Article Contents
Article Contents

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

• * Corresponding author: Radouane Yafia
• 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..

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

 Citation:

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

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 $R$

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 $R$

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 $R$

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 $R$

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

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 $R = 0.269$

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 $R = 8.43$

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)

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)

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)

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)

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)

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 $0$

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

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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