Controlling Wolbachia Transmission and Invasion Dynamics among Aedes Aegypti Population via Impulsive Control Strategy
Abstract
:1. Introduction
- Micro injection: In this process, Wolbachia strains are microinjected into aquatic stages such as eggs, larvae and pupae.
- Introgression: In this process, the Wolbachia strains are carried out to next generation through mating. If Wolbachia infected female mated with Wolbachia infected or uninfected male, then the produced offsprings have the Wolbachia strain (Called CI rescue). Suppose the Wolbachia uninfected female mated with a Wolbachia infected male then there is no viable progeny. Finally, if a non-Wolbachia female mated with a non-Wolbachia male then there is no Wolbachia infection in the offspring.
- A novel mathematical model, which considers the total of ten stages in Aedes Aegypti mosquitoes (combining both Wolbachia infected and Wolbachia uninfected) is proposed and the possible optimal stages to release the Wolbachia are discussed, and the most important concept of Wolbachia invasion and Wolbachia gain are adopted.
- The Wolbachia free equilibrium, Wolbachia present Equilibrium, Zero mosquitoes, and both Wolbachia and Non-Wolbachia mosquitoes co-existence equilibrium are derived. And utilizing fixed point theory results, the Existence and Uniqueness results of the Wolbachia invasive model are proposed. To attain optimal control, we utilized an impulsive control strategy.
- We perform global Mittag-Leffler stability analysis of the proposed model via Linear Matrix Inequality (LMI) theory and Lyapunov theory.
- In the end, by utilizing the data from the published literature, we have presented the numerical simulation of the proposed model using MATLAB software.
2. Preliminaries
- (1)
- There exists positive constants and
- (2)
- , , for any scalar .
- (3)
- , , .
3. Model Formulation
4. Equilibrium Points
4.1. Zero Mosquitoes
4.2. Wolbachia Infected Mosquitoes Free Equilibrium
4.3. Wild Mosquitoes Free Equilibrium
4.4. Both Wolbachia Infected Mosquitoes and Non-Wolbachia Mosquitoes Co-Existence Equilibrium
5. Wolbachia Invasion Model
6. Existence and Uniqueness of Solution
7. Stability Analysis
8. Numerical Simulation
- Case 1.
- In this case, we have analyzed the transmission dynamics of Wolbachia among Aedes Aegypti mosquitoes via substituting the values mentioned in Table 2.For this consider the system (5), with initial conditions , , , , , , , , , , total population , and the positive scalar used in Theorem 2 asThe Figure 4, Figure 5, Figure 6 and Figure 7 are depicts the dynamics of Equation (5) along with the parameters in Table 2 at various orders of such as and 1. We can observe by simulation results that, there is a notable decrease in non-Wolbachia mosquitoes and increase in Wolbachia infected mosquitoes.
- Case 2.
- In this case, we have analyzed the merits and demerits of considering the Wolbachia invasion. For this consider the system of Equation (6) with parameters mentioned in Table 2. We have plotted (6) with initial conditions and total population as considered in Case 1. Along with this, the other parameters , , , and are fitted.Figure 8, Figure 9, Figure 10 and Figure 11 are analyzed the dynamics of the system of Equation (6), with Wolbachia invasion and natural Wolbachia gain at various orders and 1. From this we can observe that, Wolbachia infected mosquitoes tends to annihilation before the eradication of non-Wolbachia mosquitoes. It will lead to the decay in natural CI rescue.
- Case 3.
- In this case, the decay due to the natural Wolbachia invasion is managed by releasing Wolbachia infected mosquitoes impulsively. For this case, along with the parameters mentioned in Table 2, we have fitted the values of impulsive control as , , , and , invasion rates are , , , and gain rates are and .Figure 12, Figure 13, Figure 14 and Figure 15 explicitly shows the dynamics of the systems of Equation (7) with impulsive control at orders and 1. From this we get that, at order the system leads to instability, when the system started to posses stable state and at the both population are annihilated at initial stage compared with Figure 7 and Figure 11.
9. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Appendix A.1. Wolbachia Infected Mosquitoes Free Equilibrium
- (i).
- By solving,We get the value of as,
- (ii).
