Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique.
ABSTRACT We propose a mathematical model to assess the effects of irradiated (or transgenic) male insects introduction 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.
- [Show abstract] [Hide abstract]
ABSTRACT: The anthropophilic and peridomestic female Aedes aegypti bites humans to suck blood to matu-rate fertilized eggs, which are laid in appropriate recipients (breeding sites). These eggs can hatch in contact with water releasing larvae, or can be stored in a dormant state (quiescence), which last for extended periods. Taking into account this ability of eggs of A. aegypti mosquitoes, mathemat-ical model is developed taking into account four successive quiescence stages. The analysis of the model shows that the ability of the eggs surviving in dormant state in adverse abiotic conditions, depending on the model parameters, can increase the fitness of mosquito population; in other words, the capacity of the mosquitoes generating offsprings is increased.Applied Mathematics 10/2014; 5(17):2696-2711. · 0.19 Impact Factor - SourceAvailable from: Tridip Sardar[Show abstract] [Hide abstract]
ABSTRACT: We propose and analyze a new compartmental model of dengue transmission with memory between human-to-mosquito and mosquito-to-human. The memory is incorporated in the model by using a fractional differential operator. A threshold quantity R0, similar to the basic reproduction number, is worked out. We determine the stability condition of the disease-free equilibrium (DFE) E0 with respect to the order of the fractional derivative $\alpha$ and R0. We determine $\alpha$ dependent threshold values for R0, below which DFE (E0) is always stable, above which DFE is always unstable, and at which the system exhibits a Hopf-type bifurcation. It is shown that even though R0 is less than unity, the DFE may not be always stable, and the system exhibits a Hopf-type bifurcation. Thus, making R0 < 1 for controlling the disease is no longer a sufficient condition. This result is synergistic with the concept of backward bifurcation in dengue ODE models. It is also shown that R0 > 1 may not be a sufficient condition for the persistence of the disease. For a special case, when $\alpha$ = 1/2 , we analytically verify our findings and determine the critical value of R0 in terms of some important model parameters. Finally, we discuss about some dengue control strategies in light of the threshold quantity R0.Communications in Nonlinear Science and Numerical Simulation 08/2013; · 2.57 Impact Factor - [Show abstract] [Hide abstract]
ABSTRACT: We present a simple mathematical model to replicate the key features of the sterile insect technique (SIT) for controlling pest species, with particular reference to the mosquito Aedes aegypti, the main vector of dengue fever. The model differs from the majority of those studied previously in that it is simultaneously spatially explicit and involves pulsed, rather than continuous, sterile insect releases. The spatially uniform equilibria of the model are identified and analysed. Simulations are performed to analyse the impact of varying the number of release sites, the interval between pulsed releases and the overall volume of sterile insect releases on the effectiveness of SIT programmes. Results show that, given a fixed volume of available sterile insects, increasing the number of release sites and the frequency of releases increases the effectiveness of SIT programmes. It is also observed that programmes may become completely ineffective if the interval between pulsed releases is greater that a certain threshold value and that, beyond a certain point, increasing the overall volume of sterile insects released does not improve the effectiveness of SIT. It is also noted that insect dispersal drives a rapid recolonisation of areas in which the species has been eradicated and we argue that understanding the density dependent mortality of released insects is necessary to develop efficient, cost-effective SIT programmes.Mathematical Biosciences 08/2014; · 1.45 Impact Factor
Page 1
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
Page 2
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
133
Page 3
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
134
L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147
Page 4
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
135
Page 5
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
136
L. Esteva, H. Mo Yang / Mathematical Biosciences 198 (2005) 132–147
Page 6
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
Page 7
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
;
138
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
Page 9
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).
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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
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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