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Analysis of a stage-structured dengue model

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.
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  • 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.

    Mathematics Subject Classification: Primary: 92D30; Secondary: 34C60, 34C23.

    Citation:

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  • 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 $M$ $H$
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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