5 Infectious Disease Modeling for Honey Bee Colonies

5

Infectious Disease Modeling for Honey Bee Colonies Hermann }. Eberl, Peter C. Kevan, and Vardayani Ratti

CONTENTS Abstract ..... .. .............................................................. ................ .................... ...... ..... 87 Keywords .. .......................... ..................................................................................... 88 5.1 Introduction .................................................................................................... 88 5.2 Mathematical iVIodels ..................................................................................... 89 5.3 lnfcctiou Disease r-.1odels .............................................................................. 91 5.-t Application to Honey Bees. Varma Mites. and the Acute Bee Paraly. is Virus .............................................................................................. 92 5.5 Computer Simulations of the Disease Model ................................................. 95 5.5.1 Computational Setup and Parameters................................................. 95 5.5.2 Illustrative Result!> .............................................................................. 96 5.5.3 Mathematical Analysis of the Constant Coefficient Case in an Attempt to Gain More Insight into Computer Simulations .............. 102 5.5.3. I One-Dimensional Healthy Bee Submodel.. ....................... 102 5.5.3.2 Two-Dimensional Bee-Mite Submodel. ............................ 103 5.5.3.3 Complete Bee-Mite-Virus Model. .................................... 106 5.6 Concluding Ren1arks .................................................................................... 106 Acknowledgment ................................................................................................... 106 References .............................................................................................................. 107

ABSTRACT ~1athematical models have been successfully used to study the progression of various infectious diseases for many year . l.n this chapter, we demonstrate how the ·usceptiblc- infected-removed (SIR) modeling framework can be applied to study disea. cs of honey bee colonies. We focu on the acute bee paralysis viru . which is transmitted by the parasitic mite \larroa destructor as a \ector. The resulting model consist_ of four nonautonomou . nonlinear ordinary differential equation ..... which we stud) with analytical and computational techniques. Our results indicate that, depending on model parameter. . in the absence of the virus. a mite infestation can

87

In Silico Bees

88

lead either to extinction of the bee colony or to an endemic infestation that allows the

bee colony to survive. However. when the mires also carry the virus. this infection is very difficult to be fought off without remedial intervention and might lead to the extinction of the colony. KEYWORDS

Acute bee paralysis virus, Honey bees. Mathematical model. Ordinary differential equations. SIR model. Varroa mite. Vector-borne disease

5.1

INTRODUCTIO N

Honey bees. like all animals. have diseases and pests that can destroy individuals or the entire colony if not treated in time. Diseases in the honey bee colony are mainly caused by pests and parasites. bacteria. fungi. and viruses llJ. This chapter focuses on viral diseases in honey bee colonies for which the parasitic mite Varroa destructor acts as a vector. Besides vectoring diseases, varroa mites also affect the life span of honey bees by feeding on the bee's hemolymph and by piercing its intersegmental membrane [2 .3]. There have been at least 14 viruses found in honey bee colonies l4.5j, which can differ in intensity of impact. virulence, etc .. for their host. For example. the acute bee paralysis virus (A BPV) affects the larvae and pupae that fail to metamorphose to adult stage. while in contrast the deformed wing virus (DWV) affect!' larvae and pupae that can survive to the adult stage [6]. Varroa mites are the main agents of transmission of viruses between bees. Transmission occurs when mites feed on bees and when a virus-carrying mite attaches to a healthy bee during the former's phorctic phase f3,7,8]. Thus, the pre\'iously uninfected bee becomes infected. A virus-free phoretic mite can begin carrying a virus after it moves from an uninfectcd to an infected bee (8,91. The relationships between honey bees, parasitic mites, and vi ruse. comprise a complex, interdependem system with nonlinear interactions. It cannot be understood by studying the three players. host. vector. and virus. individually and superimposing the findings. Instead. their interdependencies must be taken into account. For example, not only does tbe virus have an effect on the bee population, but also a decreasing bee population affects the mites. which again alters disease transmission. For several decades, mathematical models have been developed as a major rool to understand the course and complexity of infectious diseases in human populations and to aid in the development of vaccination and control strategies. ln this chapter. we describe how this highly successful methodology can be adapted to study the effect of varroa mites and their viruses in honey bee colonies. We stres that although these mathematical models are predictive, they are primarily qualitative, rather than quantitative. in nature. Their main purpose is not to predict how many bees in a colony become infected by a certain virus and when. Their real power lies in describing the fate of a disease qualitatively and to provide insight under which conditions a disease can be eradicated or when it will overcome the colony. potentially eradicating it. Such disease models can also be valuable tools for investigating rhe effectiveness of remedial and disease control strategies. But this aspect is beyond what we want to achieve in this chapter.

