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Mathematical model to assess the control of Aedes aegypti

mosquitoes by the sterile insect technique

Lourdes Estevaa,*, Hyun Mo Yangb

aDepartamento de Matema ´ticas, Facultad de Ciencias, UNAM 04510 Me ´xico, D.F., Mexico

bUNICAMP – IMECC, Departamento de Matema ´tica Aplicada, Caixa Postal 6065, CEP: 13081-970,

Campinas, SP, Brazil

Received 13 December 2004; received in revised form 15 April 2005; accepted 15 June 2005

Available online 25 August 2005

Abstract

We propose a mathematical model to assess the effects of irradiated (or transgenic) male insects intro-

duction in a previously infested region. The release of sterile male insects aims to displace gradually the

natural (wild) insect from the habitat. We discuss the suitability of this release technique when applied

to peri-domestically adapted Aedes aegypti mosquitoes which are transmissors of Yellow Fever and Dengue

disease.

? 2005 Elsevier Inc. All rights reserved.

Keywords: Mathematical modeling; Genetic control; Sterile insect release; Aedes aegypti; Threshold; Break-point

1. Introduction

Diverse disciplines of science as radiation biology, chemistry, ecology, molecular biology,

genetics and entomology have contributed to the control of insect pests. In this context, Knipling

[1] had conceived an approach to insect control in which the natural reproductive processes of

insects are disrupted by the use of mutagens such as gamma radiation thus rendering the insects

0025-5564/$ - see front matter ? 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.mbs.2005.06.004

*Corresponding author. Tel.: +52 55 56224858; fax: +52 55 56224859.

E-mail addresses: lesteva@lya.fciencias.unam.mx (L. Esteva), hyunyang@ime.unicamp.br (H. Mo Yang).

www.elsevier.com/locate/mbs

Mathematical Biosciences 198 (2005) 132–147

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sterile. These sterile insects are then released into the environment in very large numbers in order

to mate with the native insects that are present in the environment. A native female that mates

with a sterile male will produce eggs, but the eggs will not hatch (the same effect will occur for

the reciprocal cross). If there is a sufficiently high number of sterile insects then most of the crosses

are sterile, and as time goes on, the number of native insects decreases and the ratio of sterile to

normal insects increases, thus driving the native population to extinction. This insect control

method is now known as the sterile insect release method (SIT) [2].

Recently, techniques like genetic control have been considered, which consist of genetic manip-

ulation to produce transgenic insects in order to result in sterility or lethal (or deleterious) genes

[2]. The introduction of deleterious genes into the population via sperm that actually fertilizes the

egg has the advantage of being independent of the number of matings. This in turn avoids the

massive release of sterile males according to SIT.

In the traditional insecticide control, the amount of applied pesticides and the subsequent cost

are generally proportional to the area treated and not to the size of the population. However, the

damage done by a pest is proportional to the number of individual pest per unit area. Thus, the

cost/benefit ratio of insecticidal control increases as the size of the pest population increases. But,

in sterile male release programs, the number of sterile male released and their cost are propor-

tional to the size of the population. Since the cost/benefit ratio decreases with a decrease in size

of the pest population, it can be wise to integrated insecticidal methods with sterile male

technique.

Sterile male techniques were first used successfully in 1958 in Florida to control Screwworm fly

(Cochliomya omnivorax) [3,4]. About 50 million flies of both sexes were released per week over an

18 months period, in a total of 2 billion flies over 85000 square mile area. The pest was eradicated

after this period of application, in which 40 tons of ground meat were required each week and 20

aircraft to release the sterile flies. The total cost was about US$ 10000000.00.

Although the screwworm eradication program is the most successful use of SIT, other insect

species have been subjected to the release of sterile insects with varying success. Some examples

are screwworm fly in USA, Mexico and Libya; Mediterranean Fruit Fly (Ceratitis capitata Wiede-

mann) in USA and Mexico; Melon Fly (Dacus cucurbitae Coquillett) in Japan and Taiwan; Pink

Bollworm (Pectinophora gossypiella Saunders) in USA; Tsetse Fly (Glossina species) in Tanzania,

Zimbabwe and Upper Volta; Boll Weevil (Anthonomus grandis Boheman) in Southeastern USA;

Mexican Fruit Fly (Anastrepha ludens Loew) in USA and Mexico; Gypsy Moth (Lymantria dispar

Linnaeus) in USA and Canada [2].

