Dynamics of varroa-mites infested honey bee colonies model

Transcription

Talk Outline
Introduction
Model Formulation
Optimal control of the model
Numerical Simulations
Conclusion
Acknowledgement
Dynamics of varroa-mites infested honey bee
colonies model
Kazeem O. OKOSUN
Vaal University of Technology, Vanderbijlparrk, South Africa.
BIOMAT 2013: November 7, 2013
Vaal University of Technology, Vanderbijlparrk, South
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Talk Outline
◮
Introduction
◮
Model Formulation
◮
Optimal Control Analysis
◮
Numerical Results
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
◮
The global treat to insect pollinators of crops and wild plants
are becoming alarming and their exstintion may have high
impact on economic and environmental consequencies
◮
Not less than 15% of human food production relies on animal
pollinators, most especially the bees.
◮
For instance, US fruit and vegetable growers requires the
honey bees pollination services to generate $8-10 billion as
farm income annually
◮
In addition to pollination, honey bees also play significant role
as producer of honey and wax which in turn result in various
nutritional and industrial uses.
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
◮
Varroa mites are parasitic mites that survive by sucking the blood
from honey bees. The mites target the pupal bees that are sealed
inside a wax cell as the bees develop from a larva to an adult bee.
◮
Honey bee colonies with large mite infestations will be so weakened
that the entire colony will eventually die out.
◮
Colony collapse disorder (CCD), also known as honey bee colony
depopulation syndrome, is essentially the sudden disappearance of
honey bees from their colony. As more and more bees disappear,
the colony fails and ultimately dies.
◮
Varroa mites have a huge economic impact on the beekeeping
industry, they do not only feed on bees but also transmit deadly
viruses (e.g. Acute Bee Paralysis Virus (ABPV)) to the bees
◮
Much has not been studied regarding the disease dynamics of
honeybee and varroa mite population in the literature
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Figure: Varroa mites on a honey bee pupal and a varroa mite up close.
Photos: Zachary Huang and USDA-ARS
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
(a)
(b)
Figure: Frames taken from colonies suffering from colony collapse
disorder have few bees, but large numbers of developing larval and pupal
bees. Compare with a frame full of bees taken from a healthy colony.
Photos: Keith Delaplane from Oldroyd, 2007 and Reed Johnson
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Why is this study important? Honeybees are responsible for
pollinating roughly 1/3 of the ingredients found in food consumed
today.
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Parameter
Definition
β
λ
σ
δ
µi
µ4
α
ρ
ω
r
rate at which uninfected bees become infected
rates at which mites acquire virus
unheathy rate at which hives become foragers
rates at which mites kills bees
bees mortality rate (i = 1, 2, 3)
mites mortality rate
maximum rate that hive bees become foragers
number of mites that can be sustained per bee on average
