J. Math. Biol.
DOI 10.1007/s00285-011-0477-6
Mathematical Biology
Mathematical studies on the sterile insect technique
for the Chikungunya disease and Aedes albopictus
Y. Dumont · J. M. Tchuenche
Received: 31 January 2011 / Revised: 30 September 2011
© Springer-Verlag 2011
Abstract Chikungunya is an arthropod-borne disease caused by the Asian tiger
mosquito, Aedes albopictus. It can be an important burden to public health and a great
cause of morbidity and, sometimes, mortality. Understanding if and when disease
control measures should be taken is key to curtail its spread. Dumont and Chiroleu
(Math Biosc Eng 7(2):315–348, 2010) showed that the use of chemical control tools
such as adulticide and larvicide, and mechanical control, which consists of reducing
the breeding sites, would have been useful to control the explosive 2006 epidemic
in Réunion Island. Despite this, chemical control tools cannot be of long-time use,
because they can induce mosquito resistance, and are detrimental to the biodiversity. It
is therefore necessary to develop and test new control tools that are more sustainable,
with the same efficacy (if possible). Mathematical models of sterile insect technique
(SIT) to prevent, reduce, eliminate or stop an epidemic of Chikungunya are formulated
and analysed. In particular, we propose a new model that considers pulsed periodic
releases, which leads to a hybrid dynamical system. This pulsed SIT model is coupled
with the human population at different epidemiological states in order to assess its
efficacy. Numerical simulations for the pulsed SIT, using an appropriate numerical
scheme are provided. Analytical and numerical results indicate that pulsed SIT with
small and frequent releases can be an alternative to chemical control tools, but only if
it is used or applied early after the beginning of the epidemic or as a preventive tool.
Grants: “Projet TIS”, French Minister of Health.
Y. Dumont (B)
CIRAD, Umr AMAP, 34989 Montpellier, France
e-mail: [email protected]
J. M. Tchuenche
Department of Mathematics and Statistics, University of Guelph,
Guelph, ON, N1G 2W1, Canada
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Y. Dumont, J. M. Tchuenche
Keywords Impulse differential equation · Chikungunya · Vector-borne disease ·
Sterile insect technique · Vector control · Basic reproduction number · Floquet theory ·
Equilibrium · Global stability · Dynamically consistent scheme
Mathematics Subject Classification (2000)
92C60
92-08 · 92D30 · 37M05 · 65L12 ·
1 Introduction
Chikungunya is a vector-borne disease caused by Aedes albopictus. It is an uncommon
and not well-known tropical disease whose dynamics and behaviour are yet to be fully
understood. A good understanding of its transmission dynamics and ecology in emergent epidemic regions like Réunion Island can help to improve the control of future
epidemics. Mathematical models provide a quantitative and potentially valuable tool
for this purpose. Although mathematical studies of Chikungunya are very scant, none
of the earlier models focused on the Sterile Insect Technique (SIT). The ability to forecast, understand and control the spread of infectious diseases increasingly depends on
the capacity to formulate and test mathematical models capturing key mechanisms.
The present study builds on and extend two previous models of the Chikungunya
disease (Dumont et al. 2008; Dumont and Chiroleu 2010). For an overview on the
Chikungunya disease, see (Sudeep and Parashar 2008).
Chikungunya is endemic in East Africa and in Asia causing several clinical cases
and, sometimes, deaths. It emerged in developed countries like Réunion Island, in
2005, 2006 and most recently in 2010, and Italy, in 2007. The principal vector of the
Chikungunya in Réunion Island and in Italy is Aedes albopictus (sometimes called the
Asian tiger because it originated from Asia and it is an aggressive mosquito), which is
also a prospective vector for Dengue transmission. Since the beginning of the French
National Agency for Research project Entomochik, there is a tremendous progress in
our knowledge about the vector and the relationships between the virus and the vector
(Boyer et al. 2011b; Delatte et al. 2009, 2008a,b; Dubrulle et al. 2009; Lacroix et al.
2009; Martin et al. 2010; Vazeille et al. 2007, 2008, 2009). One of the first models for
the Chikungunya epidemic of 2005–2006 in Réunion Island was proposed in Dumont
et al. (2008). The focus in Dumont and Chiroleu (2010) is on the study on chemical
and mechanical tools available to stop or to control an epidemic, where it is shown
that the combination of Deltamethrin, the only authorized adulticide in the European
Union, and mechanical control, which consists in reducing the breeding sites, could
have been useful to stop the huge epidemic of 2006. A new biological fact, the second
Chikungunya strain in Réunion Island had a direct impact on the lifespan of infected
mosquitoes (Martin et al. 2010) is accounted for in Dumont and Chiroleu (2010). For
other studies see (Bacaer 2007; Moulay et al. 2011).
Despite the results obtained in Dumont and Chiroleu (2010), massive and long
time spraying of adulticide is not recommended. Indeed, in Martinique, located in the
French West Indies, massive spraying of Deltamethrin have led to increasing the resistance of the mosquito: 60% of Aedes aegypti, the vector of Dengue disease, are now
resistant to the Deltamethrin. Thus, during the summer of 2010, in Martinique and
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Sterile insect technique for the Chikungunya disease and Aedes albopictus
in Guadeloupe, two French overseas departments, the health authorities were unable
to use adulticide against Aedes aegypti. Consequently, this underscores the need and
necessity to study and to consider new control tools, like SIT. This biological control
tool was proposed in the early fifties by Knipling (1955). The technique is fairly simple: it consists to sterilize male eggs using irradiation and to release a large number
of irradiated males such that they will mate wild females, that will have no viable
offspring. If the release is sufficient and the duration of the treatment sufficiently long,
it would be possible to eradicate the native pests or mosquitoes. Conceptually, the
Sterile Insect Technique (by irradiation) appears to be very simple, but in reality it is
not. First, it is necessary to have a very good knowledge of the biology of the mosquito
to be eradicated (eliminated). Second, the irradiation dose needs to be adapted upon
the mosquito: the irradiation dose is not the same for an Anopheles mosquito or for an
Aedes mosquito. If the irradiation dose is too strong it can greatly affect the lifespan
and the competitivity of the male. That is, the dose may be just enough to have a 100%
sterility. The first attempt of the Sterile Insect Technique against the Chikungunya,
was made by Bellini et al. (2007), in Italy, after some cases of Chikungunya were
recorded in the district of Ravenna in North-Eastern Italy.
This study focuses on the sterile Insect Technique by irradiation, but other Sterile
Insect Techniques exist: the RIDL technique (Release of Insects carrying a Dominant
Lethal), and the Wolbacchia technique (see Alphey et al. 2010 for an overview and
further explanations/comments about Sterile Insect Techniques). The aim of the present work is to study mathematically different SIT models and to assess their potential
as control tools. A brief overview of the different approaches classically used in the
literature provides the context. We will first consider two standard models: when the
release of sterile males is proportional to the wild males, and when the release is continuous in time (Esteva 2005; Thomé et al. 2010). Despite the simplicity of the previous
approaches, they are not realistic in the field. Thus, we consider periodic or “pulsed”
releases, coupled with the epidemiological model studied in Dumont et al. (2008);
Dumont and Chiroleu (2010). We wish to assess the efficacy of the SIT approach in
comparison with chemical control tools considered in Dumont and Chiroleu (2010).
Also, we take into account the mechanical control in this study because it is easy to
handle and, more importantly, it is a biological control tool, which has no negative
impact (effect) on the environment, contrary to insecticides and larvicides.
The full model is formulated in Sect. 1. In Sect. 2, the sub-model related to the wild
mosquito population is presented. Using the theory of cooperative systems, we derive
some theoretical results: existence and uniqueness of solution, existence and stability/instability of equilibria. Based on Dumont et al. (2008) and Dumont and Chiroleu
(2010), the system “Wild Mosquito–Human population” is analysed in Sect. 3. Various
approaches for the SIT accounting for different epidemiological states for the mosquito and the Human populations is considered in Sect. 4. For each model, we show
that the system is well-posed, that there exists one or two disease-free equilibrium
point(s). Whenever possible, an analytic expression of the basic reproduction number
is derived, and some results related to the stability/instability of the disease-free equilibrium are provided. Finally, after this theoretical analysis, we construct a numerical
scheme that replicates most of the qualitative properties of the continuous system. We
present numerical approximation of the basic reproduction number R0, pulse related
123
Y. Dumont, J. M. Tchuenche
to the pulsed SIT, showing the impact of the periodicity and the rate of the release on
R0, pulse . Then, considering scenarios like those in Dumont and Chiroleu (2010), we
present some numerical simulations and discuss the results.
2 The epidemiological model
We transform the so-called L-SEIR model proposed and studied in Dumont et al.
(2008) and Dumont and Chiroleu (2010) by considering a compartmental model that
classifies hosts (humans) into three epidemiological states according to individuals disease status: susceptible (or non-immune), Sh ; infectious , Ih ; and resistant (or immune),
Rh . As a first approach, we assume that the total population Nh is constant.
The mosquito population is subdivided into several compartments: the aquatic stage
Am (including eggs, larvae, and pupae), that becomes either male mosquitoes, Mm ,
or immature female mosquitoes, Ym . After mating with wild males Mm , immature
females can become mature females, also subdivided into two epidemiological states:
susceptible, Sm and infectious, Im . Both humans and mosquitoes are assumed to be
born susceptible. An infected human is infectious during η1h days, called the viremic
period, and then becomes resistant or immune.
Cross-infection between humans and vectors is modeled by the mass-action principle normalized by the total population of humans. Every day, each mosquito bites,
on average, B times. βmh is the probability that a bite will lead to a host infection,
which implies that Bβmh represents the contact rate between infectious mosquitoes
and susceptible hosts. Similarly, Bβhm is the contact rate between infectious hosts
and susceptible mosquitoes, and the “conservation of bites”, which is a consistency
condition, is satisfied. K is the carrying capacity of breeding sites. α is a parameter
that represents the mechanical control: when α = 1, there is no mechanical control.
β is the rate at which immature females become reproducing (susceptible) females.
μb is the number of eggs at each deposit per mosquito and per day. μ A is the larval
mortality rate, η A is the larval development rate, and v is the sex ratio.
The mean lifespan for susceptible mosquitoes is 1/μm , and the mean adult lifespan
for infected mosquitoes is 1/μmoi . The last assumption is quite new in the modeling of vector-borne diseases. Indeed, for other vector-borne diseases it has never
been observed that the virus influences the lifespan of an infected mosquito. But in
Réunion Island, it was recently proven that the lifespan of the infected mosquito is
almost halved, which influences the dynamic of the disease (Martin et al. 2010). Thus,
we have 1/μmoi < 1/μm .
The population density of sterile males is denoted by M S M . We assume that M S M
compete with Mm to mate the immature females Ym : it is also assumed that the probm
. The average lifespan of the sterile
ability for a female to meet a wild male is MmM+M
T
mosquitoes is 1/μ S M . We shall consider periodic release of sterile male not accounted
for in Esteva (2005) and Thomé et al. (2010), i.e.,
M S M nT + = M S M (nT ) + r f S M ,
r is a parameter that may represent the quality of the release: if the release is far
from the females, then sterile males will have lower chance to mate wild females. f
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Sterile insect technique for the Chikungunya disease and Aedes albopictus
Fig. 1 A Pulsed SIT compartmental model for Aedes albopictus and the Chikungunya Disease
represents the competitivity of the sterile male, T is the periodicity of the release, and
S M represents the rate of release of sterile male. Thus, r f S M may represent the
release of “efficient” sterile males. However, the proposed model implicitly assumes
that if a female mates with a sterile male, then it will have no offspring, thus reducing
the next generation’s population (i.e. the sterile male also impact the number of new
born mosquito larva).
