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arXiv:1505.03742v1 [math.DS] 14 May 2015

Disease dynamics of Honeybees with Varroa destructor as parasite and

virus vector

Yun Kang1, Krystal Blanco2, Talia Davis 3, and Ying Wang 4

Abstract

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.

Keywords: Allee Eﬀects; Honeybees; Extinction; Virus; Parasite; Colony Collapse Disorder (CCD)

1. Introduction

Honeybees are the world’s most important pollinators of food crops. It is estimated that one third of food

that we consume each day mainly relies on pollination by bees. For example, in the United States, honeybees

are major pollinators of alfalfa, apples, broccoli, carrots and many other crops, and hence are of economic

importances. Honeybees have an estimated monetary value between $15 and $20 billion dollars annually as

commercial pollinators in the U.S [23]. There are growing concerns both locally and globally that despite

a 50% growth in honeybee stocks, the supply cannot keep up with the over 300% increase in agricultural

demands [66]. Therefore, the recent sharp declines in honeybee populations have been considered as a global

crisis. The most recent data from the 2012-2013 winter has shown an average loss of 44.8% of hives in the

U.S., and a total of 30.6% loss of commercial hives [55]. Some beekeepers have reported a lost of as many

1Sciences and Mathematics Faculty, College of Letters and Sciences, Arizona State University, Mesa, AZ 85212, USA

(yun.kang@asu.edu)

2Simon A. Levin Mathematical and Computational Modeling Sciences Center, Arizona State University, Tempe, AZ 85281,

USA

3School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85281, USA

4School of Life Sciences, Arizona State University, Tempe, AZ 85281, USA.

Preprint submitted to XXX May 16, 2015

as 90% of their hives [18,39].

Between 1972 and 2006, the wild honeybee populations declined severely and are now considered vir-

tually nonexistent [38,63]. Hence the use of commercial honeybees for pollination is extremely important.

Beginning in 2006, beekeepers began to report an unusual phenomenon in dying bee colonies. Worker bees

would leave the colony to forage and never return, leaving the queen and the young behind to die. No

dead worker bees were found at the nest sites; they simply disappear [11,58]. This phenomenon is known

as Colony Collapse Disorder (CCD), which is a serious problem threatening the health of honey bees and

therefore the economic stability of commercial beekeeping and pollination operations.

The exact causes and triggering factors for CCD have not been completely understood yet. Researchers

have proposed several possible causes of CCD including stress on nutritional diet, harsh winter conditions,

lack of genetic diversity, exposure to certain pesticides, diseases, and parasitic mites Varroa destructor which

are also vectors of viral diseases of honeybees [23,44]. Even before CCD was detected in honeybee colonies,

studies showed that most of the loses could be generally attributed to two main causes: the vampire mite,

Varroa destructor, which feeds on host haemolymph, weakens host immunity and exposes the bees to a

variety of viruses, and the tracheal mite, which infests the breathing tubes of the bee, punctures the tracheal

wall and sucks the bee’s blood and also exposes the bee to a variety of viruses [47,53,34]. Since then,

Varroa mites have been implicated as the main culprit in dying colonies. For example, in Canada, Varroa

mites have been found to be the main reason behind wintering losses of bee colonies [21], and more generally

studies have shown that if the mite population is not properly controlled, the honeybee colony will die [52].

Recent studies also suggest that the Varroa mite could be a contributing cause of CCD since they not only

ectoparasitically feed on bees, but also vertically transmit a number of deadly viruses to the bees [29,28].

There have been at least 14 viruses found in honeybee colonies [2,29], which can diﬀer in intensity of impact,

virulence, etc. for their host. For example, the Acute Bee Paralysis Virus (ABPV) aﬀects the larvae and

pupae which fail to metamorphose to adult stage, while in contrast the Deformed Wing Virus (DWV) aﬀects

larvae and pupae, which can still survive to the adult stage [59].

Mathematical models are powerful tools that could help us obtain insights on potential ecological processes

that link to CCD, and important factors that contribute to the mortality of honeybees. Few sophisticated

mathematical models of honeybee populations have been previously developed. DeGrandi-Hoﬀman et al.

[14] produced the ﬁrst time-based honeybee colony growth model. Martin [33] developed a simulation model

consisting of ten components, which linked together various aspects of mite biology using computer software

(ModelMaker); and Martin [34] later extended this model by including a bee model adapted from [14] to

explain the link between the Varroa mite and collapse of the host bee colony. Wilkinson and Smith [65]

proposed a diﬀerence equation model of Varroa mites reproducing in a honeybee colony. Their study focused

on parameter estimations and sensitivity analysis. Simulation models are useful but may be too complex to

study mathematically and obtain general predictions.

More recently, mathematical models have been formulated to explore potential mechanisms causing CCD

to the honeybee colony. Sumpter and Martin [56] modeled the eﬀects of a constant population of Varroa

mites on the brood and on adult worker bees, and found that suﬃciently large mite infestations may make

hives vulnerable to collapse from viral epidemics. Eberl et al. [16] developed a model connecting Varroa

mites to CCD by including brood maintenance terms which reﬂect that a certain number of worker bees

is always required to care for the brood in order for them to survive. They found an important threshold

for the number of hive worker bees needed to maintain and take care of the brood. Khoury et al. [26,25]

developed diﬀerential equations models to study diﬀerent death rates of foragers and the impact it had on

colony growth and development. They then linked their results to CCD. Betti et al. [5] studied a model

that combines the dynamics of the spread of disease within a bee colony with the underlying demographic

dynamics of the colony to determine the ultimate fate of the colony under diﬀerent scenarios. Their results

suggest that the age of recruitment of hive bees to foraging duties is a good early marker for the survival or

2

collapse of a honeybee colony in the face of infection. Kribs-Zaleta et al. [27] created a model to account for

both healthy hive dynamics and hive extinction due to CCD, modeling CCD via a transmissible infection

brought to the hive by foragers. Perry et al. [43] examined the social dynamics underlying the dramatic

colony failure with an aid of a honeybee population model. Their model includes bee foraging performance

varying with age, and displays dynamics of colony population collapse that are similar to ﬁeld reports of

CCD. These models, no doubt, are insightful and provide us a better understanding on the potential mecha-

nisms that link to CCD. However, most of these models only account for the honeybee population dynamics

with mites or viruses but not both.

The host-parasite relationship between honeybees and Varroa mites has been complicated by the mite’s

close association with a wide range of honeybee viral pathogens. In order to understand how Varroa mite in-

festations and the related viruses transmitted to honeybees aﬀect honeybee population dynamics, and which

may link to CCD, there is a need to develop realistic and mathematically tractable models that include

both mite and pathogen population dynamics. The goal of our work is to develop a useful honeybee-mite-

virus system to obtain better understanding on the synergistic eﬀects of honeybee-mite interactions and

honeybee-virus interactions on the honeybee populations dynamics, thus develop good practices to control

these parasites to maintain or increase honeybee population. The most relevant modeling papers for our

study purposes are by Sumpter and Martin [56], and Ratti et al. [44] whose work examined the transmission

of viruses via Varroa mites, using the susceptible-infectious (SI) disease modeling framework with mites as

vectors for transmission. However, Sumpter and Martin assumed that the mites’ population is constant

while Ratti et al. took no account of the fact that virus transmissions occur at diﬀerent biological stages of

Varroa mites and honeybees.

