Dynamics of malaria transmission model with sterile mosquitoes

In this paper, a malaria transmission model with sterile mosquitoes is considered. We first formulate a simple SEIR malaria transmission model as our baseline model. Then sterile mosquitoes are introduced into the baseline model. We consider the case that the release rate of sterile mosquitoes is proportional to the wild mosquito population size. To investigate the impact of releasing sterile mosquitoes on the malaria transmission, the dynamics of the baseline model and the models with the sterile mosquitoes are discussed. We derive formulas of the reproductive numbers and explore the existence of endemic equilibrium as the reproductive number is more than unity for these models. It is shown that both the baseline model and the models with the sterile mosquitoes undergo backward bifurcations. Based on theoretical analysis and numerical simulation, we investigate the impact of releasing sterile mosquitoes on malaria transmission.     

Periodic solution of a chemostat model with Monod growth rate and impulsive state feedback control

In this paper, we develop a mathematical model concerning a chemostat with impulsive state feedback control to investigate the periodicity of bioprocess. By the existence criteria of periodic solution of a general planar impulsive autonomous system, the conditions under which the model has a periodic solution of order one are obtained. Furthermore, we estimate the position of the periodic solution of order one and discuss the existence of periodic solution of order two. The theoretical results and numerical simulations indicate that the chemostat system with impulsive state feedback control either tends to a stable state or has a periodic solution, which depends on the feedback state, the control parameter of the dilution rate and the initial concentrations of microorganisms and substrate.     

一类带Monod增长率及脉冲状态反馈控制的微生物杀虫剂模型的周期解

研究了一类带Monod增长率及脉冲状态反馈控制的微生物杀虫剂模型．证明了无脉冲系统的负向全局渐近稳定性及带有脉冲状态反馈控制系统具有阶一周期解,并且给出阶一周期解存在和稳定的充分条件．数值模拟验证了理论结果．     

Periodic solution of a chemostat model with Beddington-DeAnglis uptake function and impulsive state feedback control

In this paper, a chemostat model with Beddington-DeAnglis uptake function and impulsive state feedback control is considered. We obtain sufficient conditions of the global asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable.     

Two models of interfering predators in impulsive biological control

In this paper, we study the effects of Beddington-DeAngelis interference and squabbling, respectively, on the minimal rate of predator release required to drive a pest population to zero. A two-dimensional system of coupled ordinary differential equations is considered, augmented by an impulsive component depicting the periodic release of predators into the system. This periodic release takes place independently of the detection of the pests in the field. We establish the existence of a pest-free solution driven by the periodic releases, and express the global stability conditions for this solution in terms of the minimal predator rate required to bring an outbreak of pests to nil. In particular, we show that with the interference effects, the minimal rate will only guarantee eradication if the releases are carried out frequently enough. When Beddington-DeAngelis behaviour is considered, an additional constraint for the existence itself of a successful release rate is that the pest growth rate should be less than the predation pressure, the latter explicitly formulated in terms of the predation function and the interference parameters.     

Strategies for introducing Wolbachia to reduce transmission of mosquito-borne diseases

Certain strains of the endosymbiont Wolbachia have the potential to lower the vectorial capacity of mosquito populations and assist in controlling a number of mosquito-borne diseases. An important consideration when introducing Wolbachia-carrying mosquitoes into natural populations is the minimisation of any transient increase in disease risk or biting nuisance. This may be achieved by predominantly releasing male mosquitoes. To explore this, we use a sex-structured model of Wolbachia-mosquito interactions. We first show that Wolbachia spread can be initiated with very few infected females provided the infection frequency in males exceeds a threshold. We then consider realistic introduction scenarios involving the release of batches of infected mosquitoes, incorporating seasonal fluctuations in population size. For a range of assumptions about mosquito population dynamics we find that male-biased releases allow the infection to spread after the introduction of low numbers of females, many fewer than with equal sex-ratio releases. We extend the model to estimate the transmission rate of a mosquito-borne pathogen over the course of Wolbachia establishment. For a range of release strategies we demonstrate that male-biased release of Wolbachia-infected mosquitoes can cause substantial transmission reductions without transiently increasing disease risk. The results show the importance of including mosquito population dynamics in studying Wolbachia spread and that male-biased releases can be an effective and safe way of rapidly establishing the symbiont in mosquito populations.     

