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The classic Susceptible-Infected-Susceptible epidemic model is the following:

Each node is in one of the two states: Susceptible or Infected:

Susceptible->Infected->Recovered.

Let s and i respectively represent the percentage of susceptible and infected nodes, $\beta$ represent the infection rate, and $gamma$ represent the recover rate. The differential equation system for the model is:

\begin{equation} \begin{aligned} \dfrac{ds}{dt}=\gamma i - \beta s i \\ \dfrac{di}{dt}=\beta s i - \gamma i \end{aligned} \end{equation}

with \begin{equation} s+i=1 \end{equation}

Given the value of $\beta$, $\gamma$, and the initial value of $i$ and $s$, we can numerical calculate the $s$, and $i$ as a function of $t$. Here is a typical result:

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The epidemic spreading gradually progresses to a stable fraction of infected nodes in the end. My question what decides the time to progress to the stable state? That is how the epidemic parameters affect the length of $Ts$.

It would be create if there is an analytical equation for $Ts$. If it is not possible, it would be help as well if you could talk about how $Ts$ changes with other epidemic parameters.

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Since $s+i=1$ you have to solve only $\frac{ds}{dt} = \gamma(1-s)-\beta s(1-s)$ or $\frac{ds}{\beta s^2 -(\gamma + \beta)s + \gamma} = dt$ which is $F(s):=\frac{\log(s-1) - \log(\beta s - \gamma)}{\beta - \gamma} = t$. Your time is then $F(s)-F(s_0) = T$ and probably you want $|\frac{ds}{dt}|$ small (say $<1/100$) to be in a region of stability.

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  • $\begingroup$ Do you know how the value of $Ts$ qualitatively changes with the change of $\beta$ and $\gamma$. For example, when $\beta$ increases, or when $\gamma$ increases, how would $Ts$ change? $\endgroup$ May 12, 2015 at 9:38
  • $\begingroup$ Is $Ts$ determined by $\beta/\gamma$? $\endgroup$ May 12, 2015 at 9:38
  • $\begingroup$ I gave you the explicit formula. $\endgroup$ May 12, 2015 at 11:03

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