In this article, we follow both approaches of Sumpter and Martin [56] and Ratti et al. [44], and propose a

honeybee-mite-virus model that incorporates (1) parasitic interactions between honeybee and Varroa mites;

(2) diﬀerent virus transmission terms that account for the virus transmission among honeybees, between

honeybees and mites at diﬀerent stages of Varroa mites; and (3) Allee eﬀects in the honeybee population

generated by the internal organization of honeybees, including division of labor. Our proposed model will

allow us explore the following questions:

1. What are the dynamics of a system only consisting of honeybees and the disease?

2. What are the dynamics of a system only consisting of honeybees and Varroa mites?

3. What are the synergistic eﬀects of Varroa mites and the disease on the honeybee population, and how

may these synergistic eﬀects contribute to CCD?

4. How can we maintain honeybee populations?

The structure of the remaining article is organized as follows: In Section 2, we ﬁrst provide the biological

background of honeybees, Varroa mites, and the associated virus transmission routes in the honeybee-mite

system; then we derive our SI-type model for honeybees co-infected with the mite and virus. In Section

3, we perform local and global analysis of the proposed model and the related subsystems. The results

from the analysis are then connected to biological contexts and implications. Additionally, we also explore

numerical simulations of the subsystems and the full system to obtain the eﬀects of each parameter in our

system. In Section 4, we summarize our results and the related biological implications of our studies in

ﬁnding potential causes of Colony Collapse Disorder. We also provide potential projects for future work.

The detailed mathematical proofs of our theoretical results are provided in the last section.

3

2. Biological background and model derivations

Honeybee colony: During the spring and summer, a honeybee colony typically consists of a single

reproductive queen, 20,000 – 60,000 adult worker bees, 10,000 – 30,000 individuals at the brood stage (egg,

larvae and pupae) and up to hundreds of male drones. During the winter, the colony typically reduces in size

and consists of a single queen and somewhere between 8,000 – 15,000 worker bees [34]. A large population

of workers carry out the tasks of the bee colony, which include foraging, pollination, honey production and,

in particular, caring for the brood and rearing the next generation of bees. The queen is the only fertile

individual of the colony and has an average life span of 2 – 3 years [56]. During the peak season (in the

summer), the queen lays up to 2000 eggs per day, where fertilized eggs produce female worker bees, or much

more rarely queens, while drones develop from non-fertilized eggs [5]. The bees go through the following

stages in development: egg (about 3 days), larvae (about 7 days), pupae (about 14 days), and adult. The

life span of an adult worker bee also depends on the season. Workers usually have a lifespan of 3 – 6 weeks

during the spring and summer, and are reported to live as long as 4 months during the winter [40]. The

adult drone life span is typically 20 – 40 days, with reports of drone living up to 59 days under optimal

colony conditions [40,22].

Let Nh(t) be the total number of honeybees in the colony, including the larvae, pupae and adult bee

(both hives and foragers) at time t. Let us assume that the honeybee colony has (1 −ξh)Nhpopulation of

brood, i.e., the ratio of adult honeybees to the brood is captured by ξh

1−ξhwhere ξh∈[0,1]. In general, the

ratio ξhvaries with time, and the successful honeybee colony should have ξh

1−ξh>2 [49]. For convenience,

we assume that ξhis a ﬁxed constant. This implies that the honeybee colony has ξhNhadult honeybees and

(1 −ξh)Nhbrood. In the absence of mites and virus, the population dynamics of Nh(t) is described by the

following nonlinear equation:

N′

h=r(ξhNh)2

K+ (ξhNh)2−dhNh(1)

where ris the maximum birth rate, speciﬁed as the number of worker bees born per day; the parameter

√Kis the size of the bee colony at which the birth rate is half of the maximum possible rate; and dhis

the average death rate of the worker honeybees. The term (ξhNh)2

K+(ξhNh)2describes that the successful survival

of an egg which will develop into a worker bee needs the care of adult honeybees (ξhNh) inside the colony

and also needs food brought in by the honeybee foragers. This approach follows the modeling idea in [16]

for honeybee diseases and in [24] for the population of leaf-cutter ants. This term implicitly includes the

internal organization of the honeybee population, such as division of labor.

Varroa mites: Varroa mites were ﬁrst reported in Kentucky in 1991. They have since spread to become

a major pest of honeybees in many states [4]. Varroa mites are external honeybee parasites that attack

both adult honeybees and brood, with a distinct preference for drone brood [41]. They suck the blood from

both the adults and the developing brood, weakening them and shortening the life span of the bees which

they feed on. Emerging brood may be deformed, and may be born with missing legs or wings. Untreated

infestations of Varroa mites can cause honeybee colonies to collapse [32].

The mites go through a series of stages: larva, protonymph, deutonymph and then adult. Adult females

undergo two phases in their life cycle, the phoretic and reproductive phases. During the phoretic phase,

female Varroa feed on adult bees and are passed from bee to bee as they pass one another in the colony.

During the phoretic phase, the female Varroa mites live on adult bees and can usually be found between the

abdominal segments of the bees. The mites puncture the soft tissue between the segments and feed on bee

hemolymph, harming the host [46,6]. Mite reproduction can occur only if honeybee brood is available. A

female mite enters the brood cell about one day before capping and will be sealed in with the larva. After

the capping of the cell, it lays a single male egg and several female eggs at 30-hour intervals [60], and the

4

mite feeds and develops on the maturing bee larva. When the host bee leaves the cell, the mature female

mites leave the cell. The male mite dies after mating with his sisters, and if immature female mites are

present they die as they come out of the cell, as they cannot survive once outside the cell. The adult female

mite begins searching for other bees or larvae to parasitize.

The phoretic period of the mite appears to contribute to the mite’s reproductive ability, which may last

4.5 to 11 days when brood is present in the hive or as long as ﬁve to six months during the winter when little

no brood is present in the hive. Consequently, female mites living when brood is present in the colony have

an average life expectancy of 27 days, yet in the absence of brood, they may live for many months. In the

average temperate climate, mite populations can increase 12-fold in colonies which have brood half of the

year and 800-fold in colonies which have brood year-round. This makes the mites very diﬃcult to control,

especially in warmer climates where colonies maintain brood year-round [19].

Let Nm(t) be the number of adult female Varroa mites in the honeybee colony in the absence of a virus.