The first releases of transgenic mosquitoes: an argument for the sterile insect technique

Potential applications for reducing transmission of mosquito-borne diseases by releasing genetically modified mosquitoes have been proposed, and mosquitoes are being created with such an application in mind in several laboratories. The use of the sterile insect technique (SIT) provides a safe programme in which production, release and mating competitiveness questions related to mass-reared genetically modified mosquitoes could be answered. It also provides a reversible effect that would be difficult to accomplish with gene introgression approaches. Could new technologies, including recombinant DNA techniques, have improved the success of previous mosquito releases? Criteria for an acceptable transgenic sterile mosquito are described, and the characteristics of radiation-induced sterility are compared with that of current transgenic approaches. We argue that SIT using transgenic material would provide an essentially safe and efficacious foundation for other possible approaches that are more ambitious.     

Optimal barrier zones for stopping the invasion of Aedes aegypti mosquitoes via transgenic or sterile insect techniques

Biological invasions have dramatically altered the natural world by threatening native species and their communities. Moreover, when the invading species is a vector for human disease, there are further substantive public health and economic impacts. The development of transgenic technologies is being explored in relation to new approaches for the biological control of insect pests. We investigate the use of two control strategies, classical sterile insect techniques and transgenic late-acting bisex lethality (Release of Insects carrying a Dominant Lethal), for controlling invasion of the mosquito Aedes aegypti using a spatial stage-structured mathematical model. In particular, we explore the use of a barrier zone of sterile/transgenic insects to prevent or impede the invasion of mosquitoes. We show that the level of control required is not only highly sensitive to the rate at which the sterile/transgenic males are released in the barrier zone but also to the spatial range of release. Our models characterise how the distribution of sterile/transgenic mosquitoes in the barrier zone can be controlled so as to minimise the number of mass-produced insects required for the arrest of species invasion. We predict that, given unknown rates of mosquito dispersal, management strategies should concentrate on larger release areas rather than more intense release rates for optimal control.     

Modeling and dynamics of Wolbachia-infected male releases and mating competition on mosquito control

Despite centuries of continuous efforts, mosquito-borne diseases (MBDs) remain enormous health threat of human life worldwide. Lately, the USA government has approved an innovative technology of releasing Wolbachia-infected male mosquitoes to suppress the wild mosquito population. In this paper we first introduce a stage-structured model for natural mosquitos, then we establish a new model considering the releasing of Wolbachia-infected male mosquitoes and the mating competition between the natural male mosquitoes and infected males on the suppression of natural mosquitoes. Dynamical analysis of the two models, including the existence and local stability of the equilibria and bifurcation analysis, reveals the existence of a forward bifurcation or a backward bifurcation with multiple attractors. Moreover, globally dynamical properties are further explored by using Lyapunov function and theory of monotone operators, respectively. Our findings suggest that infected male augmentation itself cannot always guarantee the success of population eradication, but leads to three possible levels of population suppression, so we define the corresponding suppression rate and estimate the minimum release ratio for population eradication. Furthermore, we study how the release ratio of infected males and natural ones, mating competition, the rate of cytoplasmic incompatibility and the basic offspring number affect the suppression rate of natural mosquitoes. Our results show that the successful eradication relies on assessing the reproductive capacity of natural mosquitoes, a selection of suitable Wolbachia strains and an appropriate release amount of infected males. This study will be helpful for public health authorities in designing proper strategies to control vector mosquitoes and prevent the epidemics of MBDs.

     

Field performance of engineered male mosquitoes

Dengue is the most medically important arthropod-borne viral disease, with 50-100 million cases reported annually worldwide. As no licensed vaccine or dedicated therapy exists for dengue, the most promising strategies to control the disease involve targeting the predominant mosquito vector, Aedes aegypti. However, the current methods to do this are inadequate. Various approaches involving genetically engineered mosquitoes have been proposed, including the release of transgenic sterile males. However, the ability of laboratory-reared, engineered male mosquitoes to effectively compete with wild males in terms of finding and mating with wild females, which is critical to the success of these strategies, has remained untested. We report data from the first open-field trial involving a strain of engineered mosquito. We demonstrated that genetically modified male mosquitoes, released across 10 hectares for a 4-week period, mated successfully with wild females and fertilized their eggs. These findings suggest the feasibility of this technology to control dengue by suppressing field populations of A. aegypti.     

