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From an epidemiological model with differential equation, we can compute the basic reproductive number R0 (the number of expected secondary case per primary case in a disease free population).

Biologicaly (from this site):

  1. R0 < 1: Each existing infection is causing less than one new infection. In this case, the disease will decline and eventually die out.
  2. R0 > 1: Each existing infection is causing more than one new infection. The disease will spread between people and there may be an outbreak or an epidemic.
  3. R0 = 1: Each existing infection is causing one new infection. The disease will stay alive without epidemic.

There is many mathematical methods to calculate the R0 from ODE models. One is based on the Disease-Free Equilibrium (DFE). From what I understood:

  1. if 0 < R0<1: DFE is stable (LAS), no epidemic

  2. if R0> 1: DFE is unstable (LAS), there is an epidemic: the introduction of an infected may lead to an endemic point (if this point is stable)

  3. But, what happen if R0=1 ?

I read this is a "central variety", and it's impossible to know what happen in term of stability of the DFE.

Is there any paper talking about endemicity from the mathematical and biological point of view ?

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  • $\begingroup$ Some of your intuition is only true for specific models - for example, what happens with R0 > 1 will change in a closed population vs. one with births and deaths. $\endgroup$ – Fomite Sep 30 '15 at 23:40
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First, you may be better off asking this question on one of the math oriented Stack Exchange sites - CrossValidated is fairly statistics heavy, and dynamic models like this are conventionally not in the domain of statisticians. However, since I work on them, I'll try to give you an answer.

There are a number of papers discussing endemicity as a concept, and the consequences of the basic reproductive number. I'd second the book recommendation made there.

Now, to turn to what happens when R0 = 1. It should be noted that these statements are only true for the deterministic forms of these models - stochastic effects in small populations play merry hell with statements like there "Will" or "Will not" be an epidemic. Also note that your answer to your question will change whether or not you have birth and death.

R0 = 1 is indeed the situation where a disease becomes endemic - each infected replaces itself and, barring other factors, you're at a steady state that is not disease free.

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$R_0$ means no exponential growth. It can be some stable periodic solution, or a constant proportion. There are so many models in epidemiology, that $R_0$ is easy to define for only the simplest ones. Here's a book on the subject with easy math treatment and a ton of Excel examples: an introduction to infectious disease modelling

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