Varroa mites feed on the haemolymph of brood and adult honeybees, and their reproduction depends on the

availability of the brood and the population of the reproductive female Varroa mites. Similarly, we deﬁne a

ﬁxed ξm∈[0,1] ratio of phoretic stage of Varroa mites to their whole population. We model the parasitic

interactions between Varroa mites and honeybees using the Holling Type I functional responses, i.e.:

ˆα

|{z}

The parasitism rate

(1 −ξh)Nh

|{z }

The honeybee brood p opulation

(1 −ξm)Nm

| {z }

The reproductive female Varroa mites population

=αNhNm

where α= ˆα(1 −ξh)(1 −ξm). This implies that the virus-free system of Varroa mites and the honeybee

population can be described by the following two nonlinear equations:

N′

h=r(ξhNh)2

K+(ξhNh)2−dhNh−αNhNm

N′

m=cαNhNm−dmNm

(2)

where the parameter αmeasures the parasitic rate of Varroa mites; cis the conversion rate from honeybee

consumption to sustenance for reproduction; and dmis the death rate of Varroa mites. Model (2) implies

that Varroa mites population Nmgoes extinct if the population of honeybees Nhgoes extinct.

Varroa mites as a disease-vector for virus transmissions: Varroa mites not only feed on host

haemolymph and weaken host immunity, but they also expose honeybee colonies to at least 14 diﬀerent

viruses including deadly viruses such as ABPV and DWV. Varroa mites can transmit viruses in their repro-

duction phase to honeybee brood and during the phoretic phase to adult honeybees. To model the virus

transmission between Varroa mites and honeybees during these two phases, we let Sh(t), Sm(t) be the suscep-

tible population of honeybees and Varroa mites, respectively; and Ih(t), Im(t) be the infected population of

honeybees and Varroa mites, respectively. Then the total population of honeybees is Nh(t) = Sh(t) + Ih(t),

and the total population of Varroa mites is Nm(t) = Sm(t) + Im(t).

The virus transmission between female Varroa mites and honeybees can occur in the following two phases

of the Varroa mite life cycle:

1. The honeybee colony has ξhShsusceptible adult honeybees; ξhIhinfected adult honeybees, ξmSm

susceptible phoretic female Varroa mites; and ξmIminfected phoretic female Varroa mites. In the

phoretic phase, female Varroa mites move between adult bees both spontaneously and just prior to

the death of their host bee [56]. Following the approach of [34], we assume that virus transmission is

frequency dependent, i.e.,

•The rate at which susceptible adult honeybees are infected by the infected phoretic female Varroa

5

mites (IPFM) is:

βmh

|{z}

probability being infected after contacts

×ξhSh

|{z}

population of healthy adult honeybees

×ξmIm

ξmSm+ξmIm

|{z }

probability being contacted by IPFM

=βmhξhShIm

Sm+Im.

•The rate at which susceptible phoretic female Varroa mites (SPFM) are infected by the infected

adult honeybees (IAH) is:

βhm

|{z}

probability being infected after contacts

×ξmSm

|{z}

population of SPFM

×ξhIh

ξhSh+ξhIh

|{z }

probability being contacted by IAH

=βhmξmSmIh

Sh+Ih.

2. The honeybee colony has (1 −ξh)Shsusceptible honeybee brood; (1 −ξh)Ihinfected honeybee brood,

(1 −ξm)Smsusceptible reproductive female Varroa mites; and (1 −ξm)Iminfected reproductive female

Varroa mites. Chen et al. found that there is a direct relationship between virus frequency and the

number of mites to which honeybee brood were exposed, i.e., the more donor mites that were introduced

per cell, the greater the incidence of virus that was detected in the honeybee brood [8,9]. This implies

that the virus transmission rate between Varroa mites and the honeybee brood during the reproductive

phase of mites is density dependent, i.e., similar to the term that describes the parasitic interaction

between mites and honeybee. Therefore, we have as follows:

•A newborn honeybee becomes infected if it is parasitize by the infected reproductive female Varroa

mites. Thus, the rate at which susceptible honeybee brood is infected by the infected reproductive

female Varroa mites (IRFM) is:

βmh2

|{z}

probability being infected after contacts

×(1 −ξh)Sh

| {z }

population of healthy honeybee brood

×ˆα(1 −ξm)Im

| {z }

parasitism by IRFM

=βmh2αShIm.

•The reproduction of Varroa mites depends on honeybee brood. The newborn Varroa mites become

infected if either the brood is infected or the female Varroa mites is infected. Thus, based on the

formulation of the host-parasite interaction model (2), the rate at which infected newborn female

Varroa mites (INFM) become infected depends on the parasitic interaction between mites and

honeybees which can be described as follows:

cα [Ih(Sm+Im) + ShIm]

|{z }

the infected newborn Varroa mites with virus

.

The virus transmission among honeybees: The proportion of honeybees which can infect themselves

is also dependent on the total number of susceptible and infected bees present in the colony, and hence

frequency-dependent transmission is used [34], which is described as follows:

6

βh

|{z}

probability being infected after contacts

×Sh

|{z}

the healthy honeybee p opulation

×Ih

Sh+Ih

|{z }

probability of contacting or being contacted by infected honeybees

=βhShIh

Sh+Ih.

The reduced ﬁtness of honeybees due to virus infections: The parasitic Varroa mites have been

shown to act as a vector for a number of viruses including DWV, ABPV, Chronic Bee Paralysis Virus

(CBPV), Slow Bee Paralysis Virus (SPV), Black Queen Cell Virus (BQCV), Kashmir Bee Virus (KBV),

Cloudy Wing Virus (CWV), and Sacbrood Virus (SBV) [33,34,35,28]. These virus infections contribute

to morphological deformities of honeybees such as small body size, shortened abdomen and deformed wings,

which reduce vigor and longevity, and they can also inﬂuence ﬂight duration and the homing ability of

foragers [28]. In our model, we assume that the infected adult honeybee population ξhIhcontributes to the

reproduction of honeybees with a reduced rate ρ∈(0,1), therefore, the healthy honeybee population Shcan

be modeled as follows,

S′

h=rξ2

h(Sh+ρIh)2

K+ξ2

h(Sh+ρIh)2

|{z }

reproduction of honeybee s

−βhShIh

Sh+Ih

|{z }

honey bee infected by themselves

−αSh(Sm+Im)

|{z }

parasitism by mites

−βmh(ξhSh)Im

Sm+Im

|{z }

adult honeybees infected by the phoretic mite s

−βmh2αShIm

|{z }

honeybee broo d infected by the reproductive mites

−dhSh

.(3)

And the infected honeybee population can be modeled by the following equation,

I′

h=ShhβhIh

Sh+Ih+βmhξhIm

Sm+Im+βmh2αImi−αIh(Sm+Im)

|{z }

Consumed by mites

−(dh

|{z}

natural honeybee mortality rate

+µh

|{z}

additional death due to virus infections

)Ih

.(4)

Let µmbe the additional death rate of Varroa mites due to virus infections. Then the population of

healthy Varroa mites Smand the infected Varroa mites Imcan be described by the following set of nonlinear

equations:

S′

m=Sm

cαSh−βhmξmIh

Sh+Ih

|{z }

the phoretic mites infected by adult honeybees

−dm

|{z}

natural mortality rate of mi tes

I′

m=cα [Ih(Sm+Im) + ShIm]

|{z }

mites born with virus infectio ns

+βhmIh(ξmSm)

