Applied Mathematical Sciences, Vol. 9, 2015, no. 74, 3659 - 3668 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.52150
Population Dynamics of Aedes aegypti Considering Quiescence John Faber Arredondo Montoya, Anibal Mu˜ noz Loaiza and Carlos Alberto Abello Mu˜ noz Universidad del Quind´ıo Grupo de Modelaci´on Matem´atica en Epidemiolog´ıa Armenia Quind´ıo, Colombia
c 2015 John Faber Arredondo Montoya et al. This article is distributed under Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract Abstract: We show three non-linear differential equation models to depict the dynamics of growth of the Aedes aegypti (dengue transmitter) population using a time-dependent developing rate. We proposed a model considering embryonic stages of the vector under adverse environmental conditions, these vectors are able to adapt and remain without biological activity for long-time periods (quiescence). These characteristics allow us to use previously reported parameters in the simulations. In the last model we do not use quiescence in order to compare and figure out the principal differences between these models.
Keywords: Aedes aegypti, dengue, quiescence, simulations
1
Introduction
Mathematical models used in dengue have contributed on the implementation of strategies against virus propagation, through setting several models involving virus dynamics and vector population growth (Aedes aegypti ). Without favorable breeding conditions, i.e., in absence of appropriate recipients, fumigation, or eradication, the eggs will not hatch. So then, the A.
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aegypti eggs can stand in a quiescent state for almost 450 days [3]. Once that these eggs are in contact with water, it is possible to reactive their biological development and they will hatch in about 10 or 15 minutes. Therefore, it is interesting to analyze the effect of quiescence on the population growth of the Aedes aegypti, trying to establish the impact of different strategies applied for the control of early and mature stages of the mosquitoes. In the 90’s it was reported a experimental work in Brazil, the title is “Influˆencia do per´ıodo de quiescˆencia dos ovos sobre o ciclo de vida de Aedes Aegypti (Linanaeus, 1762) (Diptera, Culicidae) em condi¸co˜es de laborat´orio” [3]. In that paper it was observed that this effect is highly meaningful in the larvae hatch and his average lifetime, as function of the time they are in quiescence. The statement “...altamente significativo de quiescencia sobre la eclosi´on de las larvas” [3], allow us to consider the viability of a control analysis of the vector using this factor, because it is proved that in this stage the Aedes aegypti has a highest resistance. Actually, the main efforts to control the population growth of the mosquitoes are focused on the realization of fumigations to eliminate the mature vectors. Also, trying to eliminate the possible breeding places in order to avoid the biological development of new mosquitoes. However, as in the quiescent state the eggs can live for several months in abiotic conditions, it is important to include this parameter into the dynamics of the mosquitoes. Our paper is organized as follows: In section 2 we present the preliminary notes, and in section 3 we display the main conclusions.
2
Preliminary Notes
To determine the importance of the quiescence phenomenon on the population dynamics of the A. aegypti, we propose three models, the first two models are used to study the quiescence with a constant delay and distributed delay. In the last model we do not use quiescence in order to compare with the previous ones.
2.1
Constant delay model
In this model we use three differential ordinary equations to set the dynamics of the Aedes aegypti growth dividing the population of early stages of the mosquitoes in viable eggs y and larvae-pupa z at time t, considering a delaying time τ as the transition time (egg to larvae-pupa).
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The dynamics is represented in the following compartmental diagram βz
{
x
x
φx(1− y+z K )
/
y
αy
θy(t−τ )
/
z
δz
Then, the nonlinear system that plays the dynamics is dx = βz − x dt dy y+z = φx 1 − − αy − θy(t − τ ) dt K dz = θy(t − τ ) − (δ + β)z dt
(1) (2) (3)
where x is the average number of mature mosquitoes at time t, y is the average number of viable eggs at time t, z represents the average of immature stages (larvae-pupa) at time t, β is the developing rate of the immature stages transforming to the adult stage, is the mortality rate of the mature mosquitoes, φ is the oviposition rate of the mosquitoes, K the charge capacity of the breeding places, α is the elimination rate of the eggs, θ is the egg-tolarvae transformation rate, τ is defined as the delaying time (time used in the transformation of an egg to larvae-pupa considering quiescence). Finally, δ defines the death rate of the immature stages (larvae-pupa) owed to environmental factors. The biological sense region of the system is defined in Γ = {(x, y, z) : x, y, z ≥ 0, 0 ≤ y + z ≤ K}, namely, all the non-negative population and the immature stages (egg-larvae-pupa) do not exceed the charge capacity of the environment. 2.1.1
Simulations
For the system showed in equations (1 - 3) and using the parameters defined in the table 1, we obtain the simulations presented in the figures 1 and 2. Figure 1 shows that in the first 50 days (delay time to simulate the quiescent phenomenon) the population increase until a specific value, but after the
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Figure 1: Population dynamics of the mature stages (Aedes aegypti ), viable eggs and larvae-pupa plotted as function of the time t. a) left sub-figure obtained using initial conditions x0 = 50, y0 = 300 and z0 = 100. b) right sub-figure with initial conditions x0 = 2000, y0 = 2500 and z0 = 2300.