The reasons for success in some cases and little or not success in others are due to limitations on

the technique. These limitations include lack of competitive ability of sterile insects, lack of com-

plete sterilization, insufficient sterile population, inadequate spraying, multiple matings, etc. A

number of mathematical models have been done to assist the effectiveness of the SIT (see, e.g.,

[1,5–10]). Some of these models contemplate combination of SIT with other control measures

as pesticides [11] or release of parasitoids [12].

Much research was carried out about 30 years ago, especially in India and El Salvador, on the

application of SIT to mosquitoes. Unfortunately this technique virtually ends in the mid-1970s,

not because the method was a technical failure, but because of political problems in India and

intensifying civil wars in Central America [13]. There is now a revival of interest especially in

the use of transgenesis to improve sex separation so that only non-biting males are released

L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

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and to ensure that their female progeny die without the need for radiation or chemosterilization

[14]. Therefore, with some improvements through the use of currently available transgenic tech-

nologies, SIT could become a mainstay for public health control of specific vector-borne diseases.

In this paper we are concerned about the application of the SIT for the control of Aedes aegypti

mosquitoes, which is the principal transmissor of Yellow Fever and Dengue disease. This pest spe-

cies have more than one life stage, which is known that complicates the outcome of a sterile re-

lease program [15]. In order to shed further light on the effectiveness of the technique for control

of mosquitoes we propose a general model that incorporates two life stages of the pest population.

A question to be addressed is how the recruitment of the sterile males affects the efficiency of the

SIT. We also consider two issues of mosquito behavior that are specially relevant to SIT, namely,

(a) mating competitiveness of artificially reared sterilized males; (b) dispersal ability to ensure that

sterile males are released near enough to all emerging wild females so that they have a fair chance

of mating with them.

2. The model

Aiming the description of the dynamics of biological control, the life cycle of an insect is di-

vided in two stages: the immature (eggs, larvae and pupae) and the adult one. In the case of mos-

quitoes the immature phase occurs in water. We denote by A the population size of the immature

phase of the insect at time t. For the adult form we consider the following compartments: females

before mating (singles), I; mating fertilized females, F; mating unfertilized females, U; and male

insects, M. The population size of sterile (irradiated or transgenic) insects at time t is denoted

by MT.

The per capita mortality rates of the immature form, unmating females, mating fertilized fe-

males, mating unfertilized females, natural (or wild) and sterile male insects are denoted by lA,

lI, lF, lU, lMand lT, respectively.

The net oviposition rate per female insect is proportional to their density, but it is also regulated

by a carrying capacity effect depending on the occupation of the available breeder sites. In this

model we assume that the per capita oviposition rate is given by / 1 ?A

capacity related to the amount of available nutrients and space, and / is the intrinsic oviposition

rate. The immature population becomes adult insects at a per capita rate c; a proportion r of such

are females and 1 ? r, males.

Flows from I to F and U compartments depend mainly on the number of encounters of females

with native and sterile males, and on the correspondingly mating rates. Here we assume that the

probability of an encounter of a female with natural insects is given by

rate at which female insects are fertilized is

Since sterile (irradiated or transgenic) insects are placed artificially, it is natural to think that the

probability of an encounter of a sterile male with a female depends not only on the number of

such males

MþMT

that this net probability is given by

of sterile insects that are sprayed in the adequate places. Also, the effective fertilization during the

mating could be diminished due to the sterilization, which leads us to assume that the effective

mating rate of sterile insects is given by qb, with 0 6 q 6 1. Putting together the assumptions

C

??, where C is the carrying

M

MþMT. Then the per capita

bM

MþMT, where b is the mating rate of natural insects.

MT

??

, but also on how far they are placed from the breeding sites. We will assume

pMT

MþMT, where the parameter p, with 0 6 p 6 1, is the proportion

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L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

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above we get that

sperm, where bT= pqb. In some extent, the parameter p is related to the effectiveness of sterile

male introduction regarded to the spatial distribution of female insects, while q can be thought

as a physiological modification induced by the sterilization technique. The parameter p plays

an important role in the case of insects whose reproduction depends strongly on the spatially

non-homogeneous distribution of breeding sites, like mosquitoes A. aegypti.