Sh+Ih−(dm+µm

|{z}

additional death due to virus infectio ns

)Im

.(5)

Based on the discussions above, the full model of honeybee-mites-virus population dynamics is therefore

7

modeled by the following system of diﬀerential equations:

S′

h=rξ2

h(Sh+ρIh)2

K+ξ2

h(Sh+ρIh)2−dhSh−βhShIh

Sh+Ih−βmh(ξhSh)Im

Sm+Im

−βmh2αIm−αSh(Sm+Im)

I′

h=ShhβhIh

Sh+Ih+βmhξhIm

Sm+Im+βmh2αImi−αIh(Sm+Im)−(dh+µh)Ih

S′

m=SmhcαSh−βhmξmIh

Sh+Ih−dmi

I′

m=cα [Ih(Sm+Im) + ShIm] + βhmIh(ξmSm)

Sh+Ih−(dm+µm)Im

.(6)

For convenience, let ˆ

K=K

ξ2

h

,ˆ

βmh =βmhξh,˜

βmh =βmh2α, ˆ

βhm =βhmξm.Then the full model (6) can

be rewritten as the following model

S′

h=r(Sh+ρIh)2

ˆ

K+(Sh+ρIh)2−dhSh−βhShIh

Sh+Ih−ˆ

βmhShIm

Sm+Im−˜

βmhShIm−αSh(Sm+Im)

I′

h=ShhβhIh

Sh+Ih+ˆ

βmhIm

Sm+Im+˜

βmhImi−αIh(Sm+Im)−(dh+µh)Ih

S′

m=SmhcαSh−ˆ

βhmIh

Sh+Ih−dmi

I′

m=cα [Ih(Sm+Im) + ShIm] + ˆ

βhmIhSm

Sh+Ih−(dm+µm)Im

(7)

where α, ρ, c ∈[0,1] and the virus transmission rates βh,ˆ

βmh,ˆ

βhm,˜

βmh ∈(0,1). In summary, the full

honeybee-mite-virus model (7) incorporates (1) Allee eﬀects of honeybees due to the cooperation of the in-

ternal organization; (2) parasitism interactions between honeybee and mites; (3) the vertical disease transmis-

sion mode modeled by the frequency-dependent disease transmission function during Varroa mites’ phoretic

phase; (4) the horizontal disease transmission mode modeled by the density-dependent disease transmission

function during Varroa mites’ reproductive phase; and (5) the reduced ﬁtness of honeybees due to virus

infections. The rest of this manuscript will focus on the dynamics of Model (7).

3. Mathematical analysis

Let ShIh

Sh+IhSh=Ih=0 = 0 and Im

Sm+ImSm=Im=0 = 0. Deﬁne X={(Sh, Ih, Sm, Im)∈R4

+:Sh+Ih>

0 and Sm+Im>0}, then Xcan be considered as the state space of our model (7). To continue the analysis,

let us deﬁne Nh=Sh+Ih, Nm=Sm+Imas the total population of honeybees and mites, respectively; and

the following notations:

Nc=

r

d−q(r

d)2−4ˆ

K

2, N∗=

r

d+q(r

d)2−4ˆ

K

2

Sc

h=

r

dh−rr

dh2

−4ˆ

K

2, S∗

h=

r

dh+rr

dh2

−4ˆ

K

2

Nc

h=

r

dh+µh+αN∗−rr

dh+µh+αN∗2

−4ˆ

K/ρ2

2, N∗

h=

r

dh+µh+αN∗+rr

dh+µh+αN∗2

−4ˆ

K/ρ2

2

8

ˆ

Sc

h=

r

dh+βh+ˆ

βmh+( ˜

βmh+α)N∗−sr

dh+βh+ˆ

βmh+( ˜

βmh+α)N∗2

−4ˆ

K

2,

ˆ

S∗

h=

r

dh+βh+ˆ

βmh+( ˜

βmh+α)N∗+sr

dh+βh+ˆ

βmh+( ˜

βmh+α)N∗2

−4ˆ

K

2

˜

Sc

h=

r

dh+βh+ˆ

βmh+( ˜

βmh+α)(N∗−N∗

h)−sr

dh+βh+ˆ

βmh+( ˜

βmh+α)(N∗−N∗

h)2

−4ˆ

K

2,

˜

S∗

h=

r

dh+βh+ˆ

βmh+( ˜

βmh+α)(N∗−N∗

h)+sr

dh+βh+ˆ

βmh+( ˜

βmh+α)(N∗−N∗

h)2

−4ˆ

K

2

where d= min{dh, dm}. Deﬁne fb(x, y) =

r

x+q(r

x)2−4ˆ

K/y

2and fb(x, y) =

r

x−q(r

x)2−4ˆ

K/y

2, then we have

∂f b(x, y)

∂x <0,∂f b(x, y)

∂y <0,∂fb(x, y)

∂x >0,∂fb(x, y)

∂y >0

which imply the following inequalities

Sc

h<˜

Sc

h<ˆ

Sc

h<ˆ

S∗

h<˜

S∗

h< S∗

h, Nc≤Sc

h< Nc

h,and N∗

h< S∗

h≤N∗.

Theorem 3.1 (Basic dynamical properties).Assume that all parameters are strictly positive and ρ, c ∈

[0,1]. The model (7)is positively invariant and bounded in the state space X, which is attracted to the

following compact set

C={(Sh, Ih, Sm, Im)∈R4

+: 0 ≤(Sh+Ih) + (Sm+Im) = Nh+Nm≤N∗}

provided that r

d>2pˆ

Kand time is large enough. Moreover, the following statements hold for Model (7):

•If r

2√ˆ

K> dh, then the total population of honeybees Nhis bounded by S∗

h, i.e.,

lim sup

t→∞

Nh(t)≤S∗

h.

If r

2√ˆ

K>dh+µh+αN∗

ρand Nh(0) > Nc

hhold, then the total population of honeybees Nhis persistent,

i.e.,

N∗

h≤lim inf

t→∞ Nh(t)≤lim sup

t→∞

Nh(t)≤S∗

h≤lim sup

t→∞

N(t) = N∗.

•If one of the following inequalities hold

1. r

2√ˆ

K> dh+βh+ˆ

βmh + ( ˜

βmh +α)N∗and Sh(0) >ˆ

Sc

h,or

2. r

2√ˆ

K>max ndh+βh+ˆ

βmh + ( ˜

βmh +α)(N∗−N∗

h),dh+µh+αN∗

ρowith Nh(0) ≥Sh(0) >˜

Sc

h,

then Shis persistent with the following properties:

ˆ

S∗

h≤˜

S∗

h≤lim inf

t→∞ Sh(t)≤lim inf

t→∞ Nh(t)≤lim inf

t→∞ N(t)≤lim sup

t→∞

N(t)≤N∗.

•The extinction equilibrium E0= (0,0,0,0) is always local stable. Moreover, the system (7)converges

to E0globally if dh>r

2√ˆ

Kholds; and it converges to E0locally if the initial population satisﬁes either

9

N(0) < Ncor Nh(0) < Sc

h.