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Figure 2: Population dynamics of the adult stages, viable eggs and larvae-pupa as function of the time t, using initial conditions x0 = 3500, y0 = 4150 and z0 = 3950.
quiescent time the eggs begin to hatch (also the latent-state eggs) generating a decrement in the egg’s population but increasing the remaining ones. In figure 2 we can see that this behavior is repeated every 100 days (that is a whole time period of quiescence, beginning, middle stages, end, beginning) but it tends to stabilize with a fix value after 2200 days approximately. It is important to mention that the larvae-pupa population is highest than that of the mature population, being a consequence of the larvae-pupa rate transforming to the mature stage, where it is not showed a whole biological development of that population.
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2.2
Distributed-delay model
The delay time function in the egg stage y, transforming to the larvae-pupa z, is represented as [6]: Z t F (k) (t − τ )y(τ )dτ −∞
And the system is changed to dx = βz − x dt Z t dy y+z = φx 1 − − αy − θ F (k) (t − τ )y(τ )dτ dt K −∞ Z t dz = θ F (k) (t − τ )y(τ )dτ − (δ + β)z dt −∞
(4) (5) (6)
where F (k) is a gamma function as defined in [6] ak F (k) (τ ) = τ k−1 e−aτ , k ∈ N+ , a > 0 (k − 1)! 2.2.1
Gamma function using k = 1
In this case F (t − τ ) = ae−a(t−τ )
,
a>0
and the system under study is represented as dx = βz − x dt Z t dy y+z = φx 1 − − αy − θ ae−a(t−τ ) y(τ )dτ dt K −∞ Z t dz = θ ae−a(t−τ ) y(τ )dτ − (δ + β)z dt −∞ This system is equivalent to a high order ordinary differential equation system, that results using the substitution [2] Z t Q(t) = ae−a(t−τ ) y(τ )dτ −∞
from where dQ(t) =a dt
Z
s
−∞
∂ −a(t−τ ) d [e y(τ )]dτ + ∂t dt
then, we obtain the following system
Z
t
e s
−a(t−τ )
y(τ )dτ
= a(y(t) − Q(t))
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dx dt dy dt dz dt dQ dt
= βz − x y+z = φx 1 − − αy − θQ K = θQ − (δ + β)z
(7) (8) (9)
= a(y − Q)
(10)
with initial conditions x(0) = x0 , y(0) = y0 , z(0) = z0 y Q(0) = Q0 . 2.2.2
Equilibrium points
If we consider the equations (7-10) without population variations, it is obtained a homogeneous system, as follows: 0 = βz − x y+z 0 = φx 1 − − αy − θQ K 0 = θQ − (δ + β)z 0 = a(y − Q)
(11) (12) (13) (14)
Solving the homogeneous equations,(11) to obtain x, (13) to obtain Q, and (14) for y δ+β δ+β β z Q= z y=Q= z (15) θ θ Using (15) in (12) we obtain. φβ (δ + β + θ)z (α + θ)(δ + β) 1− z− z=0 θK θ And the equilibrium points are obtained solving the previous equation β ∗ δ+β ∗ ∗ δ+β ∗ E0 = (0, 0, 0, 0) y E1 = z , z ,z , z θ θ where x=
z∗ =
E1∗
K(α + θ)(δ + β)(ρ − 1) φβ(δ + β + θ)
with
ρ=
φθβ (α + θ)(δ + β)
Using the previous reported parameters showed in the table 1 we acquire = (1944, 2431, 1944).
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Parameters value Reference 0.1 [7] β 0.1 [1] δ 0.4 [4] K 5000 -----
Parameters value Reference φ 6 [1] α 0.2 ----θ 0.4 -----
Table 1: Previously reported parameters. 2.2.3
Simulations
The corresponding simulations for the system (7-10) using the reported data in the table 1 are depicted in figure 3. It is clear that the population tends to stabilize for the values E1∗ = (1944, 2431, 1944) (in presence of an equilibrium with coexistence), where the population sizes remain as: mature stages and larvae-pupa < eggs. Different values of the a parameter were used to evaluate the gamma function, the plotted graphics showed a similar behavior. We report the case with a = 0.1.
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Figure 3: Population dynamics of Aedes aegypti mature stages, viable eggs and larvae-pupa as function of t. a) left sub-figure obtained using initial conditions x0 = 50, y0 = 300 and z0 = 100. b) right sub-figure with initial conditions x0 = 3000, y0 = 2500 and z0 = 2850.