Based on our model description (see Fig. 1, page 5) and assumptions, we establish
the following equations:
⎧
d Am
Am
⎪
1
−
(t)
=
μ
⎪
b
dt
α K (Sm + Im ) − (η A + μ A ) Am ,
⎪
⎪
⎪
dYm
⎪
⎨ dt (t) = vη A Am − βYm ,
β Mm
Ih
d Sm
dt (t) = Mm +M S M Ym − Bβhm Nh Sm − μm Sm ,
⎪
⎪
Ih
d Im
⎪
⎪
⎪ dt (t) = Bβhm Nh Sm − μmoi Im ,
⎪
⎩ d Mm
dt (t) = (1 − v)η A Am − μ M Mm ,
dM
SM
= −μ S M M S M ,
dt (t)
M S M (nT + ) = M S M (nT ) + r f S M ,
with n = 0, 1, 2, . . . , N T ,
(1)
(2)
where N T is the maximal number of releases, and
⎧
d Sh
Im
⎪
⎨ dt (t) = μh Nh − Bβmh Nh Sh − μh Sh
d Ih
Im
dt (t) = Bβmh Nh Sh − ηh Ih − μh Ih
⎪
⎩ d Rh
dt (t) = ηh Ih − μh Rh
(3)
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Y. Dumont, J. M. Tchuenche
with Sh + Ih + Rh = Nh and initial conditions (α K , m 1 Nh , m 2 Nh , 0, m 3 Nh , 0, Nh ,
1, 0). Let k and m i (i = 1, . . . , 3) be integers, such that K = k Nh , Am (0) = α K , and
Sm (0) + Mm (0) + Ym (0) + Im (0) = (m 1 + m 2 + m 3 )Nh . Note: Equations related to
the sterile females (that will have no offsprings after mating with a sterile male) are
not considered.
3 Theoretical analysis
Prior to analysing the full model above, it is insightful to first consider the following
sub-models: the first one relates to the wild mosquito population, the second when we
consider the release of sterile males, and finally, the full epidemiogical system.
3.1 The wild mosquito dynamics
Following Fig. 2, the wild mosquito dynamic is described by the following equations
⎧
d Am
⎪
⎪
dt (t)
⎪
⎪
⎪
⎨ dYm (t)
dt
d Fm
⎪
⎪
⎪
dt (t)
⎪
⎪
⎩ d Mm
dt (t)
= μb 1 −
Am
αK
Fm − (η A + μ A ) Am ,
= vη A Am − βYm ,
= βYm − μm Fm ,
= (1 − v)η A Am − μ M Mm .
Then, (Am , Ym , Fm , Mm ) ∈ Bm , where
Bm = Am , Ym , Fm , Mm ∈ R5+ ; Am ≤ α K ,
Ym + Fm + Mm ≤ max (η A α K , Ym (0) + Fm (0) + Mm (0)) .
Fig. 2 A compartmental model for the wild mosquito population
123
(4)
Sterile insect technique for the Chikungunya disease and Aedes albopictus
Set
N =
μb vη A
.
μm (η A + μ A )
(5)
N is usually called the Basic Offspring Number. There exists one Trivial Equilibrium, T E = (0, 0, 0, 0). Since N > 1, there exists a biologically realistic equilibrium
E = (A∗ , Y ∗ , F ∗ , M ∗ ), where
N −1
αK,
N
vη A ∗
vη A N − 1
A =
αK,
Y∗ =
β
β
N
β ∗
vη A N − 1
F∗ =
Y =
αK,
μm
μm N
(1 − v)η A ∗
A .
M∗ =
μM
A∗ =
(6)
We prove the following theorem:
Theorem 1 – The trivial equilibrium T E is LAS whenever N < 1, and unstable
otherwise.
– The non trivial equilibrium E exists and is LAS when N > 1.
Proof –
The local asymptotic stability is proved using the Jacobian matrix
⎞
μb 1 − αAKm
− η A + μ A + αμKb Fm
⎟
⎜
⎟
⎜
vη A
−β
J F (Z ) = ⎜
⎟.
⎠
⎝
β
−μm
−μ M
(1 − v) η A
⎛
(7)
Thus,
⎞
μb
− (η A + μ A )
⎟
⎜
vη A
−β
⎟,
J F (T E) = ⎜
⎠
⎝
β −μm
−μ M
(1 − v) η A
⎛
J F (T E) admits a regular splitting, i.e., J F (T E) = N + M, with N =
diag(JF (T E)) and M = J F (T E) − N . A straighforward computation shows
that ρ −N −1 M < 1 if N < 1, which implies that J F (T E) is a stable Metzler matrix (recall that a matrix is called Metzler if its non-diagonal entries are
nonnegative, Jacquez and Simons 1993) if N < 1. Thus, T E is LAS.
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Y. Dumont, J. M. Tchuenche
–
We assume N > 1, and compute the Jacobian at Z = E, we have
⎞
∗
−(η A + μ A + αμKb F ∗ )
μb 1 − αAK
⎟
⎜
⎟
⎜
−β
vη A
J F (E) = ⎜
⎟
⎠
⎝
β
−μm
(1 − v)η A
−μ M
⎛
or equivalently
⎞
μb
−(η A + μ A )N
N
⎟
⎜
vη A
−β
⎟.
J F (E) = ⎜
⎠
⎝
β −μm
(1 − v)η A
−μ M
⎛
Like before, J F (E) admits a regular splitting, N + M, with N = diag(J F (E)) a
stable Metzler matrix, and M = J F (E) − N a positive matrix. Then, it is straightforward to show that ρ(−N −1 M) < 1 since N > 1, which implies that J F (E) is
a stable Metzler matrix, and therefore, E is LAS.
Local stability of the disease-free equilibrium does not guarantee complete elimination of a disease. Global stability ensures that the disease either dies out or persists.
To prove this, we employ the theory of cooperative dynamical systems (Hirsch 1990;
Smith 1995).
3.1.1 Global asymptotic stability of the various equilibria
For a better understanding of the model system, we next investigate the global stability
of the equilibria. Let us consider the following subsystem
⎧
d Am
⎪
1−
(t)
=
μ
b
⎪
⎨ dt
⎪
⎪
⎩
dYm
dt (t)
d Fm
dt (t)
Am
αK
Fm − (η A + μ A )Am ,
= vη A Am − βYm ,
(8)
= βYm − μm Fm ,
that enters the class of tridiagonal systems with “positive” feedback or more generally
cooperative systems. A tridiagonal system with feedback has the following form. In
other words, let us consider the following system of n ordinary differential equations
(ODEs)
⎧
⎨ ẋ1 = f 1 (x1 , x2 , xn ),
ẋi = f i (xi−1 , xi , xi+1 ), i = 2, . . . , n − 1
(9)
⎩
ẋn = f n (xn−1 , xn ),
where f = ( f 1 , . . . , f n ) : D → Rn is C 1 on an open set D ⊆ Rn . System (9) is called
a monotone tridiagonal feedback system if there exist scalars δi ∈ {−1, +1}, i =
1, . . . , n, such that for all 1 ≤ i ≤ n, we have
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Sterile insect technique for the Chikungunya disease and Aedes albopictus
δi
∂ f i+1 (xi , xi+1 , xi+2 )
>0
∂ xi
and
δi
∂ f i (xi−1 , xi , xi+1 )
≥ 0,
∂ xi+1
for all x ∈ D and
δn
∂ f i (xn , x1 , x2 )
> 0.
∂ xn
In our case n = 3, and we have δ1 = δ2 = δ3 = 1, which implies that (8) is a monotone tridiagonal feedback system, which is known to have the Poincaré–Bendixson
property (Mallet-Paret and Sell 1996).
Finally, we recall the following result which will be useful in the sequel.
Theorem 2 (Li and Wang 2002, Theorem 2) Let us consider the following system of
n ODEs
ẋ = f (x), x ∈ D.
(10)
If the following conditions hold:
–
–
–
–
System (10) exists on a compact absorbing set K ⊂ D,
There is a unique equilibrium point E, and it is LAS;
System (10) satisfies the Poincaré–Bendixson property;
Each periodic orbit of system (10) is asymptotically stable;
Then, E is globally asymptotically stable (GAS) in D.
We use the following result.
Theorem 3 (Muldowney 1990) A periodic orbit = { p(t) : 0 ≤ t ≤ w} of (10) is
orbitally asymptotically stable with asymptotic phase if the linear system
ż(t) = J F[2] ( p(t))z(t)
is asymptotically stable, where J F[2] is the second additive compound matrix of the
Jacobian matrix J F associated with system (10).
Consider the following linear system
⎧
μ
⎨ Ẋ = − η A + μ A + α Kb F + β X −μb 1 −
Ẏ = β X − η A + μ A + αμKb F + μm Y,
⎩
Ż = vη A Y − (β + μm ) Z
A
αK
Z,
(11)
where the right-hand side comes from the compound matrix of the Jacobian Matrix J F
J F[2] ( p (t))
⎛ − ηA + μA +
=⎝
β
μb
αK
F −β
⎞
0
−μb 1 − αAK
⎠.
− η A + μ A + αμKb F − μm
vη A
−β − μm
123
Y. Dumont, J. M. Tchuenche
Consider the following Lyapunov function
V (X, Y, Z , A, Y, F) = |X | + |Y | + |Z |.
The orbit O of the periodic solution (A(t), Y (t), F(t)) is at a positive distance from
the boundary, then, there exists c > 0 such that
V (X, Y, Z , A, Y, F) ≥ c sup {|X |, |Y |, |Z |} ,
for all (X (t), Y (t), Z (t)) ∈ R3 and (A(t), Y (t), F(t)) ∈ O. We compute the right
derivative of V along the solution paths (X, Y, Z ) and (A, Y, F):
A
μb
|Z (t)| ,
F + β |X (t)| − μb 1 −
D+ |X (t)| ≤ − η A + μ A +
αK
αK
μb
D+ |Y (t)| ≤ β |X (t)| − η A + μ A +
F + μm |Y (t)| ,
αK
D+ |Z (t)| ≤ vη A |Y (t)| − (β + μm ) |Z (t)| .
Finally, we have
μb
D+ V (t) ≤ − η A + μ A +
F + μY |X (t)|
α
K
A
+ μb 1 −
+ β + μm |Z (t)|
αK
μb
+ (1 − v)η A + μ A +
F + μm |Y (t)|
αK
≤ − ((η A + μ A + μY ) |X (t)| (β + μm ) |Z (t)|)
− ((1 − v)η A + μ A + μm ) |Y (t)|
≤ − min {(η A + μ A ) , β + μm , (1 − v)η A + μ A + μm } V (t),
which implies that V (t) → 0, and (X (t), Y (t), Z (t)) → 0 as t → ∞. Therefore, the linear system (11) is asymptotically stable, and thus the periodic solution
(A(t), Y (t), F(t)) is asymptotically orbitally stable with asymptotic phase.
Since N > 1, the behavior of the local dynamics near (0, 0, 0) implies that system
(8) is uniformly persistent in D ⊆ R3+ . Indeed, the only invariant subset on ∂ D (the
boundary of D) is (0, 0, 0) which is isolated, while the only largest compact invariant
◦
set in D (the interior of D) is the endemic equilibrium (A∗ , Y ∗ , F ∗ ) which is absorbing, and it follows from a result of Hofbauer and So (1989) that system (8) is uniformly
persistent. That is, the wild mosquitoes will persist at an endemic equilibrium level if
they initially exist, with N > 1 (and (0, 0, 0) being unstable). In this case, all solutions
starting in D and sufficiently close to (0, 0, 0) move away from (0, 0, 0).
Thus, using the previous computations and explanations, the hypotheses of Theorem 2 are fulfilled, which imply that system (8) is such that
– (0, 0, 0) is GAS whenever N < 1.
– (A∗ , Y ∗ , F ∗ ) exists and is GAS when N > 1.
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Sterile insect technique for the Chikungunya disease and Aedes albopictus
Next, we consider the following theorem
Theorem 4 (Vidyasagar 1980) Consider the following C 1 system
dx
dt = f (x),
dy
dt
= g(x, y),
(12)
with (x, y) ∈ Rn × Rm . Let (x ∗ , y ∗ ) be an equilibrium point.
If x ∗ is GAS in Rn for the system ddtx = f (x), and if y ∗ is GAS in Rn for the
system ddtx = g(x ∗ , y), then (x ∗ , y ∗ ) is (locally) asymptotically stable for system (12).
Moreover, if all trajectories of (12) are forward bounded, then (x ∗ , y ∗ ) is GAS for (12).
System (4) admits only two realistic equilibria, T E and E, and can be rewritten like
(12), with x = (A, Y, F) and y = M. Thus, using the previous result and applying
Theorem 4, we deduce.
