Team:NYMU-Taipei/Modeling/Linear epidemic model

Epidemic model

Backgroud:

Honey bee is a social insect and can be divided into several classes – queens, drones, and workers, which can further be classified into field bee (which is responsible for getting honey from the nature) and house bee (which is responsible for cleaning hives). However, a single bee (especially field bee) may fall ill to CCD when it intakes water or food contaminated by Nosema ceranae spores. What’s worse, CCD may in turn spread to other bees through exchanging substances via mouthparts or feeding food to sacbroods.

After getting into bees’ midgut, Nosema spores will germinate, elongate its polarfilament, and pierce into midgut epithelial cells to transmit its genetic material. After finishing several life cycles, the infected epithelial cells will burst, leading to the spread of Nosema spores to nearby epithelial cells.

The life cycle of Nosema Ceranae:

The spores of Nosema ceranae:

The method is let bees ingest the encapsulated Bee. coli, which is suspended in sugar solution and will proliferate in bees’ midgut to build up bees’ immunity.

The purpose of this modeling:

• How many encapsulated Bee. coli does a bee need to get immunized and be effective to spread our Bee. coli to other bees and let the whole hive be immunized.
• How much time does the immunization requires and to see if it can save the whole society in time.

The assumption of infection and cure process:

It is assumed that the whole colony will only get into two consequences – one is dying out (once latent bees turn to be infected bees), and the other is survive (once latent or suspected bees becomes ingested capsule bees). As soon as suspected bees are infected with Nosema, they will move onto the latent stage. Latent bees contain low-dose Nosema Ceranae so they will not spread Nosema Ceranae to other bees and are curable by ingesting the capsule carrying Beecoli. However, if no measures are taken to prevent proliferation of Nosema in latent bees, latent bees will move onto the infected stage. Because the number of Nosema in bees is too high to be killed thoroughly, it is no use having infected bees been cured by capsule carrying Beecoli, which means they are doomed to death. What’s worse, in the infected stage, Nosema may spread from infected bees to other bees in the colony, leading to accelaration of Nosema spreading. Another assumption is that the capsule carrying Beecoli is not all effective. Neverthless, if the capsule is effective, then bees will move onto ingested capsule stage, which means they will definitely move onto the immunized stage and be cured eventually.

Equation:

Parameters:

N = total population

S = suspected

E = latent (eminent)

I = infected

C = ingested capsule

R = immunized (recovery)

α= suspected bees/ latent beesingested capsule bees rate constant

γ= suspected bees latent bees rate constant

β= ingested capsule bees immunized bees rate constant

ε= latent bees infected bees rate constant

μ= infected bees  dead bees rate constant

Explanation:

In this equation, S represents the population of the bees free of Nosema ceranae and at the same time, has not ingested the capsule carrying Beecoli. Its changing rate is composed of decreasing rate only. In this section, -γSE represents the rate of bees contaminated by other bees carrying Nosema spores and fall into latent period, and this infection rate is a function of the suspected bees and the infectious bees (γ is the rate constant that the suspected bees exchange body fluid with the infectious bees); -αS represents the rate of bees ingesting capsules containing Beecoli (α is the rate the bees ingest capsules from water source.)

In this equation, E represents the population of bees infected with a low-dose Nosema Ceranae and thus, will not spread Nosema to other bees and are curable by ingesting the capsule carrying Beecoli. The change rate of latent bees comprises of both increasing and decreasing rate. As mentioned above, γSE represents the increasing rate of bees contaminated by other bees carrying Nosema spores, which transfer suspected bees to latent bees; -εE represents the decreasing rate of bees moving from latent stage to infected stage after Nosema proliferates too much to have the bees be cured (ε is the rate constant that the latent bees transferring to infected bees without timely treatment); αEmeans the rate of latent bees ingesting capsules containing Beecoli (α is the rate the bees ingest capsules from water source.)

In this equation, I represents the population of bees infected with Nosema Ceranae after a period of time that the population of Nosema Ceranae have grown too high to be killed by the capsule carrying Beecoli and thus the bees are incurable. Its changing rate is composed of both increasing and decreasing rate. As mentioned above, εE represents the increasing rate of bees moving from latent stage to infected stage after Nosema proliferates too much to have the bees be cured; -μI represents the decreasing rate of bees from infested stage to dead stage (μ means the morality rate constant for infected bees, which are incurable due to high Nosema population and are doomed to death).

In this equation, C represents the population of bees ingested the capsule carrying Beecoli but the time is too short for the capsule to be digested and in effective action. The change rate of ingested capsule bees comprises of both increasing and decreasing rate. In this section, αS represents the rate of bees ingesting capsules containing Beecoli; αE represents the rate of latent bees ingesting capsules containing Beecoli; -βC means the decreasing rate of ingested capsule bees moving to the immunized stage after Beecoli-containing capsule is digested and kicks in (β means the rate bees get immunized after Beecoli plays the role).

In this equation, R represents the population of bees ingested the capsule carrying Beecoli and are totally cured after the ingested capsule are digested and in effective action. Its changing rate is composed of increasing rate only. βC, as mentioned, is the rate of ingested capsule bees moving to the immunized stage after Beecoli-containing capsule is digested and kicks in.

Final objective

Our Aim is to know the relationships between the population scale of bees in different Nosema invasion stages (suspected, latent, infected, ingested capsule, immunized). That is to say, we want to know the percentages of the bees cured by Beecoli and thus survived from Nosema invasion eventually.

Results:

It is assumed that the whole bee colony is infected by Nosema, which is the most severe case. That is, bees are either latent or infected. According to our experiment, bees intaking sugar solution which contains capsules in a sufficient concentration have 100％ of Bee. coli releasing from capsules. Here we discuss how much time it needs for the whole hive to recover from Nosema infection with different ratio of infected and latent stage.

1. E=10％, I= 90％

2. E=30％, I=70％:

3. E=70％, I=30％:

4. E=90％, 10％:

5. E=100％, I=0％:

As the pictures shows, the bigger the ratio of latent bees are, the more population of recovery bees are; the bigger the ratio of latent bees are, the lower the peak of infected bees’ population are; the bigger the ratio of latent bees are, the higher the peak of ingested capsule bees’ population are.

No matter which ratio of latent and infected bees is, it all shows that the whole bee colony will survive eventually, which fits our assumption that the whole colony will only get into two consequences – one is dying out (once the “latent bees” turn to be the “infected bees”), and the other is survive (once the “latent” or “suspected bees” becomes “ingested capsule bees”).

Parameters: