A Multiobjective Optimization Application to Control the <i>Aedes Aegypti</i> Mosquito using a Two-Dimensional Diffusion-Reaction Model

LIMA, J. S.; VASCONCELOS, A. S. V.; CARDOSO, R. T. N.

doi:10.5540/tcam.2023.024.04.00779

J. S. LIMA ^**Corresponding author: Josenildo Silva de Lima - E-mail: nildo2802@gmail.com

Postgraduate Program in Mathematical and Computational Modeling, PPGMMC, Federal Center of Technological Education of Minas Gerais, Avenida Amazonas 47675, 30510-000 Belo Horizonte, MG, Brazil - E-mail: nildo2802@gmail.com

http://orcid.org/0000-0003-2229-0806

A. S. V. VASCONCELOS

Postgraduate Program in Mathematical and Computational Modeling, PPGMMC, Federal Center of Technological Education of Minas Gerais, Avenida Amazonas 47675, 30510-000 Belo Horizonte, MG, Brazil - E-mail: amaliasv@hotmail.com

http://orcid.org/0000-0002-8789-614X

R. T. N. CARDOSO

Department of Mathematics, Federal Center of Technological Education of Minas Gerais, Avenida Amazonas 47675, 30510-000, Belo Horizonte, MG, Brazil - E-mail: rodrigocardoso@cefetmg.br

http://orcid.org/0000-0002-6784-2929

*Corresponding author: Josenildo Silva de Lima - E-mail: nildo2802@gmail.com

Parameters	Description	Values	Unity	Reference
ε	Fraction of eggs hatching	0.5	day⁻¹	¹1 J.H. Arias, H.J. Martinez, L.S Sepúlveda-Salcedo & O. Vasilieva.. Predator-prey model for analysis of Aedes aegypti population dynamics in Cali, Colombia. International Journal of Pure and Applied Mathematics, 4(105) (2015), 561-597. doi: 10.12732/ijpam.v105i4.2. https://doi.org/10.12732/ijpam.v105i4.2....
ϕ	Oviposition rate	1.060327 − 8.294997	day⁻¹	⁴²42 H.M. Yang . Assessing the Influence of Quiescence Eggs on the Dynamics of Mosquito Aedes aegypti. Applied Mathematics, 5(17) (2014), 2696. doi: 10.4236/am.2014.517257. https://doi.org/10.4236/am.2014.517257....
C	Carrying capacity	50 − 200	dimensionless	⁴⁵45 H.M. Yang & C.P. Ferreira . Assessing the effects of vector control on dengue transmission. Applied Mathematics and Computation, 198(1) (2008), 401-413.
γ	Fraction of female mosquitoes	0.5	day⁻¹	¹1 J.H. Arias, H.J. Martinez, L.S Sepúlveda-Salcedo & O. Vasilieva.. Predator-prey model for analysis of Aedes aegypti population dynamics in Cali, Colombia. International Journal of Pure and Applied Mathematics, 4(105) (2015), 561-597. doi: 10.12732/ijpam.v105i4.2. https://doi.org/10.12732/ijpam.v105i4.2.... ^{), (} ⁴⁴44 H.M. Yang , J.L. Boldrini, A.C. Fassoni, L.F.S. Freitas, M.C. Gomez, K.K.B. de Lima, V.R. Andrade & A.R.R. Freitas. Fitting the Incidence Data from the City of Campinas, Brazil, Based on Dengue Transmission Modellings Considering Time-Dependent Entomological Parameters. PloS One, 11(3) (2016), e0152186. doi: 10.1371/journal.pone.0152186. https://doi.org/10.1371/journal.pone.015...
α	Development rate of immature to adult	0.02615 − 0.11612	day⁻¹	⁴²42 H.M. Yang . Assessing the Influence of Quiescence Eggs on the Dynamics of Mosquito Aedes aegypti. Applied Mathematics, 5(17) (2014), 2696. doi: 10.4236/am.2014.517257. https://doi.org/10.4236/am.2014.517257....
µ _A	Natural death rate of eggs, larvae and pupae	0.01397 − 0.06001	day⁻¹	⁴²42 H.M. Yang . Assessing the Influence of Quiescence Eggs on the Dynamics of Mosquito Aedes aegypti. Applied Mathematics, 5(17) (2014), 2696. doi: 10.4236/am.2014.517257. https://doi.org/10.4236/am.2014.517257....
µ _F	Natural death rate of female mosquitoes	0.028773 − 0.035859	day⁻¹	⁴³43 H.M. Yang . The transovarial transmission in the dynamics of dengue infection: Epidemiological implications and thresholds. Mathematical biosciences , 286 (2017), 1-15.