Theorem 5 – The trivial equilibrium T E is GAS whenever N < 1.
– The non trivial equilibrium E exists and is GAS when N > 1.
3.2 The wild mosquito–human epidemiological model
Based on earlier publications (Dumont et al. 2008; Dumont and Chiroleu 2010), we
now consider an epidemiological model accounting for the mosquito dynamic, i.e. we
take into account the wild males and the immature females. Hence, the model systems
become
⎧
d Am
Am
⎪
1
−
(t)
=
μ
b
⎪
dt
α K (Sm + Im ) − (η A + μ A ) Am ,
⎪
⎪
⎪
dY
⎪
m
⎪
(t) = vη A Am − βYm ,
⎪
⎨ dt
Ih
d Sm
(13)
dt (t) = βYm − Bβhm Nh Sm − μm Sm ,
⎪
⎪
⎪ d Im
I
⎪
h
⎪
⎪
dt (t) = Bβhm Nh Sm − μmoi Im ,
⎪
⎪
⎩ d Mm
dt (t) = (1 − v)η A Am − μ M Mm ,
and
⎧ dS
h
(t) = μh Nh − Bβmh NImh Sh − μh Sh ,
⎪
⎪
⎨ dt
d Ih
Im
dt (t) = Bβmh Nh Sh − ηh Ih − μh Ih ,
⎪
⎪
⎩ d Rh
dt (t) = ηh Ih − μh Rh .
(14)
The human and mosquitoes invariant regions are respectively
h = Sh , Ih , Rh ∈ R3+ ; Sh + Ih + Rh = Nh ,
and
m = Am , Ym , Sm , Im ∈ R4+ ; Am ≤ α K = αk Nh , Ym + Sm + Im ≤ m Nh ,
where k and m are positive integer.
123
Y. Dumont, J. M. Tchuenche
Let G = h ∪ m . G is positively invariant and attracting since k and m are chosen
such that
αvη A
k ≤ m,
μm
because μmoi ≤ μm .
Straightforward computations show that one trivial disease-free equilibrium
(0, 0, 0, 0,Nh , 0, 0) exists. Since N > 1, a non-trivial disease-free equilibrium
D F E ∗ = A∗m , Ym∗ , Sm∗ , 0, Nh , 0, 0 also exists, with
1
αK,
= 1−
N
1
Sm∗ =
vη A A∗m ,
μm
vη A ∗
A .
Ym∗ =
β m
A∗m
The potential intensity of transmission and the dynamics of mosquito-borne diseases
are often described in terms of the basic reproductive number. If N > 1, the next
generation matrix calculation (Van den Driessche and Watmough 2002) shows that
the reproduction number is
∗
Sm
B 2 βmh βhm
μmoi (μh + ηh ) Nh
1 αK
B 2 βmh βhm vη A
1−
=
.
μmoi (μh + ηh ) μm
N Nh
R20 =
(15)
The threshold for epidemic spread or endemic persistence of Chikungunya depends
on α K . R0 is the number of generated cases by one infective during an epidemic and
a basis for setting transmission reduction targets that must be achieved through vector
control to eliminate endemic transmission or prevent epidemics. Thus, following (Van
den Driessche and Watmough 2002), we claim the following useful result.
Theorem 6 Let N > 1. If R0 < 1, then the D F E ∗ is locally asymptotically stable,
while it is unstable if R0 > 1.
Investigating whether the disease eventually dies out when introduced into a completely naive/susceptible population is viable. The proof is based on an important
index of epidemic potential for infectious diseases, which is central in epidemiological theory. To this effect, we state and prove the following result.
Theorem 7 The D F E is GAS if and only if RG ≤ 1.
Proof To show that D F E ∗ is GAS, we use the Kamgang–Sallet approach (Kamgang
and Sallet 2008). We decompose the vector
X = (Am , Ym , Sm , Im , Sh , Ih , Rh )
123
Sterile insect technique for the Chikungunya disease and Aedes albopictus
into two sub-vectors
X s = (Am , Ym , Sm , Sh , Rh )
and
X i = (Im , Ih ),
such that
d Xs
= A1 (X ) X s − X ∗D F E,S + A12 (X )X i + B,
dt
d Xi
= A2 (X )X i,
dt
with
A1 (X )
⎛
⎞
A∗
− η A + μ A + μb αSmK
0
μb 1 − α Km
0
0
⎜
⎟
⎜
−β 0
0
0 ⎟
vη A
⎜
⎟
⎜
⎟
=⎜
0
β − Bβhm NIhh + μm
0
0 ⎟
⎜
⎟
⎜
⎟
0
0
0
− Bβmh NImh + μh
0 ⎠
⎝
0
0
0
0
−μh
⎞
⎛ ⎛
⎞
A∗m
μh N h
μb 1 − α K 0
⎟
⎜
⎜ 0 ⎟
⎜
0
0⎟
⎜
⎟
⎟
⎜
⎟
B=⎜
,
A12 (X ) = ⎜
0
0⎟
⎜ 0 ⎟,
⎟
⎜
⎝ 0 ⎠
⎝
0
0⎠
0
0
ηh
and
A2 (X ) =
Sm
−μmoi
Bβhm N
h
Sh
Bβmh Nh − (ηh + μh )
.
A1 and A2 are Metzler matrices. A2 (X ) is irreducible for all X ∈ G. Moreover, since
N > 1, the matrix A1 (X ) has real and negative eigenvalues which imply that the
equilibrium (A∗m , Ym∗ , Sm∗ , Nh , 0) of the system ddtX s = A1 (X )X s + B, reduced to the
sub-domain {X ∈ G : X i = 0}, is GAS.
Finally, let Ā2 be an upper bound matrix of the matrix A2 (X ), then, using the fact
that Sh ≤ Nh and Sm ≤ m Nh , we obtain
Ā2 =
−μmoi Bβhm m
.
Bβmh −(ηh + μh )
123
Y. Dumont, J. M. Tchuenche
We deduce that the stability modulus of Ā2 , α( Ā2 ) = maxλ∈Sp Ā2 ) Re(λ), satisfies
α( Ā2 ) ≤ 0, for all X ∈ G, if and only if R2G < 1, with
B 2 βmh βhm
m
(ηh + μh ) μmoi
m 2
μm N
R .
=
vη A N − 1 αk 0
R2G =
Thus, using the fact that
αvη A
k ≤ m,
μm
we obtain
R2G ≥
N
R2 > R20 ,
N −1 0
showing that on the compact set G, the threshold R2G is not necessarily optimal.
Remark 1 If instead of considering the compact set G, we use the compact subset
= Am , Ym , Sm , Im ∈ R4+ ; Am ≤ α K = αk Nh , Ym + Sm + Im ≤ Sm∗ ,
then, we have RG = R0 .
The implication of the result above (Theorem 7) is that Chikungunya elimination
from the affected regions is possible if the adopted control measures can reduce the
value of RG to below unity.
4 The wild mosquito–human population with sterile males
We extend the previous epidemiological model in order to account for the wild mosquito population and the release of sterile males. There are various ways to account
for the release of sterile males. The first two basic ones have been considered while
the last one is novel:
– a continuous release of sterile males such that the ratio MMST is constant for all time
t ≥ tstar t ,
– a continuous release of sterile male, with a mortality dynamic, like in Esteva (2005)
and Thomé et al. (2010)
– a periodic or pulsed release of sterile males, which seems to be the most realistic
assumption (White et al. 2010).
For each model, we carry out a theoretical study, and whenever possible, compute
the basic reproduction number (which also defines the average time it takes for an
epidemic to complete one generation; the larger it is, the shorter the generation and
the more explosive the disease transmission will be).
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Sterile insect technique for the Chikungunya disease and Aedes albopictus
4.1 Release of sterile males proportionally to wild males
From entomologists expert opinion on the SIT, if one is able to monitor continuously
the number of wild males, then assume that the release is proportional to Mm , in other
words
M S M = r f γ M,
where r ∈ [0, 1] represents the “quality” of the release: if the release is not appropriate, then r is small. If the release is optimal, then r is near 1. f is a parameter related
to the sex fitting of the sterile male. This case appears to be only interesting from
the theoretical point of view, but difficult to achieve practically. In real experiments,
entomologists try to estimate the number of wild males and then to release 10 to 15
times more sterile males, i.e; r f γ ∈ [10, 15]. Of course, in practice, this release is not
necessarily continuous, which raises questions about the feasibility of the continuous
release.
Finally, for all t ≥ tstar t , where tstar t is the starting date of the release, we obtain
a system similar to (13)–(14) except for the following equation
d Fm
dt
=
β
1+r f γ
Ym − μm Fm .
We then define a new Basic Offspring Number:
NS M =
μb vη A
1
.
1 + r f γ μm (η A + μ A )
Using (5), we have
NS M =
1
N,
1+rfγ
showing that N S M < N , where N is the Basic Offspring Number of the wild population. Thus, if r, f and γ are adequately chosen, the release of sterile insect can lower
N S M below unity, driving the wild population to extinction.
Let us now consider the full epidemiological system with the proportional release
⎧
d Am
Am
⎪
1
−
(t)
=
μ
b
⎪
dt
α K (Sm + Im ) − (η A + μ A ) Am ,
⎪
⎪
⎪
dY
⎪
m
⎪
(t) = vη A Am − βYm ,
⎪
⎨ dt
β
Ih
d Sm
dt (t) = (1+γ ) Ym − Bβhm Nh Sm − μm Sm ,
⎪
⎪
⎪
Ih
d Im
⎪
⎪
⎪
dt (t) = Bβhm Nh Sm − μmoi Im ,
⎪
⎪
⎩ d Mm
dt (t) = (1 − v)η A Am − μ M Mm ,
(16)
123
Y. Dumont, J. M. Tchuenche
and
⎧ dS
h
(t) = μh Nh − Bβmh NImh Sh − μh Sh
⎪
⎪
⎨ dt
d Ih
Im
dt (t) = Bβmh Nh Sh − ηh Ih − μh Ih
⎪
⎪
⎩ d Rh
dt (t) = ηh Ih − μh Rh
(17)
with Sh + Ih + Rh = Nh , and initial conditions.
A straightforward computation shows that there exists one trivial disease-free equilibrium (0, 0, 0, 0, Nh , 0, 0), and since N S M > 1, a non-trivial disease-free equilibrium
∗
∗
∗
∗
D F E S∗ I T = A∗m,S I T , Ym,S
I T , Sm,S I T , Mm,S I T , M S M,S I T , Nh , 0, 0 ,
also exists, with
1
αK,
NS M
vη A
1
A∗
=
,
μm 1 + r f γ m,S I T
vη A ∗
=
,
A
β m,S I T
A∗m,S I T = 1 −
∗
Sm,S
IT
∗
Ym,S
IT
and
(1−v)η A ∗
∗
Mm,S
Am,S I T ,
IT =
μM
(1−v)η A ∗
∗
M S M,S I T = r f γ μ M Am,S I T .
Finally, if N S M > 1, using the previous computations, we derive the basic reproduction number related to system (16)–(17):
B 2 βmh βhm
μmoi (μh + ηh )
NS M − 1
R20
=
N −1
R20,S I T =
vη A
1
μm 1 + r f γ
1−
1
NS M
αK
,
Nh
(18)
(19)
The basic reproduction number R20,S I T depends now on N S M , and thus on r, f, γ .
Efficacy of the SIT: if N S M > 1, we have R20,S I T < R20 , showing the positive effect
of the SIT on the basic reproduction number. Using again (Van den Driessche and
Watmough 2002), we claim the following result.
Theorem 8 Let N S M > 1. If R0,S I T < 1 then D F E S∗ I T is locally asymptotically
stable, while it is unstable if R0,S I T > 1.
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Sterile insect technique for the Chikungunya disease and Aedes albopictus
In particular, if R20 > 1, it suffices to choose the parameters r, f and γ such that
R20 − 1
R2
0
1 + N −1
<rfγ
(20)
to get R20,S I T < 1, and then to converge, at least locally, to D F E S∗ I T . Equivalently,
N −1
let 0 < ε < N −1+
, such that N (1 − ε) > 1. Then, choosing r, f , and γ such that
R2
0
r f γ = N (1 − ε) − 1,
gives R0,S I T < 1 and A∗m = εα K . In some sense, we do not only reduce the threshold, but we are able to control the vector population and to maintain it at a low level,
without collapse of the wild population.