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5.2 MATHEMATICAl MODELS ~lathematicalmodels

are (sometimes simple) repre!>entatioru, of complex real-world problems. They can increase our understanding of the procc~ses involved and of their interplay. Mathematical models have a long tradition in mo!>t disciplines of cience and engineering. The aim of all cientific tudy is to derive theorie that enable us to make predictions. A mathematical model is the formulation of such a theory. Mathematical modeling requires a good command of mathematical tools and an understanding of the underlying phy ical or biological phenomena. The formulation of a mathematical model is a careful trade-off between complexity. difficulty. and accuracy. Increased complexity ideally also increases accuracy. but always at the expen e of added difficulty. Difficulty can take several form~. Foremost, this refers to the difficulty in obtaining a solution of the mathematical model. either analyticall) or computationally. Equally imponanl. however, is the difficulty in providing the additional information required. , uch as parameter or other input data for the model. Often. the theoretical accuracy gain of a more complex model can be complete!) 10!:>1 if the additionally required information cannot be obtained and must be substituted by crude assumption:.. Every mathematical model is spec ific to its purposes and no model fits all. ~loreover. mathematical models are not unique. Often several mathematical model can be formulated for eemingly the same. say. biological or physical proces!:>. ~lodels differ in the assumptions on which they are based and in the mathematical machinery u ed. As a generaJ rule, mathematical modeh that contain lots of details without relevance to the question at hand are not desired. In some sense. that can be understood as a consequence of Occam's razor: Simpler explanations are, other thing being equal. general!) better than more complex ones. In other words. a model should be a~ imple a~ possible and as complicated as necessary. Model formulation is an iterative. multifaceted cyclical process as shown in Figure 5.1. First. the most essential features of real-world problems are identified and described in mathematical term . Once ·uch a candidate mathematical model i~ formulated. it is analyzed with respect to its internal consistency. For most mathematical models. it is not pos ·ible to write dO\\ n the solution. In fact. the primary question a modeler has to ask is whether or not his or her model actually posse~ es

Real-world problem

I---+1

~ lathemaucal

''

Biologi of the images, the model predictions. offer potential explanations or testable hypotheses for the

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consequents of the things pictured. As such. they inform future experimental studies or uggest potemial imerventions to avoid unde irable con. equences. This, in fact. is an important area for the application of mathematical models and applies to a broad clas:-. of model:. used in science and engineering. among which are many infectious disease models.

