Figure 1.
A compartmental diagram of the dynamics of the honey bee colony combined with the dynamics of an infectious disease.
The susceptible and infected hive bees, HS and HI live within the hive. New susceptible hive bees are generated by surviving brood through the survival function, S. New infected hive bees are generated through interactions of susceptible hive bees with infected hive bees and infected foragers at rates βHH and βHF. Hive bees are recruited to foraging duties through the recruitment function R, which also allows for the reversal of duties, from foraging to hive duties. Foragers move into the infected compartment via interactions with infected hive bees and infected foragers at rates βHF and βFF. All infected bees die at rates dH or dF, and foragers die naturally at rate m.
Table 1.
Parameter values and references.
Figure 2.
Baseline demographic dynamics of the honey bee colony in the absence of disease.
Figure 3.
Scenario 1: Colony dynamics in the presence of disease with β = 5×10−5, dH = dF = 0.14.
Red arrow = onset of infection, grey shading = winter.
Figure 4.
Scenario 2: Colony dynamics under a more severe infection represented by a higher death rates from the disease, with β = 5×10−5, dH = dF = 0.56.
Red arrow = onset of infection, grey shading = winter.
Figure 5.
Scenario 3: Colony dynamics under a higher rate of transmission of the disease, with β = 5×10−3, dH = dF = 0.56.
Red arrow = onset of infection, grey shading = winter.
Figure 6.
Average age of recruitment to foraging duties (AARF) under the three scenarios in Figures 3, 4, 5.
Red arrow = onset of infection.
Figure 7.
Relationship between the rate of transmission β and disease-induced death rates dH = dF in their effects on the Average Age of Recruitment to Foraging.
The figure shows the effects of an increase of β and dH on the AARF. Note that the AARF becomes less sensitive to changes in β as β is increased. Meanwhile, for small β, an increase in dH can have a favourable effect on the AARF.
Figure 8.
Scenario 4: β = 5×10−5, dH = dF = 0.56.
Effect of the proximity of the onset of infection to the onset of winter. Red arrow = onset of infection, grey shading = winter.
Figure 9.
The expected size of the bee population at the end of winter as influenced by the severity of the disease (dH = dF) and the transmission rate of the disease (β).
For comparison, the black arrow indicates the population size at end of winter in the absence of disease (Figure 2). The figure illustrates the different sensitivity to β and dH. Note that dH has a favourable effect for small β and dH large enough.
Figure 10.
The expected size of the bee population at the end of winter as influenced by the time interval between the onset of infection and the beginning of winter (), with β = 5×10−5.
For comparison, the black arrow indicates the population size at end of winter in the absence of disease (Figure 2).
Figure 11.
The stark difference between the dynamics of Scenario 3 with an environmental hazard scenario in which the death rate is increased (by the effects of pesticides, for example) to equal the total death rate in Scenario 3.
The survival of the colony is almost guaranteed in the environmental hazard scenario while the collapse of the colony is almost guaranteed in the disease scenario.
Figure 12.
An alternative comparison of the dynamics of Scenario 3 with an environmental hazard scenario in which the comparison between the two hazards is based not on the total death rate as in Figure 11 but on the average lifespan of bees being the same in both cases.
Table 2.
Tabulated results from the model scenarios 1, 2, and 3 and experimental data from [36] and [35].