Finally, using the same approach as in the previous section, the following result
holds.
Theorem 9 D F E S∗ I T is GAS if and only if RG,S I T ≤ 1, with
R2G ≥
NS M
R2
,
N S M − 1 0,S I T
Because the proof is exactly the same as above, except for a change in the matrix
A1 (X ) where the coefficient β in line 3, row 2, is replaced by 1+rβ f γ , we omit it here
to avoid repetition.
Remark 2 We do not take advantage of the SIT for the GAS property. Two explanations are possible: First, the threshold is not the right one, or the “SIT effect” is only
local, which means that the SIT would be useful only if the system is near the D F E,
since R20 > 1. Second, the sterile insect technique is more reliable as a preventive
process and may be only efficient at the early stages of an epidemic, which in some
sense seems to be realistic.
4.2 Continuous release of sterile males
For a basis for comparison, we consider continuous release by taking into account the
mortality rate for sterile males (Esteva 2005; Thomé et al. 2010). The modified model
system now reads
⎧
d Am
Am
⎪
1
−
=
μ
b
⎪
dt
α K Fm − (η A + μ A ) Am ,
⎪
⎪
⎪
⎨ dYm = vη A − βY ,
A m
m
dt
β
M
Y
d
F
⎪
m
m
m
⎪
⎪
dt = Mm +M S M − μm Fm ,
⎪
⎪
⎩ d Mm
dt = (1 − v)η A Am − μ M Mm ,
123
Y. Dumont, J. M. Tchuenche
with
d MS M
= r f S M − μS M MS M .
dt
(21)
Remark 3 Esteva (2005) and Thomé et al. (2010) considered the following equation
for Ym
dYm
β Mm Ym
βT M S M Ym
= vη A Am −
−
,
dt
Mm + M S M
Mm + M S M
with βT = r fβ < β. After discussions with entomologists, it seems realistic to suppose that the introduction of sterile males only impacts the incoming of immature
female in the wild female compartment.
The solution of the last equation in (21) converges to the equilibrium M S∗M =
It is possible to use the previous theoretical study to make a standard analysis
of
system
(21). First,
it can be shown that there exists a trivial equilibrium T E =
0, 0, 0, 0, M S∗M . Then, assuming that N > 1, and setting
r f S M
μS M .
r f S M
μm
,
(1 − v)η A μ S M α K
(N − 1)2
SC =
,
4N
S=
(22)
(23)
we can show that if S < S C , then, there exist two non trivial viable equilibria P + and
P − such that
S N −1
+
αK
(24)
A = 1+ 1− C
S
2N
S N −1
−
A = 1− 1− C
αK.
(25)
S
2N
We have N > 1 and SSC < 1, which implies that A+ , A− > 0. Note that
1+
S
1− C
S
N −1
N −1
<
2N
N
and thus A+ ∈ (0, A∗ ). Moreover, it is obvious that A− ∈ (0, A∗ ), and from (24) and
(25), we are able to deduce the coordinates of the equilibrium P + and P − . We claim
the following results.
Theorem 10 The trivial equilibrium T E is stable. If N ≤ 1, then T E is globally
asymptotically stable.
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Sterile insect technique for the Chikungunya disease and Aedes albopictus
Theorem 11 Since N > 1 and S < S C , the equilibrium P + is stable, while P − is
unstable.
Remark 4 The proof of the above two results are long but straightforward. If S = S C ,
then there exists only one viable equilibrium P = (A∗ , Y ∗ , F ∗ , M ∗ , M S∗M ), such that
−1)
α K [compare with (6)]. S C is the threshold to maintain the wild popuA∗ = (N2N
lation. If S > S C , then there is no viable equilibrium, and the wild population will
collapse.
Following similar computations as in Esteva (2005), it can be shown for instance
that there exists N ∗ > 1, such that the wild population does not collapse if N > N ∗ ,
with
1
N ∗ = (1 + 2S) 1 + 1 −
.
((1 + 2S))2
Thus, we can summarize the previous results as follows:
0 ≤ N < N∗
Equilibrium
T E is GAS
N = N∗
P
T E is stable
N > N∗
P − is unstable
P + is stable
T E is stable
Remark 5 Following (24) and the previous table, contrary to the previous section, we
cannot really control the wild mosquito population. Whatever the control effort is,
the population at equilibrium cannot be reduced more than an half. This is a severe
drawback when the mosquito population is large.
To assess the efficacy of the SIT, choose r, f, S M and μ S M such that S > S C ,
then the system will converge to the trivial equilibrium, i.e. without wild mosquitoes.
From a practical point of view, this is not possible or only if the wild population is
small. Otherwise, when S ≤ S C , we have no idea of the impact of the SIT on the
epidemic, and, more importantly, whether the SIT is efficient or not. Therefore, it is
necessary to consider the full epidemiological model.
4.3 The full epidemiological model with continuous release of sterile males
Consider the full epidemiological model given by
⎧
d Am
Am
⎪
1
−
(t)
=
μ
b
⎪
dt
α K (Sm + Im ) − (η A + μ A ) Am ,
⎪
⎪
⎪
dY
⎪
m
⎪
(t) = vη A Am − βYm ,
⎪
⎨ dt
β Mm
Ih
d Sm
dt (t) = Mm +M S M Ym − Bβhm Nh Sm − μm Sm ,
⎪
⎪
⎪
Ih
d Im
⎪
⎪
⎪
dt (t) = Bβhm Nh Sm − μmoi Im ,
⎪
⎪
⎩ d Mm
dt (t) = (1 − v)η A Am − μ M Mm ,
(26)
123
Y. Dumont, J. M. Tchuenche
d MS M
(t) = r f S M − μ S M M S M
dt
(27)
and
⎧ dS
Im
h
⎪
⎪ dt (t) = μh Nh − Bβmh Nh Sh − μh Sh
⎨
d Ih
Im
dt (t) = Bβmh Nh Sh − ηh Ih − μh Ih
⎪
⎪
⎩ d Rh
dt (t) = ηh Ih − μh Rh
(28)
with Sh + Ih + Rh = Nh .
Using the previous result, if N > 1 and SSC ≤ 1, then there exists one disease-free
equilibrium
+
+
+
+
+ r f S M
, Nh , 0, 0 ,
D F EC = A , Y , S , M ,
μS M
with
+
A = 1+
S
1− C
S
N −1
αK,
2N
vη A
1
A+ ,
μm 1 + r f γ
vη A +
A ,
=
β
S+ =
Y+
and
M+ =
(1 − v)η A +
A .
μM
Since N > 1 and SSC ≤ 1, using the same method from previous sections, we derive
the basic reproduction number related to the epidemiological system with continuous
release of sterile males
vη A
B 2 βmh βhm
S N −1
2
αK,
R0,S I T C =
1+ 1− C
μmoi (μh + ηh ) μm
S
2N
1
S
=
(29)
1 + 1 − C R20 .
2
S
The basic reproduction number R20,S I T C strongly depends on SSC , and in some sense
the result is not so interesting, because it shows that if we release sterile males such
that SSC = 1, then R20,S I T C = 21 R20 . This means that if R20 > 2, then R20,S I T C > 1,
and thus the system will not converge to the D F E C+ . Therefore, we are unable to
control the epidemic. This is a severe drawback to the use of the SIT: it means that if
the basic reproduction number is too large, the only option is to release sterile males
to collapse the wild population or to reduce it drastically.
123
Sterile insect technique for the Chikungunya disease and Aedes albopictus
Therefore, if R20 > 2, the only option is to release sterile males such that
S
> 1,
SC
and then the system will converge mathematically to the trivial equilibrium, i.e., without wild mosquitoes. It suffices then to choose S M , r, f , and μ S M such that
r f S M
(N − 1)2 (1 − v)η A
>
αK,
μS M
4N
μm
to always collapse the wild mosquito population and thus to lower the entomogical
risk. Finally, from a practical point of view, we summarize the previous results in the
following proposition.
Proposition 1 Consider system (26)–(27)–(28).
–
if R20 > 2, then we can choose S M , r, f , and μ S M such that
r f S M
(N − 1)2 (1 − v)η A
>
αK.
μS M
4N
μm
–
(30)
Thus, the system will converge, at least locally, to the trivial equilibrium.
If 0< R20 < 2, then we can choose S M , r, f , and μ S M such that
r f S M
(N − 1)2 (1 − v)η A
=
αK,
μS M
4N
μm
in which case the system will converge, at least locally, to the disease-free equilibrium.
The modeling of continuous release is not very realistic from a practical point of
view: in real experiment it is not possible to release sterile males continuously, because
most often than not, there are many places needing such release. Thus, it seems more
realistic to consider periodic releases or the so-called pulsed releases, like in Integrative Pest Management. For a recent study on pulsed releases for Aedes aegypti, see
(White et al. 2010).
4.4 Periodic or pulsed releases of sterile males
In this section, we consider periodic or pulsed releases which leads to the following
coupled hybrid subsystems
⎧
d Am
Am
⎪
1
−
=
μ
⎪
b
dt
α K Fm − (η A + μ A )Am ,
⎪
⎪
⎪
⎨ dYm = vη A − βY ,
A m
m
dt
⎪ d Fm = β Mm Ym − μ F ,
⎪
m m
⎪
dt
Mm +M S M
⎪
⎪
⎩ d Mm
dt = (1 − v)η A Am − μ M Mm ,
(31)
123
Y. Dumont, J. M. Tchuenche
dM
SM
dt
MS M
= −μ
S M MS M ,
nT + = M S M (nT ) + r f S M ,
with n = 0, 1, 2, . . . , N T .
The last condition implies the release of r f S M sterile males every T period and at
most N T times. Altogether, this leads to the efficient number of sterile males given by
r f S M .
Before going further, let us consider the general impulse differential system
ẋ = f (t, x),
t = kT,
(32)
x(t) = x(t + ) − x(t) = Ik , t = τk = kT, k = 1, 2, . . .
with initial condition x(t0+ ) = x0 . Then, we recall the following theorem.
Theorem 12 (Lakshmikantham et al. 1989) Let the function f : R× → Rn be continuous in the set (τk , τk+1 ]× and for each k ∈ Z and x ∈ suppose that there exists
a finite limit of f (t, y) as (t, y) → (τk , x), t > τk . Then for each (t0 , x0 ) ∈ R × ,
there exist β > t0 and a solution x : (t0 , β) → Rn of the initial value problem (32).
If moreover, the function f is locally lipschitz continuous with respect to x in R×,
then this solution is unique.
It is possible to rewrite the previous system in the following way
⎧
d X1
X1
⎪
1
−
=
μ
⎪
b
⎪
dt
α K X 3 − (η A + μ A ) X 1 ,
⎪
⎪
⎪
d
X
2
⎪
⎪ dt = vη A X 1 − β X 2 ,
⎨
β X4
d X3
dt = X 4 +X 5 X 2 − μm X 3 ,
⎪
⎪
⎪
⎪
⎪ ddtX 4 = (1 − v)η A X 1 − μ M X 4 ,
⎪
⎪
⎪
⎩ d X5
dt = −μ S M X 5 .
(33)
for all t ∈ (tk , tk+1 ], with X = (Am , Ym , Fm , Mm , M S M ), and the additional ”impulse
assumption”
X = In (t),
with t = tn , tn+1 − tn = T,
and
n = 0, 1, 2, . . . , N T , (34)
where
X i (tn ) = X i+ (tn ) − X i (tn ),
and In = (0, 0, 0, 0, r f S M )T . We have X ∈ B, where
Bm = (Am , Ym , Fm , Mm , M S M ) ∈ R5+ ; Am ≤ α K ,
Ym + Fm + Mm ≤ max(vη A α K , Ym (0) + Fm (0) + Mm (0)) ,
and M S M ≤
r f S M
1−e−μ S M T
.
The right-hand side of system (33) is locally lipschitz continuous on B. Thus, using
the previous existence theorem, system (33)–(34) has a unique solution.
