We incorporate a mathematical model of Varroa destructor and the Acute Bee Paralysis Virus with an existing model for a honeybee colony, in which the bee population is divided into hive bees and forager bees based on tasks performed in the colony. The model is a system of five ordinary differential equations with dependent variables: uninfected hive bees, uninfected forager bees, infected hive bees, virus-free mites and virus-carrying mites. The interplay between forager loss and disease infestation is studied. We study the stability of the disease-free equilibrium of the bee-mite-virus model and observe that the disease cannot be fought off in the absence of varroacide treatment. However, the disease-free equilibrium can be stable if the treatment is strong enough and also if the virus-carrying mites become virus-free at a rate faster than the mite birth rate. The critical forager loss due to homing failure, above which the colony fails, is calculated using simulation experiments for disease-free, treated and untreated mite-infested, and treated virus-infested colonies. A virus-infested colony without varroacide treatment fails regardless of the forager mortality rate.

A mathematical model for the honeybee-varroa mite-ABPV system is proposed in terms of four differential equations for the: infected and uninfected bees in the colony, number of mites overall, and of mites carrying the virus. To account for seasonal variability, all parameters are time periodic. We obtain linearized stability conditions for the disease-free periodic solutions. Numerically, we illustrate that, for appropriate parameters, mites can establish themselves in colonies that are not treated with varroacides, leading to colonies with slightly reduced number of bees. If some of these mites carry the virus, however, the colony might fail suddenly after several years without a noticeable sign of stress leading up to the failure. The immediate cause of failure is that at the end of fall, colonies are not strong enough to survive the winter in viable numbers. We investigate the effect of the initial disease infestation on collapse time, and how varroacide treatment affects long-term behavior. We find that to control the virus epidemic, the mites as disease vector should be controlled.

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 of Borrelia 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.

A model is developed of malaria (Plasmodium falciparum) transmission in vector (Anopheles gambiae) and human populations that include the capacity for both clinical and parasite suppressing immunity. This model is coupled with a population model for Anopheles gambiae that varies seasonal with temperature and larval habitat availability. At steady state, the model clearly distinguishes uns hypoendemic transmission patterns from stable hyperendemic and holoendemic patterns of transmission. The model further distinguishes hyperendemic from holoendemic disease based on seasonality of infection. For hyperendemic and holoendemic transmission, the model produces the relationship between entomological inoculation rate and disease prevalence observed in the field. It further produces expected rates of immunity and prevalence across all three endemic patterns. The model does not produce mesoendemic transmission patterns at steady state for any parameter choices, leading to the conclusion that mesoendemic patterns occur during transient states or as a result of factors not included in this study. The model shows that coupling the effect of varying larval habitat availability with the effects of clinical and parasite-suppressing immunity is enough to produce known patterns of malaria transmission.

<abstract><p>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.</p></abstract>

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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.

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