saturation constant
mites birth rate
Table: (1)
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
H+F
F
= L ω+H+F − H α − σ F +H − µ1 H
d
F
− µ2 F − βFm
dt F = H α − σ F +H
F +I − δMF
d
dt H
d
dt I
=
βFm
F +I
− µ3 I − δMI
(1)
I
= β2 (M
+ FλIM
+I − µ4 m
− m) F +I M
d
λIM
dt M = rM 1 − ρ(F +I ) − F +I − µ4 M
d
dt m
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Stability of the disease-free equilibrium (DFE)
E0 = (H ∗ , F ∗ , I ∗ , M ∗ , m∗ ) = H ∗ , F ∗ , 0, 0, M ∗
(2)
The linear stability of E0 can be established using the next generation
operator method. It follows that the reproduction number of the model is
given by
R0 =
s
βρβ2 (r − µ4 )
,
(r µ3 + F ∗ δρ(r − µ4 ))(β + µ4 )
(3)
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Existence of endemic equilibrium: The endemic equilibrium satisfies
the following two polynomial scenario. We consider the bee endemic
population expressed in terms of the varrao mites virus population in the
first case, that is,
2
P(I ∗ ) = I ∗ [AI ∗ + BI ∗ + C ] = 0
(4)
where,
A = δρ(r − µ4 )(β2 + µ4 ),
B = [Ω ∗ δρ[r (β + β2 ) + 2µ4 ] + r µ3 (β2 + µ4 )][1 − Rk2 ],
(5)
C = Ω ∗ (r µ3 + Ω ∗ δρ(r − µ4 ))(β + µ4 )[1 − R20 ],
Rk =
Ω ∗ δρµ4 (β2 + βδ + µ4 )
+ β2 ) + 2µ4 ] + r µ3 (β2 + µ4 )
Ω ∗ δρ[r (β
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Proposition
1. If r > µ4 , Rk ≤ 1 then system (1) exhibits a transcritical
bifurcation.
2. If r > µ4 , Rk ≥ 1 then system (1) exhibits a backward
bifurcation.
3. If r < µ4 , Rk ≤ 1 then system (1) exhibits a backward
bifurcation.
4. If r < µ4 , Rk ≥ 1 then system (1) exhibits a transcritical
bifurcation.
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Proof:
1. For Rk ≤ 1 and r > µ4 , A > 0 we obtain when R0 > 1 that C < 0. This implies
that system (1) has a unique endemic steady state. If R0 ≤ 1, then C ≥ 0 and
B ≥ 0. In this case system (1) has no endemic steady states.
ii. If R0 ≤ 1, then C ≥ 0 and r < µ4 , then A < 0, the discriminant of (4),
∆(R0 ) := B 2 − 4AC > 0, then there exists R0c such that ∆(R0c ) = 0,
∆(R0 ) < 0 for 1 < R0 < R0c and ∆(R0 ) > 0 for R0c > R0 . One endemic steady
state when R0 = R0c and two endemic steady states when 1 < R0 < R0c .
iii. If R0 ≤ 1 and r = µ4 , the A = 0, C > 0 and B > 0 the system has no
endemic equilibrium. If R0 > 1, there exist a unique equilibrium.
2. For Rk ≥ 1 we discuss the following cases:
i. If R0 > 1 and r > µ4 in this case C < 0 and system (1) has a unique
endemic steady state. If r < µ4 , so A < 0 then the system has no endemic state
ii. If R0 ≤ 1 and r > µ4 in this case C > 0 and B < 0, while the discriminant
of (4), ∆(R0 ) := B 2 − 4AC , can be either positive or negative. We have
∆(1) = B 2 > 0 and ∆(1) = −4AC < 0, then there exists R0c such that
∆(R0c ) = 0, ∆(R0 ) < 0 for R0c < R0 < 1 and ∆(R0 ) > 0 for R0c < R0 . One
endemic steady state when R0 = R0c and two endemic steady states when
R0c < R0 < 1.
iii. If R0 ≤ 1 and r = µ4 , the A = 0, C > 0 and B < 0 there exists has a
unique endemic equilibrium. If R0 > 1, the system has no endemic equilibrium.
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Rk<1, δ>1, µ4>r
Rk>1, δ>1, r>µ4
1.8
1.4
1.6
1.2
1.4
Infected Foragers
Infected Foragers
1
1.2