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Sterile insect technique for the Chikungunya disease and Aedes albopictus
Remark 6 Note that the third equation in (33) can be replaced by
X5
d X3
X 2 − μm X 3 ,
=β 1−
dt
X4 + X5
Let us compute
the periodic
solution for the sterile males hybrid equation. Consider
the interval nT + , (n + 1) T , then for all t ∈ (nT, (N + 1) T ), we have
+
M S M (t) = M S M nT + e−μT (t−nT ) .
Then, using the fact that M S M nT + = M S M (nT ) + r f T , we have
M S M (n + 1) T + = M S M nT + e−μT T + r f S M .
From the previous relation, because e−μT T < 1, we deduce that M S M nT + converges to an equilibrium since n → +∞, that is:
M S M nT + → M S∗M,max =
r f S M
1 − e−μ S M T
(35)
M S M ≤ M S∗M . Finally, we obtain the following periodic solution for sterile male:
M S∗M (t) =
r f S M
e−μ S M mod(t−nT,T ) .
1 − e−μ S M T
(36)
4.4.1 Periodic equilibrium
Straightforward calculations yield the following periodic free-wild mosquito only
solution
T E(t) = (0, 0, 0, 0, M S∗M (t)),
where M S∗M (t) is given in (36). Next, we shall apply the separation principle. Using
(36), we now consider the following system,
⎧
d Am
Am
⎪
1
−
=
μ
⎪
b
α K Fm − (η A + μ A ) Am ,
⎪ dt
⎪
⎪
⎨ dYm = vη A − βY ,
A m
m
dt
β M m Ym
d Fm
⎪
⎪
⎪
dt = Mm +M S∗ M − μm Fm ,
⎪
⎪
⎩ d Mm
dt = (1 − v)η A Am − μ M Mm ,
(37)
dX
= A(X, t)X,
dt
(38)
or equivalently
123
Y. Dumont, J. M. Tchuenche
with
⎛
− (η A + μ A )
⎜
⎜
vη A
A(X, t) = ⎜
⎝
0
(1 − v)η A
0
−β
β Mm
Mm +M S∗ M (t)
0
⎞
0
μb 1 − αAKm
⎟
0
0 ⎟
⎟,
−μm
0 ⎠
0
−μ M
(39)
where A(X, t) is a Metzler matrix for all X ∈ R4+ , and t ≥ 0. In particular, noting that
β Mm Ym
d Fm
=
− μm Fm
dt
Mm + M S∗M
≤ βYm − μm Fm ,
the solutions of (37) are bounded, below by 0, and above by the solution of system
(4), showing that the release of sterile males has a positive effect, i.e., it reduces the
number of fertile females, thereby reducing the next generation of new offsprings.
The local asymptotic stability of the limit cycle is relatively obvious. It suffices to
compute the Jacobian Matrix of (4), that is
⎛
− ηA + μA +
⎜
⎜
vη A
J F (E)(t) = ⎜
⎜
0
⎝
μb
αK
(1 − v)η A
Fm
0
−β
μb 1 −
0
Am
αK
β Mm
Mm +M S∗ M
−μm
0
0
0
0
(
⎞
⎟
⎟
⎟. (40)
⎟
) ⎠
β M S∗ M Ym
2
Mm +M S∗ M
−μ M
Thus, since E 0 = (0, 0, 0, 0), it leads to the following linear constant system which
is equivalent to
dY
= J F (E 0 )Y
dt
with
⎞
0
0
μb
− (η A + μ A )
⎜
− (β + μY ) 0
vη A
0 ⎟
⎟.
J F (E 0 ) = ⎜
⎝
0
0
−μm 0 ⎠
0
0 −μ M
(1 − v)η A
⎛
We deduce that all eigenvalues λi of J F (E 0 ) are real and negative, for i = 1, . . . , 4,
and the monodromy matrix related to the system is simply e JF (E 0 )T , such that its Floquet multipliers are e−λi T < 1, for i = 1, . . . , 4. Using Floquet’s theorem, we deduce
that (0, 0, 0, 0) is a locally asymptotically stable equilibrium, which implies that the
trivial periodic solution T E(t) is always LAS. Note here that the previous result holds
whatever the value taken by the offspring number N .
123
Sterile insect technique for the Chikungunya disease and Aedes albopictus
From the computations in the previous subsection, we have
M S∗M,min < M S∗M (t) ≤ M S∗M,max
with
M S∗M,max =
r f S M
r f S M
=
−μ
T
1 − e SM
μS M
μS M
1 − e−μ S M T
,
and
M S∗M,min
r f S M
r f S M
=
e−μ S M T =
−μ
T
S
M
1−e
μS M
μS M
.
eμ S M T − 1
Using the previous upper and lower bound for M ∗ , we now consider
matrices
⎛
− (η A + μ A )
0
0
μb 1 − αAKm
⎜
⎜
−β
0
0
vη A
Amax (X, t) = ⎜
β Mm
⎜
0
−μ
0
∗
m
⎝
Mm +M
S M,min
(1 − v)η A
the following
0
0
−μ M
0
−β
μb 1 − αAKm
0
−μm
0
0
0
0
0
−μ M
⎞
⎟
⎟
⎟,
⎟
⎠
and
⎛
− (η A + μ A )
⎜
⎜
vη A
Amin (X, t) = ⎜
⎜
0
⎝
(1 − v)η A
β Mm
M+M S∗ M,max
⎞
⎟
⎟
⎟.
⎟
⎠
Then, let us consider the following systems
dX
= Amax (X , t)X
dt
(41)
dX
= Amin (X , t)X ,
dt
(42)
and
with X (0) = X (0) = X (0) = X 0 ∈ R4+ . Amax and Amin being Metzler matrices, we
have existence and uniqueness of X and X in R4+ .
We now state and prove the following lemma which will be used in the sequel.
Lemma 1 Let systems (39), (41), and (42), have the same initial condition X (0) =
X (0) = X (0) = X 0 ≥ 0. Then, for all t ≥ 0, we have
0 ≤ X (t) ≤ X (t) ≤ X (t).
123
Y. Dumont, J. M. Tchuenche
Proof Consider Y = X − X . Then, using equations (38) and (41), we have
dY
= B(t)Y + F(t),
dt
with B(t) =
⎞
F
− η A + μ A + μb α Km
0
μb 1 − αAKm
0
⎜
⎟
⎜
⎟
vη A
−β
0
⎜
0
⎟
⎜
⎟,
βY m
M
β Mm
⎜
⎟
1 − M +M ∗
−μm
0
⎝
⎠
Mm +M S∗ M (t)
Mm +M S∗ M (t)
m
S M,max
0
0
−μ M
(1 − v)η A
⎛
and
⎛
⎞
0
0
⎜
⎟
⎟
⎜
∗
∗ (t)
⎜
⎟.
βY
M
−M
M
m m
F(t) = ⎜
S M,max
SM
⎟
⎝ ( Mm +M ∗ (t)) M +M ∗
⎠
m
SM
S M,max
0
For all t ≥ 0, B(t) is Metzler and F3 (t) ≥ 0 for all t > 0, which implies that if
Y (0) ≥ 0, then Y (t) ≥ 0. Thus, we deduce X (t) ≥ X (t) for all t ≥ 0. Using the same
type of reasoning, we show that X (t) ≤ X (t) for all t ≥ 0. This completes the proof.
Finally, assuming that N > 1, and using the same computations as in the continuous
+
−
release case, we can show that there exist two equilibria P and P . Moreover, considering system (41), we can define the following threshold that ensures the existence
of a non trivial equilibrium for all T > 0
S=
r f S M
μm
(1 − v)η A μ S M α K
μS M
1 − e−μ S M T
=S
μS M
1 − e−μ S M T
,
(43)
where S is the threshold given in (22). From inequality (43), we immediately deduce
the following result.
Proposition 2 If T > 0 is chosen such that
T >
1
μS M
ln
1
1 − μS M
,
(44)
then S < S.
Remark 7 The inequality (44) gives an interesting relationship between the periodicity
of the releases, T , and μ S M , the mean lifespan of the sterile male.
123
Sterile insect technique for the Chikungunya disease and Aedes albopictus
Remark 8 From (44), we can choose T > 0, such that we always have S < S, indicating that the constraint to have existence of non trivial periodic equilibrium, i.e.
S < S C , is weaker than the constraint in the continuous case, i.e. S < S C .
Following Sect. 4.2, page 17, we can show that there exist two non trivial viable
equilibria P + and P − if S < S C , and such that for the aquatic stage at equilibrium,
we have
⎛
⎞
N −1
S
αK
(45)
A+ = ⎝1 + 1 − C ⎠
S
2N
⎛
A− = ⎝1 −
⎞
S ⎠N −1
αK.
1− C
S
2N
(46)
Similarly as in Sect. 4.2, we claim the following result.
Theorem 13 Since N > 1 and S < S C , the equilibrium P
unstable.
+
is stable, while P
−
is
Finally, using (45) and following Sect. 4.2, page 17, we deduce the same severe
restriction like in the continuous release case, namely A+ ∈ ( 21 A∗ , A∗ ).
Thus, even in the case of periodic pulsed releases, it is not possible to maintain
the wild population at a low level. The only possibility is to consider r, f, μ S M , T ,
and S M such that S > S C . The higher S is, the faster the convergence of the wild
population to the periodic trivial equilibrium T E. The aim, of course is not to drive
the wild mosquito population to extinction, but to lower it, such that we may reduce
the number of female mosquitoes per human, in such a way that the dynamics of the
outbreak is broken. The release will take place as long as the risk of a new outbreak is
still there. These releases will then be stopped, and following the GAS result given in
Sect. 3.1 (Theorem 5), the wild population will converge to the non trivial equilibrium.
Using Knobloch (1962), it is possible to show that a T -periodic solution p ∗ exists.
Then, to show that p ∗ is locally asymptotically stable, we may have two possibilities:
–
–
either, prove that the periodic orbit = { p(t) : 0 ≤ t ≤ T } is asymptotically
stable, that is, we have to show that the linear system ż(t) = J F[2] ( p ∗ (t))z(t) is
asymptotically stable (Muldowney 1990). This is not straightforward and seems
not possible in our case.
or, consider Floquet’s theory. Using (40), we compute J F ( p ∗ (t)) and we consider the monodromy matrix J ( p∗ (T )) related to the T -periodic system dY
dt =
J F ( p ∗ (t))Y . Floquet’s theorem says that p is LAS if ρ( J ( p∗ (T )) ) < 1. Unfortunately, it is not possible to have an analytic expression for J ( p∗ (t)) , except for
some J ( p ∗ (t)). Thus, we need to compute J ( p∗ (T )) numerically and to deduce
its spectral radius.
We summarize the previous results below
123
Y. Dumont, J. M. Tchuenche
Proposition 3 System (33)–(34) admits two periodic equilibria
T E = (0, 0, 0, 0, M S∗M )T that is locally asymptotically stable,
p ∗ (t) that is locally asymptotically stable (at least numerically).
–
–
4.5 Combination with the epidemiological system: the full system
Define
m = Am , Ym , Sm , Im , Mm ∈ R5+ ; Am ≤ α K , Ym + Fm + Mm ≤ m Nh ,
and
h = Sh , Ih , Rh ∈ R3+ ; Sh + Ih + Rh = Nh .
Let G = h ∪ m . G is positively invariant and attracting since k and m are chosen
such that
ηAα
k ≤ m,
μmin
with μmin = min μm , μmoi , μ Mm . The model systems (1) and (3) can be written in
matrix form as follows:
dx
= A(x)x + B
dt
(47)
where x = (Am , Ym , Sm , Im , Mm , Sh , Ih , Rh , MT )T , B = (0, 0, 0, 0, 0, Nh , 0,
0, T )T and
⎛
⎞
M(x) 0
0
A(x) = ⎝ 0 E(x) 0 ⎠
0
0 −μ S M
with M(x) =
m
− η A + μ A + μb Smα+I
K
⎜
⎜
r
η
A
⎜
⎜
0
⎜
⎜
⎝
0
(1 − r )η A
⎛
123
0
−β
β Mm
Mm +M S M
0
0
μb
0
− Bβhm NIhh + μm
Bβhm NIhh
0
μb
0
0
0
0
0
−μmoi 0
0 −μ M
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
Sterile insect technique for the Chikungunya disease and Aedes albopictus
and
⎛
⎜
E(x) = ⎝
− Bβmh NImh + μh
Bβmh NImh
0
0
0
⎞
⎟
− (ηh + μh ) 0 ⎠
−μh
ηh
Altogether A(x) is a Metzler matrix for all x ∈ R8+ and thus R8+ is invariant under system (47). The model system is well-posed mathematically and biologically realistic.
