Applied Mathematical Sciences, Vol. 12, 2018, no. 22, 1085 - 1097HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ams.2018.7111

A Model for Aedes aegypti Control (Diptera: culicidae)

with Resistance to Chemicals and Phytocompounds

Juan C. Jamboos T.

Departamento de Salud, El Colegio de la Frontera Sur (ECOSUR)San Cristóbal de las Casas-Chiapas, México

Oscar A. Manrique A., Steven Raigosa O., Dalia M. Muñoz P.,Anibal Muñoz L.,Valentina Zuluaga Z., Mauricio Ropero P.,

Francisco Betancourt B. and Hans Meyer C.

Grupo de Modelación Matemática en Epidemiología (GMME)Facultad de Educación, Maestría en Biomatemáticas

Facultad de Ciencias Básicas y Tecnologías, Universidad del QuindíoQuindío-Armenia, Colombia

Copyright c© Juan C. Jamboos T. et al. This article is distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduc-tion in any medium, provided the original work is properly cited.

AbstractIt is formulated a stability analysis of a mathematical model to in-

terpret the dynamics of the population growth including resistance tochemicals and phytocompounds. The threshold of the population growthof A. aegypti is determined. A sensitivity analysis and simulations ofthe model were developed. We conclude that the control focused in thenon-resistant mosquitoes lead to a decrease in the resistant mosquitoesas well as the immature stages.

Keywords: Aedes aegypti, population growth threshold, stability analysis,sensitivity analysis,insecticide resistance, resistance to chemicals

1 IntroductionAedes aegypti, is a mosquito which belongs to Culicidae family. Its prefer envi-ronments are tropical areas in Asia, Africa, the Pacific Latin America and the

1086 Juan C. Jamboos T. et al.

Caribbean [17]. Also, it is an arthropod insect what transmit diseases such as:Dengue, Chikungunya and Zika Virus among others, having great impact inPublic Health [19].

The number of notified cases of Dengue around world passed from 2,2 mil-lions in 2010 to 3,2 millions in 2015. In the Americas, in 2015 were notifiedto Panamerican Organization of Health (OPS) 693489 suspected cases fromwhich 37480 were confirmed. Most of those cases belongs to Colombia, with356079 suspected cases. This number is lower than presented in 2014 and in2016 continue decreasing the cases but to sorrow this tendency, Chikungunyafever currently threat the region and finally, 539791 cases of Zika Virus werereported in the Americas and Non-Latin Caribbean Islands [4].

Due to the above statements, it is important control the mosquito and somemechanisms are: Reduction of breeding places (favorable places for the ovipo-sition and development of the aquatic stages) [19], and information campaigns.

In the 1980s was observed a great increase in the A. aegypti infestation indexas the ones registered in the 50s and 60s [17]. For this reason, in the 1990s, toreduce the epidemic presence, several countries of Latin America applied dif-ferent insecticides, such as: temephos (which is used for the eradication of thelarvae in a focused treatment), fention, fenitrothion, malatration, delamethrinand permethrin (the last ones are used in the control of adult stages) [16].However, the mosquitoes and other species have developed resistance to someof these insecticides [17].

Resistance is the ability of a population of insects to tolerate insecticide dosesthat could be lethal to other individuals in the population. Due to this andsome other factors, such as: genetic (related to the frequency and dominanceof resistance alleles), biological (related to the life cycle, number of descen-dants per generation and rates of genetic flows) and operational (related totime, doses and formulation of the insecticides), the vectors have been able tospread to another regions around the word [19].

2 The modelWe set a model based in non-linear ordinary differential equations to explainthe dynamics of the A. aegypti growth, taking into account resistance to chem-icals (including insecticides) and phytocompounds. The variables of the modelare: x(t) : average number of immature stages (eggs, larvae and pupae), y(t) :average number of non-resistant mosquitoes, z(t) : average number of resistantmosquitoes at time t. The parameters of the model are: φ : mosquitoes ovipo-

A model for Aedes aegypti control (Diptera: culicidae) with ... 1087

sition rate, k : charge capacity of immature states, η : development rate fromimmature to adult stages, π : natural death rate of immature stages (eggs,larvae and pupae), ε : natural death rate of resistant and non-resistant adultmosquitoes, u : death fraction of non resistant adult mosquitoes by insecticideapplication, f : fraction of resistant immature stages which develop to resistantadult mosquitoes, g : fraction of eggs that hatch as female mosquitoes, 1− x

k:

regulatory fraction of immature stages.

gφ(y(t) + z(t))(1− x(t)k)

��

x(t)(1−f)ηx(t) //

��

fηx(t)

%%

y(t)

��

z(t)

��πx(t) (ε+ u)y(t) εz(t)

Figure 1: Diagram of the dynamics including resistance and control.

dx(t)

dt= gφ(y(t) + z(t))

(1− x(t)

k

)− (η + π)x(t) ≡ F (1)

dy(t)

dt= (1− f)ηx(t)− (ε+ u)y(t) ≡ G (2)

dz(t)

dt= fηx(t)− εz(t) ≡ H (3)

Where g, φ, k, η, π, ε > 0, 0 ≤ f < 1, 0 ≤ u < 1 and initial conditions x(0) =x0, y(0) = y0 y z(0) = z0.

