EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1613-1622 ISSN 1307-5543 – ejpam.com Published by New York Business Global Discrete Wolbachia Diffusion in Mosquito Populations with Allee Effects Unal Ufuktepe College of Engineering and Technology, American University of the Middle East, Egaila 54200, Kuwait Abstract. We study stability analysis of a discrete-time dynamical system of Wolbachia diffusion in mosquito populations with Allee effects on the wild mosquito population. We analyze the competition between released mosquitoes and wild mosquitos. We show local and global stabilities of the fixed points, and type of bifurcations concerning parameters. The results are verified by numerical simulations. 2020 Mathematics Subject Classifications: 39A28, 39A30, 39A33, 39A60 Key Words and Phrases: Competition model, discrete dynamical systems, stability, bifurcation, fixed point, manifolds 1. Introduction Malaria, dengue fever, West Nile virus, chikungunya, and Zika virus are well-known Mosquito-borne diseases. More than a billion people are at risk of these diseases all around the world. It has been estimated that 3.9 billion people are at risk of infection [1, 2]. The human viruses including dengue, Zika, chikungunya, and yellow fever are transmitted primarily by Aedes aegypti mosquitoes. Due to the lack of vaccines and efficient clinical cures [3], the only effective control strategy seems to be controlling the population of mosquitoes that transmit human viruses. Since massive spraying of insecticides and elimination of mosquito breeding sites are not sustainable 4400 to reduce mosquito density and might also lead to serious environmental problems, a promising strategy is the Wolbachia approach: releasing male and female Aedes aegypti mosquitoes with Wolbachia so that these mosquitoes can breed with the wild mosquito population and pass Wolbachia to the entire mosquito population. On one hand, the ability to transmit viruses to humans for mosquitoes with Wolbachia is greatly reduced [4]. On the other hand, since the Wolbachia infection often induces cytoplasmic incompatibility (CI), which leads to early embryonic death when Wolbachia-infected males mate with uninfected females [5], the Wolbachia approach would greatly reduce the density DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4524 Email address: unal.ufuktepe@aum.edu.kw (U. Ufuktepe) https://www.ejpam.com 1613 © 2022 EJPAM All rights reserved. U. Ufuktepe / Eur. J. Pure Appl. Math, 15 (4) (2022), 1613-1622 1614 of the mosquito population and can thus potentially eliminate the mosquito population and thus eradicate the mosquito-borne infectious diseases. The sterile mosquitoes technique in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population has been used in preventing malaria transmission. To study the impact of releasing sterile mosquitoes on malaria transmission, In [9], they formulate a simple SEIR (susceptible-exposed-infected-recovered) malaria transmission model as our baseline model, to derive a formula for the reproductive number of infections, and determine the existence of endemic equilibria. They then include sterile mosquitoes in the baseline model and consider the case of constant releases of sterile mosquitoes. They examine how the releases affect the reproductive numbers and endemic equilibria for the model with interactive mosquitoes and investigate how releasing sterile mosquitoes affects malaria transmission. The use of the Wolbachia strategy to suppress vector populations is a novel approach, which has the potential to reduce mosquito populations and the risk of mosquito-borne disease transmission [1, 2]. This approach, commonly known as the Incompatible Insect Technique (IIT), is a species-specific and benign approach for controlling vector popu- lations. Eggs produced from the successful mating between released male Wolbachia- carrying Aedes aegypti (Wolbachia-Aedes) mosquitoes and urban female Aedes aegypti mosquitoes in the environment (without Wolbachia) are non-viable, due to Cytoplasmic Incompatibility (CI), therefore