4.6 Disease-free equilibrium and the basic reproduction number for the impulse
system, R0, pulse
Using the approach given in Van den Driessche and Watmough (2002) and Wang
and Zhao (2008), and assuming that the sterile insects have reached their equilibrium
M S∗M (t), we rewrite the full model system in the following way
ẋ = F(x) − V(t, x),
(48)
where x = (Ih , Im , Am , Ym , Sm , Mm , Sh , Rh ), Fi (x) is the input rate of newly
infected individuals in the ith compartment, Vi+ (t, x) is the input rate of individuals by other means, Vi− (t, x) is the rate of transfer of individuals out of compartment
i, that is
⎛
0
⎜ Bβ Sm
⎜ hm Nh
⎜
⎜
⎜
F(x) = ⎜
⎜
0
⎜
⎜
⎜
⎝
0
Bβmh NShh 0 0
0
00
00
00
0
0
0
0
0
0
0
⎞
⎞⎛
000
Ih
⎜
⎟
0 0 0⎟
⎟ ⎜ Im ⎟
⎜
⎟
0 0 0 ⎟ ⎜ Am ⎟
⎟
⎟ ⎜ Ym ⎟
⎟
⎟⎜
⎟ ⎜ Sm ⎟ ,
0
⎟
⎟⎜
⎟ ⎜ Mm ⎟
0
⎟
⎟⎜
0 0 ⎠ ⎝ Sh ⎠
Rh
00
and −V (x, t) =
⎛
⎞
− (ηh + μh )
0
0
0
0
0
0
0
⎛
Ih
⎜
0
−μmoi 0
0
0
0 ⎟
⎜
⎟⎜ I
0
0
⎜
⎟⎜ m
Am
Am
0
μb 1 − α K
0
0
0 ⎟⎜
μb 1 − α K − (η A + μ A )
⎜
⎜
⎟ ⎜ Am
⎜
⎟⎜
vη
−β
0
A
⎜
⎟ ⎜ Ym
⎜ Bβ Sm
⎟⎜
β Mm
−μ
0
⎜
⎟ ⎜ Sm
m
∗ (t)
hm Nh
M
+M
m
⎜
⎟⎜
SM
⎜
⎟ ⎜ Mm
−μ M
(1 − v)η A
⎜
⎟⎝
⎜
⎟ Sh
Sh
⎝
Bβmh N
−μh 0 ⎠
h
Rh
0 −μh
ηh
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
We now follow Wang and Zhao (2008), and verify the following assumptions:
123
Y. Dumont, J. M. Tchuenche
(A1) For each 1 ≤ i ≤ 7, Fi (x) , Vi+ (x) , and Vi− (x) are nonnegative and continuous
on (0, ∞) × B, and continuously differentiable.
(A2) There is a real number T > 0 such that Fi (x) , Vi+ (x), and Vi− (x) are T −periodic in t.
(A3) If xi = 0, then Vi− = 0. In particular if x ∈ X S = {B, such that Ih = Im = 0}
then Vi− = 0 for i = 1, 2.
(A4) Fi (x) for i > 2.
(A5) If x ∈ X S , then Fi (x) =Vi+ (x) for i = 1, 2.
In particular, setting
D F E 0 = (0, 0, 0, 0, 0, Nh , 0, 0) and P D F E = 0, 0, A∗m , Ym∗ ,
Sm∗ , Mm∗ , Nh , 0 , we compute the Jacobian matrix
Dx F (P D F E) =
F(t) 0
0 0
and Dx V (P D F E) =
V
0
J −M(t)
with
F=
S ∗ (t)
Bβhm mNh
Bβmh
0
,
V =
ηh + μh 0
0
μmoi
and
⎛ ⎞
μ F∗
A∗
− η A + μ A + αb Km
0
μb 1 − α Km
0
0
0
⎜
⎟
⎜
⎟
vη A
−β
0
⎜
⎟
∗
∗
⎜
⎟
β MS M
β Mm
∗
⎜
⎟
−μ
−
Y
m
2 m
Mm∗ +M S∗ M
M (t) = ⎜
∗ +M ∗
⎟.
M
m
SM
⎜
⎟
⎜
⎟
(1
−
v)η
−μ
A
M
⎜
⎟
⎝
−μh 0 ⎠
0 −μh
Let M (T ) be the monodromic matrix related to the linear T -periodic system ddtX =
M (t) X . We assume that the periodic equilibrium A∗m , Ym∗ , Sm∗ , Mm∗ , Nh , 0 is locally
asymptotically stable, that is
(A6) ρ ( M (T )) < 1.
Of course, due the size of our matrix and the fact that we have no analytical solution
of the periodic equilibrium, we can only verify this assumption through numerical
simulations.
We consider the monodromic matrix −V (T ) related to the linear system ddtX =
−V X . Because V is a constant matrix, then
−V (T ) =
which implies immediately that
(A7) ρ (−V (T )) < 1.
123
0
e−(ηh +μh )T
0
e−μmoi T
,
Sterile insect technique for the Chikungunya disease and Aedes albopictus
Thus, system (48) verifies the assumptions (A1)–(A7), which leads to
Proposition 4 (Wang and Zhao 2008)
–
–
–
R0, pulse = 1 if and only if ρ( F−V (T )) = 1.
R0, pulse < 1 if and only if ρ( F−V (T )) < 1.
R0, pulse > 1 if and only if ρ( F−V (T )) > 1.
Thus, PDFE is asymptotically stable if R0, pulse < 1, and unstable if R0, pulse > 1.
Unfortunately, in case F and V are not constant or diagonal matrices, it is not possible to obtain a explicit analytic expression (see Wang and Zhao 2008). Fortunately,
following Theorem 2.1 in (Wang and Zhao 2008), we are able to compute an approximation of R0, pulse (see the numerical part). Indeed, we consider the following linear
w-periodic system
F
dw
= −V +
w,
dt
λ
(49)
with parameter λ ∈ (0, ∞). Let W (t, sλ), t ≥ s, s ∈ R be the evolution operator of
the system (49). Then, we have
Theorem 14 (Wang and Zhao 2008) Let assumptions (A1)–(A7) hold. Then the following statement are valid
– If ρ(W (t, sλ) = 1 has a positive solution λ0 , then λ0 is an eigenvalue of L, and
hence, R0, pulse > 1.
– If R0, pulse > 0, then λ = R0, pulse is the unique solution of ρ(W (t, sλ) = 1.
– R0, pulse = 0 if and only if ρ(W (t, sλ) < 1, for all λ > 0.
This theorem will be useful to obtain numerical approximation of R0, pulse in the
next section (see also Klausmeier 2008). Without sterile insects, we have Sm∗ (t) = Sm∗ ,
and we then recover the basic reproduction number given in (15). Our dynamical system is relatively complex, which implies that there is no analytic formula for R0, pulse .
We will estimate R0, pulse numerically, and present simulations according to scenarios
previously considered in Dumont and Chiroleu (2010), to assess the efficacy of the
pulsed SIT.
5 The numerical scheme and simulations
5.1 A dynamical consistent scheme
In this section, building on the approach in Dumont et al. (2008) and Dumont and
Chiroleu (2010), a numerical scheme that preserves as much as possible all the qualitative properties of the continuous problem is presented. We consider the nonstandard
finite difference approach. It is shown in Anguelov et al. (2009b, 2011a) that standard methods applied to epidemiological models can lead to negative solutions or
to spurious dynamic behaviors, like, for instance, converging to the D F E even if
123
Y. Dumont, J. M. Tchuenche
R0 > 1. Hence, the need to develop a reliable scheme that handle much of the qualitative properties shown in the previous section. The nonstandard approach relies on the
following important rules: the standard denominator t in each discrete derivative is
replaced by a time-step function 0 < φ(t) < 1, such that φ(t) = t + O(t 2 );
the nonlinear terms are approximated in a non local way; for instance the nonlinear
term Sm (tn )Ih (tn ) in the continuous problem can be approximated by Smn Ihn+1 . For an
overview and some applications of the nonstandard finite difference method (NSFD),
see (Anguelov et al. 2009a,b, 2011b; Dumont and Lubuma 2005, 2007; Mickens 1994,
2005).
Let x n be an approximation of x(tn ), where tn = nt, n ∈ {N } and t > 0. A
nonstandard approximation for system (1)–(3) is given by
⎧ n+1 n
Am −Am
⎪
⎪
φ(t)
⎪
⎪
⎪
Ymn+1 −Ymn
⎪
⎪
⎪
φ(t)
⎨ n+1
n
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
= μb (Smn+1 + Imn+1 ) −
μb
n
α K (Sm
+ Imn ) + η A + μ A An+1
m
n+1
= r η A An+1
m − βYm
Sm −Sm
φ(t)
Imn+1 −Imn
φ(t)
Mmn+1 −Mmn
φ(t) (t)
=
=
=
In
β Mm Ymn+1
− Bβhm Nhh Smn+1 − μm Smn+1 ,
Mmn +MTn
In
Bβhm Nhh Smn+1 − μmoi Imn+1 ,
n+1
(1 − r )η A An+1
m − μ M Mm ,
n
M Sn+1
M −M S M
φ1 (t)
M Sn±
M =
(50)
= −μ S M M Sn+1
M ,
M Sn M + S M
and
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Shn+1 −Shn
φ(t)
Ihn+1 −Ihn
φ(t)
Rhn+1 −Rhn
φ(t)
In
= μh Nh − Bβmh Nmh Shn+1 − μh Shn+1
In
= Bβmh Nmh Shn+1 − ηh Ihn+1 − μh Ihn+1
(51)
= ηh Ihn+1 − μh Rhn+1
T
Thus in other words, setting X n = An , Y n , S n , I n , M n , MTn , Shn , Ihn , Rhn , following
(47), we have
I d − φ (t) A X n X n+1 = X n ,
(52)
with A (X n ) a Metzler matrix. Thus, choosing φ (t) such that I d − φ (t) A (X n )
is an M-matrix, ensures that the positivity property is preserved if the initial condition
X 0 is nonnegative, because the inverse of an M-matrix is a positive matrix. A possible
choice for φ (t) is
φ (t) =
123
1 − e−Mt
,
M
(53)
Sterile insect technique for the Chikungunya disease and Aedes albopictus
with
αr η A K
, β, μh + ηh , μh
M = max μT , μ M , μmoi , μm + Bβhm
Nh
μb
+ ηA + μA .
+ Bβmh ,
αK
(54)
Summing the three equations in (51) yields Shn + Ihn + Rhn = Nh . This conservative
property is also preserved by the scheme. Finally, a last but important property to hold
is (at least) the local stability properties of the various equilibria.
Then, knowing the exact solution for the sterile males, we are able to construct the
so-called exact scheme, i.e. M Sn M = M S M (tn ) for all time tn and all time-step t > 0.
Following Mickens (1994), it suffices to consider
φ1 (t) =
1 − e−μ S M t
.
μS M
Theorem 15 The nonstandard finite difference scheme (50)–(51) is elementary stable
whenever φ(t) is chosen according to (53), (54), and
M ≥ max{|λ|2 /2|Reλ|},
(55)
where λ traces the eigenvalues of the Jacobian matrices of the right-hand side of model
(1)–(3) at any equilibrium.
Proof Let X ∗ an equilibrium of problem (47), i.e. A (X ∗ ) X ∗ = 0. This is equivalent
to
I d − φ (t) A X ∗ X ∗ = X ∗ ,
which implies that (52) and (47) have the same equilibrium. Moreover, let J be the
i
Jacobian matrix of A(X ) computed at equilibrium X ∗ , i.e. Ji j = ∂∂ A
X j . J is in practice
diagonalizable, using the factorization
−1 J = diag(λ1 , λ2 , . . . , λn ),
where is a transition matrix. Thus, setting ε = X − X ∗ , the linearization of the
system (1)–(3) at X ∗ reads
dε
= J ε,
dt
(56)
which is equivalent to the uncoupled system
dη
= diag(λ1 , λ2 , . . . , λn )η.
dt
(57)
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Y. Dumont, J. M. Tchuenche
Thus, applying the NSFD scheme (52) to the system (56) or (57), we obtain the linearized schemes
εk+1 = (I − φ(t)J )−1 εk
or
ηk+1 = diag
1
1
,...,
1 − φ(t)λ1
1 − φ(t)λn
ηk .