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.8
1
1.2
1.4
1.6
1.8
2
Reproductive Number
2.2
2.4
0
0.95
2.6
1
1.05
1.1
1.15
Reproductive Number
(a)
1.2
1.25
(b)
Rk<1, δ<1, µ4>r
r=µ4
0.7
4.5
4
0.6
3.5
Infected Foragers
Infected Foragers
0.5
0.4
0.3
3
2.5
2
1.5
0.2
1
0.1
0
0.5
0
0.2
0.4
0.6
Reproductive Number
(c)
0.8
1
0
0
0.5
1
Reproductive Number
1.5
2
(d)
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
R0 contour plot
1
0.3
0.9
0.8
0.8
0.3
0.3
0.7
0.6
0.3
δ
0.2
0.5
4
0.
0.4
0.4
0.3
0.3
0
1
0.4
0.2
0.8
0.5
0.1
0.6
0.1
0.4
δ
0
0.2
0
0
r
0.5
0.2
1
0
0.6
0.3
0.2
0.4
R
0
0.6
0.4
0.5
0.4
0.6
0.8
1
r
(e)
(f)
R0 contour plot
0.4
1
0.25
0.9
0.2
0.2
0.0
5
0.1
0.15
0.15
0.8
0.1
0.7
0.3
4
0.05
0.05
5
0.1
0.5
0.0
0.1
0.4
0.1
5
µ
R
0
0.6
0.2
0.1
0.1
0.3
0.2
0.6
0.0
0.1
0.15
5
1
0.8
0.5
0.1
0
1
0.4
µ4
0
0.2
0
(g)
0
r
0
0.2
0.4
0.6
0.8
1
r
(h)
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Rk
8
2
0.8
k
1.
4
1
1.2
1.2
0.6 0.8
1
1
0.8
0.5
0.2 0.4
0.3
R
6
1.
1
1.4
1.5
0.5
0.4
2.5
1.8
0.8
0.2
0.4
0.6
1.2
0.6
0.7
1.2
1.
6
1.
4
1.
1.6
0.8
µ4
2.2
2
2
1.4
1
0.4
0.2
1
0.9
0.6
0.6
0.2
0
1
0.4
1
0.4
0.1
0
0.2
0.4
0.8
0.5
0.2
0.2
0
0.6
0.4
0.6
0.8
1
µ4
r
(i)
0
0.2
0
r
(j)
Rk
1
2
0.5
2
3
2.5
1
0.5
0.8
1.5
0.9
1.5
1
2.5
0.7
2
0.5
k
R
0.5
1
1.5
2
δ
0.5
1.5
0.6
1
1.5
1
0.4
0.5
0.3
1
0.5
0
1
0.5
0.2
1
0.1
0.8
0.5
0.6
0.4
0
0
0.2
0.4
µ4
(k)
0.6
0.8
1
δ
0
0.2
0
µ4
(l)
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Summary of Results
R0
R0
R0
R0
< 0,
< 0,
< 0,
< 0,
Rk
Rk
Rk
Rk
<1
<1
<1
<1
50% of µ4 , 10% of r
5% of µ4 , 50% of r
10% of δ, 0 < µ4 < 40%
10% of δ, 60% < µ4 < 100%
R0
R0
R0
R0
< 0,
< 0,
< 0,
< 0,
Rk
Rk
Rk
Rk
>1
>1
>1
>1
50%
33%
25%
25%
of
of
of
of
µ4 , 30% of r
µ4 , 50% of r
δ, 0 < µ4 < 30%
δ, 70% < µ4 < 100%
Table: (2)
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Given the objective function J, as:
J(u1 , u2 ) =
Z
tf
0
[AI + Bm + CM + Du12 + Eu22 ]dt
(6)
where U = {(u1 , u2 ) such that u1 and u2 are measurable with 0 ≤ u1 ≤ 1 and
0 ≤ u2 ≤ 1, for t ∈ [0, tf ]} is the control set.
Subject to equations (1), we apply Pontryagin’s Maximum Principle
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
The co-state variable associated with the system is represented by
G (t), the current value Hamiltonian is then written as
Ha = AI + Bm + CM + Du12 + Eu22
H+F
F
+GH L ω+H+F − H α − σ F +H − µ1 H
F
+GF H α − σ F +H − µ2 F −
+GI
n
βFm
F +I (1
βFm
F +I (1
− u2 ) − µ3 I − u1 δMI
o
− u2 ) − u1 δMF
o
n
I
+ FλIM
(1
−
u
)
−
µ
m
+Gm β2 (M − m) F +I
2
4
+I
M
λIM
+GM rM 1 − ρ(F +I ) − F +I (1 − u2 ) − µ4 M
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Theorem
Given optimal controls u1∗ , u2∗ and solutions H, F , I , M, m of the
corresponding state system (1) and (6) that minimize J(u1 , u2 )
over U. Then there exists adjoint variables GH , GF , GI , GM , Gm
satisfying
∂Ha
−dGi
=
dt
∂i
where i = H, F , I , M, m and with transversality conditions
GH (tf ) = GF (tf ) = GI (tf ) = GM (tf ) = Gm (tf ) = 0
(8)
(9)
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
βFm(GF − GI ) + λIM(GM − Gm )
u1∗ = min 1, max 0,
,
2D(F + I )
δM(FGF − IGI )
u2∗ = min 1, max 0,
,
2E
(10)
(11)
By standard control arguments involving the bounds on the
controls, we conclude


