# American Institute of Mathematical Sciences

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  November 2018 Early access  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
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##### References:
The transmission diagram of dengue virus between mosquitoes and humans
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
Backward bifurcation diagram. Parameter values used are: $b = 0.14$, $d_h = 1/25000$, $\beta_h = 0.75$, $\psi_h = 0.01$, $\gamma_h = 1/5$, $\delta_h = 0.02$, $\eta_h = 1/3$, $p_A = 5e+10$, $\gamma_a = 1/20$, $\kappa = 0.01$, $\beta_m = 0.75$, $\gamma_m = 0.1$, $d_a = 0.2$, $d_m = 0.03$. $\xi_2+\xi_4-\xi_3 = -1302.986 <0$, $R_c = 0.511.$
The curves of the growth rate function
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
The local sensitivity indices of the six model parameters with respect to the basic reproductive number $R_0$
The contour plot of the basic reproduction number, $R_0$ as a function of $\alpha_J$ and $\alpha_v$
The contour plot of the basic reproduction number, $R_0$ as a function of $\alpha_b$ and $\alpha_v$
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 $R_0 = 0.527$, $\xi_2+\xi_4-\xi_3 = 42.97$
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 $R_0 = 0.527$, $\xi_2+\xi_4-\xi_3 = 42.97$
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 $d_a$
State variables
 State variables Mosquito Human 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 variables Mosquito Human Aquatic $A$ Susceptible $M_S$ $H_S$ Exposed $M_E$ $H_E$ Infectious adults $M_I$ $H_I$ Recovered $H_R$ Total adults $M$ $H$
Parameter definitions and values
 Interpretation Parameter Range Reference 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 $\psi_{h}$ 0.01 Assumed 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
 Interpretation Parameter Range Reference 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 $\psi_{h}$ 0.01 Assumed 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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