(58)
Now, if X ∗ is asymptotically stable for (1)–(3), then the real parts of all the eigenvalues
λi are negative and it follows from (58) that
1
ρ (I − (t)J )−1 = max
;1 ≤ i ≤ n
|1 − (t)λi |
1
= max ;1 ≤ i ≤ n
1 + 2(t)|Reλi | + 2 (t)|λi |2
< 1,
which shows that X ∗ is asymptotically stable for the scheme (52).
Suppose now that X ∗ is unstable for (1)–(3), then there exists at least one eigenvalue
of J , say λ1 , with positive real part. We then have
1
1
=
2
|1 − (t)λ1 |
1 − 2(t)Reλ1 + 2 (t)|λ1 |2
>1
whenever the requirement (55) is met. Therefore, X ∗ is unstable for the scheme (52).
Thus, at least locally, we are sure that the scheme (52) has the same dynamic as the
model system (1)–(3).
5.2 Numerical simulations
It is generally unrealistic to expect that estimates of any single quantity such as the basic
reproduction number derived from a particular data source will agree with estimates
derived from other data sources, and here we ask: which biological interpretations
are relevant in the context of the parameter used to populate the model? That is to
what features of the data are the parameter estimates sensitive? To this end, various
numerical simulations are carried out. The model parameters are described in Table 1,
page 33.
The parameters f and μ S M can change depending on the level of radiation. In
Table 1, we consider results obtained by Bellini et al. (2007) for standard radiation
level that ensures 100% sterility.
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Sterile insect technique for the Chikungunya disease and Aedes albopictus
Table 1 Epidemiological and entomological parameters
Parameters Description
Average value or range of values
B
Average daily biting (per day)
1
βmh
βhm
Transmission probability from Im (per bite)
Transmission probability from Ih (per bite)
≈ 0.365 (0.95)
1/μh
Average lifespan of humans (in days)
78 × 365
1/ηh
Average viremic period (in days)
3
1/ηm
Extrinsic incubation period (in days)
2
r
Ratio female-male
0.5
1/μm
Average lifespan of adult female mosquitoes (in days)
15
1/μmoi
Average lifespan of infected female mosquitoes (in days) 9
1/μ M
Average lifespan of male mosquitoes (in days)
13
1/μ S M
Average lifespan of sterile male mosquitoes (in days)
8
β
Mating rate of wild male
0.7
f
Relative Competitivity
0.7
≈ 0.5 × 0.365
μb
Number of eggs at each deposit per capita (per day)
14
μA
Natural mortality of larvae (per day)
3
ηA
Maturation rate from larvae to adult (per day)
≈ 0.05
Fig. 3 A compartmental model for the wild mosquito population and Pulsed releases of sterile males
5.2.1 Simulations without SIT control
Here, using the above parameters, we present the results obtained for Saint Denis,
without any control, i.e. S M = 0 (see Figs 2 and 3).
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Y. Dumont, J. M. Tchuenche
Evolution of Infected people in Saint Denis per week
700
I_h
I_Data
600
Infected People
500
400
300
200
100
0
0
5
10
15
20
25
30
Weeks
Fig. 4 Evolution of the infected population per week in Saint-Denis in 2005; comparison of simulated data
with real data (dashed line)
Evolution of Infected People in Saint Denis per Week
Number of Infected People
14000
12000
10000
8000
6000
Change of Strain
4000
2000
0
0
50
100
150
Week
Fig. 5 Simulation of the evolution of the infected population per week in Saint-Denis from 2005 till 2008
Using the above 2005 parameters, the basic reproduction number R20 is estimated to
be around 0.84, while in November 2005, after the mutation of the virus (βhm moved
from the value 0.365 to the value 0.95), we obtain R20 ≈ 2.20. This partly explains
the explosive epidemic of 2006, after the outbreak of 2005, Fig. 4 and 5 (see Dumont
and Chiroleu 2010).
5.2.2 Simulations with pulsed SIT and mechanical control: assessing the efficacy
To assess the efficacy of the control tools (also see Dumont and Chiroleu
we
N 2010),
n
compare the cumulative number of infected humans, i.e. C H =
n=0 Ih , over a
certain period of time [0, T ], with and without control, by considering the following
fraction:
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Sterile insect technique for the Chikungunya disease and Aedes albopictus
F0c = 100
c
CH
0
CH
,
0 and C c are the cumulative numbers of infected humans without and with
where C H
H
control, respectively. Hence, F0c indicates the efficacy of the control tools to reduce the
number of infected humans over a certain period of time: the lower F0c is, the better
the control tool.
Remark 9 In the following simulations, we assume that the disease is eradicated as
soon as the number of infected humans per week is less than 0.75. It is possible to
consider another threshold, like 0.5 or 0.25. We have verified that our simulations are
not sensitive to the choice of the threshold, i.e. whatever the threshold, we obtained
the same kind of results.
We test different possible scenarios to control or eradicate the disease. In particular,
we considered the following cases:
–
–
–
–
–
–
the number of sterile male S M per release,
different periodicities for the release,
different quality of the release: some places are not necessarily easily reachable
and this can affect the ”quality of the release” which will also have an effect on the
mating rate between sterile male and wild female. It is also necessary to simulate
different scenarios in order to take into account the case when the release is not
optimal (an optimal release may be near breeding sites, such that sterile males
could breed with immature females, immediately after their emergence).
the starting date of the release, ti , corresponds to the time lag between the emergence of the first case, in March 2005 in Saint-Denis, and the beginning of the
treatment. We mainly consider three start dates, namely ti = 100 days, which
corresponds to a couple of days after the peak of the outbreak of 2005, and ti =
200 days, which corresponds approximately to the beginning of the rainy season,
and ti = 300 days, which corresponds approximately to the beginning of the
vector control by the health authorities in 2006.
the duration of the SIT treatment.
The influence of a combination of SIT treatment and mechanical control, with
α = 0.75. This type of intervention was conducted in 2006 (it was called ”Kass
Moustik”’). It consists in reducing the number of breeding sites, at least near
inhabited areas. Because of the work load required by this intervention, it is necessary to involve the local population: the aim being to encourage inhabitants to
keep their gardens and neighborhood clean and in particular to reduce the number
of breeding sites. It is now admitted that Aedes albopictus stays in the area of its
birth place if it has suitable conditions to develop and to survive (blood and sugar
meals).
Using the parameters given in Table 1, page 33, we obtain R20 2.20, indicating that
an epidemic may occur. From the previous computations, the impact of the SIT can
be quite different depending on the values of S M , γ , r, f, μ S M , and T .
Remark 10 Following model (16)–(17), with continuous releases proportional to the
wild males population, with f being fixed by the irradiation protocol, and r being
123
8
1.
2
10
8
1.6
1.4
6
1.2
1
0.2
0.6 0.4
4
0.8
2
0.2 0.4 0.6 0.8
1
Λ
1.2 1.4 1.6 1.8 2
5
x 10
0
14
12
10
8
1.
4
12
SIT with Mechanical control (α=0.75). Evolution of R with respect to Λ and T
1.6
0
T: Periodicity of the Release (in days)
SIT without Mechanical control. Evolution of R with respect to Λ and T
14
2.2
T: Periodicity of the Release (in days)
Y. Dumont, J. M. Tchuenche
0.6
2
1.
6
0.8
0.4
1
0.2
4
2
0.2 0.4 0.6 0.8
1
Λ
1.2 1.4 1.6 1.8 2
5
x 10
1
2
1.
1.4
1.8
1.6
8
6
4
2
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8 2
5
x 10
Λ
0.
2
4
0.
0.
6
12
10
0.8
0.8
1
0.6
10
1.2
12
14
1.6
0.2
0.4
SIT with Mechanical control (α=0.75). Evolution of R0 with respect to Λ and T
1.4
0
14
T: Periodicity of the Release (in days)
SIT without Mechanical control. Evolution of R with respect to Λ and T
2
T: Periodicity of the Release (in days)
Fig. 6 Evolution of the basic reproduction number since r = 0.05: a without mechanical control b with
mechanical control (α = 0.75)
8
6
4
2
0.2 0.4 0.6 0.8 1
Λ
1.2 1.4 1.6 1.8 2
5
x 10
Fig. 7 Evolution of the basic reproduction number when r = 0.2: a without mechanical control b with
mechanical control (α = 0.75)
imposed by the environment, it suffices to choose γ to verify inequality (20). Since
R20 > 2, following Proposition 1, page 20, model (26)–(28) with continuous releases
is absolutely not useful, except if it is coupled with mechanical control or if a large
number of sterile males is continuously released, such that the wild population will be
driven to extinction. In particular, μ S M and f being fixed by the irradiation protocol,
it suffices to choose S M according to r to verify inequality (30).
Graphical representations depicting the evolution of R0, pulse with respect to T and
S M follow. We first consider a constant release rate S M whatever the periodicity
of the release, for different release qualities, with and without mechanical control.
In Fig. 6, it is quite clear that the combination SIT-Mechanical control is very powerful and leads to a drastic decay of R0, pulse . The smaller the period between two
releases is, the more efficient the SIT is. This result was somehow expected.
Figure 7 yields the same kind of results, i.e. the combination SIT-Mechanical control gives the best result. Moreover, comparing Figs. 6 and 7, the release quality appears
to be an important parameter.
123
0.5
1
1.5
2
2.5
3
3.5
SIT with Mechanical control (α=0.75). Evolution of R0 with respect to Λ and T
0.5
4
1
1
1.5
2
4
Λ
0.8
0.6
0.
4
0.
1.2
1.4
4
1.6
60
55
50
45
40
35
30
25
20
15
10
5
0.
2
0.
8
0.
0.4
0.
2
1
6
1.2
1.4
1.6
1.8
0
T: Periodicity of the Release (in days)
SIT without Mechanical control. Evolution of R with respect to Λ and T
60
55
50
45
40
35
30
25
20
15
10
5
2
T: Periodicity of the Release (in days)
Sterile insect technique for the Chikungunya disease and Aedes albopictus
2.5
3
3.5
4
4
Λ
x 10
x 10
1
1.5
2
Λ
2.5
3
3.5
4
4
x 10
1
0.8
0.6
0.4
0
0.2
0.2
0.5
SIT with Mechanical control (α=0.75). Evolution of R with respect to Λ and T
60
55
50
45
40
35
30
25
20
15
10
5
1.4
1.2
0.4
6
0.
0.8
1
1.2
1.4
1.6
1.8
55
50
45
40
35
30
25
20
15
10
5
0.4
0
T: Periodicity of the Release (in days)
SIT without Mechanical control. Evolution of R with respect to Λ and T
2
T: Periodicity of the Release (in days)
Fig. 8 Evolution of the basic reproduction number when r = 0.05: a without mechanical control, b with
mechanical control (α = 0.75)
0.5
1
1.5
2
2.5
3
3.5
4
4
x 10
Λ
Fig. 9 Evolution of the basic reproduction number when r = 0.2: a without mechanical control, b with
mechanical control (α = 0.75)
Now, instead of considering that at each T time we release r f S M “efficient”
sterile males, we consider r f S M T , in such a way that the number of sterile males
that are released may depend on a release rate and the periodicity. In other words, we
would like to compare small and frequent releases with large and rare releases, i.e.,
M S M (nT + ) = M S M (nT ) + r f S M T,
with n = 0, 1, 2, . . . , N T .
We obtain very interesting results, that are far different from those obtained in Figs. 6
and 7.
In Fig. 8, like in Figs. 6 and 7, the combination SIT-Mechanical control gives the
best results. As expected, if the release rate is not sufficiently large, the periodicity of
the release does not change R0, pulse . But if S M reaches a threshold value ∗ , then it
appears clearly that small and frequent releases may be more efficient than large and
rare releases. The threshold ∗ may vary upon mechanical control: compare Fig. 8a
and b.