0 If ξ2∗ ≤ 0
0 If ξ1∗ ≤ 0








ξ2∗ If 0 < ξ2∗ < 1
ξ1∗ If 0 < ξ1∗ < 1 ; u ∗ =
u1∗ =
2






 1 If ξ2∗ ≥ 1
 1 If ξ1∗ ≥ 1
ξ1∗ =
βFm(GF −GI )+λIM(GM −Gm )
2D(F +I )
and
ξ2∗ =
δM(FGF −IGI )
2E
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Proof.
Corollary 4.1 of Fleming and Rishel [8] gives the existence of an optimal control due to
the convexity of the integrand of J with respect to optimal pair u1 , u2 , a priori
boundedness of the state solutions, and the Lipschitz property of the state system
with respect to the state variables. Then the adjoint equations can be written as
−
dGH
=
dt
Lω
− (ω+H+F
G + α(GH − GF ) +
)2 H
−
dGF
=
dt
Lω
G +
− (ω+H+F
)2 H
− GI )
µ2 GF +
GM ) −
−
dGI
=
dt
dGm
=
dt
dGM
−
=
dt
− GH ) + µ1 GH
βmI (1−u )
σH 2
(GF − GH ) + (F +I )2 2 (GF
(F +H)2
β2 (M−m)I
λIM(1−u2 )
u1 δMGF + (F +I )2 Gm + (F +I )2 (Gm −
βFm(1−u2 )
(GI
(F +I )2
βF (1−u2 )
(GF
F +I
β2 (M−m)F
Gm
(F +I )2
λFM(1−u2 )
(G
M −
(F +I )2
M2r
ρ(F +I )2 GM
− GF ) +
+µ3 GI + u1 δMGI +
−
σF 2
(GF
(F +H)2
− GI ) +
u1 δ(FGF − IGI ) −
β2 I
F +I
β2 I
F +I
Gm ) −
rM 2
G
ρ(F +I )2 M
Gm + µ4 Gm
Gm +
λI (1−u2 )
(GM
F +I
− Gm ) + µ4 GM +
2rM
G
ρ(F +I ) M
(12)
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Efforts on prevention of Foragers from infection (u1 ) only
0.25
0.88
u = u =0
1
u = u =0
2
1
u1 ≠ 0, u2 = 0
0.87
MITES INFESTED WITH VIRUS
0.86
MITES (VIRUS FREE)
2
u1 ≠ 0, u2 = 0
0.2
0.85
0.84
0.83
0.82
0.15
0.1
0.05
0.81
0
0.8
0
5
10
15
Time (Months)
20
25
30
0
5
10
(m)
15
Time (Months)
20
25
30
(n)
u1= u2=0
IINFESTED FORAGER WITH VIRUS
0.14
u1 ≠ 0, u2 = 0
0.12
0.1
0.08
0.06
0.04
0.02
0
5
10
15
Time (Months)
20
25
30
(o)South
Vaal University of Technology, Vanderbijlparrk,
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Efforts on prevention of Foragers from mites attacks (u2 ) only
0.25
0.88
u = u =0
1
1
2
u1= 0, u2 ≠ 0
MITES (VIRUS FREE)
u = u =0
2
u1 = 0, u2 ≠ 0
0.86
0.84
0.82
0.8
0.78
0.2
0.15
0.1
0.05
0.76
0.74
0
5
10
15
Time (Months)
20
25
0
30
0
5
10
(a)
15
Time (Months)
20
25
30
(b)
0.152
0.151
u1= u2=0
u1 = 0, u2 ≠ 0
0.15
0.149
0.148
0.147
0.146
0
5
10
15
Time (Months)
20
25
30
Vaal University of Technology, Vanderbijlparrk,(c)
South
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Prevention of Foragers from infection (u1 ) and Protection of Foragers (u2 ) from
mites attacks
0.25
0.88
1
2
1
u1 ≠ 0, u2 ≠ 0
2
u1= u2=0
MITES (VIRUS FREE)
u ≠ 0, u ≠ 0
u = u =0
0.86
0.84
0.82
0.8
0.78
0.2
0.15
0.1
0.05
0.76
0.74
0
5
10
15
Time (Months)
20
25
0
30
0
5
10
(a)
15
Time (Months)
20
25
30
(b)
u = u =0
1
0.14
2
u1 ≠ 0, u2 ≠ 0
0.12
0.1
0.08
0.06
0.04
0.02
0
5
10
15
