Fig 1.
Upper: Core model. Lower: Extended model. The diagram illustrates class (horizontal arrows) and state (vertical arrows) transitions of honeybees. The queen produces new brood (B). As bees are older, they change their classes from brood (B), nurses (N), nectar-receivers (R0 and R1), to foragers (F0 and F1). Bees can change into an infection state (iB, iN, iR0, iR1) with a certain probability if they receive nectar from infected ones. Nectar-receivers and foragers can also change their nectar-loaded state between unloaded (subscript 0) and loaded (subscript 1). φ represents ‘death’. See the main text for a more detailed description of each transition. The full model is composed of both core and extended parts.
Table 1.
Model parameters.
Fig 2.
Role of the hygienic behavior (krem).
A: Healthy brood and nurses in a colony with the hygienic behavior are maintained against an infection. The simulation is based on Eqs 1–4 with model parameters listed in Table 1 with l0 = 120 and krem = 2.5 × 10−3. We treat pt1 ⋅ kRN ⋅ iR1 as a parameter, which is equal to 5 × 10−4. The infection is introduced (pt1 ⋅ kRN ⋅ iR1 = 5 × 10−4) at day 20 (vertical purple line). B: All brood and nurses in a colony without the hygienic behavior become infected. The simulation setting is the same as in panel A, except that krem = 0. C: The steady-state number of healthy nurses is plotted as krem is varied while other parameters are fixed. Bold lines represent a stable steady-state and a thin line represents an unstable steady-state. D: The 2-parameter bifurcation diagram is plotted as both krem and pt0 are varied. The diagram is divided into three regions depending on the state of the colony (healthy, bistable, and infected).
Fig 3.
Role of the egg-laying rate (l0).
A: All brood and nurses in a small-size colony (e.g., colony with a low egg-laying rate) become infected even with the hygienic behavior response. The simulation setting is the same as in Fig 2A, except that l0 = 70. The infection is introduced (pt1 ⋅ kRN ⋅ iR1 = 5 × 10−4) at day 20 (vertical purple line). B: The two-parameter bifurcation diagram is plotted between l0 and pt0 while other parameters are fixed. The red letters a and b mark the critical values of l0, below which the colonies are always infected when pt0 is 0.3 and 0.5, respectively. C: Simulation of three colonies with the same parameter set from Fig 2A, but the simulation begins with 2400 initial healthy brood and different initial numbers of healthy nurses. The infection is introduced (pt1 ⋅ kRN ⋅ iR1 = 5 × 10−4) at day 0. D: Simulation setting is the same as that of panel C, except that the initial number of healthy brood is 600.
Fig 4.
Role of the probability of nurses being infected from infected brood (pt,rem) and the death rate of infected bees (kd).
The two-parameter bifurcation diagrams are plotted between A: krem and pt,rem, B: l0 and pt,rem, C: krem and kd, and D: l0 and kd. The diagrams are divided into regions corresponding to the state of the colony (healthy, bistable, and infected).
Fig 5.
Simulations of the full model.
The full model is simulated with parameters listed in Table 1 with l0 = 120 (panels A and B) and l0 = 70 (panels C and D). One infected forager returning from nectar-collecting (iF1) is introduced at day 20 (iF1(20) = 1) (vertical purple line).
Fig 6.
Simulations of the full model under seasonal effects.
The full model is simulated with parameters listed in Table 1 under seasonal effects. During winter (grey stripes), which lasts five months a year, we set l0 = 0, kr = 0, and nN, nR, and nF are increased 4-folds. A and B: l0 during non-winter months = 300. C and D: l0 during non-winter months = 1000. One infected forager returning from nectar-collecting (iF1) is introduced at day 360 (vertical purple line).
Fig 7.
Stochastic simulations of the full model under seasonal effects.
Stochastic simulations of two independent colonies under an identical condition are shown in upper and lower panels, respectively. Seasonal effects are implemented, as described in Fig 6 and the main text. Both colonies are simulated with the same parameter set (l0 during non-winter months = 1000 and other parameters from Table 1). One infected forager returning from nectar-collecting (iF1) is introduced at day 360 (vertical purple line).
Fig 8.
Percentages of survived colonies from stochastic simulations.
Percentages of survived colonies are calculated from 1000 repeats of stochastic simulations. Model parameters are from Table 1, except those that are listed in each figure panel. Seasonal effects are implemented as described in the main text. A: kr during non-winter months (black line) or psurv (red line) is varied. B and C: l0 during non-winter months is varied. D: The timing of an infection (t0) is varied in relation to the onset of spring (i.e., iF1(t0) = 1).