3 Stability analysisFirst of all, we have calculated the equilibrium points with and without pop-ulations, those are determined by applying the variation of system 1 - 3 equalto zero and resolving the non-linear algebraic system for each variable of thepopulation, with x(t) = x, y(t) = y y z(t) = z:

0 = gφ(y + z)(1− x

k

)− (η + π)x

0 = (1− f)ηx− (ε+ u)y

0 = fηx− εz

1088 Juan C. Jamboos T. et al.

the following equilibrium points are determined E0 = (0, 0, 0) y E1 = (x, y, z),where;

x = k

(1− 1

ξ(u)

), y =

(1− f)ηk(ε+ u)

(1− 1

ξ(u)

), z =

fηk

ε

(1− 1

ξ(u)

)

ξ(u) =gφη(fu+ ε)

ε(η + π)(ε+ u)(4)

ξ(u) is the population growth threshold of mosquitoes with resistance and con-trol.

In the linearization process of the non-linear equations system (1)-(3) was cal-culated the Jacobian matrix at the generic equilibrium point E = (x∗, y∗, z∗).

J(E) =

Fx Fy FzGx Gy Gz

Hx Hy Hz

where, Fx = −gφ(y∗+z∗)

k− (η + π), Fy = gφ

(1− x∗

k

), Fz = gφ

(1− x∗

k

), Gx =

(1− f)η,Gy = (ε+ u), Gz = 0, Hx = fη,Hy = 0 y Hz = −ε.

Evaluating the Jacobian matrix at the equilibrium point E0,

J(E0) =

−(n+ π) gφ gφ(1− f)η −(ε+ u) 0fη 0 −ε

and the characteristic equation is given by

p(λ) = λ3 + a1λ2 + a2λ+ a3 = 0

from where

a1 = 2ε+ u+ η + π

a2 = ε2 + (u+ 2π + 2η)ε+ (u− gφ)η + πu

a3 = 1− ξ(u)

so, the main minors of the Hurwitz matrix are:

41 = a1, 42 =

[a1 1a3 a1

]43 =

a1 1 0a3 a2 a10 0 a3

following the Routh-Hurwitz criterion, the equilibrium point E0 is local andasymptotically stable if the following inequalities are satisfied a1 > 0, a3 > 0

A model for Aedes aegypti control (Diptera: culicidae) with ... 1089

for a1a2 > a3.

Now, evaluating the Jacobian matrix at the equilibrium point E1

J(E1) =

Fx Fy FzGx Gy Gz

Hx Hy Hz

p(λ) = λ3 + a1λ

2 + a2λ+ a3 = 0

with,

a1 = −(Fx +Gy +Hz)

a2 = (Gy +Hz)Fx +HzGy − (GxFy +HxFz +HyGz

a3 = (−FxGy + FyGx)Hz − FyGzHx + (−GxHy +GyHx)Fz + FxGzHy

then, if the main minors of the Hurwitz matrix

41 = a1, 42 =

[a1 1a3 a1

]43 =

a1 1 0a3 a2 a10 0 a3

for a1 > 0, a3 > 0 y a1a2 > a3, the equilibrium point E1 is local and asymp-totically stable, otherwise is unstable.

4 Numerical stability analysisThe numerical stability analysis is carried out using the parameter values inthe Table 1. Free of infection and prevalence equilibrium points were calcu-lated using the Maple software. In the linearization process are evaluated the

Table 1: Estimated parameters of the model.Par. g φ η π ε k u fVal. 0.6 4 0.904 0.096 0.1 5000 0 - 0.3 - 0.6 0 - 0.2 - 0.5 - 0.7

equilibrium points to determined their stability. By using the information inTable 1 and the previously calculated equilibrium points is carried out thelocal stability analysis. Moreover, using the Jacobian matrix are obtained theresult in Tables 2, 3 and 4

• for u = 0 and f = 0 we have

E0 = (0, 0, 0) , E1 = (4770, 43117, 0)

• for u = 0 and f = 0.2 we have

E0 = (0, 0, 0) , E1 = (4770, 34493, 8623)

1090 Juan C. Jamboos T. et al.

• for u = 0 and f = 0.5 we have

E0 = (0, 0, 0) , E1 = (4770, 21558, 21558)

• for u = 0 and f = 0.7 we have

E0 = (0, 0, 0) , E1 = (4770, 12935, 30182)

