Dynamics of varroa-mites infested honey bee colonies model Kazeem O. OKOSUN

Transcription

1 Dynamics of varroa-mites infested honey bee colonies model Kazeem O. OKOSUN Vaal University of Technology, Vanderbijlparrk, South Africa. BIOMAT 23: November 7, 23

2 Talk Outline Optimal Control Analysis Numerical Results

3 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 5% 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- 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.

4 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

5 Figure: Varroa mites on a honey bee pupal and a varroa mite up close. Photos: Zachary Huang and USDA-ARS

6 (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, 27 and Reed Johnson

7 Why is this study important? Honeybees are responsible for pollinating roughly /3 of the ingredients found in food consumed today.

8 Parameter Definition β rate at which uninfected bees become infected λ rates at which mites acquire virus σ unheathy rate at which hives become foragers δ rates at which mites kills bees µ i bees mortality rate (i =,2, 3) µ 4 mites mortality rate α maximum rate that hive bees become foragers ρ number of mites that can be sustained per bee on average ω saturation constant r mites birth rate Table: ()

9 ( d dt H = L H+F ω+h+f ) ( d dt F = H α σ F F+H ( H d dt I = βfm F+I µ 3 I δmi α σ F F+H ) µ H ) µ 2 F βfm F+I δmf () d dt m = β 2(M m) I ( d dt M = rm M ρ(f+i) ) F+I µ 4 m λim F+I µ 4 M F+I + λim

10 Stability of the disease-free equilibrium (DFE) ) E = (H, F, I, M, m ) = (H, F,,, M (2) The linear stability of E can be established using the next generation operator method. It follows that the reproduction number of the model is given by βρβ 2 (r µ 4 ) R = (rµ 3 + F δρ(r µ 4 ))(β + µ 4 ), (3)

11 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, where, P(I ) = I [AI 2 + BI + C] = (4) A = δρ(r µ 4 )(β 2 + µ 4 ), B = [Ω δρ[r(β + β 2 ) + 2µ 4 ] + rµ 3 (β 2 + µ 4 )][ R 2 k ], C = Ω (rµ 3 + Ω δρ(r µ 4 ))(β + µ 4 )[ R 2 ], (5) R k = Ω δρµ 4 (β 2 + βδ + µ 4 ) Ω δρ[r(β + β 2 ) + 2µ 4 ] + rµ 3 (β 2 + µ 4 )

12 Proposition. If r > µ 4, R k then system () exhibits a transcritical bifurcation. 2. If r > µ 4, R k then system () exhibits a backward bifurcation. 3. If r < µ 4, R k then system () exhibits a backward bifurcation. 4. If r < µ 4, R k then system () exhibits a transcritical bifurcation.

13 Proof: Talk Outline. For R k and r > µ 4, A > we obtain when R > that C <. This implies that system () has a unique endemic steady state. If R, then C and B. In this case system () has no endemic steady states. ii. If R, then C and r < µ 4, then A <, the discriminant of (4), (R ) := B 2 4AC >, then there exists R c such that (R c ) =, (R ) < for < R < R c and (R ) > for R c > R. One endemic steady state when R = R c and two endemic steady states when < R < R c. iii. If R and r = µ 4, the A =, C > and B > the system has no endemic equilibrium. If R >, there exist a unique equilibrium. 2. For R k we discuss the following cases: i. If R > and r > µ 4 in this case C < and system () has a unique endemic steady state. If r < µ 4, so A < then the system has no endemic state ii. If R and r > µ 4 in this case C > and B <, while the discriminant of (4), (R ) := B 2 4AC, can be either positive or negative. We have () = B 2 > and () = 4AC <, then there exists R c such that (R c ) =, (R ) < for R c < R < and (R ) > for R c < R. One endemic steady state when R = R c and two endemic steady states when R c < R <. iii. If R and r = µ 4, the A =, C > and B < there exists has a unique endemic equilibrium. If R >, the system has no endemic equilibrium.

14 .8 R k <, δ>, µ 4 >r.4 R k >, δ>, r>µ 4 Infected Foragers Infected Foragers Reproductive Number (a) Reproductive Number (b).7 r=µ R k <, δ<, µ 4 >r Infected Foragers Infected Foragers Reproductive Number (c) Reproductive Number (d)

15 δ.5 R contour plot.9 R δ.2.4 r r.6 (e) (f) R contour plot R.2..5 µ r.6.8 µ r (g) (h)

16 δ Talk Outline.5 2 R k µ r R k µ r.6.8 (i) (j) R k R k µ 4.5 δ.2.4 µ (k) (l)

17 Summary of Results R <, R k < 5% of µ 4, % of r R <, R k < 5% of µ 4, 5% of r R <, R k < % of δ, < µ 4 < 4% R <, R k < % of δ, 6% < µ 4 < % R <, R k > 5% of µ 4, 3% of r R <, R k > 33% of µ 4, 5% of r R <, R k > 25% of δ, < µ 4 < 3% R <, R k > 25% of δ, 7% < µ 4 < % Table: (2)

18 Given the objective function J, as: J(u,u 2 ) = Z tf [AI + Bm + CM + Du 2 + Eu2 2 ]dt (6) where U = {(u,u 2 ) such that u and u 2 are measurable with u and u 2, for t [,t f ]} is the control set. Subject to equations (), we apply Pontryagin s Maximum Principle