5.3

INFECTIOUS DISEASE MODELS

The course of infectious di~ea e can be predicted using mathematical models that have been developed 0\·er many years and ha'e experienced a huge surge in activity in the last decade [12-14]. While most of these research effons are driven by diseases of human populations. the underlying concepts can be adapted to diseases of animals as well. Because parameters change temporally and geographically. an accurate quantitative prediction of future epidemics over everal years normally cannot be expected. especially for newly emerging diseases. Consequently. al o a quantitative validation is only possible after the fact. However, instead of focusing on quantitative predictions. Lhe real strength and power of the mathematical approach to understand disease dynamics arc the qualitative insight that it gives into the complex interpla~ between host and disea e. In lhis light. the success of infectious di ease models has been phenomenal. even in the ab ence of accurate data that describe a particular disea. e quantitatively. We want to make use of Lhis successful strategy for honey bee diseases. The traditional modeling framework that we have adapted for viru infections in honey bee population . . is the so-called susceptible-infected-removed (SIR) models in which the population i de~cribed in terms of usceptible and infected individual. : further particularities depend on the disea~c under consideration. For viruses in honey bees. several of the simplifying assumptions often made in infectious disease model. for humans are not possible. For example. because the bee population size fluctuates greatly with the seasons and because viral disease can reduce the population con iderably. to the point of extinction. the usual assumption that the overall population size remains relatively unaffected by the virus and can be con idered a constant is not po ible, leading to more algebraically involved models. Moreover, because the characteristic timescale of the disea. e i:, comparable to or even bigger than the characteri. tic timescale of bee biology. birth and natural death of bee cannot be neglected and modeb nlU~t be developed that account for the panicularitie of bee reproduction. Furthermore. ABPV and other mrroa-associated virusc. arc to be modeled a vector-borne diseases: a possible extension of the SIR modeling framework for vector-borne diseases is Ross- Macdonald modcll15j. The basic. traditional SLR model divide the host population into susceptible (S). infected (1). and removed or recovered (R) individual . The susceptible are individuals that arc not currently. but may become. infected. and the infected group consists of individuals that arc currently infected. Dependjng on the disease. the R group can indicate those individuals who were infected but have now recovered from the disease or those who were removed. for example. by death. The SIR model i conceptualited in Figure 5.2. Each compartment in this model i!) described by a relatively simple ordinary differential equation that con titutcs

In Silica Bees

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Susceptible

Recovered

Infected

FIGURE 5.2 Conceptualization of the SIR modeling iramework: After infection. susceptible individuals lca"c the S compartment and enter the I companmcm. After reco\'ery/ removal. they lem·e the I compartment and enter the R compartment. The infection rate can depend on hot h. that is. the number of aIread)• infected and susceptible indiYiduals.

Susceptible bees

!

}-or ~

' ' I

Infected bees -~

I

'

I

I ·~..

I

'·· I I

I

Mites car· rying \'irus

y'

~·lites not carrying viru~

FIGURE 5.3 Role of mite~ in the . pread of infection in a honey bee colony: Virus-carrying mite' affect the transmi-;sion rate from susceptible to infected bees. All mite~.' irus carrying or not. affect mortality of susceptible and infected bees.

a ··mass balance ... that is. a balance of gain and loss in the compartment. The rate at which susceptible individuals become infected, that is. the rate of transfer from the S into the I compartment. depends on the number of already infected individuals but can also depend on the number of susceptible i ndivid u a l ~. In the Iiterature. the SIR model has been adapted to describe a variety of diseases. For example. it can be considered that recovered individuals become susceptible again. An extension of the model to vector-borne diseases can be achie,·ed by making the infection rate dependent on the number of vector!. that carry the virus. In many in tances. this require the inclusion of additional differential equations that describe how tbe vector population e\'olves and hO\\ \'ector acquire the virus (see. e.g .. Figure 5.3).

5.4

APPLICATION TO HONEY BEES, VARROA MITES, AND THE ACUTE BEE PARALYSIS VIRUS

We apply the traditional Sl R modeling framework ro describe the fate of a honey bee colony that is exposed to the ABPV. ABPV. together with the Kashmir bee virus. black queen cell virus. and the I raeli acute paralysis virus, belongs to the family Dicistroviridae. These viru e share a number of biological characteristic , sucb as principal transmission routes and primary host life stages fl6). ABPV b a common infective agent of honey bee that i. frequently detected in apparently healthy colonics. Bees affected by thi. viru are unable to fly. lo. c hair from their bodies. and

