Simulating the Effect of Aedes aegypti by the Acquired Resistance to Chemicals

Transcription

1 Applied Mathematical Sciences, Vol. 12, 2018, no. 29, HIKARI Ltd, Simulating the Effect of Aedes aegypti by the Acquired Resistance to Chemicals Carlos A. Abello M., Humberto Colorado T., Eliecer Aldana B. Grupo de Modelación Matemática en Epidemiología (GMME) Facultad de Educación, Universidad del Quindío Armenia, Quindío Colombia Copyright 2018 Carlos A. Abello M. et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract It is presented a population simulation model with ordinary nonlinear differential equations for the dynamics of the effect that female mosquitoes have on chemical control, for which the numerical analysis of local stability is performed, showing that the stationary solution in coexistence of mosquitoes resistant to chemicals is local and asymptotically stable for the set of calculated parameters, and the study is complemented by simulating the system in Maple. Keywords: Simulation model, population dynamics, chemical control, local stability, resistance to chemicals 1 Introduction Dengue is a viral infection transmitted to humans (host) by a female mosquito, member of the genus Aedes and belonging to the family Culicidae, known in the scientific community with the name of Aedes aegypti. This disease is endemic in tropical and subtropical regions, with a wide range of adaptation to climatic conditions (temperature, humidity, precipitation). There are multiple limiting factors in its control, such as the transport (mobility) of susceptible and infectious people, who can spread the virus, the dispersion of non-carrier female mosquitoes as carriers of the virus: the latency

2 1434 Carlos A. Abello M. et al. period (diapause), time in which the biological activity stops immature states due to inappropriate weather conditions; the asymptomatic state of the infection that spreads the disease in a silent way, indicating percentages of 70% and 80% of silent incidence; the different types of domiciliary breeding sites, potential sources of vector reproduction. In the same way, the delay times in the application of the different controls; the coinfection with the presence of chikungunya, zika virus, mayaro virus and the vertical transmission of the virus in the female mosquito. Another limiting factor is the acquired resistance of the mosquito to the effect of the chemicals used in its control. Experimentally tested populations of Aedes aegypti exhibit high levels of resistance to pyrethroids [1, 2, 3, 4, 5, 6, 7, 8]. Aedes aegypti resistance has also been found to the insecticide cyclodienes [9]. The great epidemiological impact of the infections transmitted by the vector Aedes aegypti, suggests the implementation of integral control programs, including the different mathematical models that interpret the transmission dynamics and control of the disease [10, 11, 12, 13, 14]. 2 The model A simulation model is formulated based on ordinary nonlinear differential equations to analyze the effect of the resistance acquired by the Aedes aegypti mosquito to the chemicals. The model presents the following variables at a time t, x 1 (t) x 1 : average number of female mosquitoes not resistant to the effect of some chemical products, x 2 (t) x 2 : average number of immature stages (eggs, larvae, pupae) not resistant to the effect of chemical products, used in their control, y 1 (t) y 1 : average number of female mosquitoes resistant to the chemical effect andy 2 (t) y 2 : average number of immature states (eggs, larvae, pupae) resistant to chemicals. The model parameters are ω: development rate of pupal stage to adult mosquitoes both non-resistant and resistant, ɛ: death rate due to environmental conditions of adult mosquitoes, both non-resistant and resistant, π: death rate due to environmental conditions of the immature stages (eggs, larvae, pupae), both nonresistant and resistant, φ: rate of oviposition of non-resistant female mosquitos as resistant, k: carrying capacity of both resistant and non-resistant immature stages, f = 0.5: fraction of eggs (progeny) that produce female mosquitoes that are not resistant as resistant and g: fraction of progeny that acquires resistance to chemicals.

3 Simulating the effect of Aedes aegypti population 1435 The flow diagram of the dynamics with resistance and non-resistance is shown in Figure 1, ωx 2 x 1 (0.5)φx 1 (1 x 2 +y 2 k ) x 2 ɛx 1 πx 2 ϱ y 1 (0.5)φy 1 (1 x 2 +y 2 k ) y 2 ωy 2 ɛy 1 πy 2 Figure 1: Dynamics of population growth of Aedes aegypti with resistance to chemical control, where ϱ = (1 g)fφx 1 (1 x 2+y 2 k ). corresponding to the dynamic system (1)-(4): dx 1 dt = ωx 2 ɛx 1 (1) dx 2 dt = g(0.5)φx 1(1 x 2 + y 2 ) (π + ω)x 2 k (2) dy 1 dt = ωy 2 ɛy 1 (3) dy 2 dt = (1 g)(0.5)φx 1(1 x 2 + y 2 ) + (0.5)φy 1 (1 x 2 + y 2 ) (π + ω)y 2 k k (4) ç with initial conditions x 1 (0) = x 10, x 2 (0) = x 20, y 1 (0) = y 10 and y 2 (0) = y 20 ; ω, ɛ, π, φ, k > 0 y 0 < g < 1. In addition, the region of biological sense where non-resistant and resistant population trajectories are non-negative and continuous is, Ω = {(x 1, x 2, y 1, y 2 ) R 4 + : x 1 0, y 1 > 0, 0 x 2 + y 2 k}. 3 Stability analysis We start the analysis of local and numerical stability, with the values of the parameters of the model indicated in Table 1, and calculating the stationary solutions, constant solutions of the system of equations (1)-(4), that is, where dx 1 = 0, dx 2 = 0, dy 1 = 0 and dy 2 = 0. For dt dt dt dt