suppression of mosquito populations could be achieved with regular releases over time. Wolbachia technology is a novel vector control approach that can reduce mosquito populations and the risk of mosquito-borne diseases, which has recently gained popularity amongst countries. In 2016, Singapore embarked on a multi-phased field study named Project Wolbachia – Singapore, to evaluate the use of Wolbachia technology as an Aedes aegypti mosquito population suppression tool to fight dengue. Due to the novelty of this technology in Singapore, this study aims to understand the public’s acceptance and sentiments towards the use of Wolbachia technology [10]. Threshold values for the releases of sterile mosquitoes are derived for all of the models that determine whether the wild mosquitoes are wiped out or coexist with the sterile mosquitoes. Numerical examples are given to demonstrate the dynamics of the models. 2. Stability Analysis of The Discrete Model There are many population models about mosquitoes [7–9]. Below the discrete model of competition between two species is given by [6]. xt+1 = b1xt 1 + α(xt + yt) + (1− d1)xt, yt+1 = b2yt 1 + β(xt + yt) yt (xt + yt) + (1− d2)yt, (1) U. Ufuktepe / Eur. J. Pure Appl. Math, 15 (4) (2022), 1613-1622 1615 where xt represents the number of mosquito population infected with Wolbachia, and yt represents the number of uninfected mosquito population at time t. The parameters b1, b2 > 0 denote the birth rate of xt, yt, respectively. α and β denote the competition coefficients within or between species, respectively. 0 < d2 < d1 < 1 are mortality rates of yt and xt. We add the Allee effect to uninfected mosquitoes in this model. xt+1 = b1xt 1 + α(xt + yt) + (1− d1)xt, yt+1 = b2yt 1 + β(xt + yt) yt (xt + yt) + (1− d2)yt yt c+ yt) , (2) where c > 0 is the Allee effect constant. The study of the dynamical properties of this map allows us to have information about the future behavior of mosquitoes populations. The positive invariant region of our model is as follows. Let a = 1− d2: yt+1 = b2yt 1 + β(xt + yt) yt (xt + yt) + (1− d2)yt yt (c+ yt) (3) We get the following inequality from this equation yt+1 ≤ b2 β + (1− d2)yt. (4) By t iteration of this recurrence inequality, we get yt+1 − aty0 ≤ b2 β (1 + a+ a2 + · · ·+ at−1) (5) since 0 < a < 1, then lim t→∞ yt ≤ b2 d2β . Similarly, lim t→∞ xt ≤ b1 d1α . Therefore, there is a positive invariant for our mapping. U. Ufuktepe / Eur. J. Pure Appl. Math, 15 (4) (2022), 1613-1622 1616 3. Fixed Points and their stability In this section, We investigate the fixed points of the map (2) and their stability conditions[11, 12]. Let xt+1 = f(xt, yt) = b1xt 1 + α(xt + yt) + (1− d1)xt, yt+1 = g(xt, yt) = b2yt 1 + β(xt + yt) yt (xt + yt) + (1− d2)yt yt (c+ yt) , (6) where we assumed g(0, 0) = 0. Then the solution of the following system of equations gives us the fixed points: x = b1x 1 + α(x+ y) + (1− d1)x, y = b2y 1 + β(x+ y) y (x+ y) + (1− d2)y y (c+ y) . The fixed points are F0(0, 0), F1( b1−d1 αd1 , 0), F2(0, y ∗) where y∗ = √ (βc+b2c−d2)2+4c(b2−βd2)−βc+b2c−d2 2(βd2−b2) or y∗ = √ (−βc+b2c−d2)2+4c(b2−βd2)+βc−b2c+d2 2(b2−βd2) , and F3(A,B) where A = 1 α2b2d21 (a+ b) (a = αb1b2d1 + alphab1d1 − alphab2d 2 1 − alphad21 + βb21 − 2βb1d1 + βd21, and b = α2b2cd 2 1 − αb1d1d2 + αd21d2 − βb21d2 + 2βb1d1d2 − βd21d2) B = 1 α2b2d21 (m+ n). (m = −α2b2cd 2 1−αb1d1+αd21−βb21+2βb1d1−βd21 and n = βb21d2+αb1d1d2−2βb1d1d2− αd21d2 + βd21d2 The Jacobian matrix of the map (2) is: J(x, y) = ( fx fy gx gy ) , where fx(x, y) = b1 (1 + α(x+ y)) − αb1x (1 + α(x+ y))2 + (1− d1) U. Ufuktepe / Eur. J. Pure Appl. Math, 15 (4) (2022), 1613-1622 1617 , fy(x, y) = − ab1x (a(x+ y) + 1)2 , gx(x, y) = − b2y 2 (x+ y)2(1 + β(x+ y)) − βb2y 2 (x+ y)(1 + β(x+ y))2 + 2b2y (x+ y)(1 + β(x+ y)) − (1− d2)y 2 (c+ y)2 + 2(1− d2)y c+ y , and gy(x, y) = − b2y 2 (x+ y)2(1 + β(x+ y) + 1) − βb2y 2 (x+ y)(β(x+ y) + 1)2 + 2b2y (x+ y)(β(x+ y) + 1) − (1− d2)y 2 (c+ y)2 + 2(1− d2)y c+ y , For the fixed point F0(0, 0) the Jacobian matrix is ; J(0, 0) = ( 1− d1 + b1 0 0 0 ) . The eigenvalues are λ1 = 1 − d1 + b1 and λ2 = 