Can we model honey be health?

Transcription

1 Can we model honey be health? Ingemar Fries, Eva Forsgren, Department of Ecology Swedish Univdersity of Agricultural Sciences Uppsala, Sweden Apimondia, Argentina, 2011

2 MODELS AND MODELING - HOW AND FOR WHAT? Models can be used to sell things or to illustrate things or to attract attention! Some of this is also true for mathematical modeling!!

3 Models are estimates of the real world. Most often models deal with a small subsection - an extract from the world around us

4 Building a mathematical model is like building other kinds of models - for example, visual models (pictures), verbal models (speech, articles,or books), and three dimensional models (sculpture). In contrast to for exampel a picture model, mathematical models often try to describe series of events and flows of entities over time We begin by looking at the real world and then we make a rough sketch. Then we go back to the real world, comparing what we see with our model so far.

5 We build up the model step-by-step, starting with the real world and repeatedly comparing our model to the real world. In the process we often learn more about the real world. Features of the real world that don't immediately draw our attention often become more noticeable as we draw our picture. Because we need to formalize and quantify relationships to define our model (to describe our world) building models will unravel where more information is needed for a formal description. Even if the final model produces results of no use, the construction itself may produce important results and initiate vital experiments.

6 Mathematics, like English or French, is a language -- a language that we use to express ourselves, to communicate with others, and, perhaps most importantly, to think -- to manipulate ideas. Mathematical modeling is about how we use the language of mathematics to express, communicate, and think about the real world. Modern modeling tools allow non-mathematicians to use advanced mathematics.

7 Simple Building Blocks Simple blocks can be used to describe incredibly complex and often beautiful things. Our basic building blocks are Numbers - things, reproductive rate, height, weight etc. Addition - combining two disjoint sets of things Subtraction - combining two disjoint sets of things Multiplication - repeating a unit a defined number of times Division - dividing a unit in a defined number of subsections Functions - relationships between two or more different quantities For modeling purposes you will also use two more sophisticated building blocks. Integration Differentiation

8 Integration Multplication and integration are closely related. Integration can be thought of as generalized multiplication. For a rectangle the area is height multiplied by width or H(b - a). If the height of a figure varies, according to the function h(x) the area is given by the integral

9 Differentiation Differentiation and division are closely related. They can both be used to determine the rate at which one quantity is changing compared to another -- for example, we often use them to determine the velocity of a moving object, or the rate at which its location is changing compared to time. When this rate is constant, simple division will do the job. When the rate of change is not constant then simple division must be replaced by differentiation -- partial differential equations.

10 Kinds of Mathematical Models Static models Discrete dynamical systems Continuous dynamical systems Optimization models Probabilistic models

11 Static models

12 Dynamic models

13 Discrete dynamical systems are very useful for three reasons. 1. In many situations change really is discrete -- it occurs at welldefined time intervals. Examples of this kind of change include farm prices and the population growth of temperate zone insects. 2. Continuous change can often be approximated very well by discrete change. Discrete change is often easier to work with and is conceptually more simple. Thus, even when a continuous model is better, a discrete model may work almost as well and may be more widely understood. 3. Data is usually discrete rather than continuous. So even when the underlying model is continuous we may be forced to work with a discrete model because of the limitations of the available data.

14 Vito Volterra Born: 3 May 1860 in Ancona, Papal States (now Italy) Died: 11 Oct 1940 in Rome, Italy Volterra published papers on partial differential equations, particularly the equation of cylindrical waves. His most famous work was done on integral equations. He published many papers on what is now called 'an integral equation of Volterra type'.

15 A classical use of differential equations is the Lotka-Volterra model describing interactions between two species in an ecosystem, a predator and a prey. Two graphs plotted using the same model parameters. The only difference is in initial density of prey. Pop pulation size H=Prey P=Predator Relative changes in prey predator density for both initial conditions. Trajectories are closed lines.

16 There are many tools available for mathematical modeling and simulations. One of the most commonly used is STELLA or ithink from Mac or PC

17 Deer Deer Deer Populat ion Deer Populat ion birt hs birt hs deat hs birt h fract ion birt h fract ion deat h fract ion You start by drawing your conception of the system and defining how events are interconnected. Then parameter values must be defined. For each time step in the simulation a new value is calculated based on the definitions made. Output can be graphed or tabulated and the actual equations can be viewed in a separate mode.