Notes: The positive invariance and boundedness results from Theorem 3.1 imply that our model is well-

deﬁned biologically. In addition, Theorem 3.1 indicate follows:

1. Initial conditions are important for the persistence of honeybees.

2. The inequality r

2√ˆ

K> dhis a necessary condition for honeybee persistence, i.e., the large intrinsic

growth rate r, small half saturation ˆ

K, and the small death rate of honeybees dh.

3. The small values of disease transmission rates βh,ˆ

βmh,˜

βmh; and small values of mite attacking rate

αare also important for the persistence of the healthy honeybee population Sh.

Recall that d= min{dh, dm}, N c=

r

d−q(r

d)2−4ˆ

K

2, N∗=

r

d+q(r

d)2−4ˆ

K

2and

N∗

h=

r

dh+µh+αN∗+rr

dh+µh+αN∗2−4ˆ

K/ρ2

2.

Theorem 3.1 implies that under proper initial conditions, honeybees can persist if r

2√ˆ

K>dh+µh+αN∗

ρand Nh(0) >

Nc

hholds. Notice that ξhis the ratio of adult honeybees in the colony, and

r

2pˆ

K

=rξh

2√K>dh+µh+αN∗

ρ⇔rξh

2√K−

αqr

d2−4K

ξ2

h

2ρ>dh+µh+αr

2d

ρ

and rξh

2√K−

αr(r

d)2−4K

ξ2

h

2ρis an increasing function of ξh. This implies that the large hives to brood ratio ξh

is important for the persistence of honeybees.

Theorem 3.2 (Persistence and extinction of disease or mites).The following statements hold

•If N∗<dm

αc , then the total population of mite Nmgoes extinct, i.e.,

lim sup

t→∞

Nm(t) = 0

where system (7)is attracted to the mite-free invariant set MF ={(Sh, Ih, Sm, Im)∈R4

+:Sm+Im=

0}, and its dynamics is equivalent to the following two-D model (8)

S′

h=r(Sh+ρIh)2

ˆ

K+(Sh+ρIh)2−dhSh−ShβhIh

Sh+Ih

I′

h=ShβhIh

Sh+Ih−(dh+µh)Ih

.(8)

•If r

2√ˆ

K> dh, S∗

h<dm

αc ,and Sh(0) > Sc

h, then the total population of honeybees persists while the

healthy mite population Smgoes extinct, i.e.,

lim sup

t→∞

Sm(t) = 0

10

where the system (7)is attracted to the healthy-mite-free invariant set H M F ={(Sh, Ih, Sm, Im)∈

R4

+:Sm= 0}and its dynamics is equivalent to the following three-D system (9):

S′

h=r(Sh+ρIh)2

ˆ

K+(Sh+ρIh)2−dhSh−βhShIh

Sh+Ih−ˆ

βmhSh−˜

βmhShIm−αShIm

I′

h=ShhβhIh

Sh+Ih+ˆ

βmh +˜

βmhImi−αIhIm−(dh+µh)Ih

I′

m=cαImhIh+Sh−dm+µm

cα i.

(9)

•Assume that r

2√ˆ

K> dh+βh+ˆ

βmh + ( ˜

βmh +α)N∗and Sh(0) >ˆ

Sc

h.Then the disease I=Ih+Im

persists if the following condition holds:

min nβh,˜

βmh +cα ˆ

S∗

ho

max n(dh+µh),(dm+µm)o≥1.

Notes: The results of the reduced dynamics in Theorem 3.2 can be easily obtained by the theory of asymp-

totically autonomous systems [7]. The detailed proof of our results are provided in the last section.

If d= min{dm, dh}=dm, then we have N∗=

r

dm+q(r

dm)2−4ˆ

K

2<dm

αc ⇔pαc(αcK −r)< dm<r

2√K.

If dm>r

2√K, then the system (7) converges to the extinction equilibrium E0according to Theorem 3.1.

On the other hand, if d= min{dm, dh}=dh, then according to Theorem 3.2, the condition S∗

h=N∗=

r

dh+rr

dh2

−4ˆ

K

2<dm

αc can lead to the extinction of the mite population. Therefore, we can conclude that

large values of the death rate of mites, dm, can lead to the extinction of the whole colony; and large values

of the death rate of mites, dm, small values of mite attacking rate, α, and its energy conversion rate, c, can

lead to either its own extinction or the extinction of the healthy mite population Sm. Here we would like

to point out that it is possible to have the persistence of infected mites while the healthy mite goes extinct

(see the resulting dynamics (9) when the healthy mite goes extinct).

Let Nh= lim inft→∞ Nh(t) and Nm= lim inf t→∞ Nm(t).

1. Assume that Nh>0 under proper initial conditions (see suﬃcient conditions provided in Theorem

3.1). If Nh>dm+µm

cα and N(0) > Nhhold, then the total population of mites Nmis persistent, i.e.,

lim inf

t→∞ Nm(t)≥Nh−dm+µm

cα .

2. Assume that r

2√ˆ

K>dh+µh+αN∗

ρ, Nh(0) > Nc

hand Nm>0 (under proper initial conditions). If the

following inequality holds, i.e.,

max nβh+ˆ

βhmN∗

ˆ

S∗

h

+cαS∗

h,ˆ

βmhS∗

h

Nm+˜

βmh +cαS∗

ho

min n(αNm+dh+µh),(dm+µm)o<1,

then the system (7) is attracted to the disease-free invariant set DF ={(Sh, Ih, Sm, Im)∈R4

+:

11

Ih+Im= 0}and its dynamics is equivalent to the following two-D model (10)

S′

h=rS2

h

ˆ

K+S2

h−dhSh−αShSm

S′

m=cαShSm−dmSm

.(10)

A detailed argument supporting the statement above has been provided in the last section, which suggest

that small values of its natural death rate, dm, and the additional death rate due to the virus, µm, can

promote the persistence of the mite population.

The results of Theorem 3.2 also suggest that: 1. the persistence of the virus requires a large value for

the disease transmission rate between adult honeybees, βh, or that the disease transmission rate between

honeybee brood and reproductive mites, ˜

βmh; or small values of total death rates of honeybees, dh+µh, and

mites, dm+µm;2. the extinction of the virus requires small values of all disease transmission rates, i.e.,

small values of βh,ˆ

βmh,˜

βmh,ˆ

βhm; or large values of total death rates of honeybees and mites.

In the following three subsections, we explore the global dynamics of the mite-free subsystem (8), the disease-

free subsystem (10), and the healthy-mite-free subsystem (9).

3.1. Dynamics of the mite-free subsystem

First, let us deﬁne the following notations:

a=1

βh

dh+µh−1,˜

d= (a+ 1)dh+µh=dhβh

dh+µh

βh

dh+µh−1+µh,

I1

h=

r

˜

d−r(r

˜

d)2−4ˆ

K

(a+ρ)2

2, I2

h=

r

˜

d+r(r

˜

d)2−4ˆ

K

(a+ρ)2

2, Sk

h=aIk

h, k = 1,2.