2.3
System without delay
We set a model with immature stages (larvae-pupa) transforming to the adult stage at constant developing rate, without presence of the quiescence phenomenon, the obtained equations system is:
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dx = βz − x dt dy y+z = φx 1 − − (α + θ)y dt K dz = θy − (δ + β)z dt 2.3.1
(16) (17) (18)
Equilibrium points
Setting to zero the equations (16-18), we obtain the following system %vspace1cm
0 = βz − x y+z 0 = φx 1 − − (α + θ)y K 0 = θy − (δ + β)z
(19) (20) (21)
solving (19) to obtain x and (21) for z, we have β z these equations are equivalent to x=
x=
βθ z (δ + β)
z=
θ δ+β
z=
θ δ+β
Evaluating these results in (17) we have φβθ δ+β+θ 1− − (α + θ) y = 0 (δ + β) K(δ + β) Equilibrium with no coexistence E0 = (0, 0, 0) is obtained using y = 0 and also equilibrium with coexistence E1 , where φβθ δ+β+θ 1− − (α + θ) (δ + β) K(δ + β) this is δ + β + θ K(δ + β) y = 1− K(δ + β) δ + β + θ ∗
so then
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E1 = [1 − ρ1 ]
K(δ + β) Kθ Kβθ , [1 − ρ1 ] , [1 − ρ1 ] (δ + β + θ) δ+β+θ δ+β+θ
δ+β+θ and using the parameters reported in the table 1, we K(δ + β) obtain E1∗ = (1945, 2431, 1945). where ρ1 =
2.3.2
Simulations
The simulations for the system (16-18) using the reported parameters in the table 1 are plotted in figure 4.
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Figure 4: Population dynamics of the adult stages, viable eggs and larvae-pupa as function of the time t. a) letf sub-figure obtained using initial conditions x0 = 50, y0 = 300 and z0 = 100. b) right sub-figure obtained using initial conditions x0 = 3000, y0 = 2500 and z0 = 2850.
We observe that population tends to stabilize immediately in the equilibrium values E1∗ , these values are higher than that of the ones obtained with quiescence.
3
Main Results
The previous simulations showed that the quiescence phenomenon decreases in the mature and immature (larvae-pupa) population, these results are in good agreement with the ones acquired in Brazil [3]. Those studies showed that even if the quiescence remains for a long time, not all the eggs in a quiescent state are able to hatch and the average lifetime is also affected negatively because of quiescence.
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Considering a constant delay, the population tends to stabilize in a oscillatory form by intervals, where, the egg’s population is reduced an the remaining ones are increased (and inversely). This fact emulates the quiescence phenomenon on the population dynamics. The previous statements indicate that control strategies such as, breeding place elimination works to eliminate the places for a future oviposition and helps to diminish the immature stages population, even if the oviposition of the eggs was made. Acknowledgements. We acknowledge to GMME group for the support to develop this investigation. The autors thank M.E. Dalia Marcela Mu˜ noz P., and M.C. J. Guerrero-S´anchez (IFUAP-BUAP) for useful discussions.
References [1] Y. Dumont, F. Chiroleu, C. Domerg, On a temporal model for the Chikungunya disease: Modeling, theory and numerics. Mathematical Biosciences 213 (2008) 80-91. http://dx.doi.org/10.1016/j.mbs.2008.02.008 [2] M. Fargue, R´educibilit´e des systems hereditaires a des systemes dynamiques. En: C. R. Acad. Sci. Paris Ser. 277 (1973), p. 471-473. [3] H. Garc´ıa, I. Garc´ıa, Influencia do per´ıodo de quiescencia dos ovos sobre o ciclo de vida de Aedes Aegypti (Linanaeus, 1762) (Diptera, Culicidae) em condicoes de laborat´orio. Revista da Sociedade Brasileira de Medicina Tropical. 349-355. http://dx.doi.org/10.1590/s0037-86821999000400003 [4] C. Mar´ın, A. Mu˜ noz, H. Toro, L. Restrepo, Modelado de estrategias para el control qu´ımico y biol´ogico del Aedes aegypti (Diptera: Culicidae). Escuela Regional de Matem´aticas Universidad del Valle - Colombia. Vol. XIX, No 1, Junio (2011) p. 63-78. [5] A. Salvatella, Aedes aegypti (Linnaeus,1762) (Diptera, Culicidae), el vector del dengue y la fiebre amarilla. [6] Y. Takeuchi, W. Ma, Delayed SIR Epidemic Models for Vector Diseases. National Natural Science Foundation of China; Foundation of USTB for WM and the Ministry; Science and culture in Japan. [7] J. Thirion, El Mosquito Aedes aegypti y el dengue en M´exico. Bayer Environmental Science. Bayer de M´exico, S.A. de C.V. (2003). Received: March 5, 2015; Published: May 4, 2015
