November  2018, 23(9): 4045-4061. doi: 10.3934/dcdsb.2018125

Analysis of a stage-structured dengue model

1. 

Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

2. 

Department of Mathematics and School of Mechanical Engineering, Purdue University, West Lafayette, IN 47906, USA

3. 

Department of Mathematics, Purdue University, West Lafayette, IN 47906, USA

* Corresponding author: wanh2046@163.com

Received  June 2017 Revised  October 2017 Published  April 2018

Fund Project: G. Lin is supported by NSF Grants DMS-1555072 and DMS-1736364. H. Wan is Supported by Jiangsu Overseas Research and Training Program for University Prominent Young and Middleaged Teachers and Presidents, the NSF of the Jiangsu Higher Education Committee of China (15KJD110004, 17KJA110002) and A Project Funded by PAPD of Jiangsu Higher Education Institutions

In order to study the impact of control measures and limited resource on dengue transmission dynamics, we formulate a stage-structured dengue model. The basic investigation of the model, such as the existence of equilibria and their stability, have been proved. It is also shown that this model may undergo backward bifurcation, where the stable disease-free equilibrium co-exists with an endemic equilibrium. The backward bifurcation property can be removed by ignoring the disease-induced death in human population and the global stability of the unique endemic equilibrium has been proved. Sensitivity analysis with respect to $R_0$ has been carried out to explore the impact of model parameters. In addition, numerical analysis manifests that the more intensive control measures in targeting immature and adult mosquitoes are both effective in preventing dengue outbreaks. It is also shown that the earlier the control intervention begins, the less people would be infected and the earlier dengue would be eradicated. Even later epidemic prevention and control can also effectively reduce the severity of pandemic. Moreover, comprehensive control measures are more effective than a single measure.

Citation: Jinping Fang, Guang Lin, Hui Wan. Analysis of a stage-structured dengue model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4045-4061. doi: 10.3934/dcdsb.2018125
References:
[1]

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N. Becker, Mosquitoes and Their Control, Kluwer Academic/Plenum, New Nork, 2003.Google Scholar

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L. BillingsI. B. SchwartzL. B. ShawM. McCraryD. S. Burke and D. A. T. Cummings, Instabilities in multiserotype disease models with antibody-dependent enhancement, J. Theore. Biolo., 246 (2007), 18-27. doi: 10.1016/j.jtbi.2006.12.023. Google Scholar

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J. Fang and H. Wan, Analysis of a stage-structured mosquito population model, Journal of Shanghai Normal University, 46 (2017), 417-421. Google Scholar

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J. Z. FarkasS. A. GourleyR. Liu and A. Yakubu, Modelling Wolbachia infection in a sex-structured mosquito population carrying West Nile virus, J. Math. Biol., 75 (2017), 621-647. doi: 10.1007/s00285-017-1096-7. Google Scholar

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N. FergusonR. Anderson and S. Gupta, The effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens, Proc. Natl. Acad. Sci. USA, 96 (1999), 790-794. doi: 10.1073/pnas.96.2.790. Google Scholar

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S. M. GabraA. 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

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Guangzhou Statistics Bureau, 2016. Available from: http://www.gzstats.gov.cn.Google Scholar

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S. MungaN. Minakawa and G. Zhou, Survivorship of immature stages of Anopheles gambiae s.l. (Diptera: Culicidae) in natural habitats in western Kenya highlands, Journal of Medical Entomology, 44 (2007), 58-764. Google Scholar

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E. A. C. Newton and P. Reiter, A model of the transmission of Dengue fever with an evaluation of the impact of ultra-low volume (ULV) insecticide applications on Dengue epidemics, American Journal of Tropical Medicine and Hygene, 47 (1992), 709-720. doi: 10.4269/ajtmh.1992.47.709. Google Scholar

[26]

A. SaltelliP. AnnoniI. AzziniF. CampolongoM. Ratto and S. Tarantola, Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index, Computer Physics Communications, 181 (2010), 259-270. doi: 10.1016/j.cpc.2009.09.018. Google Scholar

[27]

F. SarrazinF. Pianosi and T. Wagener, Global Sensitivity Analysis of environmental models: Convergence and validation, Environmental Modelling & Software, 79 (2016), 135-152. doi: 10.1016/j.envsoft.2016.02.005. Google Scholar