According to Theorem 3.2, if N∗<dm

αc , then the dynamics of (7) is equivalent to the following mite-free

dynamics (8)

S′

h=r(Sh+ρIh)2

ˆ

K+(Sh+ρIh)2−dhSh−ShβhIh

Sh+Ih

I′

h=ShβhIh

Sh+Ih−(dh+µh)Ih

whose dynamics can be summarized by the following theorem:

Theorem 3.3 (Dynamics of the mite-free subsystem).The mite-free subsystem (8)can have one, three,

or ﬁve equilibria. Deﬁne RV

0=βh

dh+µh. The existence and stability conditions for these equilibria are listed

in Table 1. In addition, the global dynamics of the mite-free subsystem (8)can be summarized as follows:

1. The trajectory of (8)converges to extinction (0,0) for all initial conditions in R2

+if one of the following

conditions hold:

•r

2√ˆ

K< dh, or

•RV

0>1and dh<r

2√ˆ

K<˜

d

a+ρ.

2. The trajectory of (8)converges to either (0,0) or (S∗

h,0) for almost all initial conditions in R2

+if the

inequalities RV

0<1and r

2√ˆ

K> dh.

3. The trajectory of (8)converges to either (0,0) or (S2

h, I2

h)for almost all initial conditions in R2

+if the

inequalities RV

0>1,r

2√ˆ

K>max ndh,˜

d

a+ρohold.

12

Equilibria Existence Condition for Existence Stability Condition

(0,0) Always exists Always locally stable

(Sc

h,0) r

2√ˆ

K> dhSaddle if RV

0<1; Source if RV

0>1

(S∗

h,0) r

2√ˆ

K> dhSink if RV

0<1; Saddle if RV

0>1

(S1

h, I1

h)RV

0>1 and r

2√ˆ

K>˜

d

a+ρAlways a saddle.

(S2

h, I2

h)RV

0>1 and r

2√ˆ

K>˜

d

a+ρAlways locally asymptotically stable

Table 1: The existence and stability of equilibrium for the mite-free subsystem (8). We have RV

0=βh

dh+µh,a=1

RV

0−1,˜

d=

(a+ 1)dh+µh=dhRV

0

RV

0−1+µh, Sc

h=

r

dh

−rr

dh2−4ˆ

K

2, S∗

h=

r

dh+rr

dh2−4ˆ

K

2,and I1

h=

r

˜

d

−rr

˜

d2−4ˆ

K

(a+ρ)2

2, I2

h=

r

˜

d+rr

˜

d2−4ˆ

K

(a+ρ)2

2, Sk

h=aIk

h, k = 1,2.

Notes: Theorem 3.3 implies that the mite-free subsystem (8) has relatively simple dynamics, i.e., no limit

cycle. The results show the following interesting ﬁndings:

1. Honeybees can persist with proper initial conditions if the virus transmission rate among honeybees

βhis not large, i.e., RV

0<1.

2. Both honeybees and the virus can coexist if βhis in the medium range, i.e. RV

0>1 and ˜

d

a+ρ<r

2√ˆ

K

3. However, the large virus transmission rate among honeybees βhcan drive honeybees to extinction.

This occurs when the inequalities RV

0>1 and dh<r

2√ˆ

K<˜

d

a+ρhold.

3.2. Dynamics of the healthy-mite-free subsystem

To continue the study, let us deﬁne the following notations:

ˆa=(µh+dh−βh−ˆ

βmh) + q(βh+ˆ

βmh −µh−dh)2+ 4 ˆ

βmh(dh+µh)

2ˆ

βmh

,ˆ

d= (ˆa+ 1)dh+µh,

and

ˆ

I1

h=

r

ˆ

d−r(r

ˆ

d)2−4ˆ

K

(ˆa+ρ)2

2,ˆ

I2

h=

r

ˆ

d+r(r

ˆ

d)2−4ˆ

K

(ˆa+ρ)2

2,ˆ

Sk

h= ˆaˆ

Ik

h, k = 1,2.

According to Theorem 3.2, if r

2√ˆ

K> dh, S∗

h<dm

αc ,and Sh(0) > Sc

h, then the total population of honeybees

persists while the healthy mite population Smgoes extinct, i.e.,

lim sup

t→∞

Sm(t) = 0

where the system (7) is attracted to the healthy-mite-free invariant set HM F ={(Sh, Ih, Sm, Im)∈R4

+:

Sm= 0}and its dynamics is equivalent to the following three-D system (9):

S′

h=r(Sh+ρIh)2

ˆ

K+(Sh+ρIh)2−dhSh−βhShIh

Sh+Ih−ˆ

βmhSh−˜

βmhShIm−αShIm

I′

h=ShhβhIh

Sh+Ih+ˆ

βmh +˜

βmhImi−αIhIm−(dh+µh)Ih

I′

m=cαImhIh+Sh−dm+µm

cα i

.

13

The condition of S∗

h<dm

αc is not the necessary condition for lim supt→∞ Sm(t) = 0 in the full system (7). As

an illustration, we can see that the healthy mites go extinct for the following set of parameter values when

S∗

h>dm

αc :

r= 3.5, ρ =.8,ˆ

K= 20, dh=.15, βh= 0.05,ˆ

βmh =.13,˜

βmh = 0.05,

and

α=.1, µh=.12, c = 0.3, βhm = 0.025, dm=.145, µm= 0.005.

The dynamics of the healthy-mite-free subsystem (9) can be summarized by the following theorem:

Theorem 3.4 (Dynamics of the healthy-mite-free subsystem).If r

2√ˆ

K> dhand S∗

h<dm+µm

cα , then the

population of infected mites goes extinct in the subsystem (9), i.e.,

lim sup

t→∞

Im(t) = 0

which leads to the following mite-free model (11):

S′

h=r(Sh+ρIh)2

ˆ

K+(Sh+ρIh)2−dhSh−βhShIh

Sh+Ih−ˆ

βmhSh

I′

h=ShhβhIh

Sh+Ih+ˆ

βmhi−(dh+µh)Ih.

(11)

The dynamics of the mite-free model (11)can be summarized as follows:

1. If r

2√ˆ

K<ˆ

d

ˆa+ρ, then the mite-free model (11)has only the extinction equilibrium (0,0) which is globally

stable.

2. If r

2√ˆ

K>ˆ

d

ˆa+ρ, then the mite-free model (11)has the extinction equilibrium (0,0) and two interior

equilibria (ˆaˆ

Ik

h,ˆ

Ik

h), k = 1,2where (ˆ

S1

h,ˆ

I1

h) = (ˆaˆ

I1

h,ˆ

I1

h)is always a saddle; and both (0,0) and (ˆ

S2

h,ˆ

I2

h) =

(ˆaˆ

I2

h,ˆ

I2

h)are always locally asymptotically stable.