18 As your perception of the study system increases, subsystems can be linked to the main system Deer Deer Populat ion birt hs deat hs birt h fract ion deat h fract ion ~ Plant Lif e consuming Veget at ion regenerat ing veget at ion per deer regenerat ion per plant

19 Deer Deer_Population(t) = Deer_Population(t - dt) + (births - deaths) * dt INIT Deer_Population = 100 INFLOWS: births = Deer_Population*birth_fraction OUTFLOWS: deaths = Deer_Population*death_fraction birth_fraction =.2 death_fraction = GRAPH(Vegetation) (0.00, 1.00), (100, 0.815), (200, 0.61), (300, 0.48), (400, 0.36), (500, 0.26), (600, 0.165), (700, 0.11), (800, 0.055), (900, 0.03), (1000, 0.02) Plant Life Vegetation(t) = Vegetation(t - dt) + (regenerating - consuming) * dt INIT Vegetation = 3500 INFLOWS: regenerating = Vegetation*regeneration_per_plant OUTFLOWS: consuming = Deer_Population*vegetation_per_deer regeneration_per_plant =.5 vegetation_per_deer = 15

20 Vegetation regeneration = 1 (everything eaten is regenerated) 1: Deer Populat ion 1 : : : Graph 1: p2 (Unt it led Graph) Years

21 Vegetation regeneration = 0,5 (half of what is eaten is regenerated) 1: Deer Populat ion 2: Deer Populat ion 1 : : : Graph 1: p2 (Unt it led Graph) Years

22 We have made an attempt to model the population dynamics of varroa mites (Varroa destructor) in the honey bee system +

23 The model attempts to describe the system before the parasite exerts a negative feedback on the host vitality.

24 Basic elements in flow diagram for Varroa population dynamics model Worker brood Reproductive rate 4 7 Phoretic mites Cell entry rate Mite mortality at bee emergence 8 Phoretic mite mortality rate Drone brood 1 2 Reproductive rate 5 From Calis, Fries & Ryrie,

25 Basic elements in flow diagram for Varroa population dynamics model Entry rate 1. Number of available drone/worker cells 2. Colony size Reproductive rate 3. Adult female progeny per mother mite and cell entry in drone/worker cells 4. Mite fertility in drone/worker cells 5. Duration of the postcapping period for drone/worker cells 6. Cells with male or female offspring only 7. Removal rate of infested brood in drone/worker cells Mortality rates 8. Mite mortality at emergence from drone/worker cells 9. Phoretic mite mortality in summer 10. Phoretic mite mortality in winter From Calis, Fries & Ryrie, 1999

26 Mite population model

27 Colony model

28 Mite invasion rates 6.49*Available_drone_cells/colony_weight 0.56*Available_worker_cells/colony_weight

29 Mite offspring model

30 Phoretic mite mortality model From Calis, Fries & Ryrie, 1999

31 1: t ot al live mit es 1 : Total live mites 1 : : Unrest rict ed growt h (Tot al live mit es) Days

32 Brood data switches Brood Data Models No worker cells ~ Allen data W ~ Yucatan data W Brood data switch Yucatan data D day of year Allen data D No Drone cells precull ~

33 PARAMETER VALUES USED FOR SENSITIVITY ANALYSIS LOW MODEL HIGH Drone Worker Drone Worker Drone Worker Invasion rate equations into drone cells = 6.49*Available drone cells/colony weight (brood attractivity), Boot et al., 1995 into worker cells = 0.56*Available worker cells/colony weight

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35 We can model Varroa mite population dynamics Can we also model honey bee health in general? Undoubtedly YES!! in principle But the short version of this talk is NO!! The fundamental lack of information on pathogen interactions makes an attempt very difficult An attempt to mathematically formalize the relations between the honey bee host, all available pathogens, environmental and genetic parameters, will reveal our ignorance in honey bee epidemiology

36 Realistic models of the host population dynamics already exist Sub-models of individual pathogens can probably be developed for some of them (AFB?) and reveal were data is lacking for others This can generate research to fill data gaps The big challenge is to model when pathogens (or environment or genetic composition) start interacting with the host population dynamics, and with each other with unpredictable patterns So tools for modeling honey bee health exist but we lack sufficient know-how on parameter interactions

37 Thank you for your attention!!