[28]

L. B. ShawL. Billings and I. B. Schwartz, Using dimension reduction to improve outbreak predictability of multistrain diseases, J. Math. Biol., 55 (2007), 1-19. doi: 10.1007/s00285-007-0074-x. Google Scholar

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C. Shekhar, Deadly dengue: New vaccines promise to tackle this escalating global menace, Chemistry and Biology, 14 (2007), 871-872. doi: 10.1016/j.chembiol.2007.08.004. Google Scholar

[30]

I. M. Sobol, Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates, Mathematics and Computers in Simulation, 55 (2001), 271-280. doi: 10.1016/S0378-4754(00)00270-6. Google Scholar

[31]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[32]

H. Wan and J. Cui, A model for the transmission of dengue, IEEE Xplore, (2009). doi: 10.1109/ICBBE.2009.5162187. Google Scholar

[33]

H. Wan and H. Zhu, A new model with delay for mosquito population dynamics, Math. Biosci. Eng., 11 (2014), 1395-1410. doi: 10.3934/mbe.2014.11.1395. Google Scholar

[34]

L. XuL. C. StigeK. ChanJ. ZhouJ. YangS. SangM. WangZ. YangZ. YanT. JiangL. LuY. YueX. LiuH. LinJ. XuQ. Liu and N. C. Stenseth, Climate variation drives dengue dynamics, Proc. Natl. Acad. Sci. U S A., 114 (2017), 113-118. doi: 10.1073/pnas.1618558114. Google Scholar

show all references

References:
[1]

S. AiJ. Li and J. Lu, Mosquito-stage-structured malaria models and their global dynamics, SIAM J. Appl. Math., 72 (2012), 1213-1237. doi: 10.1137/110860318. Google Scholar

[2]

N. Becker, Mosquitoes and Their Control, Kluwer Academic/Plenum, New Nork, 2003.Google Scholar

[3]

L. BillingsA. Fiorillo and I. B. Schwartz, Vaccinations in disease models with antibody-dependent enhancement, Math. Biosci., 211 (2008), 265-281. doi: 10.1016/j.mbs.2007.08.004. Google Scholar

[4]

L. BillingsI. B. SchwartzL. B. ShawM. McCraryD. S. Burke and D. A. T. Cummings, Instabilities in multiserotype disease models with antibody-dependent enhancement, J. Theore. Biolo., 246 (2007), 18-27. doi: 10.1016/j.jtbi.2006.12.023. Google Scholar

[5]

Q. Cheng, Q. Jing, R. C. Spear, J. M. Marshall, Z. Yang and P. Gong, Climate and the Timing of Imported Cases as Determinants of the Dengue Outbreak in Guangzhou, 2014: Evidence from a Mathematical Model, PLoS. Negl. Trop. Dis., 10 (2016), e0004417. doi: 10.1371/journal.pntd.0004417. Google Scholar

[6]

G. ChowellP. Diaz-DuenasJ.C. MillerA. Alcazar-VelazcoJ. M. HymanP. W. Fenimore and C. Castillo Chavez, Estimation of the reproduction number of dengue fever from spatial epidemic data, Mathematical Biosciences, 208 (2007), 571-589. doi: 10.1016/j.mbs.2006.11.011. Google Scholar

[7]

F. A. B. CoutinhoM. N. BurattiniL. F. Lopez and E. Massad, Threshold conditions for a non-Autonomous epidemic system describing the population dynamics of dengue, Bulletin of Mathematical Biology, 68 (2006), 2263-2282. doi: 10.1007/s11538-006-9108-6. Google Scholar

[8]

M. Derouich, A. Boutayeb and E. H. Twizell, A model of dengue fever, BioMed. Eng. OnLine, 2 (2003).Google Scholar

[9]

M. Derouich and A. Boutayeb, Dengue fever: Mathematical modelling and computer simulation, Applied Mathematics and Computation, 177 (2006), 528-544. doi: 10.1016/j.amc.2005.11.031. Google Scholar

[10]