Parameters	Values	Reference
Mutation rate	5%	³²32 R.H.C. Takahashi, J.A. Vasconcelos, J.A. Ramírez & L. Krahenbuhl. A multiobjective methodology for evaluating genetic operators. IEEE Transactions on Magnetics, 39(3) (2003), 1321-1324. ^{) (} ³⁶36 A.S. Vasconcelos , L.S. Silva, R.T. Cardoso & J.L. Acebal . Optimization of a rainfall dependent model for the seasonal Aedes aegypti integrated control: a case of Lavras/Brazil. Applied Mathematical Modelling , 90 (2021), 413-431.
Recombination rate	90%	³²32 R.H.C. Takahashi, J.A. Vasconcelos, J.A. Ramírez & L. Krahenbuhl. A multiobjective methodology for evaluating genetic operators. IEEE Transactions on Magnetics, 39(3) (2003), 1321-1324. ^{) (} ³⁶36 A.S. Vasconcelos , L.S. Silva, R.T. Cardoso & J.L. Acebal . Optimization of a rainfall dependent model for the seasonal Aedes aegypti integrated control: a case of Lavras/Brazil. Applied Mathematical Modelling , 90 (2021), 413-431.
Biased crossover probability	30%	³²32 R.H.C. Takahashi, J.A. Vasconcelos, J.A. Ramírez & L. Krahenbuhl. A multiobjective methodology for evaluating genetic operators. IEEE Transactions on Magnetics, 39(3) (2003), 1321-1324. ^{) (} ³⁶36 A.S. Vasconcelos , L.S. Silva, R.T. Cardoso & J.L. Acebal . Optimization of a rainfall dependent model for the seasonal Aedes aegypti integrated control: a case of Lavras/Brazil. Applied Mathematical Modelling , 90 (2021), 413-431.
Simulations	30	³²32 R.H.C. Takahashi, J.A. Vasconcelos, J.A. Ramírez & L. Krahenbuhl. A multiobjective methodology for evaluating genetic operators. IEEE Transactions on Magnetics, 39(3) (2003), 1321-1324. ^{) (} ³⁶36 A.S. Vasconcelos , L.S. Silva, R.T. Cardoso & J.L. Acebal . Optimization of a rainfall dependent model for the seasonal Aedes aegypti integrated control: a case of Lavras/Brazil. Applied Mathematical Modelling , 90 (2021), 413-431.

Parameters	Values	Reference
Mutation rate	5%	⁸8 K. Deb, A. Pratap, S. Agarwal & T. Meyarivan. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE transactions on evolutionary computation, 6(2) (2002), 182-197.
Recombination rate	90%	⁸8 K. Deb, A. Pratap, S. Agarwal & T. Meyarivan. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE transactions on evolutionary computation, 6(2) (2002), 182-197.
Simulations	30	³⁶36 A.S. Vasconcelos , L.S. Silva, R.T. Cardoso & J.L. Acebal . Optimization of a rainfall dependent model for the seasonal Aedes aegypti integrated control: a case of Lavras/Brazil. Applied Mathematical Modelling , 90 (2021), 413-431.

\{\begin{cases} \frac{\partial A (x, y, t)}{\partial t} = κ_{1} (\frac{\partial^{2} A}{\partial x^{2}} + \frac{\partial^{2} A}{\partial y^{2}}) + ϵ ϕ (d) (1 - \frac{A}{C (r)}) F - (α (d) + μ_{A} (r, d) + u_{A}) A, \\ \frac{\partial F (x, y, t)}{\partial t} = κ_{2} (\frac{\partial^{2} F}{\partial x^{2}} + \frac{\partial^{2} F}{\partial y^{2}}) + γ α (d) A - (μ_{F} (d) + u_{F}) F, \end{cases}

\{\begin{cases} \frac{\partial \tilde{A} (x, y, t)}{\partial t} = κ_{1} (\frac{\partial^{2} \tilde{A} (x, y, t)}{{\partial x}^{2}} + \frac{\partial^{2} \tilde{A} (x, y, t)}{{\partial y}^{2}}), \\ \frac{\partial \tilde{F} (x, y, t)}{\partial t} = κ_{2} (\frac{\partial^{2} \tilde{F} (x, y, t)}{{\partial x}^{2}} + \frac{\partial^{2} \tilde{F} (x, y, t)}{{\partial y}^{2}}), \end{cases}