Notes: Theorem 3.4 implies that, under the condition of S∗

h<dm+µm

cα , the infected mite population Im

goes extinct in the healthy-mite-free subsystem (9) which reduces to the new mite-free subsystem (11). The

subsystem (11) occurs from the healthy-mite-free subsystem (9) when increases the value of dm+µm

cα , e.g.,

increasing the value of the natural death rate of mites dm, or increasing the additional death rate of mites

caused by virus infections, or decreasing values of the energy conversion rate c, or decreasing the value of

mite attacking rate α.

Synergistic eﬀects of parasitic mites and virus infections: We have two types of honeybee-virus

interaction subsystems (i.e., no mites): if the colony has no mites at all, then we have the reduced subsystem

(8); if the colony has mites initially but healthy mites go extinct ﬁrst, and then the infected mites go extinct

during the process due to changes in environment, then we have the limiting subsystem consisting of only

honeybees and the virus (11). From Theorem 3.3, we can conclude that the colony can have persistence of

honeybees while the virus goes extinct when the inequalities RV

0>1,r

2√ˆ

K>max ndh,˜

d

a+ρohold. However,

according to Theorem 3.4, the subsystem (11) has persistence of disease whenever r

2√ˆ

K>ˆ

d

ˆa+ρholds. This

implies that the existence of both mites and virus initially could promote the persistence of disease.

The results from Theorem 3.4 provide us a general view of the dynamics of the healthy-mite-free subsystem

(9). In the rest of this subsection, we will explore the existence condition for the interior equilibrium of (9).

14

An interior equilibrium (Sh, Ih, Im) of the healthy-mite-free subsystem (9), satisﬁes the following equations:

cαImhIh+Sh−dm+µm

cα i= 0 ⇒Nh=Ih+Sh=dm+µm

cα

r(Sh+ρIh)2

ˆ

K+(Sh+ρIh)2−dhNh−αNhIm−µhIh= 0 ⇒r(Sh+ρIh)2

ˆ

K+(Sh+ρIh)2−dhNh−αNhIm−µhIh= 0

⇒r(Nh−Ih+ρIh)2

ˆ

K+(Nh−Ih+ρIh)2−dhNh−αNhIm−µhIh= 0

⇒Im=

r(dm+µm

cα −Ih+ρIh)2

ˆ

K+(dm+µm

cα −Ih+ρIh)2−dhdm+µm

cα −µhIh

α(dm+µm

cα )=f1(Ih)

(12)

ShhβhIh

Sh+Ih+ˆ

βmh +˜

βmhImi=αIhIm+ (dh+µh)Ih

dm+µm

cα −Ihh βhIh

dm+µm+ˆ

βmh +˜

βmhImi=αIhIm+ (dh+µh)Ih

⇒Im=(dm+µm

cα −Ih)hβhIh

dm+µm+ˆ

βmhi−(dh+µh)Ih

αIh−(dm+µm

cα −Ih)˜

βmh =f2(Ih)

.(13)

The equations above imply that the interior equilibrium of (9) is the positive intercept of f1(Ih) and f2(Ih)

subject to 0 < Ih<dm+µm

cα . The expression for the function f1(Ih) implies that the subsystem (9) has no

interior equilibrium if

r < dh(dm+µm)

cα ⇔rcα < dh(dm+µm)

since Im=f1(Ih)<0 when this inequality holds. Therefore, we could expect the extinction of Imfor small

values of r, c, α and large values of dh, dm, µm. This has been conﬁrmed by numerical simulations. Our

simulations provide important insights on the complicated dynamic of (9), which suggest the following:

•The population of honeybees and infected mites in (9) experiences sudden collapse when we increase

the values of c, α, ˆ

K, and the disease transmission rates, and when we decrease the values of dh, ρ, r, µm.

This is due to the fact that increasing or decreasing the values of these parameters destabilizes the

system and generates ﬂuctuated dynamics. The destabilizing eﬀects generate unstable oscillations. The

amplitudes of oscillations increase until they touch the stable manifold of the extinction equilibrium,

which cause the collapse of the whole colony. The destabilizing eﬀects of c, α, ˆ

K, dmcan be explained

through the dynamics of the virus free subsystem (10) that we will investigate in the next subsection.

•Decreasing the values of c, α, ˆ

Kcan stabilize the system; small values of c, α can cause the extinction

of the infected mite population Si, and lead to the coexistence of Shand Ih.

•Increasing the value of µhcan stabilize the system but large values of µhcan cause extinction of the

whole colony due to the initial oscillations.

•Decreasing µmcan destabilize the system; while increasing it can stabilize the system; large values of

µmcan lead to the extinction of Miand the persistence of Hs, Hi.

•Decreasing the value of ρcould destabilize the system, thus causing the extinction of the colony.

•Increasing the virus transmission rates (i.e., βh,ˆ

βhm,˜

βmh,ˆ

βhm) can stabilize the system, while decreas-

ing their values can destablize the system and cause the extinction of all species.

15

3.3. Dynamics of the disease-free subsystem

Theorem 3.2 in previous section provides suﬃcient conditions that lead to the following disease-free

dynamics (10)

S′

h=rS2

h

ˆ

K+S2

h−dhSh−αShSm

S′

m=cαShSm−dmSm

whose dynamics can be summarized by the following theorem:

Theorem 3.5 (Dynamics of the disease-free subsystem).Let H∗=dm

αc , M∗=1

αhrH∗

ˆ

K+(H∗)2−dhi.

The disease-free subsystem (10)can have one, three, or four equilibria. The existence and stability conditions

for these equilibria are listed in Table 2. The global dynamics of the disease-free subsystem (10)can be

Equilibria Existence Condition Stability Condition

(0,0) Always exists Always locally stable

(Sc

h,0) r

2√ˆ

K> dhSaddle if Sc

h<dm

αc =H∗; Source if Sc

h>dm

αc =H∗

(S∗

h,0) r

2√ˆ

K> dhSink if S∗

h<dm

αc =H∗; Saddle if S∗

h>dm

αc =H∗

(H∗, M ∗)Sc

h<dm

αc =H∗< S∗

hSink if H∗>pˆ

K; Source if H∗<pˆ

K.

Table 2: The existence and stability of equilibrium for the disease-free subsystem (10), where Sc

h=

r

dh

−rr

dh2−4ˆ

K

2, S∗

h=

r

dh+rr

dh2−4ˆ

K

2and H∗=dm

αc , M∗=1

αhrH ∗

ˆ

K+(H∗)2

−dhi.

summarized as follows:

1. The system (10)converges to extinction (0,0) for almost all initial conditions if r

2√ˆ

K< dhor dm

αc < Sc

h.

2. If S∗

h<dm

αc , depending on initial condition, the trajectory of (10)converges to either (0,0) or (S∗

h,0).

3. If Sc

h<dm

αc < S∗

hthen Model (10)has a unique interior equilibrium (H∗, M∗)which is locally asymp-

totically stable when dm

αc >pˆ

Kand is a source when dm

αc <pˆ

K.