R. A. EricksonS. M. PresleyL. J. S. AllenK. R. Long and S. B. Cox, A dengue model with a dynamic Aedes albopictus vector population, Ecological Modelling, 221 (2010), 2899-2908. Google Scholar

[11]

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

[12]

L. Esteva and C. Vargas, A model for dengue disease with variable human population, J. Math. Biol., 38 (1999), 220-240. doi: 10.1007/s002850050147. Google Scholar

[13]

L. Esteva and C. Vargas, Influence of vertical and mechanical transmission on the dynamics of dengue disease, Mathematical Biosciences, 167 (2000), 51-64. doi: 10.1016/S0025-5564(00)00024-9. Google Scholar

[14]

L. Esteva and C. Vargas, Coexistence of different serotypes of dengue virus, J. Math. Biol., 46 (2003), 31-47. doi: 10.1007/s00285-002-0168-4. Google Scholar

[15]

J. FangS. A. Gourley and Y. Lou, Stage-structured models of intra-and inter-specific competition within age classes, J. Differential Equations, 260 (2016), 1918-1953. doi: 10.1016/j.jde.2015.09.048. Google Scholar

[16]

J. Fang and H. Wan, Analysis of a stage-structured mosquito population model, Journal of Shanghai Normal University, 46 (2017), 417-421. Google Scholar

[17]

J. Z. FarkasS. A. GourleyR. Liu and A. Yakubu, Modelling Wolbachia infection in a sex-structured mosquito population carrying West Nile virus, J. Math. Biol., 75 (2017), 621-647. doi: 10.1007/s00285-017-1096-7. Google Scholar

[18]

Z. 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

[19]

N. FergusonR. Anderson and S. Gupta, The effect of antibody-dependent enhancement on the transmission dynamics and persistence of multiple-strain pathogens, Proc. Natl. Acad. Sci. USA, 96 (1999), 790-794. doi: 10.1073/pnas.96.2.790. Google Scholar

[20]

S. M. GabraA. 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

[21]

Guangzhou Statistics Bureau, 2016. Available from: http://www.gzstats.gov.cn.Google Scholar

[22]

J. Li, Malaria model with stage-structured mosquitoes, Math. Biosci. Eng., 8 (2011), 753-768. doi: 10.3934/mbe.2011.8.753. Google Scholar

[23]

M. Li, G. Sun, L. Yakob, H. Zhu, Z. Jin and W. Zhang, The driving force for 2014 dengue outbreak in Guangdong, China, PLoSONE, 11 (2016), e0166211. doi: 10.1371/journal.pone.0166211. Google Scholar

[24]

S. MungaN. Minakawa and G. Zhou, Survivorship of immature stages of Anopheles gambiae s.l. (Diptera: Culicidae) in natural habitats in western Kenya highlands, Journal of Medical Entomology, 44 (2007), 58-764. Google Scholar

[25]

E. A. C. Newton and P. Reiter, A model of the transmission of Dengue fever with an evaluation of the impact of ultra-low volume (ULV) insecticide applications on Dengue epidemics, American Journal of Tropical Medicine and Hygene, 47 (1992), 709-720. doi: 10.4269/ajtmh.1992.47.709. Google Scholar

[26]

A. SaltelliP. AnnoniI. AzziniF. CampolongoM. Ratto and S. Tarantola, Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index, Computer Physics Communications, 181 (2010), 259-270. doi: 10.1016/j.cpc.2009.09.018. Google Scholar

[27]

F. SarrazinF. Pianosi and T. Wagener, Global Sensitivity Analysis of environmental models: Convergence and validation, Environmental Modelling & Software, 79 (2016), 135-152. doi: 10.1016/j.envsoft.2016.02.005. Google Scholar

[28]

L. B. ShawL. Billings and I. B. Schwartz, Using dimension reduction to improve outbreak predictability of multistrain diseases, J. Math. Biol., 55 (2007), 1-19. doi: 10.1007/s00285-007-0074-x. Google Scholar

[29]

C. Shekhar, Deadly dengue: New vaccines promise to tackle this escalating global menace, Chemistry and Biology, 14 (2007), 871-872. doi: 10.1016/j.chembiol.2007.08.004. Google Scholar

[30]

