Honeybees have an irreplaceable position in agricultural production and the stabilization of natural ecosystems. Unfortunately, honeybee populations have been declining globally. Parasites, diseases, poor nutrition, pesticides, and climate changes contribute greatly to the global crisis of honeybee colony losses. Mathematical models have been used to provide useful insights on potential factors and important processes for improving the survival rate of colonies. In this review, we present various mathematical tractable models from different aspects: 1) simple bee-only models with features such as age segmentation, food collection, and nutrient absorption; 2) models of bees with other species such as parasites and/or pathogens; and 3) models of bees affected by pesticide exposure. We aim to review those mathematical models to emphasize the power of mathematical modeling in helping us understand honeybee population dynamics and its related ecological communities. We also provide a review of computational models such as VARROAPOP and BEEHAVE that describe the bee population dynamics in environments that include factors such as temperature, rainfall, light, distance and quality of food, and their effects on colony growth and survival. In addition, we propose a future outlook on important directions regarding mathematical modeling of honeybees. We particularly encourage collaborations between mathematicians and biologists so that mathematical models could be more useful through validation with experimental data.

Warmer temperatures are expected to increase the incidence of Lyme disease through enhanced tick maturation rates and a longer season of transmission. In addition, there could be an increased risk of disease export because of infected mobile hosts, usually birds. A temperature-driven seasonal model ofBorrelia burgdorferi(Lyme disease) transmission among four host types is constructed as a system of nonlinear ordinary differential equations. The model is developed and parametrized based on a collection of lab and field studies. The model is shown to produce biologically reasonable results for both the tick vector (Ixodes scapularis) and the hosts when compared to a different set of studies. The model is used to predict the response of Lyme disease risk to a mean annual temperature increase, based on current temperature cycles in Hanover, NH. Many of the risk measures suggested by the literature are shown to change with increased mean annual temperature. The most straightforward measure of disease risk is the abundance of infected questing ticks, averaged over a year. Compared to this measure, which is difficult and resource-intensive to track in the field, all other risk measures considered underestimate the rise of risk with rise in mean annual temperature. The measure coming closest was “degree days above zero.” Disease prevalence in ticks and hosts showed less increase with rising temperature. Single field measurements at the height of transmission season did not show much change at all with rising temperature.

The literature on Lyme disease includes a lively debate about the paradoxical role of changing deer populations. A decrease in the number of deer will both (1) reduce the incidence of Lyme disease by decreasing the host populations for ticks and therefore tick populations, and (2) enhance the incidence of Lyme disease by offering fewer reservoir-incompetent hosts for ticks, forcing the vector to choose reservoir-competent, and therefore possibly diseased, hosts to feed on. A review of field studies exploring the net impact of changing deer populations shows mixed results. In this manuscript, we investigate the hypothesis that the balance of these two responses to changing deer populations depends on the relative population sizes of reservoir-competent vs. reservoir-incompetent hosts and the presence of host preference in larval and adult stages. A temperature driven seasonal model of Borrelia burgdorferi sensu stricto (cause of Lyme disease) transmission among three host types (reservoir-competent infected and uninfected hosts, and reservoir-incompetent hosts) is constructed as a system of nonlinear ordinary differential equations. The model, which produces biologically reasonable results for both the tick vector Ixodes scapularis Say 1921 and the hosts, is used to investigate the effects of reservoir-incompetent host removal on both tick populations and disease prevalence for various relative population sizes of reservoir-competent hosts vs. reservoir-incompetent hosts. In summary, the simulation results show that the model with host preference appears to be more accurate than the one with no host preference. Given these results, we found that removal of adult I. scapularis(Say) hosts is likely to reduce questing nymph populations. At very low levels questing adult abundance may rise with lack of adult hosts. There is a dilution effect at low reservoir-competent host populations and there is an amplification effect at high reservoir-competent host populations.

Rocky Mountain spotted fever (RMSF) is a significant health problem in Sonora, Mexico. The tick vector, Rhipicephalus sanguineus, feeds almost exclusively on domestic dogs that, in this region, also serve as the reservoir for the tick-borne pathogen, Rickettsia rickettsii. A process-based mathematical model of the life cycle of R. sanguineus was developed to predict combinations of insecticidal dog collars and long-lasting insecticidal wall treatments resulting in suppression of indoor tick populations. Because of a high burden of RMSF in a rural community near the Sonora state capital of Hermosillo, a test area was treated with a combination of insecticidal dog collars and long-lasting insecticidal wall treatments from March 2018 to April 2019, with subsequent reduction in RMSF cases and deaths. An estimated 80% of the dogs in the area had collars applied and 15% of the houses were treated. Data on tick abundance on walls and dogs, collected during this intervention, were used to parameterize the model. Model results show a variety of treatment combinations likely to be as successful as the one carried out in the test community.