Finally, we assume that sterile insects are recluted and sprayed at a constant rate denoted by a.

According to the assumptions above the model is given by

A0¼ / 1 ?A

C

bTMT

MþMTis the per capita rate at which female insects are fecundated with sterile

??

F ? ðc þ lAÞA;

I0¼ rcA ?

bMI

M þ MT

bMI

M þ MT

M0¼ ð1 ? rÞcA ? lMM;

M0

?bTMTI

M þ MT

? lII;

F0¼? lFF;

T¼ a ? lTMT

ð1Þ

and the remaining decoupled equation for mating unfertilized females is

bTMTI

M þ MT

U0¼? lUU:

3. Equilibrium points

The population of sterile insects approaches the equilibriuma

lT, independently of the initial con-

?

satisfy the following relations:

ditions. Then, system (1) has a trivial equilibrium P0¼ 0;0;0;0;a

to the state where natural insects are absent, and there is only a constant population of sterile in-

sects. The non-trivial steady states

A;?I;F;M;a

lT

?

ðlIþ bÞM þ ðlIþ bTÞa

F ¼ðc þ lAÞCA

/ðC ? AÞ

M ¼ð1 ? rÞcA

lM

where A is a solution of the second degree equation

lT

?

, with U ¼ 0, corresponding

??

?I ¼

rcA M þa

lT

?

lT

;

;

;

ð2Þ

pðAÞ ¼ aA2þ bA þ c ¼ 0;

ð3Þ

L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

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with coefficients

a ¼1

C

/rcb

ðc þ lAÞðb þ lIÞlF

b ¼ 1 ?

ðc þ lAÞðb þ lIÞlF

c ¼

ðb þ lIÞð1 ? rÞclT

The remaining decoupled mating unfertilized females is given by

abT?I

lUðlTM þ aÞ:

From the expression for F it follows that the positive non-trivial equilibrium solutions must sat-

isfy 0 < A < C. Now, at the endpoints of this interval

;

/rcb

;

ðbTþ lIÞlMa

:

U ¼

pð0Þ ¼ c > 0;

pðCÞ ¼ C þ c > 0;

dp

dAðCÞ ¼

It follows that p(A) has one or two roots inside the interval (0,C) if and only if (i)dp

(ii) b2? 4ac > 0. Let us define

R ¼

ðlAþ cÞðb þ lIÞlF

and

/rcb

ðlAþ cÞðb þ lIÞlF

þ 1 > 0:

dAð0Þ < 0 and

/rcb

ð4Þ

S ¼

ðbTþ lIÞlMa

ðb þ lIÞð1 ? rÞcClT

Then, Eq. (3) can be written as

pðAÞ ¼R

and the conditions for biological existence of the non-trivial equilibria become

R > 1

:

ð5Þ

CA2? ðR ? 1ÞA þ CS ¼ 0

ð6Þ

ð7Þ

and

S 6ðR ? 1Þ2

4R

? Sc:

ð8Þ

When both conditions hold, the solutions are given by

A?¼ðR ? 1Þ

2R

C 1 ?

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðR ? 1Þ2

1 ?

4RS

s

"#

ð9Þ

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L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

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and

Aþ¼ðR ? 1Þ

2R

C 1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðR ? 1Þ2

1 ?

4RS

s

"#

:

ð10Þ

Therefore, under conditions (7) and (8), system (1) has two positive equilibria P1?and P1þ, cor-

respondingly to A?and Aþ. If equality holds in (8) then P1?and P1þcollapse to an equilibrium P1

with A ¼ðR??1Þ

equilibria given by

2R? C, which provides minimum R*that guarantees the existence of the non-trivial

R?¼ ð1 þ 2SÞ 1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 þ 2SÞ2

1 ?

1

s

"#

:

ð11Þ

Notice that R*> 1.

The parameter R can be interpreted biologically as follows: since

vival of the immature phase of the insect,1

the fraction of immature forms that become females, then

succeed to become a female insect. Arguing in the same way,

will be fertilized, and finally

Thus, the product of these three quantities, which is equal to R, is the average number of second-

ary female insects produced by a single female insect. In demographic terms R is the basic offspring

number of insect population (equivalent to basic reproductive number in the epidemiological con-

text). For natural (or wild) insects to maintain in nature, condition R > 1 is necessary. However, it

is reasonable to assume that, in the presence of sterile insects, this could not be a sufficient con-

dition since a proportion of the females are not fertilized.

The expression R can be rewritten as R ¼

/th¼ðlaþ cÞðb þ lIÞlF

rcb

is the threshold oviposition rate. Hence, for / > /thwe have the infestation of mosquito popula-

tion at a level that depends on /; otherwise, we have the mosquito population going to extinction.

The model considers a carrying capacity, which limits the mosquito population. Hence, we have

the limiting threshold oviposition rate /th¼ðlaþcÞlF

/th! 1 and R < 1 for finite oviposition rate.

The parameter S can be written as the ratio between s1and s2, where

s1¼ ðbTþ lIÞrcC

lI

lT

The quantities s1and s2represent the maximal mating rates among female insects and, respec-

tively, sterile and natural male insects; then S measures the number of mated but not fertilized

female insects with respect to the fertilized ones. Also, when b = bT, S measures the ratio between

the number of sterile males and the number of natural insects in equilibrium. If S is sufficiently

high (S P Sc), the next generation of wild insects would be lower than the actual one since a pro-

portion of eggs would not hatch. If sterile male insects are sprayed for a long period of time, this

pattern would drive the natural insect population to zero.

1

cþlAis the average time of sur-

cis the average time of its permanence as such, and r is

rc

cþlAis the probability that an egg will

b

bþlIis the probability that a female

lFis the average number of eggs oviposited by each fertilized female.

/

/

/th, where

;

rc

for b ! 1. Note that when b ! 0, we have

a

and

s2¼ ðb þ lIÞrcC

lI

ð1 ? rÞcC

lM

:

L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

137

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Therefore, whenever S P Sc, we have the displacement of the natural insects. The critical num-

ber of mated but not fertilized insects with respect to the fertilized ones Sc, increases with the basic

offspring number R, with bounds 0;R

4

?

bigger during the rainy season than in the dry season. Then, the number of mosquitoes that

should be sprayed, and consequently the cost of SIT will depend heavily upon the time of the year

when this control technique is applied.

?, for R = 1 and R ? 1 (for / ? 1), respectively. In the case

of mosquito population, the value of R has great variations during the year, being much more

4. Stability of equilibria

In this section we analyze conditions for stability of the equilibrium points. For this, we calcu-

late the eigenvalues regarded to the Jacobian of system (1), given by

J ¼

?/

CF ? ðlAþ cÞ

rc

0

/ 1 ?A

C

??

00

?lI?bMþbTMT

bM

MþMT

0

0

MþMT

0

ðbT?bÞMT?I

ðMþMTÞ2

bMT?I

ðMþMTÞ2

?lM

0

ðb?bTÞM?I

ðMþMTÞ2

?

0

?lT

0

?lF

0

0

bM?I

ðMþMTÞ2

ð1 ? rÞc

0

0

B

B

B

@

B

B

B

B

B

B

1

C

C

C

A

C

C

C

C

C

C

;

ð12Þ

evaluated at the equilibrium points. The eigenvalues of J at the equilibrium P0= (0,0,0,0,a/lT)

are ?(c + lA), ?(b + lI), ?lF, ?lM, and ?lT; therefore P0is always locally asymptotically stable.

Global stability can be proved for R 6 1 using the function V : R5

rc

lAþ cA þ I þ

whose orbital derivative

rc/

ðlAþ cÞC

is less or equal to zero for R 6 1. From inspection of system (1) it can be seen that the maximal

invariant set contained in_V ¼ 0 is P0. Then, from La-Salle Lyapunov Theorem [16], P0is globally

asymptotically stable for R 6 1.

Now, we will analyze the stability of the equilibria P1?and P1þ. For these points it is clear that

?lTis an eigenvalue of the Jacobian. To obtain the characteristic equation for the other four

eigenvalues, we use the identities

þ! R given by

V ¼

/rc

ðlAþ cÞlF

F;

ð13Þ

_V ¼ ?

??

AF ?ðb þ lIÞ

M þ MTð1 ? RÞIM ?ðbTþ lIÞIMT

M þ MT

;

ð14Þ

/F

C

rc/ 1 ?A

ð1 ? rÞc/ 1 ?A

þ lAþ c ¼ ðlAþ cÞ

?

?

C

C ? A;

¼ lFðlAþ cÞ lIþbM þ bTMT

bMT?I

ðM þ MTÞ2¼lFlMðlAþ cÞMT

C

?

bM

M þ MT

?

M þ MT

??

;

C

M þ MT

;

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L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

Page 8

derived from system (1) in equilibrium. After some manipulation it can be seen that the eigen-

values are the roots of the polynomial

k4þ a1k3þ a2k2þ a3k þ a4;

ð15Þ

with

a1¼ ðlAþ cÞ

C

C ? Aþ lIþbM þ bTMT

C

C ? A

lIþbM þ bTMT

M þ MT

a3¼ lIþbM þ bTMT

M þ MT

M þ MT

þ lFþ lM;

a2¼ ðlAþ cÞ

??

lIþbM þ bTMT

M þ MT

?

?

CM þ MTA

ðC ? AÞðM þ MTÞ

þ lFþ lM

??

þ

?

lFþ lM

ð Þ þ lFlM;

?

lFðlAþ cÞ

A

C ? A

?

??

þ lMðlAþ cÞ

C

C ? A

??

þ lFlM

??

þ ðlAþ cÞlFlM

?

;

a4¼lFlMðc þ lAÞðbTþ lIÞa

ðM þ MTÞðC ? AÞlTCSðA

By the Routh–Hurwitz criteria, the roots of a polynomial of order four have negative real parts if

and only if ai> 0, with i = 1,...,4, and ða1a2? a3Þa3> a2

(15) the coefficients a1, a2and a3are positive. Moreover, after some tedious calculations, it can

be seen that the last condition is fulfilled for all positive A. Therefore, the stability of both

P1?and P1þis given by the sign of the coefficient a4, which is positive if and only if

sðAÞ ¼ A

is greater than zero. Notice that sðAÞ has a unique positive root A ¼ A?¼ CS ?1 þ

from (8), it can be seen that it satisfies the inequality

A?>2CS

R ? 1:

Now, we evaluate the polynomial (6) at A*. Using (16), and inequalities (8) and (17), we obtain

pðA?Þ ¼R

< ?ð2RS þ R ? 1Þ2CS

4RS

ðR ? 1Þ2? 1

according to the biological existence conditions (7) and (8). The inequality p(A*) < 0 implies

A?< A?< Aþ due to the positively defined coefficient of A2. Since sðAÞ is such that sðAÞ < 0

2þ 2CSA ? C2SÞ:

1a4. It is clear that for the polynomial

2þ 2CSA ? C2S;

ð16Þ

and,

ffiffiffiffiffiffiffiffiffiffi

1 þ1

S

q

??

ð17Þ

Cð?2CSA?þ C2SÞ ? ðR ? 1ÞA?þ CS ¼ ?ð2RS þ R ? 1ÞA?þ ðR þ 1ÞCS

R ? 1þ ðR þ 1ÞCS ¼ ?4RCS2

¼ ðR ? 1ÞCS

R ? 1þ ðR ? 1ÞCS

!

< 0;

L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

139

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for A < A?and sðAÞ > 0 for A > A?, we have a4< 0 for A?and a4> 0 for Aþ. Therefore P1?is

always unstable and P1þ, locally asymptotically stable.

We summarize the above in the following:

?

P1þ, stable.

According to Theorem 1, for a fixed R > 1, if S is above Sc¼ðR?1Þ2

insects by sterile male release, independently of their initial population size.

Let us discuss the above results in terms of the equilibrium points and their respective stability

regions. In order to do this we analyze the number of immature phase in equilibrium A as a func-

tion of the basic offspring number R and the recruitment rate a.

In the bifurcation diagram given by Fig. 1, the trivial equilibrium P0is represented by the

R-axis. The stability of this point is global for 0 6 R < R*, and local for R > R*, where R*is

given by Eq. (11). When R = R*, the turning equilibrium point P1appears, which is given by

P1¼ A;?I;F;M;a

A above. For R > R*and a > 0, P1bifurcates to the non-trivial equilibrium points P1?and P1þ,

which are unstable and locally stable, respectively. We call R*the threshold value since it sepa-

rates the region where we have only sterile insects (R < R*) from the region where natural and

sterile mosquitoes coexist at two different levels (R > R*).

The threshold value R*depends on S, which is a linear function of a, according to Eq. (5), with

S(0) = 0 and S(a ! 1) ! 1. Therefore, the bounds for the threshold value are R*(0) = 1 and

R*(S ! 1) ! 1.

Theorem 1. The equilibrium P0¼ 0;0;0;0;a

ðR?1Þ2

lT

?

of the system (1) is always stable. When R > 1 and

4RS> 1, the non-trivial equilibria, P1?and P1þ, are feasible. In this case P1?is always unstable and

4R

then it is possible to control

lT

??

, where A ¼ðR??1Þ

2R? C, and?I, F and M are obtained from Eq. (2) substituting

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2040 6080100120

A/C

R

R*

P1+

P1_

P0

P1

Fig. 1. Bifurcation diagram of system (1) with respect to R. The value R*= 12.16 corresponds to S = 2.56 in Eq. (11).

140

L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

Page 10

Notice that in the absence of sterile insects (a = 0), the condition for existence of natural insects

is R > 1. If this is not fulfilled, then we have the eradication of the insects, P0= (0,0,0,0,0). At

R = 1 we have the bifurcation value (see bifurcation diagram given in Fig. 2). However, when

the sterile insects are released (a > 0), the threshold value situates at R = R*(>1), indicating that

one of the conditions to the existence of natural insects is furnished by increasing the basic off-

spring number. This is due to the decreasing in the mating fertilized females by the sterile male

insects resulting in diminishing the net production of offsprings.

Now, let us consider the interval R > R*. Recall that in this interval we have, besides the equi-

librium point P0the equilibrium points P1?and P1þwhich are represented respectively by the

lower and upper branches of the parabola in Fig. 1. We call break-point the unstable equilibrium

point P1?¼ A?;?I?;F?;M?;a

by A?given by Eq. (9). The small root A?of Eq. (6) forms the decreasing branch of the poly-

nomial?s solution, which has the maximum value at R = R*A ¼ðR??1Þ

tote A ¼ 0, when R ! 1. For R > R*this decreasing branch separates two attracting regions

containing one of the equilibrium points P0and P1þ. In other words, we have a hyper-surface gen-

erated by the coordinates of the equilibrium point P1?, e.g., f A?;?I?;F?;M?;a

one of the equilibrium points P0and P1þis attractor depending on the relative position of the ini-

tial conditions supplied to the dynamical system (1) with respect to the hyper-surface. The term

?break-point? was used by Macdonald to refer to the critical level for successful introduction of

infection in terms of unstable equilibrium point [17].

For fixed R > 1, the bifurcation diagram with respect to a is illustrated in Fig. 3. When a = a*,

where a*is the value for which (R ? 1)2/4RS = 1, or equivalently

lT

??

, where?I?, F?and M?are obtained from Eq. (2) substituting A

2R? C

??

and tends to the asymp-

lT

??

¼ 0, such that

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

02468 10

A/C

R

P0

P1

Fig. 2. Bifurcation diagram of system (1) with respect to R when a = 0.

L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

141

Page 11

a?¼ðb þ lIÞð1 ? rÞclTC

ðbTþ lIÞlM

P1?and P1þcollapse to the equilibrium P1, and become infeasible for a > a*. In the last case P0

is globally asymptotically stable. Then, eradication of the pest is obtained beyond the threshold

value a*, disregarding the initial conditions. However, for a less than a*control of the pest is

not necessarily reached since the outcome will depend on the initial conditions. The same figure

shows that for small a, the attracting region of P1þincreases.

ðR ? 1Þ2

4R

:

ð18Þ

5. Numerical results

In this section we present some numerical results for the model. The values of the parameters

are given in Table 1. The vital variables /, lA, lI, lF, lM, c and r are according with values re-

ported in the literature for A. aegypti mosquitoes (e.g., [18]), and will remain fixed otherwise indi-

cated. We assume thatbT

due to physiological modifications and failures related to spatial distribution. Finally, we assume

b¼ 0:7, which means that sterile males loss 30% of the mating capacity

0

0.2

0.4

0.6

0.8

1

0 20406080

α

100 120 140160

A/C

P1+

P1-

P1

α∗

Fig. 3. Bifurcation diagram of system (1) with respect to a when R > R*. The value of a*in this diagram is ?145

mosquitoes/day according to values listed in Table 1 of Section 5 and assuming C = 600.