I. M. Sobol, Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates, Mathematics and Computers in Simulation, 55 (2001), 271-280. doi: 10.1016/S0378-4754(00)00270-6. Google Scholar

[31]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[32]

H. Wan and J. Cui, A model for the transmission of dengue, IEEE Xplore, (2009). doi: 10.1109/ICBBE.2009.5162187. Google Scholar

[33]

H. Wan and H. Zhu, A new model with delay for mosquito population dynamics, Math. Biosci. Eng., 11 (2014), 1395-1410. doi: 10.3934/mbe.2014.11.1395. Google Scholar

[34]

L. XuL. C. StigeK. ChanJ. ZhouJ. YangS. SangM. WangZ. YangZ. YanT. JiangL. LuY. YueX. LiuH. LinJ. XuQ. Liu and N. C. Stenseth, Climate variation drives dengue dynamics, Proc. Natl. Acad. Sci. U S A., 114 (2017), 113-118. doi: 10.1073/pnas.1618558114. Google Scholar

Figure 1.  The transmission diagram of dengue virus between mosquitoes and humans
Figure 2.  The time courses of $H_I(t)$ with different initial values. Parameter values used are: $b = 0.14$, $\Psi_h = 454$, $d_h = 1/25000$, $\beta_h = 0.75$, $\psi_h = 0.01$, $\gamma_h = 1/5$, $\delta_h = 0.02$, $\eta_h = 1/3$, $p_A = 5e+10$, $\gamma_a = 1/20$, $\kappa = 0.01$, $\beta_m = 0.75$, $\gamma_m = 0.1$, $d_a = 0.2$, $d_m = 0.03$. $R_0 = 0.513$, $R_c = 0.511$, $\xi_2+\xi_4-\xi_3 = -1302.986$. With different initial values, one curve (red) tends to the value of 0, and the other curve (blue) tends to the value of 19767
Figure 3.  Backward bifurcation diagram. Parameter values used are: $b = 0.14$, $d_h = 1/25000$, $\beta_h = 0.75$, $\psi_h = 0.01$, $\gamma_h = 1/5$, $\delta_h = 0.02$, $\eta_h = 1/3$, $p_A = 5e+10$, $\gamma_a = 1/20$, $\kappa = 0.01$, $\beta_m = 0.75$, $\gamma_m = 0.1$, $d_a = 0.2$, $d_m = 0.03$. $\xi_2+\xi_4-\xi_3 = -1302.986 <0$, $R_c = 0.511.$
Figure 4.  The curves of the growth rate function
Figure 5.  The global Sobol sensitivity indices of the six model parameters with respect to the basic reproductive number $R_0$ in the stage-structured dengue model
Figure 6.  The local sensitivity indices of the six model parameters with respect to the basic reproductive number $R_0$
Figure 7.  The contour plot of the basic reproduction number, $R_0$ as a function of $\alpha_J$ and $\alpha_v$
Figure 8.  The contour plot of the basic reproduction number, $R_0$ as a function of $\alpha_b$ and $\alpha_v$
Figure 9.  The cumulative number of infected human hosts according to different control intervention time. Parameter values used when there is no intervention control measures are: $b = 0.14$, $\Psi_h = 454$, $d_h = 1/25000$, $\beta_h = 0.75$, $\psi_h = 0.01$, $\gamma_h = 1/5$, $\delta_h = 0.001$, $\eta_h = 1/3$, $p_A = 5e+10$, $\gamma_a = 1/20$, $\kappa = 0.01$, $\beta_m = 0.75$, $\gamma_m = 0.1$, $d_a = 0.2$, $d_m = 0.03$ and $R_0 = 0.527$, $\xi_2+\xi_4-\xi_3 = 42.97$
Figure 10.  The prevailing time according to different control intervention time. Parameter values used when there is no intervention control measures are: $b = 0.14$, $\Psi_h = 454$, $d_h = 1/25000$, $\beta_h = 0.75$, $\psi_h = 0.01$, $\gamma_h = 1/5$, $\delta_h = 0.001$, $\eta_h = 1/3$, $p_A = 5e+10$, $\gamma_a = 1/20$, $\kappa = 0.01$, $\beta_m = 0.75$, $\gamma_m = 0.1$, $d_a = 0.2$, $d_m = 0.03$ and $R_0 = 0.527$, $\xi_2+\xi_4-\xi_3 = 42.97$