Notes: Theorem 3.5 provides us a global picture on the dynamics of the disease-free subsystem (10), i.e.,

the honeybee colony only infected with mites but not the virus. By applying the results in [57,61], we can

conclude that the disease-free subsystem (10) undergoes a subcritical Hopf-bifurcation at dm

αc =pˆ

K. The

subsystem (10) has a unique unstable limit cycle around (H∗, M ∗) whenever dm

αc <pˆ

K. In this case, the

periodic orbits expand until it touches the stable manifold of the boundary equilibrium (Sc

h,0) which leads

to the extinction of both honeybees and the parasitic mites. We refer to this phenomena as a catastrophic

event which could be linked to CCD. Our theoretical results also suggest that a small death rate for mites

and a large parasitism rate can destabilize the system.

Linking to CCD: To illustrate the catastrophic event, we use reasonable parameters from [56,44]. Let the

reproduction of egg per day during summer be r= 1500; and the population size of the honeybee colony

at which the birth rate is half of the maximum possible rate be pˆ

K= 2000; the natural death rate of

honeybees is dh= 0.01; the parasitism rate is α= 0.005; the energy conversion rate is c= 0.01; and the

natural death rate of mites is dm= 0.1. This set of parameter values gives dm

αc <pˆ

Kwhich implies that

acatastrophic event will occur (see Figure 1; the population of honeybees is in red and collapses around

16

0 20 40 60 80 100 120 140 160 180 200

0

1000

2000

3000

4000

5000

6000

time

Sh

Sm

r = 1500; K= 4000000; dh =0.01;

α=0.005; c = 0.01;dm=0.1;

Sh(0)=H*+1;Sm(0)=50

Figure 1: Time series of Model (10) when r= 1500, α = 0.005, c = 0.01, dh= 0.01, dm= 0.1: population of honeybees is in

red while Varroa mites is in black.

time=200).

Note that ˆ

K=K

ξ2

h

and α= ˆα(1 −ξh)(1 −ξm) where ξh, ξmare ratio of adult bees and ratio of phoretic

stage of Varroa mites in honeybee colony, respectively. The catastrophic event occurs when

dm

αc <pˆ

K⇔dm

ˆα(1 −ξm)c√K<(1 −ξh)

ξh⇔ξh

(1 −ξh)<ˆα(1 −ξm)c√K

dm

which implies that low hive to brood ratio can also destabilize the system and cause the sudden extinction

of honeybees.

3.4. Dynamics of the full system

Recall that the full system (7) of honeybee-mite-virus interactions can be described by the following set

of equations:

S′

h=r(Sh+ρIh)2

ˆ

K+(Sh+ρIh)2−dhSh−βhShIh

Sh+Ih−ˆ

βmhShIm

Sm+Im−˜

βmhShIm−αSh(Sm+Im)

I′

h=ShhβhIh

Sh+Ih+ˆ

βmhIm

Sm+Im+˜

βmhImi−αIh(Sm+Im)−(dh+µh)Ih

S′

m=SmhcαSh−ˆ

βhmIh

Sh+Ih−dmi

I′

m=cα [Ih(Sm+Im) + ShIm] + ˆ

βhmIhSm

Sh+Ih−(dm+µm)Im

.

The results from the previous section provide us a complete picture of the dynamics of the subsystems of

the full system (7). In this subsection, we explore the dynamics of the full system. Here we start with the

17

persistence of honeybees and the extinction of disease and mites. From Theorem 3.1,3.2,3.3,3.4, and 3.5,

we have the following corollary:

Corollary 3.1 (The persistence of honeybees).Assume that r

2√ˆ

K> dh. If N∗<dm

αc and RV

0=βh

dh+µh<

1, the full system (7)converges to the disease-mite-free set DM F ={(Sh, Ih, Sm, Im)∈R4

+:Sm+Ih+Im=

0}where the system (7)is reduced to the following one-D system (14):

S′

h=rS2

h

ˆ

K+S2

h−dhSh(14)

whose dynamics can be summarized as follows:

1. If the inequality r

2√ˆ

K< dhholds, then (14)converges to 0.

2. If the inequalities r

2√ˆ

K> dhand Sh(0) > Sc

hhold, then (14)converges to S∗

h. If the initial condition

falls below Sc

h, i.e., Sh(0) < Sc

h, then (14)also converges to 0.

Moreover, if one of the following inequalities hold

1. r

2√ˆ

K> dh+βh+ˆ

βmh + ( ˜

βmh +α)N∗and Sh(0) >ˆ

Sc

h,or

2. r

2√ˆ

K>max ndh+βh+ˆ

βmh + ( ˜

βmh +α)(N∗−N∗

h),dh+µh+αN∗

ρowith Nh(0) ≥Sh(0) >˜

Sc

h,

then Shis persistent with the following properties:

ˆ

S∗

h≤˜

S∗

h≤lim inf

t→∞ Sh(t)≤lim inf

t→∞ Nh(t)≤lim inf

t→∞ N(t)≤lim sup

t→∞

N(t)≤N∗.

Notes: Based on Theorem 3.1 and Theorem 3.5, we can also conclude that the extinction of disease occurs

when all values of all disease transmission rates, βh,ˆ

βmh,˜

βmh,ˆ

βhm are small; with the consequence that the

full system (7) converges to either (0,0,0,0) or (S∗

h,0,0,0) when RM

0=S∗

h

H∗<1 while (7) converges to either

(0,0,0,0) or (H∗,0, M ∗,0) when 1 <RM

0<S∗

h

Sc

h.

The dynamics of the full system (7) can be extremely complicated. We ﬁrst look at the existence of

its interior equilibrium. An interior equilibrium (Sh, Ih, Sm, Im) of the system (7), satisﬁes the following

equations:

0 = SmhcαSh−ˆ

βhmIh

Sh+Ih−dmi

⇒Ih=ShcαSh−dm

dm+ˆ

βhm−cαSh,dm

cα < Sh<dm+ˆ

βhm

cα

⇒Nh=Sh+Ih=ˆ

βhmSh

dm+ˆ

βhm−cαSh=g1(Sh)

(15)

0 = r(Sh+ρIh)2

ˆ

K+(Sh+ρIh)2−dh(Sh+Ih)−α(Sh+Ih)(Sm+Im)−µhIh

⇒Nm=Sm+Im=

r(Sh+ρIh)2

ˆ

K+(Sh+ρIh)2−dh(Sh+Ih)−µhIh

α(Sh+Ih)

⇒Nm=rSh(ˆ

βhm+(dm−cαSh))( ˆ

βhm+(dm−cαSh)(1−ρ))2

αˆ

βhmhˆ

K(ˆ

βhm+(dm−cαSh))2+S2

h(ˆ

βhm+(dm−cαSh)(1−ρ))2i−dh

α−µh(cαSh−dm)

ˆ

βhm =g2(Sh)

(16)

18

0 = cα(Sh+Ih)(Sm+Im)−dm(Sm+Im)−µmIm

⇒Im=Nm(cα(Sh+Ih)−dm)

µm=Nm[cα ˆ

βhmSh−dm(ˆ

βhm+(dm−cαSh))]

µm(ˆ

βhm+(dm−cαSh))=Nm(cαSh