Dynamics of interactive wild and sterile mosquitoes with time delay

Mathematical modelling and release thresholds of transgenic sterile mosquitoes

We formulate simple differential equation models to study the impact of releases of transgenic sterile mosquitoes carrying a dominant lethal on mosquito control based on the modified sterile insects technique. The early acting bisex, late acting bisex, early acting female-killing, and late acting female-killing lethality strategies are all considered. We determine release thresholds of the transgenic sterile mosquitoes, respectively, for these models by investigating the existence of positive equilibria and their stability. We compare the model dynamics, in particular, the thresholds of the models numerically. The late acting lethality strategies are generally more effective than their corresponding early acting lethality strategies, but the comparison between the late acting bisex and early acting female-killing lethality strategies depends on different parameter settings.

The impact of predators of mosquito larvae on Wolbachia spreading dynamics

Dengue fever creates more than 390 million cases worldwide yearly. The most effective way to deal with this mosquito-borne disease is to control the vectors. In this work we consider two weapons, the endosymbiotic bacteria Wolbachia and predators of mosquito larvae, for combating the disease. As Wolbachia-infected mosquitoes are less able to transmit dengue virus, releasing infected mosquitoes to invade wild mosquito populations helps to reduce dengue transmission. Besides this measure, the introduction of predators of mosquito larvae can control mosquito population. To evaluate the impact of the predators on Wolbachia spreading dynamics, we develop a stage-structured five-dimensional model, which links the predator-prey dynamics with the Wolbachia spreading. By comparatively analysing the dynamics of the models without and with predators, we observe that the introduction of the predators augments the number of coexistence equilibria and impedes Wolbachia spreading. Some numerical simulations are presented to support and expand our theoretical results.

Uniqueness and stability of periodic solutions for an interactive wild and Wolbachia-infected male mosquito model

We investigate a mosquito population suppression model, which includes the release of Wolbachia-infected males causing incomplete cytoplasmic incompatibility (CI). The model consists of two sub-equations by considering the density-dependent birth rate of wild mosquitoes. By assuming the release waiting period T is larger than the sexual lifespan T¯ of Wolbachia-infected males, we derive four thresholds: the CI intensity threshold sh∗, the release amount thresholds g∗ and c∗, and the waiting period threshold T∗. From a biological view, we assume sh>sh∗ throughout the paper. When g∗<c<c∗, we prove the origin E0 is locally asymptotically stable iff T<T∗, and the model admits a unique T-periodic solution iff T≥T∗, which is globally asymptotically stable. When c≥c∗, we show the origin E0 is globally asymptotically stable iff T≤T∗, and the model has a unique T-periodic solution iff T>T∗, which is globally asymptotically stable. Our theoretical results are confirmed by numerical simulations.

Existence and stability of two periodic solutions for an interactive wild and sterile mosquitoes model

In this paper, we study the periodic and stable dynamics of an interactive wild and sterile mosquito population model with density-dependent survival probability. We find a release amount upper bound G∗, depending on the release waiting period T, such that the model has exactly two periodic solutions, with one stable and another unstable, provided that the release amount does not exceed G∗. A numerical example is also given to illustrate our results.

Impact of releasing period and magnitude on mosquito population in a sterile release model with delay

Assuming that there are multiple batches of sterile males reared and released during the maturation period, we derive a switching delay differential model to study the fate of wild females under an impulsive and periodic release of sterile males. For the release magnitude of each batch c, we find two threshold values [Formula: see text] and [Formula: see text], and prove that when [Formula: see text], the model admits exactly two periodic solutions, among which one is asymptotically stable and the other is unstable. The trivial equilibrium, corresponding to the elimination of wild females, is locally asymptotically stable, and it becomes globally asymptotically stable when [Formula: see text]. One key step is to prove that every solution is sandwiched between two "good" solutions.

Periodic Orbits of a Mosquito Suppression Model Based on Sterile Mosquitoes

In this work, we investigate the existence and stability of periodic orbits of a mosquito population suppression model based on sterile mosquitoes. The model switches between two sub-equations as the actual number of sterile mosquitoes in the wild is assumed to take two constant values alternately. Employing the Poincaré map method, we show that the model has at most two T-periodic solutions when the release amount is not sufficient to eradicate the wild mosquitoes, and then obtain some sufficient conditions for the model to admit a unique or exactly two T-periodic solutions. In particular, we observe that the model displays bistability when it admits exactly two T-periodic solutions: the origin and the larger periodic solution are asymptotically stable, and the smaller periodic solution is unstable. Finally, we give two numerical examples to support our lemmas and theorems.

