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

show all references

##### 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 $\overline\psi>A_2^*$
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
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
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 $5.49 \times 10^8$
 [1] V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066 [2] Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 [3] Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008 [4] Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023 [5] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [6] Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53 [7] Xiaoyi Zhou, Tong Ye, Tony T. Lee. Designing and analysis of a Wi-Fi data offloading strategy catering for the preference of mobile users. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021038 [8] Brandy Rapatski, James Yorke. Modeling HIV outbreaks: The male to female prevalence ratio in the core population. Mathematical Biosciences & Engineering, 2009, 6 (1) : 135-143. doi: 10.3934/mbe.2009.6.135 [9] Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195 [10] Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181 [11] Mikhail Gilman, Semyon Tsynkov. Statistical characterization of scattering delay in synthetic aperture radar imaging. Inverse Problems & Imaging, 2020, 14 (3) : 511-533. doi: 10.3934/ipi.2020024 [12] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [13] Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931 [14] Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027 [15] Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206 [16] Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 [17] Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 [18] Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068 [19] Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213 [20] Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409

2019 Impact Factor: 1.338