# American Institute of Mathematical Sciences

June  2020, 40(6): 3467-3484. doi: 10.3934/dcds.2020042

## A stage structured model of delay differential equations for Aedes mosquito population suppression

 1 School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, China 2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 3 Center for Applied Mathematics 4 College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China

* Corresponding author: Moxun Tang

Dedicated to Prof. Wei-Ming Ni on the occasion of his 70th birthday

Received  February 2019 Published  October 2019

Tremendous efforts have been devoted to the development and analysis of mathematical models to assess the efficacy of the endosymbiotic bacterium Wolbachia in the control of infectious diseases such as dengue and Zika, and their transmission vector Aedes mosquitoes. However, the larval stage has not been included in most models, which causes an inconvenience in testing directly the density restriction on population growth. In this work, we introduce a system of delay differential equations, including both the adult and larval stages of wild mosquitoes, interfered by Wolbachia infected males that can cause complete female sterility. We clarify its global dynamics rather completely by using delicate analyses, including a construction of Liapunov-type functions, and determine the threshold level $R_0$ of infected male releasing. The wild population is suppressed completely if the releasing level exceeds $R_0$ uniformly. The dynamical complexity revealed by our analysis, such as bi-stability and semi-stability, is further exhibited through numerical examples. Our model generates a temporal profile that captures several critical features of Aedes albopictus population in Guangzhou from 2011 to 2016. Our estimate for optimal mosquito control suggests that the most cost-efficient releasing should be started no less than 7 weeks before the peak dengue season.

Citation: Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042
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Dedicated to Prof. Wei-Ming Ni on the occasion of his 70th birthday

##### References:
The dynamical complexity of (3) not covered by Theorems 2.4 and 2.5. The parameter values are specified in (25) and $K_L = 5\times 10^5$. $\overline{R} = 10^5\in (0, R_0)$ in A-C, and $\overline{R} = R_0 = 1.073\times 10^6$ in D. A. The solution with $\overline\phi = 1.1\times 10^5>L_1$ and $\overline\psi = 7.5\times 10^3<A_1$ rotates around the unstable equilibrium point $E_1$ before moving towards $E_0$. B and C. Either $E_0$ or $E_2$ may attract solutions with $\overline\phi>L_1$ and $\overline\psi<A_1$, or $\overline\phi<L_1$ and $\overline\psi>A_1$; see (26) for the values of $L_1$ and $A_1$. D. Either $E_0$ or $E^*_2$ may attract solutions with $\overline\phi>L_2^*$ and $\overline\psi<A_2^*$, or solutions with $\overline\phi<L_2^*$ and $\overline\psi>A_2^*$
">Figure 2.  The temporal profiles of wild Aedes albopictus population in Guangzhou from 2011 to 2016. The profiles were simulated by (3) with $\overline{R}\equiv 0$, supplemented by the temperature-dependent rates estimated from (27) and Table 2
. It exhibits a quasi-periodicity as temperature annually, and peaks from July to October during the high-risk season of dengue fever">Figure 3.  The temperature dependency of the threshold releasing level $R_0$. The figure was simulated by (24) with the daily rates estimated from (27) and Table 2. It exhibits a quasi-periodicity as temperature annually, and peaks from July to October during the high-risk season of dengue fever
into (3). The mosquito population is reduced more than $>95\%$ on September 12, 2012, and the same low level is kept in phase Ⅱ of the five weeks behind">Figure 4.  Suppression of the wild Aedes albopictus population during the high-incidence season of dengue fever in 2012. The curves were generated by substituting the same parameter values and the initial data described in Table 2 into (3). The mosquito population is reduced more than $>95\%$ on September 12, 2012, and the same low level is kept in phase Ⅱ of the five weeks behind
The life table of Aedes albopictus. The parameters are adapted to Aedes albopictus population in subtropical monsoon climate as in Guangzhou
 Para. Definition value Reference $N$ Number of eggs laid by a female 200 [20,22,39] $\mu_E$ Hatch rate of egg (day$^{-1}$) Dependent of T [6,17,25] $\beta$ Mean larvae produced by a female (day$^{-1}$) $\beta=2N\mu_E/\tau_A$ $m$ Minimum larva mortality rate (day$^{-1}$) Dependent of T [4,5,25] $\mu$ Pupation rate (day$^{-1}$) Dependent of T [6,17,25] $\bar{\alpha}$ Pupa survival rate (day$^{-1}$) Dependent of T [6,17,25] $\delta$ Adult female mortality rate (day$^{-1}$) Dependent of T [4,5,25] $\tau_E$ Development period of egg (days) (3.7, 18.3) [16,18,22,38] $\tau_L$ Development period of larva (days) (5.2, 27.7) [16,18,22,38] $\tau_P$ Development period of pupa (days) (1.5, 8.6) [16,18,22,38] $\tau_A$ Mean longevity of female (days) (4.8, 40.9) [16,18,22,38]
 Para. Definition value Reference $N$ Number of eggs laid by a female 200 [20,22,39] $\mu_E$ Hatch rate of egg (day$^{-1}$) Dependent of T [6,17,25] $\beta$ Mean larvae produced by a female (day$^{-1}$) $\beta=2N\mu_E/\tau_A$ $m$ Minimum larva mortality rate (day$^{-1}$) Dependent of T [4,5,25] $\mu$ Pupation rate (day$^{-1}$) Dependent of T [6,17,25] $\bar{\alpha}$ Pupa survival rate (day$^{-1}$) Dependent of T [6,17,25] $\delta$ Adult female mortality rate (day$^{-1}$) Dependent of T [4,5,25] $\tau_E$ Development period of egg (days) (3.7, 18.3) [16,18,22,38] $\tau_L$ Development period of larva (days) (5.2, 27.7) [16,18,22,38] $\tau_P$ Development period of pupa (days) (1.5, 8.6) [16,18,22,38] $\tau_A$ Mean longevity of female (days) (4.8, 40.9) [16,18,22,38]
The parameter values of (27) and the temperature-dependent mortality rates
 Para. Value Reference $\mu_E$ $\rho_E = 0.24, a_E = 10798, b_E = 100000, c_E = 14184$ [6,17,25] $\mu = \mu_L$ $\rho_L = 0.2088, a_L = 26018, b_L = 55990, c_L = 304.6$ [6,17,25] $\bar{\alpha} = \mu_P$ $\rho_P = 0.384, a_P = 14931, b_P = -472379, c_P = 148$ [6,17,25] $m$ $\left\{ \begin{array}{ll} 0.0000866 T^2-0.00368 T+0.09, \, \, T\ge12.5 ^oC, \\0.5, \, \, {\rm else}. \end{array} \right.$ [4,5,25] $\delta$ $\left\{ \begin{array}{ll} 0.000114 T^2-0.00427 T+0.1278, \, \, T\ge 15 ^oC, \\0.5, \, \, {\rm else}. \end{array} \right.$ [4,5,25]
 Para. Value Reference $\mu_E$ $\rho_E = 0.24, a_E = 10798, b_E = 100000, c_E = 14184$ [6,17,25] $\mu = \mu_L$ $\rho_L = 0.2088, a_L = 26018, b_L = 55990, c_L = 304.6$ [6,17,25] $\bar{\alpha} = \mu_P$ $\rho_P = 0.384, a_P = 14931, b_P = -472379, c_P = 148$ [6,17,25] $m$ $\left\{ \begin{array}{ll} 0.0000866 T^2-0.00368 T+0.09, \, \, T\ge12.5 ^oC, \\0.5, \, \, {\rm else}. \end{array} \right.$ [4,5,25] $\delta$ $\left\{ \begin{array}{ll} 0.000114 T^2-0.00427 T+0.1278, \, \, T\ge 15 ^oC, \\0.5, \, \, {\rm else}. \end{array} \right.$ [4,5,25]
The least releasing levels $r_1$ in phase I, $r_2$ in phase II, and the total numbers in the two phase to reduce $>95\%$ of wild Aedes albopictus on September 12, 2012, and keep the same low level in phase II of the five weeks behind. The same parameter values and the initial data in Figure 2, except $\overline R(t) = r_1$ in phase Ⅰ, and $\overline R(t) = r_2$ in phase Ⅱ, were inserted into (3) for simulations
 Phase Ⅰ Phase Ⅱ (5 weeks) Total number Suppression period, $r_1$, Release times $r_2$, Release times 5 weeks, $4.4 \times 10^9$, 12 $1.24 \times 10^7$, 12 $5.29 \times 10^{10}$ 6 weeks, $5.11 \times 10^8$, 14 $8.48 \times 10^6$, 12 $7.26 \times 10^9$ 7 weeks, $1.48 \times 10^8$, 16 $8.42 \times 10^6$, 12 $2.47 \times 10^9$ 8 weeks, $7.65 \times 10^7$, 19 $5.89 \times 10^6$, 12 $1.52 \times 10^9$ 9 weeks, $4.06 \times 10^7$, 21 $4.63 \times 10^6$, 12 $9.08 \times 10^8$ 10 weeks, $2.1 \times 10^7$, 23 $5.48 \times 10^6$, 12 $5.49 \times 10^8$
 Phase Ⅰ Phase Ⅱ (5 weeks) Total number Suppression period, $r_1$, Release times $r_2$, Release times 5 weeks, $4.4 \times 10^9$, 12 $1.24 \times 10^7$, 12 $5.29 \times 10^{10}$ 6 weeks, $5.11 \times 10^8$, 14 $8.48 \times 10^6$, 12 $7.26 \times 10^9$ 7 weeks, $1.48 \times 10^8$, 16 $8.42 \times 10^6$, 12 $2.47 \times 10^9$ 8 weeks, $7.65 \times 10^7$, 19 $5.89 \times 10^6$, 12 $1.52 \times 10^9$ 9 weeks, $4.06 \times 10^7$, 21 $4.63 \times 10^6$, 12 $9.08 \times 10^8$ 10 weeks, $2.1 \times 10^7$, 23 $5.48 \times 10^6$, 12 $5.49 \times 10^8$
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