subject to : \{\begin{cases} \frac{\partial A (x, y, t)}{\partial t} = κ_{1} (\frac{\partial^{2} A}{\partial x^{2}} + \frac{\partial^{2} A}{\partial y^{2}}) + ϵ ϕ (d) (1 - \frac{A}{C (r)}) F - (α (d) + μ_{A} (r, d) + u_{A}) A, \\ \frac{\partial F (x, y, t)}{\partial t} = κ_{2} (\frac{\partial^{2} A}{\partial x^{2}} + \frac{\partial^{2} A}{\partial y^{2}}) + γ α (d) A - (μ_{F} (d) + u_{F}) F, \\ A_{x} (0, 0, t) = 0, A_{x} (L, L, t) = 0, t > 0, \\ A_{y} (0, 0, t) = 0, A_{y} (L, L, t) = 0, t > 0, \\ F_{x} (0, 0, t) = 0, F_{x} (L, L, t) = 0, t > 0, \\ F_{y} (0, 0, t) = 0, F_{y} (L, L, t) = 0, t > 0, \\ A (x, y, 0) = g_{1} (x, y), \\ F (x, y, 0) = g_{2} (x, y) \\ 0 ⩽ u_{A_{1}} ⩽ 0.50, \\ 0 ⩽ u_{A_{2}} ⩽ 0.50, \\ 0 ⩽ u_{F} ⩽ 0.21, \\ 2 ⩽ t_{A_{1}} ⩽ 20, \\ 30 ⩽ t_{A_{2}} ⩽ 50, \\ 2 ⩽ t_{F_{1}} ⩽ 50, \end{cases}

\{\begin{cases} \min_{u_{A}, u_{F}, t_{A}, t_{F}} J_{1} = C_{1} ∭_{Ω \times I} u_{A} d x d y d t + C_{2} ∭_{Ω \times I} u_{F} d x d y d t \\ \min_{u_{A}, u_{F}, t_{A}, t_{F}} J_{2} = C_{3} ∭_{Ω \times I} F d x d y d t \end{cases}

subject to : \{\begin{cases} \frac{\partial A (x, y, t)}{\partial t} = κ_{1} (\frac{\partial^{2} A}{\partial x^{2}} + \frac{\partial^{2} A}{\partial y^{2}}) + ϵ ϕ (d) (1 - \frac{A}{C (r)}) F - (α (d) + μ_{A} (r, d) + u_{A}) A, \\ \frac{\partial F (x, y, t)}{\partial t} = κ_{2} (\frac{\partial^{2} A}{\partial x^{2}} + \frac{\partial^{2} A}{\partial y^{2}}) + γ α (d) A - (μ_{F} (d) + u_{F}) F, \\ A_{x} (0, 0, t) = 0, A_{x} (L, L, t) = 0, t > 0, \\ A_{y} (0, 0, t) = 0, A_{y} (L, L, t) = 0, t > 0, \\ F_{x} (0, 0, t) = 0, F_{x} (L, L, t) = 0, t > 0, \\ F_{y} (0, 0, t) = 0, F_{y} (L, L, t) = 0, t > 0, \\ A (x, y, 0) = g_{1} (x, y), \\ F (x, y, 0) = g_{2} (x, y) \\ 0 ⩽ u_{A_{1}} ⩽ 0.50, \\ 0 ⩽ u_{A_{2}} ⩽ 0.50, \\ 0 ⩽ u_{F} ⩽ 0.21, \\ 2 ⩽ t_{A_{1}} ⩽ 20, \\ 30 ⩽ t_{A_{2}} ⩽ 50, \\ 2 ⩽ t_{F_{1}} ⩽ 50, \end{cases}

\{\begin{cases} A (x, y, 0) = 10 {(\sin (π (\frac{x - 1}{19 - 1}) (\frac{y - 1}{19 - 1})))}^{100}, if 1 \leq (x, y) \leq 19, \\ F (x, y, 0) = 10 {(\sin (π (\frac{x - 1}{19 - 1}) (\frac{y - 1}{19 - 1})))}^{100}, if 1 \leq (x, y) \leq 19, \\ A (x, y, 0) = F (x, y, 0) = 0, if 0 ⩽ (x, y) < 1 or 19 < (x, y) ⩽ 20 . \end{cases}

[1] *Corresponding author: Josenildo Silva de Lima - E-mail: nildo2802@gmail.com

Parameters	$u_{A_{1}}$	$t_{A_{1}}$	$u_{A_{2}}$	$t_{A_{2}}$	u _F	$t_{F_{1}}$	J	Ef. Values
	47%	7	46%	33	3%	31	2906	13%

Brasil

Brasil

A Multiobjective Optimization Application to Control the Aedes Aegypti Mosquito using a Two-Dimensional Diffusion-Reaction Model

ABSTRACT