Time (Months)
20
25
30
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
◮
A mathematical model for the dynamics of varroa-mites infested
honey bees colonies is formulated and analyzed.
◮
It is found that honey bees disease free equilibrium only exists in the
absence of varroa mites and the infected bees population increases
with increase in the varroa mites population (leading to collony
collapse disorder)
◮
The model is also found that the model exhibit multiple equilibria.
◮
Incorporating time dependent controls, using Pontryagin’s Maximum
Principle to derive necessary conditions for the optimal control of
the disease, the numerial results suggest that the comibination of
prevention control (u1 ) on foragers from infection and protection of
foragers (u2 ) from mites attacks is the most effective strategy to
prevent Colony Collapse Disorder in honeybee colony.
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Acknowledgement
◮
Vaal University of Technology, Vanderbijlpark, South Africa
◮
Organizers of BIOMAT 2013
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Thank You
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa
Talk Outline
Introduction
Model Formulation
Conclusion
Acknowledgement
Oldroyd, B.P. 2007. What’s Killing American Honey Bees? PLoS Biology. 5(6):
e168.
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mishchenko, The
mathematical theory of optimal processes. Wiley, New York, (1962)
Van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and
sub-threshold endemic equilibria for compartmental models of disease
transmission. Math. Biosci. 180: 29-48.
Dornberger, L. Mitchell, C. Hull, B. Ventura, W. Shopp, H. Kribs-Zeleta, C. And
Grover, J. 2012. Death of the bees: A mathematical model of colony collapse
disorder. Technical report. The University of Texas, ARLINGTON.
www.uta.edu/math/preprint.
Ratti, V. Kevan, P.G. Eberl, H.J. Amathematical model for population dynamics
of honeybee colonies with Varroa destructor and Acute Bee Paralysis Virus.
Engelsdorp D., Underwood R., Caron D. and Hayes J. 2007. An Estimate of
Managed Colony Losses in the Winter of 20062007: A Report Commissioned by
the Apiary Inspectors of America. American Bee Journal. 147: 599-603.
Cox-Foster D., et al. 2007. A Metagenomic Survey of Microbes in Honey Bee
Colony Collapse Disorder. Science 318: 283-287.
Dynamics
Africa.
of varroa-mites
BIOMAT 2013:
infested
November
honey bee
7, colonies
2013 modelKa

Dynamics of varroa-mites infested honey bee colonies model

Transcription

Similar documents

The PTB primary clocks CS1 and CS2 Andreas Bauch Physikalisch