- By solvingWe get the value of as,Substitute the value of from (i),
- (iii).
- By solvingWe get the value of as,Substitute the value of from (ii),
- (iv).
- By solvingWe get the value of as,Substitute the value of from (ii),
- (v).
- By solvingWe get the value of as,Substitute the value of and from (iii) and (iv),
Appendix A.2. Wild Mosquitoes Free Equilibrium
- (i)
- By solvingWe get,
- (ii)
- By solvingWe get,
- (iii)
- By solvingWe get,
- (iv)
- By solvingWe get,Substitute the value of from (iii),
- (v)
- By solving,Put ,
Appendix A.3. Both Wolbachia and Non-Wolbachia Mosquitoes Co-Existence Equilibrium
- (i)
- (ii)
- (iii)
- Let
- (iv)
- Where, ;
- (v)
- Where,
- (vi)
- (vii)
- (viii)
- (ix)
- (x)
- The above equation is a quadratic equation on . That is,
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Parameter | Description |
---|---|
, | Reproduction rate of non-Wolbachia mosquitoes |
and Wolbachia infected mosquitoes respectively | |
The natural death rate of eggs without Wolbachia infection | |
The natural death of larvae without Wolbachia infection | |
The natural death of pupae without Wolbachia infection | |
The natural death of adult female mosquitoes without Wolbachia infection | |
The natural death of adult male mosquitoes without Wolbachia infection | |
The natural death of eggs with Wolbachia infection | |
The natural death of larvae with Wolbachia infection | |
The natural death of pupae with Wolbachia infection | |
The natural death of adult female mosquitoes with Wolbachia infection | |
The natural death of infected adult male mosquitoes with Wolbachia infection | |
The rate at which the fraction of non-Wolbachia eggs matured into non-Wolbachia larvae | |
The rate at which the fraction of non-Wolbachia larvae matured into non-Wolbachia pupae | |
The rate at which the fraction of non-Wolbachia pupae matured into non-Wolbachia | |
immature female or male | |
The rate at which the fraction of the Wolbachia infected mosquito eggs | |
matured into Wolbachia infected or uninfected larvae | |
The rate at which the fraction of the Wolbachia infected mosquito larvae | |
matured into Wolbachia infected or uninfected pupae | |
The rate at which the fraction of the Wolbachia infected mosquito pupae | |
matured into Wolbachia infected or uninfected adults | |
The probability of having male or female mosquitoes |
Parameters | Description | Data |
---|---|---|
Reproduction rate of Wolbachia uninfected mosquitoes | 1.25/day [52] | |
, , | The death rate of aquatic Wolbachia uninfected mosquitoes | /day [53] |
The Maturation rate of Wolbachia uninfected mosquitoes | /day [54] | |
The death rate of adult Wolbachia uninfected mosquitoes | /day [53] | |
The death rate of aquatic Wolbachia infected mosquitoes | /day [53] | |
The death rate of adult Wolbachia infected mosquitoes | /day [24] | |
Reproduction rate of Wolbachia infected mosquitoes | [52] | |
The maturation rate of Wolbachia infected mosquitoes | /day [24] |
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Dianavinnarasi, J.; Raja, R.; Alzabut, J.; Niezabitowski, M.; Bagdasar, O. Controlling Wolbachia Transmission and Invasion Dynamics among Aedes Aegypti Population via Impulsive Control Strategy. Symmetry 2021, 13, 434. https://doi.org/10.3390/sym13030434
Dianavinnarasi J, Raja R, Alzabut J, Niezabitowski M, Bagdasar O. Controlling Wolbachia Transmission and Invasion Dynamics among Aedes Aegypti Population via Impulsive Control Strategy. Symmetry. 2021; 13(3):434. https://doi.org/10.3390/sym13030434
Chicago/Turabian StyleDianavinnarasi, Joseph, Ramachandran Raja, Jehad Alzabut, Michał Niezabitowski, and Ovidiu Bagdasar. 2021. "Controlling Wolbachia Transmission and Invasion Dynamics among Aedes Aegypti Population via Impulsive Control Strategy" Symmetry 13, no. 3: 434. https://doi.org/10.3390/sym13030434