19 The co-state variable associated with the system is represented by G(t), the current value Hamiltonian is then written as H a = AI + Bm + CM + Du 2 + Eu2 2 ( ) +G H {L H +G F {H H+F ω+h+f ( α σ F F+H ( α σ F F+H ) µ 2 F βfm +G I { βfm F+I ( u 2) µ 3 I u δmi +G m {β 2 (M m) I +G M {rm ( M ρ(f+i) F+I + λim ) } µ H } F+I ( u 2) u δmf } } F+I ( u 2) µ 4 m ) } λim F+I ( u 2) µ 4 M

20 Theorem Given optimal controls u, u 2 and solutions H,F,I,M,m of the corresponding state system () and (6) that minimize J(u,u 2 ) over U. Then there exists adjoint variables G H,G F,G I,G M,G m satisfying dg i = H a dt i where i = H, F, I, M, m and with transversality conditions (8) G H (t f ) = G F (t f ) = G I (t f ) = G M (t f ) = G m (t f ) = (9)

21 ( ( ))} u {, = min βfm(gf G I ) + λim(g M G m ) max,, 2D(F + I) ( ( ))} () u2 {,max = min δm(fgf IG I ),, () 2E By standard control arguments involving the bounds on the controls, we conclude If ξ If ξ2 u = ξ If < ξ < ; u2 = ξ2 If < ξ2 < If ξ If ξ2 ξ = βfm(g F G I )+λim(g M G m) 2D(F+I) and ξ2 = δm(fg F IG I ) 2E

22 Proof. Corollary 4. 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 u,u 2, 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 dg H dt dg F dt = Lω (ω+h+f) 2 G H + α(g H G F ) + σf2 (F+H) 2 (G F G H ) + µ G H = Lω (ω+h+f) 2 G H + σh2 (F+H) 2 (G F G H ) + βmi( u 2) (F+I) 2 (G F G I ) µ 2 G F + u δmg F + β 2(M m)i (F+I) 2 G m + λim( u 2) (F+I) 2 (G m G M ) M 2 r ρ(f+i) 2 G M dg I dt = βfm( u 2 ) (F+I) 2 (G I G F ) + β 2(M m)f (F+I) 2 G m +µ 3 G I + u δmg I + λfm( u 2) (F+I) 2 (G M G m) rm2 ρ(f+i) 2 G M dgm dt dg M dt = βf( u 2) (G F+I F G I ) + β 2I F+I Gm + µ 4G m = u δ(fg F IG I ) β 2I F+I Gm + λi( u 2) (G F+I M G m) + µ 4 G M + 2rM ρ(f+i) G M (2)

23 Efforts on prevention of Foragers from infection (u ) only.88 u = u 2 =.25 u = u 2 =.87 u, u 2 =.2 u, u 2 = MITES (VIRUS FREE) MITES INFESTED WITH VIRUS Time (Months) (m) Time (Months) (n).4 u = u 2 = u, u 2 = IINFESTED FORAGER WITH VIRUS Time (Months) Vaal University of Technology, Vanderbijlparrk, (o) South Dynamics Africa. of varroa-mites BIOMAT 23: infested November honey bee 7, colonies 23 modelka

24 Efforts on prevention of Foragers from mites attacks (u 2 ) only.88 u = u 2 =.25 u = u 2 =.86 u =, u 2.2 u =, u 2 MITES (VIRUS FREE) MITES INFESTED WITH VIRUS Time (Months) (a) Time (Months) (b).52 IINFESTED FORAGER WITH VIRUS u = u 2 = u =, u Time (Months) Vaal University of Technology, Vanderbijlparrk, (c) South Dynamics Africa. of varroa-mites BIOMAT 23: infested November honey bee 7, colonies 23 modelka

25 Prevention of Foragers from infection (u ) and Protection of Foragers (u 2 ) from mites attacks.88 u = u 2 =.25 u, u 2.86 u, u 2.2 u = u 2 = MITES (VIRUS FREE) MITES INFESTED WITH VIRUS Time (Months) (a) Time (Months) (b).4 u = u 2 = u, u 2 IINFESTED FORAGER WITH VIRUS Time (Months)

26 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 (u ) on foragers from infection and protection of foragers (u 2 ) from mites attacks is the most effective strategy to prevent Colony Collapse Disorder in honeybee colony.

27 Vaal University of Technology, Vanderbijlpark, South Africa Organizers of BIOMAT 23

28 Thank You

29 Oldroyd, B.P. 27. What s Killing American Honey Bees? PLoS Biology. 5(6): e68. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mishchenko, The mathematical theory of optimal processes. Wiley, New York, (962) Van den Driessche, P. and Watmough, J. (22). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 8: Dornberger, L. Mitchell, C. Hull, B. Ventura, W. Shopp, H. Kribs-Zeleta, C. And Grover, J. 22. Death of the bees: A mathematical model of colony collapse disorder. Technical report. The University of Texas, ARLINGTON. 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. 27. An Estimate of Managed Colony Losses in the Winter of 2627: A Report Commissioned by the Apiary Inspectors of America. American Bee Journal. 47: Cox-Foster D., et al. 27. A Metagenomic Survey of Microbes in Honey Bee Colony Collapse Disorder. Science 38: