So the bee population model is more or less complete, but it still is not representative of the real world. Out here in the real world, we have the varroa mite. To properly model the growth of varroa within a colony, we first need to understand the life cycle of this critter. After reading endlessly in various literature on the subject, my conclusion is, the life cycle of the mite is fairly well understood, and is dramatically effected by what type of cell a given foundress mite enters. We need some numbers to realistically place timeframes on varroa development, a good reference is found here.

The phoretic period may last 4.5 to 11 days when brood is present in the hive or as long as five to six months during the winter when no brood is present in the hive. Consequently, female mites living when brood is present in the colony have an average life expectancy of 27 days, yet in the absence of brood, they may live for many months.

To get a handle on the reproductive success of those mites, another quote from the same article

Considering mortality in brood cells and improper mating, the average foundress mite produces about one offspring per worker cell she enters, and about two offspring per drone cell. Drones take longer to develop so more mites are produced in drone cells.

To try model these numbers is fairly strait forward. When a mite emerges we keep them in a phoretic state for 4 days. Starting on day 5, we assume that half of the mites available for going into cells will successfully find their way into a cell to try and reproduce. A fertile varroa mite going into a worker cell will produce one offspring, so two mites will emerge. That original foundress has been under the cap for 10 days and spent 5 days phoretic, so is now 15 days old, and after another 5 days of phoretic behaviour will enter another cell, so we have two fertile mites entering worker cells at this time. 10 days later we will have 4 mites emerge, but the original foundress is now fully aged and dies of age. The net result is, after two varroa brood cycles in worker brood, a single foundress mite has resulted in 3 mites in the colony. Things change when there is drone brood available, literature suggests the varroa much prefer to hop into a drone cell over a worker cell. When a varroa mite enters the drone cell, she remains under the cap for 14 days, and will produce two viable daughters. After a phoretic period, all 3 of these mites will enter drone cells, and all 3 produce 2 more viable daughters, for a total of 9 mites emerging on the second round. At this time the original foundress mite dies of age, but leaves behind 8 viable mites as daughters and grand daughters. As this math shows, there are two very distinctly different details when comparing the varroa life cycle to the honeybee life cycle. The bee population is based on a single queen laying eggs, and will grow in a linear fashion limited by the rate of egg laying of the queen when not limited by temperatures for brood incubation. The varro life cycle is shorter, and not limited by a single queen laying eggs, it grows exponentially rather than linearly because all fertile varroa mites are producing offspring. To model varroa growth, we know the mites prefer a drone cell so it would be easy to just place all of the fertile varroa into drone cells when they are available, but, this is not realistic. There are 10 worker cells open on capping day for every drone cell that is open, not all of the varroa will find a drone cell. To account for this, we have prefererred drones when it’s time for varroa to enter cells, but assume only half of them find a drone cell, the other half entering cells will end up in a worker cell. This is the basis on which the varroa growth has been incorporated into the colony growth model. Introducing varroa into the colony growth model does introduce another new concept, that of the ‘sick bee’. We know that bee virus are vectored by the varroa mite, and a colony with extremely high varroa levels will show lots of sick bees in the form of deformed wing virus and other inflictions. To account for this, a new type of bee has been incorporated into the population model, the ‘sick bee’. Any bee emerging from a cell that was populated with a varroa mite during the brood development is not placed in the normal bee population to graduate from nurse to wax maker and on up to forager. Instead, they are placed into the ‘sick bee’ population, ie, deformed wing etc. I cant find any suitable references in the literature to suggest how long a sick bee will live, but, we chose a rather arbitrary ‘it make sense to me’ way of handling the sick bees. If the bee is inflicted with deformed wing, it can still manage to crawl around on the frames, clean cells, etc. Where trouble begins for that bee is when the time comes to orient and graduate to foraging, instead of flying out of the hive, it ends up crawling out because it cant fly. Once a sick bee crawls out of the hive, it’s gone, so we have arbitrarily set the numbers so that sick bees crawl out and die at an age of 25 days, shortly after they should have graduated from being a house bee to a foraging bee. The final tweak to the handling of varroa and sick bees came from a comment I saw in an online video by Jamie Ellis from the University of Florida. His comment was, when varroa population gets large, some cells will end up with two or more varroa feeding on the pupae in that cells. With two or more varroa feeding on a pupae, that bee will be dead or very close to it when it emerges. This final detail explains one symptom we often see when a hive crashes due to varroa load, we see numerous bees that were in the process of emerging and never completely got out of the cell when doing the post mortem. After adding this little bit into the math for handling varroa populations as they explode, the graphs mimic almost perfectly what we see from hives crashing due to varroa loads.