Release of genetically engineered insects: a framework to identify potential ecological effects

Genetically engineered (GE) insects have the potential to radically change pest management worldwide. With recent approvals of GE insect releases, there is a need for a synthesized framework to evaluate their potential ecological and evolutionary effects. The effects may occur in two phases: a transitory phase when the focal population changes in density, and a steady state phase when it reaches a new, constant density. We review potential effects of a rapid change in insect density related to population outbreaks, biological control, invasive species, and other GE organisms to identify a comprehensive list of potential ecological and evolutionary effects of GE insect releases. We apply this framework to the Anopheles gambiae mosquito – a malaria vector being engineered to suppress the wild mosquito population – to identify effects that may occur during the transitory and steady state phases after release. Our methodology reveals many potential effects in each phase, perhaps most notably those dealing with immunity in the transitory phase, and with pathogen and vector evolution in the steady state phase. Importantly, this framework identifies knowledge gaps in mosquito ecology. Identifying effects in the transitory and steady state phases allows more rigorous identification of the potential ecological effects of GE insect release.     

Feasible Introgression of an Anti-pathogen Transgene into an Urban Mosquito Population without Using Gene-Drive

Background

Introgressing anti-pathogen constructs into wild vector populations could reduce disease transmission. It is generally assumed that such introgression would require linking an anti-pathogen gene with a selfish genetic element or similar technologies. Yet none of the proposed transgenic anti-pathogen gene-drive mechanisms are likely to be implemented as public health measures in the near future. Thus, much attention now focuses instead on transgenic strategies aimed at mosquito population suppression, an approach generally perceived to be practical. By contrast, aiming to replace vector competent mosquito populations with vector incompetent populations by releasing mosquitoes carrying a single anti-pathogen gene without a gene-drive mechanism is widely considered impractical.

Methodology/Principal Findings

Here we use Skeeter Buster, a previously published stochastic, spatially explicit model of Aedes aegypti to investigate whether a number of approaches for releasing mosquitoes with only an anti-pathogen construct would be efficient and effective in the tropical city of Iquitos, Peru. To assess the performance of such releases using realistic release numbers, we compare the transient and long-term effects of this strategy with two other genetic control strategies that have been developed in Ae. aegypti: release of a strain with female-specific lethality, and a strain with both female-specific lethality and an anti-pathogen gene. We find that releasing mosquitoes carrying only an anti-pathogen construct can substantially decrease vector competence of a natural population, even at release ratios well below that required for the two currently feasible alternatives that rely on population reduction. Finally, although current genetic control strategies based on population reduction are compromised by immigration of wild-type mosquitoes, releasing mosquitoes carrying only an anti-pathogen gene is considerably more robust to such immigration.

Conclusions/Significance

Contrary to the widely held view that transgenic control programs aimed at population replacement require linking an anti-pathogen gene to selfish genetic elements, we find releasing mosquitoes in numbers much smaller than those considered necessary for transgenic population reduction can result in comparatively rapid and robust population replacement. In light of this non-intuitive result, directing efforts to improve rearing capacity and logistical support for implementing releases, and reducing the fitness costs of existing recombinant technologies, may provide a viable, alternative route to introgressing anti-pathogen transgenes under field conditions.     

Impact of population age structure on Wolbachia transgene driver efficacy: ecologically complex factors and release of genetically modified mosquitoes

Wolbachia symbionts hold theoretical promise as a way to drive transgenes into insect vector populations for disease prevention. For simplicity, current models of Wolbachia dynamics and spread ignore ecologically complex factors such as the age structure of vector populations and overlapping vector generations. We developed a model including these factors to assess their impact on the process of Wolbachia spread into populations of three mosquito species (Anopheles gambiae, Aedes aegypti and Culex pipiens). Depending on the mosquito species, Wolbachia parameters, released mosquito life stage and initial age structure of the target population, the number of Wolbachia-infected mosquitoes that we predict would need to be released ranged from less than the threshold calculated by the simple model to a 10-30-fold increase. Transgenic releases into age-structured populations, which is an expectation for wild mosquitoes, will be difficult and depending on the circumstances may not be economically or logistically feasible due to the large number of infected mosquitoes that must be released. Our results support the perspective that understanding ecological factors is critical for designing transgenic vector-borne disease control strategies.     

