Global dynamics of a mathematical model for a honeybee colony infested by virus-carrying Varroa mites (original) (raw)

Importance of brood maintenance terms in simple models of the honeybee - Varroa destructor - acute bee paralysis virus complex

Electronic Journal of Differential Equations, 2010

We present a simple mathematical model of the infestation of a honeybee colony by the Acute Paralysis Virus, which is carried by parasitic varroa mites (Varroa destructor). This is a system of nonlinear ordinary dif- ferential equations for the dependent variables: number of mites that carry the virus, number of healthy bees and number of sick bees. We study this model with a mix of analytical and computational techniques. Our results indicate that, depending on model parameters and initial data, bee colonies in which the virus is present can, over years, function seemingly like healthy colonies before they decline and disappear rapidly (e.g. Colony Collapse Disor- der, wintering losses). This is a consequence of the fact that a certain number of worker bees is required in a colony to maintain and care for the brood, in order to ensure continued production of new bees.

Analysis of Varroosis Model in Honeybee Colony with Interventions

African Scientific Reports

A deterministic mathematical model is proposed and analyzed to study the transmission dynamics of Varroosis in honeybee colony with interventions. The study combined both treatment and biocontrol strategy on curbing the menace of Varroosis on honeybee colony. As such, the study established the existence of the most important four steady states that include: disease-free and infestation-free, infestation with virus-free Varroa-mites, infestation with virus-carrying Varroa-mites and endemic steady state. Moreover, the study established the existence of backward bifurcation and sensitivity analysis of the model was performed. Correspondingly, the analysis of the model reveals that, ineffective treatment can induce backward bifurcation. Furthermore, the study results indicated that, when treatment is 100% effective, the disease-free and infestation-free steady state is globally asymptotically stable for R0 < 1, whereas for R0 > 1the global stability of the endemic steady state is ...

A mathematical model of Varroa mite ( Varroa destructor Anderson and Trueman) and honeybee ( Apis mellifera L.) population dynamics

International Journal of Acarology, 2004

A mathematical model of population interactions between Varroa destructorand a honey bee colony is described. Validation tests indicate that the model generatesmite population predictions that are similar to those from actual colonies including: weekly mite-drop, daily rates of population increase, and exponential growth rates for mite populations. The modelpredicts that colony survival thresholds for mite populations and the effectiveness of miticides such as fluvalinate are dependent on climate and the yearly brood rearing cycle in a colony. Miticides applied in the late summerprovide the best chances for the sur vival ofheavily infested colonies. The model also predicts that large mite populations treated with miticides in the spring will recover by autumn to levels similarto those in untreatedcolonies. This is because in the treated colonies the surviving mites infest drone brood at lower numbers per cell and this increases repro ductive success and hence the growth rate of the mite population.

Disease dynamics of honeybees with Varroa destructor as parasite and virus vector

Mathematical Biosciences, 2016

The worldwide decline in honeybee colonies during the past 50 years has often been linked to the spread of the parasitic mite Varroa destructor and its interaction with certain honeybee viruses carried by Varroa mites. In this article, we propose a honeybee-mite-virus model that incorporates (1) parasitic interactions between honeybees and the Varroa mites; (2) five virus transmission terms between honeybees and mites at different stages of Varroa mites: from honeybees to honeybees, from adult honeybees to phoretic mites, from honeybee brood to reproductive mites, from reproductive mites to honeybee brood, and from honeybees to phoretic mites; and (3) Allee effects in the honeybee population generated by its internal organization such as division of labor. We provide completed local and global analysis for the full system and its subsystems. Our analytical and numerical results allow us have a better understanding of the synergistic effects of parasitism and virus infections on honeybee population dynamics and its persistence. Interesting findings from our work include: (a) Due to Allee effects experienced by the honeybee population, initial conditions are essential for the survival of the colony. (b) Low adult honeybee to brood ratios have destabilizing effects on the system, generate fluctuated dynamics, and potentially lead to a catastrophic event where both honeybees and mites suddenly become extinct. This catastrophic event could be potentially linked to Colony Collapse Disorder (CCD) of honeybee colonies. (c) Virus infections may have stabilizing effects on the system, and could make disease more persistent in the presence of parasitic mites. Our model illustrates how the synergy between the parasitic mites and virus infections consequently generates rich dynamics including multiple attractors where all species can coexist or go extinct depending on initial conditions. Our findings may provide important insights on honeybee diseases and parasites and how to best control them.

