Oscillations induced by quiescent adult female in a model of wild aedes aegypti mosquitoes

Ahmed Aghriche; Radouane Yafia; M. A. Aziz Alaoui; Abdessamad Tridane

doi:10.3934/dcdss.2020194

Article Contents

2020, Volume 13, Issue 9: 2443-2463. Doi: 10.3934/dcdss.2020194

This issue Previous Article Dynamics of solutions of a reaction-diffusion equation with delayed inhibition Next Article Oscillation criteria for second-order quasi-linear neutral functional differential equation

Oscillations induced by quiescent adult female in a model of wild aedes aegypti mosquitoes

1.
Ibn Zohr University, CST Campus Universitaire Ait Melloul, Agadir, Morocco
2.
Laboratory of Analysis, Geometry and Applications (LAGA), Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Campus Universitaire, BP 133, Kenitra, Morocco
3.
Normandie Univ, France; ULH, LMAH, F-76600 Le Havre; FR-CNRS-3335, ISCN, 25 rue Ph. Lebon 76600, Le Havre, France
4.
Department of Mathematical Sciences, United Arab Emirates University Al Ain, Abu Dhabi, United Arab Emirates

^* Corresponding author: Radouane Yafia
^* Corresponding author: Radouane Yafia

Received: October 31, 2018

Revised: March 31, 2019

Early access: December 2019

Published: June 2020

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Abstract

Aedes aegypti (Ae. aegypti: mosquito) is a known vector of several viruses including yellow fever, dengue, chikungunya and zika. In the current paper, we present a delayed mathematical model describing the dynamics of Ae. aegypti. Our model is governed by a system of three delay differential equations modeling the interactions between three compartments of the Ae. aegypti life cycle (females, eggs and pupae). By using time delay as a parameter of bifurcation, we prove stability/switch stability of the possible equilibrium points and the existence of bifurcating branch of small amplitude periodic solutions when time delay crosses some critical value. We establish an algorithm determining the direction of bifurcation and stability of bifurcating periodic solutions. In the end, some numerical simulations are carried out to support theoretical results..

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Mathematics Subject Classification: Primary: 39A05, 92D25; Secondary: 92D30, 39A23, 39A28.

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Figure 1. Schematic representation describing the interaction between females ($ F $), eggs ($ E $) and pupae ($ P $) of Ae. aegypti population

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Figure 2. Surfaces representing the effect of parameters (left $ \gamma $ and $ \alpha $) and (right $ \sigma $ and $ \alpha $) on the variations of the population reproduction number $ R $

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Figure 3. Surfaces representing the effect of parameters(left $ \beta $ and $ \alpha $) and (right $ \mu_{F} $ and $ \alpha $) on the variations of the population reproduction number $ R $

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Figure 4. Surfaces representing the effect of parameters (left $ \mu_{E} $ and $ \alpha $) and (right $ \mu_{P} $ and $ \alpha $) on the variations of the population reproduction number $ R $

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Figure 5. Surfaces representing the effect of parameters (left $ \mu_{P} $ and $ \mu_{F} $) and (right $ \mu_{P} $ and $ \beta $) on the variations of the population reproduction number $ R $

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Figure 6. Surface representing the effect of parameters $ \mu_{P} $ and $ \gamma $ on the variations of the population reproduction number $ R $

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Figure 7. Stability of $ E_0 $ for $ \tau = 0 $ in $ (t,FEP) $ plane (left) and in $ (F,E,P) $ space (right) and non existence of $ E_1 $ for $ \mu_F = 5 $ and $ R = 0.269 $

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Figure 8. Instability of $ E_0 $ and stability of $ E_1 $ for $ \tau = 0 $ in $ (t,FEP) $ plane (left) and in $ (F,E,P) $ space (right) with $ R = 8.43 $

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Figure 9. Instability of $ E_0 $ and stability of $ E_1 $ for $ \tau = 5 $ in $ (t,FEP) $ plane (left) and in $ (F,E,P) $ space (right)

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Figure 10. Mikhailov hodograph indicating the stability of $ E_1 $ for $ \tau = 5 $ (left) and instability of $ E_1 $ for $ \tau = 50 $ (right). Note that, The steady state $ E_1 $ is stable if the curve crosses imaginary axis above zero (at all crossing points)

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Figure 11. Periodic solution bifurcated from the steady state $ E_1 $ for $ \tau = \tau_0 = 10.35 $ in $ (t,FEP) $ plane (left) and in $ (F,E,P) $ space (right)

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Figure 12. Periodic solution bifurcated from the steady state $ E_1 $ for $ \tau = \tau_0+\epsilon = 10.85 $ with $ \epsilon = 0.5 $ in $ (t,FEP) $ plane (left) and in $ (F,E,P) $ space (right)

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Figure 13. Chaotic solution bifurcated from the steady state $ E_1 $ for $ \tau = \tau_0+\epsilon = 11.15 $ with $ \epsilon = 0.8 $ in $ (t,FEP) $ plane (left) and in $ (F,E,P) $ space (right)

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Figure 14. Existence of pair purely imaginary roots (left). The branches of bifurcation (diagram of bifurcation), branch $ 2 $ (blue line) is the branch of the periodic orbits that arise from the Hopf point, branch $ 3 $ (red line) is the branch of period doubling bifurcation point in branch $ 2 $ and branch $ 4 $ (green line) is the branch of period doubling bifurcation point in branch $ 3 $ (right). Note that, the branch $ 1 $ is the steady state $ E_1 $ with amplitude $ 0 $

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Table 1. Parameters estimation

parameter	value	reference
$ \gamma $	$ 0.90 $	Assumed
$ \alpha $	$ 0 .6 $	Assumed
$ \sigma $	$ 4 $	[29]
$ \beta $	$ 0.4 $	[29]
$ \mu_F $	$ 0.16 $	Assumed
$ \mu_E $	$ 0.15 $	Assumed
$ \mu_P $	$ 0.01 $	Assumed
$ k $	$ 500 $	Assumed

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Table 2. The sensitivity indices of the population reproduction number $ R $

Parameter	Sensitivity index	Index at parameters value
$ \gamma $	$ \frac{\mu_{E}}{\gamma+\mu_{E}} $	$ +0.1428 $
$ \alpha $	$ \frac{\mu_{P}}{\alpha+\mu_{P}} $	$ +0.366 $
$ \sigma $	$ +1 $	$ +1 $
$ \beta $	$ +1 $	$ +1 $
$ \mu_F $	$ -1 $	$ -1 $
$ \mu_E $	$ -\frac{\mu_{E}}{\gamma+\mu_{E}} $	$ -0.1428 $
$ \mu_P $	$ -\frac{\mu_{P}}{\gamma+\mu_{P}} $	$ -0.336 $

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