Dynamics of delayed mosquitoes populations models with two different strategies of releasing sterile mosquitoes (original) (raw)
Related papers
Impulsive releases of sterile mosquitoes and interactive dynamics with time delay
Journal of Biological Dynamics
To investigate the impact of periodic and impulsive releases of sterile mosquitoes on the interactive dynamics between wild and sterile mosquitoes, we adapt the new idea where only those sexually active sterile mosquitoes are included in the modelling process and formulate new models with time delay. We consider different release strategies and compare their model dynamics. Under certain conditions, we derive corresponding model formulations and prove the existence of periodic solutions for some of those models. We provide numerical examples to demonstrate the dynamical complexity of the models and propose further studies.
MODELING WOLBACHIA SPREAD IN MOSQUITOES THROUGH DELAY DIFFERENTIAL EQUATIONS
Dengue fever is the most common mosquito-borne viral disease. A promising control strategy targets the mosquito vector Aedes aegypti by releasing the mosquitoes infected by the endosymbiotic bacterium Wolbachia to invade and replace the wild population. With infection, Wolbachia reduces the mosquito's dengue transmission potential and brings infected females a reproductive advantage through cytoplasmic incompatibility. As Wolbachia often induces fitness costs, it is important to analyze how the reproductive advantage offsets the fitness costs for the success of population replacement. In this work, we develop a model of delay differential equations to study Wolbachia infection dynamics. We prove that, when the infection does not alter the mean life span, Wolbachia can spread into the whole population as long as the infection frequency stays strictly above a threshold value for a period no less than the prereproductive time τ. For the other cases, we find that such a threshold value cannot be well defined. Our numerical simulation shows that our model can generate predictions well fitting with the experimental data. It also reveals the striking phenomena that the minimal releasing of infected mosquitoes sufficient for fixation is insensitive to τ , but the waiting time increases almost linearly with τ. However, when τ is fixed but the ratio of infected males over females varies, the waiting time decreases rapidly when the ratio increases moderately, but responds rather weakly when the ratio increases further.
Discrete and Continuous Dynamical Systems - B, 2021
A two-dimensional stage-structured model for the interactive wild and sterile mosquitoes is derived where the wild mosquito population is composed of larvae and adult classes and only sexually active sterile mosquitoes are included as a function given in advance. The strategy of constant releases of sterile mosquitoes is considered but periodic and impulsive releases are more focused on. Local stability of the origin and the existence of a positive periodic solution are investigated. While mathematical analysis is more challenging, numerical examples demonstrate that the model dynamics, determined by thresholds of the release amount and the release waiting period, essentially match the dynamics of the alike one-dimensional models. It is also shown that richer dynamics are exhibited from the two-dimensional stage-structured model.
A Delayed Mathematical Model to break the life cycle of Anopheles Mosquito
2016
In this paper, we propose a delayed mathematical model to break the life cycle of anopheles mosquito at the larva stage by incorporating a time delay τ at the larva stage that accounts for the period of growth or development to pupa. We prove local stability of the system’s equilibrium and find the critical values for Hopf bifurcation to occur. Also, we find that the system’s equilibrium undergoes stability switching from stable to periodic to unstable and vice versa when the time delay τ crosses the imaginary axis from the left half plane to the right half plane in the (Re,Im) plane. Finally, we perform some numerical simulations and the results agree well with the analytical analysis. This is the first time such a model is proposed.
Mathematical modeling of sterile insect technology for control of anopheles mosquito
Computers & Mathematics with Applications, 2012
The Sterile Insect Technology (SIT) is a nonpolluting method of control of the invading insects that transmit disease. The method relies on the release of sterile or treated males in order to reduce the wild population of anopheles mosquito. We propose two mathematical models. The first model governs the dynamics of the anopheles mosquito. The second model, the SIT model, deals with the interaction between treated males and wild female anopheles. Using the theory of monotone operators, we obtain dynamical properties of a global nature that can be summarized as follows. Both models are dissipative dynamical systems on the positive cone R 4 + . The value R = 1 of the basic offspring number R is a forward bifurcation for the model of the anopheles mosquito, with the trivial equilibrium 0 being globally asymptotically stable (GAS) when R ≤ 1, whereas 0 becomes unstable and one stable equilibrium is born with well determined basins of attraction when R > 1. For the SIT model, we obtain a threshold numberλ of treated male mosquitoes above which the control of wild female mosquitoes is effective. That is, for λ >λ the equilibrium 0 is GAS. When 0 < λ ≤λ, the number of equilibria and their stability are described together with their precise basins of attraction. These theoretical results are rephrased in terms of possible strategies for the control of the anopheles mosquito and they are illustrated by numerical simulations.