0. Then if |1 − d1 + b1| < 1, then F0 is asymptotically stable fixed point. The Jacobian matrix for the fixed point F1( b1−d1 αd1 , 0) is ; J(F1) = ( 1− d1 + d21 b1 0 0 0 ) . The eigenvalues are λ1 = 1 − d1 + d21 b1 and λ2 = 0. Then if |1 − d1 + d21 b1 | < 1 the F1 is asymptotically stable fixed point otherwise unstable. The Jacobian Matrix for the fixed point F2(0, y ∗) is; J(F2) = ( 1 0 gx(F2) gy(F2) ) . The eigenvalues are λ1 = 1 and λ2 = gy(F2). We have non-hyperbolic fixed point. The Center Manifold Theorem must be applied. We explain stability of F2 in the Numerical result section and gave the center manifold curves. U. Ufuktepe / Eur. J. Pure Appl. Math, 15 (4) (2022), 1613-1622 1618 For the fixed point F3(A,B) = the Jacobian Matrix is; J(F2) = ( B11 B12 B21 B22 ) . where B12 = b(b1 − d1) 2 − αd1 ( (b1(αcd1 − d1 − 1) + b12 + d1 ) α(b21 , B11 = b21(α+ β) + d21(α(−αb1c+ b1 − 1) + β) + b1d1(α− b1) + α− 2β) αb21 , B21 = − b2(αd1 + 2β(b1 − d1)) ( β(b1 − d1) 2 − αd1(b1(αcd1 − 1) + d1) )2 αb21d1(b1 − d1)2(αd1 + β(b1 − d1))2 , and B22 = ( αd1(b1(αcd1 − b2β(b1 − d1) 2 ) G αb21d1(b1 − d1)2(αd1 + β(b1 − d1))2 where G = (α2d21 ( b31(d2 − 1)(αcd1 + 1) + b1b2(d1(2− αc) + 1) + b21(−2b2 − d1d2 + d1)− b2d1 ) + αβd1(b1−d1) ( b1b2(d1(2− 2αc) + 3) + b31(d2 − 1) + b21(−2b2 − d1d2 + d1)− 3b2d1 ) +2β2b2(b1−d1) 3) By the trace-determinant (Jury Condition), if |trJ | − 1 < detJ < 1 The positive fixed point is stable. 4. Numerical Results In this section, we verify the theoretical results of our model by numerical simulations. We use Mathematica and Sage software for these simulations. In order to investigate the impact their interaction and the Allee effect on we changed the values of b1, b2, c, α, β, and the mortality rates (d1, d2). We fix the intraspecific competition coefficients (cij = 1). The most important result we get, F1 is globally asymptotically stable, this means that whatever the initial value of Wolbachia-infected mosquito population xt, the population of uninfected mosquito population yt will extinct but xt will persist with respect to Allee effect. U. Ufuktepe / Eur. J. Pure Appl. Math, 15 (4) (2022), 1613-1622 1619 Figure 1: Phase portrait for the model (2) for F1 fixed poınt, α = β = 1.8, c = 8, d1 = 0.3, d2 = 0.3 and b1 = b2 = 1, (x0, y0) = (5, 0.1). Figure 2: Phase portrait for the model (2) for F2 fixed poınt, α = β = 0.5, c = 8, d1 = 0.25, d2 = 0.5 and b1 = 1, b2 = 5, (x0, y0) = (0.1, 2). We take α = β = .5, b1 = 1, b2 = 5, c = 8, d1 = .25, 2 = .5, we get the fixed point F2(0, 12.4), and λ1 = 0.89, λ1 = 0.51 eigenvalues. The center manifold for these parameters U. Ufuktepe / Eur. J. Pure Appl. Math, 15 (4) (2022), 1613-1622 1620 Figure 3: a) for F1 fixed poınt, α = β = 1.8, c = 8, d1 = 0.3, d2 = 0.3 and b1 = b2 = 1, (x0, y0) = (5, 0.1). b) Time series of (x, y) for the model for F1 fixed poınt, α = β = 1.8, c = 8, d1 = 0.5, d2 = 0.2 and b1 = b2 = 1, (x0, y0) = (5, 0.1). Figure 4: Phase diagram with isoclines for the model for F1 fixed poınt, α = β = 0.8, c = 1.8, d1 = d2 = 0.5, and b1 = b2 = 1, (x0, y0) = (2, 3). REFERENCES 1621 are h(x) = 20x3 − 316x2 + 17x, and (y) = 0.0001y3 − 0.04y2 (with green and blue colors in Figure 4). In Figure 1. and Figure 2. , we give the phase diagrams for the fixed points and In Figure 3. we give the time series of the model. if we let c = 0, that means there is no Allee Effect, and keeping the other parameters the same, we see that only the positive fixed point is stable. 5. Conclusion An innovative and effective method to control mosquitoes is to employ Wolbachia, which has led to a growing number of researchers building models to study the dynamics of Wolbachia transmission. Considering that the collection data of mosquitoes in the wild are discrete, we established a discrete competition model to study the conditions for Wolbachia to successful spread in mosquitoes. Numerical simulations are also provided to demonstrate these theoretical results. We mainly showed that the simulation results are consistent with the theoretical results. 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