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rrernbk uncontrollably. The virus has been suggested to be a primary cause of bee mortality. Infected pupae and adults suffer rapid death. A BPV i associated with arroa mites and ha been implicated in colony collapse disorder. Due to the imultaneous appearance of mite and virus under field conditions. it is difficult to eparate the effects of both pathogens [ 17]. Therefore, they should be studied together. and mathematical models of the disease dynamics should include both pathogens ,imultancously. In the SIR framework . susceptible individuals are healthy bees, infected mdi\'iduals arc . ick bees. and the vector is the mrroa mite. The virus is tntn miued to the bee~ via mites that feed on their hemolymph by pierci ng the intersegmental membrane. The model for honey bees is with demography. that is. the birth and the death of the bees is taken into consideration. In our ca e. the remo1•ed compartment is the deceased bees. A :-. 0 and xi> 0 exist if the maximum growth rate J.l is large enough. compared tO the death rate d 1• More specifically. after some routine algebra. the condition for the existence of these equilibria is obtained as 11

"r--;1 -J..l > - - K VII-I d1 11- J

(5.8)

that is. it depends also on the brood maintenance coefficients, that is. the number of bee required to care for the brood. We denote the larger of the two equilibria by x; and the smaller one by x;. The latter is unstable. while the former is asymptotically stable. Becau e the solution x(r) of (5.7) must be strictly monotonic between two equilibrium points, we obtain the following picture: lf initially the bee population is small, x(O) < x;. the solution cOn\'erges to the trivial equilibrium x~: that is. the colony die · out because too few worker bees are present to care for the brood. On the other hand. tf x{O) > x;, the solution converges to the asymptotically stable equilibrium x~ : that 1s. a healthy colony can establish itself attaining equilibrium For the data in Table 5.1, the nontrivial equilibria x~. 2 exi tin spring, summer. and fall , but not in winter. when no new bees arc horn. Thus. in wimer. all solutions converge LO the trivial equilibrium. However. a. it i not reached in finite time. the population can recover in ~pr ing if at the beginning of the season the bee population is tronger than the :!quilibrium obtained for that season. In the time-dependent computer imulations that were shown earlier. this corresponds to the situation found in Figure 5.-ta and c.

x;.

x;

5.5.3 .2 Two-Dimensional Bee-Mite Submodel We investigate now how the stability of the equilibria x~. 1 . ~ of (5.8) changes when oarasitic mites (but not \iru es) arc considered in the colony. To this end. we study the bee-mite subsy ·tem of (5.3) and (5A): (5.9)

dM dr ~ere .

=rJt ( I- Nf ) a..\

we again assume (5.6) for the brood maintenance function g(x).

(5.10)

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Bees

The two-dimensional model (5.9) and (5.10) can ha,·e up to five equilibrium points. that i . point ex·. ,\1[ ) for which dx/ dr =d.\1 dt =0. Sec Figure 5.7. The first one is again a trivial cqumbrium point A with (x·. M·)=(O.O) (no bees. no mites). which can be shown to be always asymptotically stable. for the same reasons that x~ was stable in the one-dimen ional bee-only model and because in the absence of bees mites do not have a host. Solutions that come close enough to this equilibrium will lead to the dying om of both bees and mites. This corresponds to our dynamic computer simulations in Figure 5.5b through d. X

4

J04

3 '2

,' ~

,'

,'

,,

,'

,'

,, - x-nullchnes --- M-nulklines

/

0 ~ ------ ------------ -------- ---------- ---- -1

-'2

-3 -4 ~-----~--~-----~-----L--~-----L---~

1.5

0.5

0

(a)

2

2.5

3

3.5

.t

X

4 10.0

x to•

·l~~--,-----~--~~---~---r----,---~---~

/

/

3

/

/

2 / L >-

~

/

/

/

- x-nullcll nes -- - M-nullclines

/

/

0 A -1

-2 -3 -4~----------~------~----~------~----~------L-----~ ? :> 0.5 1.5 •• 3.5 0 2 3

(b)

FI GU RE 5.7 :\ullclines and location of equilibria for the bee-mite model (5.9) and (5.10): Shown arc the da~hcd curve~ along" hich dM!dr = Oand the solid cunc, along which dxldr = O. T he equilibria are the intersection~ of a dashed and a solid cun·c. (a) The tri' ial equilibrium A i~ the only equilibrium: (b) only the mite-free equilibria A. B. C exist: (c). (d) two additional equilibria D. E with M · = ax'>O
> x =~n- 1 K.