Table 1

Parameter values for model (1)

/lA

lI

lF

lU

lM

lT

c

r

b

50.05 0.05 0.050.05 0.1 0.10.0750.51

Units are days?1except for r.

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L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

Page 12

that the ratio

greater than S. According to the analytical results this implies that P0and P1þare locally stable,

therefore success of sterile male insects release will depend strongly on the initial population size

of the different insects classes.

The bifurcation diagram of Fig. 1 suggests that the unstable branch corresponding to

P1?¼ A?;?I?;F?;M?;a

conditions Að0Þ;?I?;F?;M?;a

P0(resp. P1þ).

The above is confirmed by the numerical simulations as it is illustrated in Fig. 4. This figure

shows the temporal course of the proportion A/C of the trajectories with initial conditions

Að0Þ;?I?;F?;M?;a

Numerical results show that the same behavior is obtained when we let A?fixed and move

another coordinate of P1?slightly above or below its value.

We also observe that the region of stability of P0decreases in a non-linear way with respect to

R. This is illustrated in Figs. 5 and 6. In Fig. 5 we see that for R = 21.16 the initial conditions

Að0Þ;0;0;0;a

cated (R = 42.32) only the trajectories with A(0)/C < 0.1 approach P0(see Fig. 6).

When S decreases the region of stability decreases even more faster as it is shown in Fig. 7. Here

the value of R is as in Fig. 6 and S = 1.28.

Summarizing, numerical simulations indicate that when the equilibria P0and P1þcoexist, the

stability region of the first one is rather small compared with the corresponding region of P1þ.

In this situation, SIT would have a low chance to be successful, unless the immature population

C

a=lTis 0.8. For these set of parameters R = 21.16, S = 2.56 and Sc= 4.79, which is

lT

??

separates the regions of stability of P0and P1þ. More explicity, initial

?

lT

?

with Að0Þ < A?(resp. Að0Þ > A?) are in the basin of attraction of

lT

??

, where Að0Þ ¼ 0:999 ? A?and Að0Þ ¼ 1:001 ? A?, respectively.

lT

??

with A(0)/C < 0.4 are in the basin of attraction of P0. When the size of R is dupli-

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 200400 6008001000

A/C

days

a

b

Fig. 4. Numerical simulations of system (1): (a) Að0Þ ¼ 0:999 ? A?and (b) Að0Þ ¼ 1:001 ? A?. In this case R = 21.16,

S = 2.56 and Sc= 4.79.

L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

143

Page 13

would be relatively small. Application of alternative controls to decreases the number of eggs,

larvae and pupae (e.g., larvicides) would improve the effectiveness of SIT in this case.

Now, we analyze how mating capacity and dispersion affect the release of sterile males. For this,

let S ¼

a=lT

ð1?rÞcC=lMthe ratio between sterile and natural male population in equilibrium, where for

Fig. 5. Numerical simulations of system (1), for R = 21.16, S = 2.56 and Sc= 4.79. The initial conditions from button

to top are A(0)/C = 0.1,0.2,...,0.9.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

050100 150200250

A/C

days

Fig. 6. Numerical simulations of system (1), with / = 10 days?1. In this case R = 42, S = 2.56 and Sc= 10.06. The

initial conditions from bottom to top are A(0)/C = 0.1,0.2,...,0.9.

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L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

Page 14

the natural mosquitoes we made A ¼ C (see third equation of (2)). From the definition of S given

by (5) we have S ¼

mating capacity, and p, the ability of dispersion). In the best of the scenarios, we have p = q = 1,

which implies S ¼ S. Taking a basic offspring number R = 42.32 (corresponding to an ovoposition

of 10 eggs per day per female) the value of S that assures completely effectiveness of SIT should be

greater than Sc= (R ? 1)2/4R = 10, i.e., 10 sterile males for each wild male. As p or/and q decre-

ment, the value of S increases. Thus, for instance, if the mating capacity of sterile males is half of

the one for wild males (q = 0.5), or the dispersion is p = 0.5, then S should be greater than 19, i.e.,

the number of steriles for each natural mosquito should be almost doubled (here we assume the

value of lIgiven above).