Pest control by the release of insects carrying a female-killing allele on multiple loci

With recent advances in genetics, many new strategies for pest control have become feasible. This is the second article in which we model new techniques for pest control based on the mass release of genetically modified insects. In this article we model the release of insects carrying a dominant and redundant female killing or sterilizing (FK) allele on multiple genetic loci. If such insects are released into a target population, the FK allele can become widely spread in the population through the males while reducing the population each generation by killing females. We allow the number of loci used to vary from 1 to 20. We also allow the FK allele to carry a fitness cost in males due to the gene insertions. Using a model, we explore the effectiveness and optimal strategies for such releases. In the most ideal circumstances (no density-dependence and released insects equal in fitness to wild ones), FK releases are several orders of magnitude more effective than equal sized sterile male releases. For example, a single release of 19 FK-bearing males for every two wild males, with the released males carrying the FK allele on 10 loci, reduces the target population to 0.002% of no-release size. An equal sized sterile release reduces the target population to 5% of no-release size. We also show how the effectiveness of the technique decreases as the fitness cost of the FK alleles in males increases. For example, the above mentioned release reduces the target population to 0.7% of no-release size if each FK allele carries a fitness cost in males of 5%. Adding a simple model for density-dependence and assuming that each of the released males carries the FK allele on six loci, we show that the release size necessary to reduce the target population to 1/100 of no-release size in 10 generations of releases varies from 0.44:1 to 4:1 (depending on parameter values). We also calculate the optimal number of loci on which to put the FK allele under various circumstances.     

The transmission dynamics of a within-and between-hosts malaria model

In this paper, we developed a novel deterministic coupled model tying together the effects of within-host and population level dynamics on malaria transmission dynamics. We develop within-host and within-vector dynamic models, population level between-hosts models, and a nested coupled model combining these levels. The unique feature of this work is the way the coupling and feedback for the model use the various life stages of the malaria parasite both in the human host and the mosquito vector. Analysis of the coupled and the within-human host models indicate the existence of locally asymptotically stable infection- and parasite-free equilibria when the associated reproduction numbers are less than one. The population-level model, on the other hand, exhibits backward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium. A global sensitivity analysis was carried out to measure the effects of the sensitivity and uncertainty in the various model parameters estimates. The results indicate that the most important parameters driving the pathogen level within an infected human are the production rate of the red blood cells from the bone marrow, the infection rate, the immunogenicity of the infected red blood cells, merozoites and gametocytes, and the immunosensitivity of the merozoites and gametocytes. The key parameters identified at the population level are the human recovery rate, the death rate of the mosquitoes, the recruitment rate of susceptible humans into the population, the mosquito biting rate, the transmission probabilities per contact in mosquitoes and in humans, and the parasite production and clearance rates in the mosquitoes. Defining the feedback functions as a linear function of the mosquito biting rate, numerical exploration of the coupled model reveals oscillations in the parasite populations within a human host in the presence of the host immune response. These oscillations dampen as the mosquito biting rate increases. We also observed that the oscillation and damping effect seen in the within-human host dynamics fed back into the population level dynamics; this in turn amplifies the oscillations in the parasite population within the mosquito-host.     

Conceptual framework and rationale

The sterile insect technique (SIT) has been shown to be an effective and sustainable genetic approach to control populations of selected major pest insects, when part of area-wide integrated pest management (AW-IPM) programmes. The technique introduces genetic sterility in females of the target population in the field following their mating with released sterile males. This process results in population reduction or elimination via embryo lethality caused by dominant lethal mutations induced in sperm of the released males. In the past, several field trials have been carried out for mosquitoes with varying degrees of success. New technology and experience gained with other species of insect pests has encouraged a reassessment of the use of the sterility principle as part of integrated control of malaria vectors. Significant technical and logistic hurdles will need to be overcome to develop the technology and make it effective to suppress selected vector populations, and its application will probably be limited to specific ecological situations. Using sterile males to control mosquito vector populations can only be effective as part of an AW-IPM programme. The area-wide concept entails the targeting of the total mosquito population within a defined area. It requires, therefore, a thorough understanding of the target pest population biology especially as regards mating behaviour, population dynamics, dispersal and level of reproductive isolation. The key challenges for success are: 1) devising methods to monitor vector populations and measuring competitiveness of sterile males in the field, 2) designing mass rearing, sterilization and release strategies that maintain competitiveness of the sterile male mosquitoes, 3) developing methods to separate sexes in order to release only male mosquitoes and 4) adapting suppression measures and release rates to take into account the high reproductive rate of mosquitoes. Finally, success in area-wide implementation in the field can only be achieved if close attention is paid to political, socio-economic and environmental sensitivities and an efficient management organization is established taking into account the interests of all potential stakeholders of an AW-IPM programme.     