Impulsive release strategies of sterile mosquitos for optimal control of wild population

To investigate the release strategies of sterile mosquitoes for the wild population control, we propose mathematical models for the interaction between two-mosquito populations incorporating impulsive releases of sterile ones. The long-term control model is first studied, and the existence and stability of the wild mosquito-extinction periodic solution are exploited. Thresholds of the release amount and release period which can guarantee the elimination of the wild mosquitos are obtained. Then for the limited-time control model, three different optimal strategies in impulsive control are investigated. By applying a time rescaling technique and an optimization algorithm based on gradient, the optimal impulsive release timings and amounts of sterile mosquitoes are obtained. Our results show that the optimal selection of release timing is more important than the optimal selection of release amount, while mixed optimal control has the best comprehensive effect.

Mosquito Control Based on Pesticides and Endosymbiotic Bacterium Wolbachia

Mosquito-borne diseases, such as dengue fever and Zika, have posed a serious threat to human health around the world. Controlling vector mosquitoes is an effective method to prevent these diseases. Spraying pesticides has been the main approach of reducing mosquito population, but it is not a sustainable solution due to the growing insecticide resistance. One promising complementary method is the release of Wolbachia-infected mosquitoes into wild mosquito populations, which has been proven to be a novel and environment-friendly way for mosquito control. In this paper, we incorporate consideration of releasing infected sterile mosquitoes and spraying pesticides to aim to reduce wild mosquito populations based on the population replacement model. We present the estimations for the number of wild mosquitoes or infection density in a normal environment and then discuss how to offset the effect of the heatwave, which can cause infected mosquitoes to lose Wolbachia infection. Finally, we give the waiting time to suppress wild mosquito population to a given threshold size by numerical simulations.

Study of the sterile insect release technique for a two-sex mosquito population model

In this paper, to study the large-scale time control and limited-time control of mosquito population in a field, a two-sex mosquito population model with stage structure and impulsive releases of sterile males is proposed. For the large-scale time control, a wild mosquito-free periodic solution is given and conditions under which it is globally stable are obtained by the use of the monotone system theory. Besides, based on the stability analysis, threshold conditions under which the wild mosquito population is eliminated or not are obtained. Then we study three different optimal release strategies for the limited-time control, which takes into account both of the population control level of wild mosquitoes and the economic input. To solve technical problems in optimal impulsive control, a time rescaling technique is applied and the gradients of cost function with respect to all control parameters are obtained. In addition, by the aid of numerical simulation, we get the optimal release amounts and release timings for each release strategy. Our study indicates that the optimal release timing control is superior to the optimal release amount control. However, simultaneous optimal selection of release amount and release timing leads to the best control performance.

Stability analysis in a mosquito population suppression model

In this work, we study a non-autonomous differential equation model for the interaction of wild and sterile mosquitoes. Suppose that the number of sterile mosquitoes released in the field is a given nonnegative continuous function. We determine a threshold [Formula: see text] for the number of sterile mosquitoes and provide a sufficient condition for the origin [Formula: see text] to be globally asymptotically stable based on the threshold [Formula: see text]. For the case when the number of sterile mosquitoes keeps at a constant level, we find that the origin [Formula: see text] is globally asymptotically stable if and only if the constant number [Formula: see text] of sterile mosquitoes released in the field is above [Formula: see text].

Impulsive releases of sterile mosquitoes and interactive dynamics with time delay

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 the suppression dynamics of Aedes mosquitoes with mating inhomogeneity

A novel strategy for controlling mosquito-borne diseases, such as dengue, malaria and Zika, involves releases of Wolbachia-infected mosquitoes as Wolbachia cause early embryo death when an infected male mates with an uninfected female. In this work, we introduce a delay differential equation model with mating inhomogeneity to discuss mosquito population suppression based on Wolbachia. Our analyses show that the wild mosquitoes could be eliminated if either the adult mortality rate exceeds the threshold [Formula: see text] or the release amount exceeds the threshold [Formula: see text] uniformly. We also present the nonlinear dependence of [Formula: see text] and [Formula: see text] on the parameters, respectively, as well as the effect of pesticide spraying on wild mosquitoes. Our simulations suggest that the releasing should be started at least 5 weeks before the peak dengue season, taking into account both the release amount and the suppression speed.

Wolbachia infection enhancing and decaying domains in mosquito population based on discrete models

In this article, we formulate and study a discrete equation model depicting the pattern of Wolbachia infection in a mosquito population. A domain in [Formula: see text] is called a Wolbachia infection enhancing (or decaying) domain if in which the Wolbachia infection frequency of the next generation is always bigger (or smaller) than that of the current generation. We first give a complete analysis of the equivalent Wolbachia infection frequency curves. And then we clearly characterize the Wolbachia infection enhancing domain and decaying domain for all of the parameters, respectively. Finally, some numerical examples are also provided to illustrate our theoretical results.