3

105

- .t- nullclines --- M - nullclines

2

::e: o

A---

J3 --------------------- ---------------------- ------

-1

-2 -3 -4

0

0.5

1.5

(c)

2

2.5

3

3.5

.t

4

X JO'l

--- --- ---

3 2

---

--~

0 -I

-·~~--~----~----~~----~-----L----~------L-----~

0

0.5

FIGURE 5.7

1.5

2

3

2.5

3.5

;o:

(d)

(continued)

Furthermore, it is easy to \'erify that if (5.8) is sa1isfied, then there exist two mitefree equilibrium points. associated with the two equilibria 2 of the aforementioned bee-only model: B: (.r; . 0) and C: (.r;. 0). The former inherits the instability of its phase portrait is an unstable node. The latter is an unstable saddle: only solutions on the line M=O (no mites in the system) approach the equilibrium: all other equilibria are repelled. The instability of these Illite-free equilibria is explained a~ follows: The growth of the mites is logistic. Therefore. as oon as mites enter the system, they stan c tablishi ng them elve:.. That explains why in our computer simulations of the model we cannot fi nd a case where the mites are fought off by the bees. Depending on parameters. there might also be two equilibria points D: (x~ .M;) and £ :(.r; .M;-) for which both .r~.E andM ~.£ are positi,·e. Here. we have

x;.

x;.

In Sifico Bees

106 x~

< x~ and M~ < M ~. Using phase plane analy is tech';(ues, we find that the equilibrium D is always unstable. while E is stable if x~ > n n -l K and unstable otherwise. See [23] for the mathematical details. The simulations in Figure 5.4b and d correspond to stable £. 5.5.3.3 Complete Bee-Mite-Virus Model We investigate now the question whether a stable. mite-infested honey bee colony can fight off the virus. To this end. we consider the complete four-dimensional models (5.1 ) through (5.4) with (5.6). It is easily verified that the equilibrium£ of the bee-mite model has a corresponding disease-free equilibrium £. : (x~ . 0. M~ . 0). Of particular interest is the case where the equilibrium E is !'table under (5.9) and (5.10). Linearization of (5.1 ) through (5.3) around this equilibrium shows that it is asymp· totically stable if the following condition on the parameters is satisfied: (5.11) Therefore. if the virus is introduced into a properly working mite-infested colony, the colony can fight off the virus if the sick bees die fast. This corresponds to the scenario we observed in our simulations in Figure 5.6b. However. as we have seen there, this required a relatively low choice for the parameter a. and. therefore. our results suggest that it might not be a frequent event that a colony can rid itself of the ABPV disease. As this equilibrium is the only stable equilibrium with a positive bee population size. the alternative seems the complete decline of the bee colony. 5.6

CONCLUDI N G REMARKS

In recent years. mathematical models have been successfully used to study the progression of infectious diseases. While typically set in the context of human populations. many of the principles are generally valid and can be applied to animal populations as well. In this chapter. we try to show how this is done for honey bee diseases in a simple example. focusing ou honey bees. ,.a,Tva mites. and the ABPV. This model, although conceptually simple. is complicated enough to not allow for a complete. rigorous mathematical analysis. Therefore, it must he studied in computer simulations. aided by analytically obtained mathematical insight. The model we describe here is but a firs t step from which many potential directions for further work origi nate. This includes more detailed descriptions of the disease. incorporation of effects such as queen failure or swarming, which we neglected. or the effect of mite control strategies on the viral disease. ACKNOWLEDGMENT

The authors would like to thank the Ontario Ministry for Agriculture. Food and Rural Affairs (OMAFRA) and the Canadian Pollination Initiative (CANPOLlN) for financial support.


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