• for u = 0.2 and f = 0 we have

E0 = (0, 0, 0) , E1 = (4770, 43117, 0)

• for u = 0.2 and f = 0.2 we have

E0 = (0, 0, 0) , E1 = (4506, 10863, 8147)

• for u = 0.2 and f = 0.5 we have

E0 = (0, 0, 0) , E1 = (4654, 7013, 21038)

• for u = 0.2 and f = 0.7 we have

E0 = (0, 0, 0) , E1 = (4712, 4260, 29817)

• for u = 0.6 and f = 0 we have

E0 = (0, 0, 0) , E1 = (3387, 4374, 0)

• for u = 0.6 and f = 0.2 we have

E0 = (0, 0, 0) , E1 = (4267, 4408, 7714)

• for u = 0.6 and f = 0.5 we have

E0 = (0, 0, 0) , E1 = (4597, 2968, 20777)

• for u = 0.6 and f = 0.7 we have

E0 = (0, 0, 0) , E1 = (4690, 1817, 29677)

f equilibrium point eigenvalues stability0 (0,0,0) (-0.1, -2.09, 0.99) unstable0 (4769.54, 43116.66, 0) (-0.1, -21.7, -0.09) stable0.2 (0,0,0) (-0.1, -2.09, 0.99) unstable0.2 (4769.54, 34493.33, 8623.33) (-0.1, -21.7, -0.09) stable0.5 (0,0,0) (-0.1, -2.09, 0.99) unstable0.5 (4769.54, 21558.33, 21558.33) (-0.1, -21.7, -0.09) stable0.7 (0,0,0) (-0.1, -2.09, 0.99) unstable0.7 (4769.54, 12935, 30181.66 ) (-0.1, -2.09, 0.99) stable

Table 2: Local stability analysis for u = 0 and different values of f .

f equilibrium point eigenvalues stability0 (0,0,0) (-0.1, -2.09, 0.99) unstable0 (4769.54, 43116.66, 0) (-0.1, -2.09, -0.09) stable0.2 (0, 0, 0) (-0.10, -2.09, 0.99) unstable0.2 (4506.16, 10862.85, 8147.14) (-0.10, -10.14, -0.07) stable0.5 (0, 0, 0) (-0.10, -2.09, 0.99) unstable0.5 (4654.31, 7012.50, 21037.50) (-0.10, -14.47, -0.08) stable0.7 (0, 0, 0) (-0.10, -2.09, 0.99) unstable0.7 (4711.92, 4259.58, 29817.08) (-0.10, -17.36, -0.09) stable

Table 3: . Local stability analysis of the system for u = 0.2 and differentvalues of f .

A model for Aedes aegypti control (Diptera: culicidae) with ... 1091

f equilibrium point eigenvalues stability0 (0,0,0) (-0.1, -2.09, 0.99) unstable0 (3386.79, 4373.80, 0) (-0.1, -3.31, -0.117) stable0.2 (0, 0, 0) (-0.10, -2.09, 0.99) unstable0.2 (4266.72, 4408.13, 7714.24) (-0.10, -6.86, -0.052) stable0.5 (0, 0, 0) (-0.10, -2.09, 0.99) unstable0.5 (4596.69, 2968.15, 20777.08) (-0.10, -12.41, -0.085) stable0.7 (0, 0, 0) (-0.10, -2.09, 0.99) unstable0.7 (4689.76, 1816.95, 29676.85) (-0.10, -16.12, -0.091) stable

Table 4: Local stability analysis of the system for u = 0.6 and different valuesof the f .

5 Sensitivity analysis of the mosquito popula-tion growth threshold

The local sensitivity is a relative measure of the change in a variable whenthe parameter changes [11, 12, 13]. That index for the threshold is calculatedemploying the next expression

Iξp =∂ξ

∂p× p

ξ

where, p is a parameter, then, the following indexes are obtained

Iξg = 1

Iξφ = 1

Iξη =π

π + η

Iξπ = − π

π + η

Iξε =−2εfu− fu2 − ε2

(ε+ u)(fu+ ε)

the results of the sensitivity analysis of the mosquito population growth thresh-old are shown in Table 5. Due to that the sensitivity index respecting to pa-

Table 5: Sensitivity index of the mosquito population growth.parameter g φ η π

Iξp 1 1 0.776 -0.776

rameter ε depends on u and f , for that parameter we have the next index:

1092 Juan C. Jamboos T. et al.

Table 6: Sensitivity index of the mosquito population growth with respect tothe parameter ε.