Modeling the Influence of Mites on Honey Bee Populations

Veterinary Sciences, 2020

The Varroa destructor mite has been associated with the recent decline in honey bee populations. While experimental data are crucial in understanding declines, insights can be gained from models of honey bee populations. We add the influence of the V. destructor mite to our existing honey bee model in order to better understand the impact of mites on honey bee colonies. Our model is based on differential equations which track the number of bees in each day in the life of the bee and accounts for differences in the survival rates of different bee castes. The model shows that colony survival is sensitive to the hive grooming rate and reproductive rate of mites, which is enhanced in drone capped cells.

Structure of the global attractors in a model for ectoparasite borne diseases

BIOMATH, 2012

BIOMATH Editor-in-Chief: Roumen Anguelov B f BIOMATH w w w. b i o m a t h f o r u m . o r g / b i o m a t h Abstract-We delineate a mathematical model for the dynamics of the spread of ectoparasites and the diseases transmitted by them. We present how the dynamics of the system depends on the three reproduction numbers belonging to three of the four possible equilibria and give a complete characterization of the structure of the global attractor in each possible case depending on the reproduction numbers.

The Role of Viral Infection in Pest Control: A Mathematical Study

Bulletin of Mathematical Biology, 2007

In this paper, we propose a mathematical model of viral infection in pest control. As the viral infection induces host lysis which releases more virus into the environment, on the average 'κ' viruses per host, κ ∈ (1, ∞), so the 'virus replication parameter' is chosen as the main parameter on which the dynamics of the infection depends. There exists a threshold value κ 0 beyond which the infection persists in the system. Still for increasing the value of κ, the endemic equilibrium bifurcates towards a periodic solution, which essentially indicates that the viral pesticide has a density-dependent 'numerical response' component to its action. Investigation also includes the dependence of the process on predation of natural enemy into the system. A concluding discussion with numerical simulation of the model is also presented.

Global stability for a virus dynamics model with nonlinear incidence of infection and removal

SIAM Journal on Applied Mathematics, 2006

Global dynamics of a compartmental model which describes virus propagation in vivo is studied using the direct Lyapunov method, where the incidence rate of the infection and the removal rate of the virus are assumed to be nonlinear. In the case where the functional quotient between the force of infection and the removal rate of the virus is a nonincreasing function of the virus concentration, the existence of a threshold parameter, i.e., the basic reproduction number or basic reproductive ratio, is established and the global stability of the equilibria is discussed. Moreover, in the absence of the above-mentioned monotonicity property, estimations for the sizes of the domains of attraction are given. Biological significance of the results and possible extensions of the model are also discussed.

Pest control through viral disease: Mathematical modeling and analysis

Journal of Theoretical Biology, 2006

This paper deals with the mathematical modeling of pest management under viral infection (i.e. using viral pesticide) and analysis of its essential mathematical features. As the viral infection induces host lysis which releases more virus into the environment, on the average 'k' viruses per host, k 2 ð1; 1Þ, the 'virus replication parameter' is chosen as the main parameter on which the dynamics of the infection depends. We prove that there exists a threshold value k 0 beyond which the endemic equilibrium bifurcates from the free disease one. Still for increasing k values, the endemic equilibrium bifurcates towards a periodic solution. We further analyse the orbital stability of the periodic orbits arising from bifurcation by applying Poor's condition. A concluding discussion with numerical simulation of the model is then presented. r