# Team:NYMU-Taipei/Modeling/Linear epidemic model/explanation

National Yang Ming University

Explanation:

\frac{d[S]}{dt}= -γSE-αS

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.)

\frac{d[E]}{dt}=γSE-εE-αE

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.)

\frac{d[I]}{dt}=εE-μI

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).

\frac{d[C]}{dt}=αS +αE-βC

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).

\frac{d[R]}{dt}= βC

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.