(u, f) (0, 0) (0, 0.2) (0, 0.5) (0, 0.7) (0.2, 0) (0.2, 0.2) (0.2, 0.5) (0.2, 0.7) (0.6, 0) (0.6, 0.2) (0.6, 0.5) (0.6, 0.7)Iξε (u, f) -1 -1 -1 -1 -0.33 -0.61 -0.83 -0.91 -0.14 -0.68 -0.89 -0.95

As observed in Table 5, the population growth threshold of the mosquito isproportionally greater to the increase of the eggs fraction which hatch as fe-male mosquitoes (g) and to the increase of the mosquitoes oviposition rate (φ).

Additionally, from Table 5 and 6, if the natural death rate of the resistant andnon-resistant adult mosquitoes increases, the epidemic threshold decreases,this effect shows that a control policy focused in the increase of the mortalityof resistant and non-resistant adult mosquitoes drives to the reduction of thetotal mosquitoes population.

Figure 2: Behavior of the mosquito population growth threshold as functionof ε with u = 0.6 and f = 0.7.

In Figure 2 the population growth threshold of the mosquitoes as function ofε is depicted, where it is evident that an increase in ε leads to a reduction inthe mosquito population growth threshold, then the mortality rate of adultmosquitoes directly affects to the threshold behavior.

A model for Aedes aegypti control (Diptera: culicidae) with ... 1093

6 Simulations

The simulations of the system were done using the MAPLE software with theparameter values of Table 1 and the next initial conditions x(0) = 500, y(0) =200 y z(0) = 100. In Figure 3 it the behavior of different stages of the

Figure 3: Population behavior for f = 0 and u = 0, 0.3, 0.6.

mosquitoes contemplated in the model (x(t), y(t) y z(t)) is observed with differ-ent value for u and the resistance fraction of the immature stages, which drivesto resistant adult mosquitoes equal to 0% resulting in z(t) being extinguishedfor this reason (f = 0), on the other hand, it is observed that the increase in thecontrol (u) affects directly to the average number of non-resistant mosquitoesand consequently inferring in the average number of immature stages, thoseones decrease considerably. In Figure 4 the resistant mosquitoes to the insec-ticide grow quickly and stabilize in an average of 7500 to 8500 with a 20%of resistance is shown, while the non-resistant population of adult mosquitoesstabilizes quickly considering control with insecticide and grow rapidly withoutcontrol. The immature stages stabilize in an average of 10 days, resulting in areduction of 100 to 500 immature stages per control.

Figure 4: Behavior of the populations for f = 0.2 and u = 0, 0.3, 0.6.

1094 Juan C. Jamboos T. et al.

Figure 5: Behavior of the populations for f = 0.5 and u = 0, 0.3, 0.6.

Figure 6: Behavior of the populations for f = 0.7 and u = 0, 0.3, 0.6.

In Figure 5 it can be seen that if the resistance increases the control and theimmature stages population gently decrease, the population of non-resistancemosquitoes noticeably decreases with each control stabilizing in an average of15 days considering control with insecticides and exponentially growing with-out consider it. Also, the resistant mosquitoes population increases quicklyand stabilizes in an average of 55 days, if 21000 individuals are considered.

Figure 6 depicts the behavior of the average number of immature stages (eggs,larvae, pupae) (x(t)), the average number of non-resistant mosquitoes (y(t))and the average number of resistant mosquitoes (z(t)), for different scenariosof u and taking into account the resistance fraction of immature stages, whichproduces resistant adult mosquitoes equal to 70%. From where, it is possible tonote that with a better control (u) the number of adult mosquitoes decreases.This effect is evident in the average number of non-resistant mosquitoes aftercontrol.

A model for Aedes aegypti control (Diptera: culicidae) with ... 1095

7 Conclusions

• The epidemic threshold of population decrease not only by the naturalmortality of mosquitoes but also by the oviposition decreasing and thenumber of female mosquitoes grow up from the hatching.

• In the same way, the population growth threshold strongly decreaseswhen the natural mortality of adult mosquitoes appears. Then, thisthreshold tends to be asymtotically stable in the time while mortalityincreases, as shown in figure 2.

• On the other hand, the behavior of each population in the time, em-phasize the sensitivity of non resistance mosquito population which fallsoutstandingly with a low control, that means the susceptible popula-tion with low concentrations of chemicals and/or phytocompounds, iscontrolled, while a higher concentration does not display an importantincrease of mortality.

In further researches, we suggest to consider some parameters closely relatedto genetical changes induced by spatial distribution of A. aegypti, the effectof insecticide applications in low concentrations and doses and the synergybetween these with alternative of control methods.

Acknowledgements. Juan Carlos Jamboos Toledo, thanks Consejo Na-cional de Ciencia y Tecnología - CONACyT of Mexico, for supporting hisdoctoral studies while the present publication was made. Thanks to Grupo deModelación Matemática en Epidemiología (GMME), Facultad de Educación,Universidad del Quindío - Colombia.

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Received: January 25, 2017; Published: September 4, 2018