4 1436 Carlos A. Abello M. et al. Tabla 1: Values assigned to the parameters. Parámetro ω ɛ π φ k g Valor this we solve the following algebraic system: (0.0823)x 2 (0.0362)x 1 = 0, (0.9)(4.6472)(0.5)x 1 (1 x 2+y ) ( )x 2 = 0, (0.0823)y 2 (0.0362)y 1 = 0, (0.1)(0.5)(4.6472)x 1 (1 x 2+y 2 )+(0.5)(4.6472)y (1 x 2+y 2 ) ( )y = 0. Obtaining two equilibrium points: without populations, E 0 = (0, 0, 0, 0), and with populations, E 1 = (0, 0, 2233, 982). In the process of linearization, we determine the Jacobian matrix J evaluated at the point of generic equilibrium E 1 = (ˆx 1, ˆx 2, ŷ 1, ŷ 2 ), J = ɛ ω 0 0 g(0.5)φ(1 ˆx 2+ŷ 2 ) g(0.5)φˆx 1 (π + ω) 0 g(0.5)φˆx 1 k k k 0 0 ɛ ω (1 g)(0.5)φ(1 ˆx 2+ŷ 2 ) (1 g)(0.5)φˆx 1 (0.5)φŷ 1 (0.5)φ(1 ˆx 2+ŷ 2 ) Γ k k k k where Γ = (1 g)(0.5)φˆx 1 (0.5)φŷ 1 (π + ω). k k With the values of the parameters of Table 1 and the equilibrium point E 1 = (0, 0, 2233, 982), we calculate the following Jacobian matrix, J E1 = (5) We calculate the characteristic equation J E1 λi 4 = 0 corresponding to the matrix (6),

5 Simulating the effect of Aedes aegypti population 1437 Figure 2: Behavior of non-resistant and resistant populations: x 1 : dotted line, x 2 : dashed line-point, y 1 continuous line and y 2 : dashed line, with g = 0.9. λ λ λ λ = 0 whose eigenvalues are: λ 1 = , λ 2 = , λ 3 = , λ 4 = Since Re(λ i ) < 0, the equilibrium point E 1 is local and asymptotically stable, for the set of parameters in Table 1. 4 Simulations The simulations of the dynamic system (1)-(4), are made with the values of the parameters in Table 1, using the Maple software. In Figure 1, it is observed that the adult mosquito population without resistance (dotted line) has a peak around 100 days next to 1500 non-resistant mosquitoes and then decreases to extinction; while the population of resistant mosquitoes grows to a stable value after 1000 days (continuous line). The populations of immature non resistant and resistant states have a behavior similar to the previous ones but in lower levels.

6 1438 Carlos A. Abello M. et al. 5 Conclusion It is concluded that for a low fraction of 10% resistance acquired by the mosquito, the population of adult mosquitos and immature resistant states is stabilized above non-resistant populations, which end up becoming extinct. This impact is due to the vertical transmission of resistance to the offspring of resistant mosquitoes. The point of stability of the populations is E 1 = (0, 0, 2233, 982) that agrees with the numerical analysis. The model studied gives an idea of the local qualitative and numerical behavior of the resistance dynamics acquired by the vector Aedes aegypti. It is recommended to propose models and analysis of greater dimension that include the human population (host) and the different stages of the life cycle of the mosquito. Acknowledgements. The authors are very grateful with Grupo de Modelación Matemática en Epidemiología (GMME) and its institution, Universidad del Quindío. References [1] M. K. Grossman, V. Uc-Puc, J. Rodriguez, D. J. Cutler, L. T. Morran, P. Manrique, G. M. Vazquez-Prokopec, Restoration of pyrethroid susceptibility in a highly resistant Aedes aegypti population, Biology Letters, 14 (2018), no. 6, [2] A. J. Cornel, J. Holeman, C. C. Nieman, Y. Lee, C. Smith, M. Amorino, F. S. Mulligan III, Surveillance, insecticide resistance and control of an invasive Aedes aegypti (Diptera: Culicidae) population in California, F1000Research, 5 (2016), [3] J. C. McAllister, M. S. Godsey, M. L. Scott, Pyrethroid resistance in Aedes aegypti and Aedes albopictus from Port Prince, Haiti, Journal of Vector Ecology, 37 (2012), no. 2, [4] I. R. Montella, A. J. Martins, P. F. Viana, J. B. P. Lima, I. A. Braga, D. Valle, Insecticide resistance mechanisms of Brazilian Aedes aegypti populations from 2001 to 2004, The American Journal of Tropical Medicine and Hygiene, 77 (2007), no. 3,

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8 1440 Carlos A. Abello M. et al. [14] A. Ahdika, N. Lusiyana, Investigation of Aedes aegypti s resistance toward insecticides exposure using Markov modeling, International Journal of Mosquito Research, 5 (2018), no. 4, Received: November 5, 2018; Published: November 26, 2018