Bulletin of Mathematical Biology, 2018
Mosquito-borne diseases remain a significant threat to public health and economics. Since mosquitoes are quite sensitive to temperature, global warming may not only worsen the disease transmission case in current endemic areas but also facilitate mosquito population together with pathogens to establish in new regions. Therefore, understanding mosquito population dynamics under the impact of temperature is considerably important for making disease control policies. In this paper, we develop a stage-structured mosquito population model in the environment of a temperature-controlled experiment. The model turns out to be a system of periodic delay differential equations with periodic delays. We show that the basic reproduction number is a threshold parameter which determines whether the mosquito population goes to extinction or remains persistent. We then estimate the parameter values for Aedes aegypti, the mosquito that transmits dengue virus. We verify the analytic result by numerical simulations with the temperature data of Colombo, Sri Lanka where a dengue outbreak occurred in 2017. Keywords Mosquito • Climate change • Periodic delay • Dengue • Basic reproduction ratio • Population dynamics This work is supported in part by the NSERC of Canada.
Mathematical insights and integrated strategies for the control of Aedes aegypti mosquito
This paper proposes and investigates a delayed model for the dynamics and control of a mosquito population which is subject to an integrated strategy that includes pesticide release, the use of mechanical controls and the use of the sterile insect technique (SIT). The existence of positive equilibria is characterized in terms of two threshold quantities, being observed that the " richer " equilibrium (with more mosquitoes in the aquatic phase) has better chances to be stable, while a longer duration of the aquatic phase has the potential to destabilize both equilibria. It is also found that the stability of the trivial equilibrium appears to be mostly determined by the value of the maturation rate from the aquatic phase to the adult phase. A nonstandard finite difference (NSFD) scheme is devised to preserve the positivity of the approximating solutions and to keep consistency with the continuous model. The resulting discrete model is transformed into a delay-free model by using the method of augmented states, a necessary condition for the existence of optimal controls then determined. The particular-ities of different control regimes under varying environmental temperature are investigated by means of numerical simulations. It is observed that a combination of all three controls has the highest impact upon the size of the aquatic population. At higher environmental temperatures , the oviposition rate is seen to possess the most prominent influence upon the outcome of the control measures.
Applied Mathematics and Computation, 2010
A delay ordinary deterministic differential equation model for the population dynamics of the malaria vector is rigorously analysed subject to two forms of the vector birth rate function: the Verhulst-Pearl logistic growth function and the Maynard-Smith-Slatkin function. It is shown that, for any birth rate function satisfying some assumptions, the trivial equilibrium of the model is globally-asymptotically stable if the associated vectorial reproduction number is less than unity. Further, the model has a non-trivial equilibrium which is locally-asymptotically stable under a certain condition. The non-trivial equilibrium bifurcates into a limit cycle via a Hopf bifurcation. It is shown, by numerical simulations, that the amplitude of oscillating solutions increases with increasing maturation delay. Numerical simulations suggest that the Maynard-Smith-Slatkin function is more suitable for modelling the vector population dynamics than the Verhulst-Pearl logistic growth model, since the former is associated with increased sustained oscillations, which (in our view) is a desirable ecological feature, since it guarantees the persistence of the vector in the ecosystem.
Mathematical Modelling of Natural Phenomena
Starting from an age structured partial differential model, constructed taking into account the mosquito life cycle and the main features of theWolbachia-infection, we derived a delay differential model using the method of characteristics, to study the colonization and persistence of theWolbachia-transinfectedAedes aegyptimosquito in an environment where the uninfected wild mosquito population is already established. Under some conditions, the model can be reduced to a Nicholson-type delay differential system; here, the delay represents the duration of mosquito immature phase that comprises egg, larva and pupa. In addition to mortality and oviposition rates characteristic of the life cycle of the mosquito, other biological features such as cytoplasmic incompatibility, bacterial inheritance, and deviation on sex ratio are considered in the model. The model presents three equilibriums: the extinction of both populations, the extinction ofWolbachia-infected population and persistence o...
Journal of Differential Equations, 2009
We present a global study on the stability of the equilibria in a nonlinear autonomous neutral delay differential population model formulated by Bocharov and Hadeler. This model may be suitable for describing the intriguing dynamics of an insect population with long larval and short adult phases such as the periodical cicada. We circumvent the usual difficulties associated with the study of the stability of a nonlinear neutral delay differential model by transforming it to an appropriate non-neutral nonautonomous delay differential equation with unbounded delay. In the case that no juveniles give birth, we establish the positivity and boundedness of solutions by ad hoc methods and global stability of the extinction and positive equilibria by the method of iteration. We also show that if the time adjusted instantaneous birth rate at the time of maturation is greater than 1, then the population will grow without bound, regardless of the population death process.