Figure 11.  The cumulative number of infected human hosts according to different control intervention time with single control intervention measure. Parameter values used are the same as that used in Fig. 9 except $d_m$, $p_A$ and $d_a$
Table 1.  State variables
State variablesMosquitoHuman
Aquatic $A$
Susceptible $M_S$ $H_S$
Exposed $M_E$ $H_E$
Infectious adults $M_I$ $H_I$
Recovered $H_R$
Total adults $M$ $H$
State variablesMosquitoHuman
Aquatic $A$
Susceptible $M_S$ $H_S$
Exposed $M_E$ $H_E$
Infectious adults $M_I$ $H_I$
Recovered $H_R$
Total adults $M$ $H$
Table 2.  Parameter definitions and values
InterpretationParameterRangeReference
The maximum value of the recruitment $\bar{p}_A$[1e+10, 5e+10]Assumed
rate of viable mosquito eggs
without intervention
Biting rate (the average number of$b$[0.14, 0.24][23]
bites per mosquito per day)
The duration of the whole cycle, $1/\gamma_a$ $[7,20]$ days[24]
from egg laying to an adult
mosquito eclosion
Mosquito incubation time$1/\gamma_m$[2, 10] days[23,25]
Natural death rate of immature $\bar{d}_a$[0.2, 0.75][15,23]
mosquitoes
Natural death rate of adult mosquitoes $\bar{d}_m$[0.02, 0.07][15,23]
Density-dependent mortality rate of $\kappa$0.01[15]
immature mosquitoes
Recruitment rate of human $\Psi_h$454[21]
Human life span $1/d_h$25000 days[25]
Human incubation time$1/\gamma_h$5 days[25]
Human infection duration$1/\eta_h$3 days[25]
Human disease-induced death rate $\delta_h$0.001, 0.02[7,20]
Transmission probability $\beta_h$0.75[25]
(from vectors to human)
Transmission probability $\beta_{m}$0.75[25]
(from human to vectors)
Progression rate from $H_R$ to $H_S$ class $\psi_{h}$0.01Assumed
Intervention parameter for adult $\alpha_v$[0, 0.04]Assumed
mosquito death rate
Intervention parameter for immature $\alpha_J$[0, 0.55]Assumed
mosquitoes death rate
Intervention parameter for immature $\alpha_b$[0, $\bar{p}_A$]Assumed
mosquitoes recruitment rate
InterpretationParameterRangeReference
The maximum value of the recruitment $\bar{p}_A$[1e+10, 5e+10]Assumed
rate of viable mosquito eggs
without intervention
Biting rate (the average number of$b$[0.14, 0.24][23]
bites per mosquito per day)
The duration of the whole cycle, $1/\gamma_a$ $[7,20]$ days[24]
from egg laying to an adult
mosquito eclosion
Mosquito incubation time$1/\gamma_m$[2, 10] days[23,25]
Natural death rate of immature $\bar{d}_a$[0.2, 0.75][15,23]
mosquitoes
Natural death rate of adult mosquitoes $\bar{d}_m$[0.02, 0.07][15,23]
Density-dependent mortality rate of $\kappa$0.01[15]
immature mosquitoes
Recruitment rate of human $\Psi_h$454[21]
Human life span $1/d_h$25000 days[25]
Human incubation time$1/\gamma_h$5 days[25]
Human infection duration$1/\eta_h$3 days[25]
Human disease-induced death rate $\delta_h$0.001, 0.02[7,20]
Transmission probability $\beta_h$0.75[25]
(from vectors to human)
Transmission probability $\beta_{m}$0.75[25]
(from human to vectors)
Progression rate from $H_R$ to $H_S$ class $\psi_{h}$0.01Assumed
Intervention parameter for adult $\alpha_v$[0, 0.04]Assumed
mosquito death rate
Intervention parameter for immature $\alpha_J$[0, 0.55]Assumed
mosquitoes death rate
Intervention parameter for immature $\alpha_b$[0, $\bar{p}_A$]Assumed
mosquitoes recruitment rate
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