastest-spreading Vector-borne Infectious Disease Worldwide, Available from: http://www.jianke.com/crbpd/1681049.html. [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. [11] X. M. Feng, S. G. Ruan, Z. 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. [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. [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. [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. [15] S. M. Garba, A. 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. [16] J. Ge, K. I. Kim, Z. 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. [17] J. Ge, Z. 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. [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. [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. [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. [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. [22] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [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. [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. [25] A. K. Tarboush, Z. 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. [26] J. J. Tewa, J. 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. [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. [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. [29] Z. Wang and X. Q. Zhao, Global dynamics of a time-delayed dengue transmission model, Can. Appl. Math. Q., 20 (2012), 89-113. [30] WHO, Fact Sheet 117: Dengue and dengue hemorrhagic fever, (2009). [31] N. Z. Xu and H. M. Yang, Mosquitoes and prevent(Chinese), Disease Monitor and Control, 4 (2010), 635. [32] H. M. Yang, M. L. G. Macoris, K. 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. [33] X. H. Zhang, S. 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.

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References:
 [1] A. Abdelrazec, J. Belair, C. 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. [2] I. Ahn, S. 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. [3] L. J. S. Allen, B. M. Bolker, Y. 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. [4] P. Álvarez-Caudevilla, Y. 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. [5] S. Bhatt, P. W. Gething and O. J. Brady, The global distribution and burden of dengue, Nature, 496 (2013), 504-507.  doi: 10.1038/nature12060. [6] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd, 2003. [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. [8] Dengue Fever, Available from: https://en.wikipedia.org/wiki/Dengue_fever. [9] Dengue Fever: The Fastest-spreading Vector-borne Infectious Disease Worldwide, Available from: http://www.jianke.com/crbpd/1681049.html. [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. [11] X. M. Feng, S. G. Ruan, Z. 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. [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. [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. [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. [15] S. M. Garba, A. 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. [16] J. Ge, K. I. Kim, Z. 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. [17] J. Ge, Z. 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. [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. [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. [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. [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. [22] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [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. [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. [25] A. K. Tarboush, Z. 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. [26] J. J. Tewa, J. 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. [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. [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. [29] Z. Wang and X. Q. Zhao, Global dynamics of a time-delayed dengue transmission model, Can. Appl. Math. Q., 20 (2012), 89-113. [30] WHO, Fact Sheet 117: Dengue and dengue hemorrhagic fever, (2009). [31] N. Z. Xu and H. M. Yang, Mosquitoes and prevent(Chinese), Disease Monitor and Control, 4 (2010), 635. [32] H. M. Yang, M. L. G. Macoris, K. 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. [33] X. H. Zhang, S. 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.
$p = 10$ 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.
$p = 5$ 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.
$p = 10$ 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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