Modelling the effects of inherited sterility for the application of the sterile insect technique

1 The sterile insect technique (SIT) involves the release of large numbers of sterile or partially‐sterile insects into a wild pest population to dilute the number of successful wild matings, with the eventual aim of eradication or area‐wide suppression. General population models, encompassing a wide range of SIT types, were used to derive principles for optimizing the success of SIT, with particular emphasis on the application of partial sterility leading to inherited sterility in the F₁ population. 2 The models show that inherited sterility can only be guaranteed to be more effective than complete sterility if matings between irradiated‐lineage partners are unsuccessful. This is widely assumed but rarely examined experimentally. 3 The models allow the critical overflooding ratio, φ_c, to be calculated for a particular target species, suggesting the release rate required to prevent population increase. Successful eradication using SIT alone should aim for a substantially higher release rate than suggested by φ_c. 4 The models show that pest populations may continue to increase in the first few generations of SIT releases, regardless of release rate, as irradiated‐lineage individuals infiltrate the population. This does not necessarily imply that the SIT programme will be unsuccessful in the longer term. 5For pests with overlapping generations, the models suggest that frequent small releases may be more effective than less frequent large releases, particularly when the average release rate is close to the critical threshold for success.     

Historical applications of induced sterilisation in field populations of mosquitoes

Research on sterile mosquito technology from 1955 to the 1980s provided a substantial body of knowledge on propagation and release of sterile mosquitoes. Radiation sterilisation and chemosterilisation have been used effectively to induce dominant lethality and thereby sterilise important mosquito vectors in the laboratory. Experimental releases of chemosterilised males provided complete control of Anopheles albimanus in a small breeding population (14-15 sq km) in El Salvador. Releases of radiation sterilised males failed to control either Aedes aegypti or Anopheles quadrimaculatus in the USA. Releases of radiation-sterilised and chemosterilised male Culex quinquefasciatus in the USA and India were successful in some instances. Development of genetic sexing systems for Anopheles and improved physical separation methods for Culex have made it possible to rear and release males almost exclusively (> 99%) minimizing the release of potential vectors, the females. Factors that affected efficacy in some field programmes included reduction of competitiveness by radiation, immigration of fertilized females from outside the release zones, and inability of laboratory-bred males to perform in the wild. Despite significant progress, institutional commitments to carry the process further were generally lacking in the late 1970s and until recently. Now, with renewed interest and support for further assessment of this technology, this paper summarizes the current knowledge base, prioritizes some areas of investigation, and challenges scientists and administrators to maintain an awareness of progress, remain realistic about the interpretation of new findings, and make decisions about the sterile insect technique on the basis of informed scientific documentation. Areas recommended for priority research status include the establishment of genetic sexing mechanisms that can be transferred to other mosquito species, re-examination of radiation sterilisation, aerial release technology and mass rearing.     

Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide

We present a mathematical model to describe the dynamics of mosquito population when sterile male mosquitoes (produced by irradiation) are introduced as a biological control, besides the application of insecticide. In order to analyze the minimal effort to reduce the fertile female mosquitoes, we search for the optimal control considering the cost of insecticide application, the cost of the production of irradiated mosquitoes and their delivery as well as the social cost (proportional to the number of fertilized females mosquitoes). The optimal control is obtained by applying the Pontryagin’s Maximum Principle.     

Assessing the Feasibility of Controlling Aedes aegypti with Transgenic Methods: A Model-Based Evaluation

Suppression of dengue and malaria through releases of genetically engineered mosquitoes might soon become feasible. Aedes aegypti mosquitoes carrying a conditionally lethal transgene have recently been used to suppress local vector populations in small-scale field releases. Prior to releases of transgenic insects on a wider scale, however, most regulatory authorities will require additional evidence that suppression will be effective in natural heterogeneous habitats. We use a spatially explicit stochastic model of an Ae. aegypti population in Iquitos, Peru, along with an uncertainty analysis of its predictions, to quantitatively assess the outcome of varied operational approaches for releases of transgenic strains with conditional death of females. We show that population elimination might be an unrealistic objective in heterogeneous populations. We demonstrate that substantial suppression can nonetheless be achieved if releases are deployed in a uniform spatial pattern using strains combining multiple lethal elements, illustrating the importance of detailed spatial models for guiding genetic mosquito control strategies.