Another main question regarding the SIT control is the role of the introduction of wild mos-

quitoes in a region that is being controlled by the release of sterile males. To analyze this situation,

let us suppose that sterile male release displaced the natural insects, so that system is in the state

P0¼ 0;0;0;0;a

toes as a non-zero initial condition in one of the mosquito stages. If the aquatic phase is trans-

ported, numerical simulations according with the parameters above indicate that the initial

condition A(0) should be above of 3:5 ? A?to initiate a re-colonization of wild mosquitoes that

will approach the equilibrium P1þ. For fertilized females F, the condition Fð0Þ P 2:1 ? F? is

necessary to initiate the re-infestation. On the other hand, waves of isolated mated and unfertil-

ized females I as well as natural males M cannot re-invade. Then, as it was expected, the intro-

duction of a sufficiently high number of fertilized females could contribute to the failure of SIT

control.

bþlI

pqbþlIS (recall that bT= pqb, where q measures the percentage of reduction of

lT

??

. Additionally, let us consider the arrive of an invasion wave of wild mosqui-

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100150 200250

A/C

days

Fig. 7. Numerical simulations of system (1), with

from bottom to top are A(0)/C = 0.1,0.2,...,0.9.

C

a=lT¼ 1:2, for R = 42, S = 1.28 and Sc= 10.07. The initial conditions

L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147

145

Page 15

6. Conclusion

In this paper we formulated a model to asses the effectiveness of the sterile insect release tech-

nique (SIT) applied to A. aegypti mosquito population. The results are given in terms of the basic

offspring of natural population R, the ratio between the unfertile and fertile matings S, and the

intrinsic rate of release of sterile males a.

Completely success of SIT can be achieved for R below the threshold R*(S) given by Eq. (11).

Splitting this condition in terms of R, we have two cases. The first case, R 6 1 does not need com-

ments. However, when R > 1, SIT control effectively works when S P Sc, where Scprovides us the

relation of sterile male insects released with respect to wild insects, that is, the displacement of

natural mosquitoes can be obtained if the release rate of irradiated mosquitoes a is beyond the

threshold a*given by Eq. (18).

The model predicts that when R is above R*, and a is below a*, extinction of the wild insects de-

pends on its initial population size. The stability diagrams given in Figs. 1 and 3, as well as the

numerical simulations given in Section 3 suggest that the region of stability of P0is small compared

with the corresponding region for P1þ, and decreases non-linearly when R increases or a decreases.

In this case, it is possible that small perturbations of initial conditions in the stability region of P0

drives the system to the coexistence equilibrium P1þ. One practical implication of this is that the

introduction of fertile females in a region that it is under treatment by SIT could initiate a re-infes-

tation of wild mosquitoes in such a way that the dynamics approach the coexistence equilibrium.

For mosquito population, mating competitiveness and dispersion of sterilized males are special

relevant to SIT. These properties are modeled here via the parameters q and p. Field experiments

using chemosterilized or chromosomally translocated and sex ratio-distorting A. aegypti have

show moderately good mating competitiveness [19,20]. Adequate large releases (12–15 steriles

for each wild male assuming a population with ovoposition rate equal to 10 eggs per day and

q = 0.8) well mixed with isolated wild populations might have been expected to yield high levels

of egg sterility. However, it seems that in the practice they frequently did not do so. This has been

attributed to an influx of already mated females from outside the sterile male released area [13].

Dispersal ability is a major concern for the SIT to ensure that sterile males are released near en-

ough to all breeding sites so that they have a chance of mating with wild females. In the model the

reduction of chance of encounters is modeled by the factor p. Thus, p will have a high value for a

highly dispersing species, meanwhile for a poorly dispersing one, p ? 0. A. aegypti is considered a

species having little dispersion and strongly localized around breeding sites, hence sterile males

would need to be released at intervals of about 50 m along urban streets to find all the local females

[21,22]. However, more recent studies [23] found that females can disperse over more than an800 m

radius. More studies should be done in order to clarify the grade of dispersion of such species.

The mathematical model presented in this paper do not include all factors affecting sterile re-

leases. Some biological details are sacrificed in order to make a model mathematically tractable.

Nevertheless, according to observations, one important point to consider is the immigration of

females that have already had fertile matings and will lay fertile eggs nullifying the effect of sterile

releases, which is left for a future work. Clarification of the role of immigrants in a population is

important not only for the SIT, but also for assessing whether local efforts at larval control with

insecticides of environmental management could have an impact on the adult population or

whether they are likely to be swamped by immigration.

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L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147