In Fig. 9, we show that if we improve the quality of the release, we improve the
efficacy of the SIT.
123
Y. Dumont, J. M. Tchuenche
Efficacy of the SIT
Efficacy of the SIT with Mechanical Control
25
30
Periodicity of the release
Periodicity of the release
30
100
20
15
80
30
20
90
10
10
70
60
5
50
25
40
20
30
20
15
70
10
10
60
5
50
40
50
100
150
200
250
50
300
100
duration of the treatment
150
200
250
300
duration of the treatment
Fig. 10 Release rate of S M = 5 × 104 sterile Males with a poor release quality r = 0.05 for various periodicity and treatment duration. The start date is day 100: a without mechanical control, b with mechanical
control (α = 0.75)
Efficacy of the SIT with Mechanical Control
Efficacy of the SIT
30
25
Periodicity of the release
Periodicity of the release
30
5
100
40
20
90
30
20
15
5
10
10
5
70
5
5
5
25
5
60
20
70
30
10
15
40
10
5
50
20
5
50
80
50
100
150
200
250
duration of the treatment
300
50
100
150
200
250
300
duration of the treatment
Fig. 11 Release rate of S M = 5 × 104 sterile males with a good release quality r = 0.2 for various periodicity and treatment duration. The start date is day 100: a without mechanical control, b with mechanical
control (α = 0.75)
In the next set of simulations, we investigate the importance of the SIT start date,
like for the standard chemical control tools (Dumont and Chiroleu 2010). In Figs. 10
and 11, we present simulations for ti = 100 days. We assume that the quality of the
release is either poor, i.e. r = 0.05 (Fig. 10) , or good, r = 0.2 (Fig. 11). We also
consider various release rates with or without control. Since r = 0.05, there is a clear
improvement when the release is combined with mechanical control; in particular for
the same result, the duration of the treatment is lower and if the treatment’s duration is
large, then only a small amount of the population becomes infected. Clearly, mechanical control increases the efficacy of the SIT, and it seems that small and frequent
releases have more impact than larger releases.
In the case of a good release quality, i.e. r = 0.2, the results are of course far
better. Mechanical control still improves the result, but its impact is not so important
like in the previous case. Anyway, the combination permits to lower the duration of
the treatment and to consider larger releases. Interestingly, without mechanical control, the best results are obtained for lower and frequent releases. In contrary, with
123
Sterile insect technique for the Chikungunya disease and Aedes albopictus
30
25
20
80
15
30
40
90
30
60
10
30
70
50
30
5
50
100
150
200
250
Periodicity of the release
Periodicity of the release
Efficacy of the SIT with Mechanical Control
Efficacy of the SIT
30
25
30
70
20
20
50
15
10
40
60
10
5
300
50
100
duration of the treatment
150
200
250
300
duration of the treatment
Fig. 12 Release rate of S M = 5 × 104 sterile males with a poor release quality r = 0.05 for various periodicity and treatment duration. The start date is day 200: a without mechanical control, b with mechanical
control (start date is day 100, and α = 0.75)
Efficacy of the SIT with Mechanical Control
Efficacy of the SIT
30
25
Periodicity of the release
Periodicity of the release
30
10
70
20
10
20
10
15
10
80
90
10
10
60
50
5
10
10
40
25
10
50
40
30
20
15
70
20
10
60
5
30
50
100
150
200
250
duration of the treatment
300
50
100
150
200
250
300
duration of the treatment
Fig. 13 Release rate of S M = 5 × 104 sterile males with a good release quality r = 0.2 for various periodicity and treatment duration. The start date is day 200: a without mechanical control, b with mechanical
control (start date is day 100, and α = 0.75)
mechanical control, the periodicity of the release is less important, and in that case,
it seems possible to consider lower releases with a large frequency. This is important
from a practical point of view. In fact, the larger the release rate and/or the quality of
the release, the better are the results, with or without mechanical control.
Let us now consider ti = 200: see Figs. 12 and 13. Clearly, for r = 0.05, the SIT is
efficient only if the treatment duration is large and the releases are relatively frequent.
If we suppose that the mechanical control began at day 100, then the combination of
SIT and mechanical control leads to a far better result: compare Fig. 12a and b.
Finally, if the quality of the release is rather good, i.e. r = 0.2, then the SIT gives
very good results and even better when combined with the mechanical control: compare Fig. 13a and b. Of course, this may be an idealized condition, where release
sites are easy to reach, and/or when the releases are made near breeding sites. Many
facts may contribute to get very low values for r . Therefore, it is preferable to present
simulations for both cases.
123
Y. Dumont, J. M. Tchuenche
Efficacy of the SIT
Efficacy of the SIT with Mechanical Control
30
Periodicity of the release
Periodicity of the release
30
90
25
20
90
90
15
90
90
10
90
5
50
100
150
200
250
40
70
40
25
70
50
20
15
70
10
40
60
5
300
50
100
duration of the treatment
150
200
250
300
duration of the treatment
Fig. 14 Release rate of S M = 5 × 104 sterile males with a poor release quality r = 0.05 for various periodicity and treatment duration. The start date is day 300: a without mechanical control, b with mechanical
control (start date is day 100, and α = 0.75)
Efficacy of the SIT
Efficacy of the SIT withMechanical Control
30
30
Periodicity of the release
Periodicity of the release
30
90
90
25
90
90
90
20
15
90
10
5
50
100
150
200
250
duration of the treatment
300
30
25
60
20
30
70
15
40
50
10
5
50
100
150
200
250
300
duration of the treatment
Fig. 15 Release rate of S M = 5 × 104 sterile males with a good release quality r = 0.2 for various periodicity and treatment duration. The start date is day 300: a without mechanical control, b with mechanical
control (start date is day 100, and α = 0.75)
Improving on the work in Dumont and Chiroleu (2010), we consider ti = 300. For
this case, the SIT is not efficient at all, even with r = 0.2: see Figs. 14a and 15a.
In some sense, because the SIT is a biological process, its effect is not instantaneous
like insecticide. Thus, if the SIT is used too late, then it is absolutely not efficient.
However, if we combine the SIT with mechanical control, then the results are clearly
better: see Figs. 14b and 15b.
Note also that we assume that the mechanical control began earlier (at day 100 after
the first case), which shows clearly the importance of this control tool and of having
support from the local population in order to lower as much as possible the number of
breeding sites, in particular during the rainy season.
In summary, various models to describe the sterile insect technique in the case of
Chikungunya disease to which the Aedes albopictus mosquito is the vector are presented using the classical S I and S I R framework. They are coupled with models for
the dynamics of aquatic stages (eggs, larvae and pupae), first proposed in Dumont et al.
(2008) and Dumont and Chiroleu (2010). Different stages for adult mosquito dynamics are considered, with a distinction made between mature and immature females,
123
Sterile insect technique for the Chikungunya disease and Aedes albopictus
and also, between wild and sterile insects. A framework is developed to examine the
influence of sterile insects as they are used to mate with females, thereby reducing the
proliferation of the species. This consideration leads to a complete model for which
the theoretical study is made step-by-step, considering for instance, only the population dynamics of wild mosquitoes, the wild mosquito–human transmission dynamic,
etc. Local and global stabilities are studied, using classical mathematical tools such
as three dimensional cooperative systems, Metzler matrix or Floquet multipliers and
the Kamgang–Sallet approach for the stability of the disease-free equilibrium. Three
approaches for the use of sterile insects are compared, namely: the case where the
release of sterile mosquitoes is proportional to the number of wild mosquitoes, the
case of a continuous release of sterile mosquitoes and finally, a periodic release of
sterile mosquitoes. When possible, the basic reproduction number is computed using
the next generation operator (Van den Driessche and Watmough 2002), and when not
possible, the disease threshold is estimated directly through entomological and epidemiological parameters. Several remarks throughout the text help explain the biological
implication of those thresholds.
6 Conclusion
This study on the dynamics of Chikungunya disease complements two previous ones
(Dumont et al. 2008; Dumont and Chiroleu 2010). Insecticide is not sustainable: for
massive spraying and long time use, it is lethal for many other species and, moreover,
it can induce a resistance effect on the mosquito population. It is therefore crucial to
explore other ways and in particular to develop biologically sustainable vector control
alternative that only focuses on the targeted mosquito population, Aedes albopictus.
Mathematical models of SIT have been known for the past 50 years (Knipling 1955),
and has been used more or less successfully many times (Alphey et al. 2010).
Various ingredients are necessary to maximize the success of SIT: first, to have a
good understanding of the biology of the virus, second, the dynamics of the vector, and,
third, to know the impact of the SIT (due to the radiation) on Aedes albopictus biology
(lifespan and competitivity). The first two conditions are almost reached (research is
still actively being conducted on Chikungunya and Aedes albopictus). The third one
which is missing is addressed herein. We used results obtained by Bellini et al. (2007)
in Italy, but because the focus is on Réunion Island, it is necessary to provide similar
results in Réunion Island. Research on this subject is being conducted at the International Atomic Energy Agency in Vienna and in Réunion Island, and preliminary
results are in good agreement with Bellini’s experiments. Nevertheless, in the case of
pulsed releases, our theoretical study and prospective numerical simulations clearly
indicate that it is better to have small and frequent releases than large and rare releases.
The computation of R0, pulse according to the periodicity and the release rate suggest
that it may be possible to control the epidemic, but only if the releases are done early
in the epidemic. An estimate of the epidemic threshold in Réunion Island conveys
important information about the prospects for effective control of future Chikungunya
outbreaks. Moreover, the analysis on the wild mosquito with pulsed releases recovers
most of the conclusions in White et al. (2010).
123
Y. Dumont, J. M. Tchuenche
The irradiation parameters are very important: the aim is to reach 100% of sterility
such that the lifespan of the sterile male and the relative competitivity may not be
too much affected. The quality of the releases, r , is also an important parameter. In
particular, we know that from place to place it is not always possible to reach a better
release quality, and our simulation show that the results are sensitive to r . Since the
SIT is not use as a preventive tool, but as a control tool (like insecticides), the start date
of the SIT campaign is very important for standard vector control tools: the earlier,
the better. Finally, like in Dumont and Chiroleu (2010), the combination of SIT and
mechanical control provides the best control strategy. Our results are in agreement with
entomological knowledges and field results, but for efficiency, mechanical control has
to be carried out daily with the help of the local population.
Aedes albopictus is now found in many countries around the world, eg. Italy and
the South of France. Dengue and even Chikungunya cases have been recorded in September 2010 in the South of France, indicating that potential for an outbreak cannot be
ruled out. Thus, Chikungunya is no longer geographically localised in tropical areas
only. The disease poses a global threat to public health. This study also underscores the
need for designing continuous and targeted epidemiologic surveillance, which constitutes an important component of the public health response to an epidemic. Epidemic
data, combined with geographic and demographic information of areas most at risk
can provide valuable insights into the nature of the disease spread, help project its time
course, and provide guidance on optimal control strategies.
This study is not complete. Despite the availability of theoretical methods and
advanced numerical algorithms to build reliable dynamic consistent schemes to
provide efficient numerical simulations, further research is needed to understand
the vector–virus–human dynamics. The spatial component is also important. Several
Mark-Release-Capture experiments have been carried out (Lacroix et al. 2009; Boyer
et al. 2011a) in order to estimate the range of displacements of the mosquito, its mean
lifespan, and to model mosquitoes displacement with respect to weather conditions
and landscape roughness. Work is ongoing on how to better handle these experiments
and to build a fairly comprehensible model accounting for spatial structure (Dufourd
and Dumont 2011; Dumont 2011).
Finally, let us mention that in the TIS-project (a project, located in Reunion Island,
dedicated to assess the feasability of SIT as a vector control tool), Anopheles mosquito
is studied too (Anguelov et al. 2011a,c).
Acknowledgments Thanks to the reviewers for insightful comments. YD is supported by the TIS project and the French Ministry of Health. JMT acknowledges with thanks the support of the Schlumberger
Foundation African Scientist Visiting Fellowship at Clare Hall, University of Cambridge (UK). AMAP
(Botany and Computational Plant Architecture) is a joint research unit which associates CIRAD (UMR51),
CNRS (UMR5120), INRA (UMR931), IRD (2M123), and Montpellier 2 University (UM27); http://amap.
cirad.fr.
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