08 May 2020 - viridi - Sparisoma Viridi
Try to understand the SI model from Albert-László Barabási’s book.
From book chapter - 10 Spreading Phenomena, it states that epidemic models
where for SI model only $S$ dan $I$ variables are considered.
In a population of $N$ individuals, each typical individual has $\lt k \gt$ contacts and disease will be transmitted from an infected individual to a susceptible one in a unit time is $\beta$, then probability, with the homogenous mixing hypothesis, the infected individual encounters susceptible one is $S(t)/N$. In a unit time there are $\lt k \gt S(t) / N$ infected individual in contact with susceptible one. Infected individual I(t) are transmitting the pathogen at rate $\beta$, which make the average number of new infection $dI$ during a time frame $dt$ is
\begin{equation} \label{egn:model-si} \frac{dI}{dt} = \beta \lt k \gt \left( \frac{S}{N} \right) I. \end{equation}
It s move convenient to work with normalized variables in a close population
\begin{equation} \label{egn:model-s-normalized} s = \frac{S}{N} \end{equation}
and
\begin{equation} \label{egn:model-i-normalized} i = \frac{I}{N}. \end{equation}
Eqns \eqref{egn:model-s-normalized} and \eqref{egn:model-i-normalized} will turn \eqref{egn:model-si} into
\begin{equation} \label{egn:model-si-normalized-1} \frac{di}{dt} = \beta \lt k \gt s i, \end{equation}
or simply
\begin{equation} \label{egn:model-si-normalized-2} \frac{di}{dt} = \beta \lt k \gt (1- i) i, \end{equation}
since
\begin{equation} \label{egn:model-si-normalized-3} s + i = 1. \end{equation}
Eqns \eqref{egn:model-si-normalized-2} can further written as
\begin{equation} \label{egn:model-si-normalized-2-solving-1} \frac{di}{(1- i) i} = \beta \lt k \gt dt, \end{equation}
where each side are already function of one variable. Left side of Eqn \eqref{egn:model-si-normalized-2-solving-1} can written in the form of
\begin{equation} \label{egn:model-si-normalized-2-solving-2} \frac{di}{(1 - i) i} = \frac{di}{1-i} + \frac{di}{i} = -\frac{d(1-i)}{1-i} + \frac{di}{i}. \end{equation}
Substitution Eqn \eqref{egn:model-si-normalized-2-solving-2} into \eqref{egn:model-si-normalized-2-solving-1} and performing integration on both sides will produce
\begin{equation} \label{egn:model-si-normalized-2-solving-3} -\ln(1-i) + \ln(1-i_0) + \ln i - \ln i_0 = \beta \lt k \gt (t - t_0), \end{equation}
which can be simplified into
\begin{equation} \label{egn:model-si-normalized-2-solving-4} \ln \left[\frac{(1-i_0) i}{(1-i) i_0} \right] = \beta \lt k \gt (t - t_0). \end{equation}
Eqn \eqref{egn:model-si-normalized-2-solving-4} dan further written as
\begin{equation} \label{egn:model-si-normalized-2-solving-5} (1-i_0) i = (1-i) i_0 e^{\beta \lt k \gt (t - t_0)} \end{equation}
and then
\begin{equation} \label{egn:model-si-normalized-2-solving-6} (1-i_0) i = i_0 e^{\beta \lt k \gt (t - t_0)} - i_0 i e^{\beta \lt k \gt (t - t_0)}, \end{equation}
\begin{equation} \label{egn:model-si-normalized-2-solving-7} i \left\{ 1 + i_0 \left[e^{\beta \lt k \gt (t - t_0)} - 1 \right] \right\} = i_0 e^{\beta \lt k \gt (t - t_0)}, \end{equation}
which leads to
\begin{equation} \label{egn:model-si-normalized-2-solving-8} i = \frac{i_0 e^{\beta \lt k \gt (t - t_0)}}{1 + i_0 \left[e^{\beta \lt k \gt (t - t_0)} - 1 \right]} \end{equation}
as the final form, where $i(t_0) = i_0$. We can choose that $t_0 = 0$ and this will turn \eqref{egn:model-si-normalized-2-solving-8} into
\begin{equation} \label{egn:model-si-normalized-solution} i = \frac{i_0 e^{\beta \lt k \gt t}}{1 + i_0 \left(e^{\beta \lt k \gt t} - 1 \right)}. \end{equation}
For very large value of $t$, Eqn \eqref{egn:model-si-normalized-solution} becomes
\begin{equation} \label{egn:model-si-normalized-solution-t-infty-1} i(\infty) \approx \frac{i_0 e^{\beta \lt k \gt t}}{1 + i_0 e^{\beta \lt k \gt t}} \end{equation}
since $e^{\beta \lt k \gt t} \gt\gt -1$ and then also
\begin{equation} \label{egn:model-si-normalized-solution-t-infty-2} i(\infty) \approx \frac{i_0 e^{\beta \lt k \gt t}}{i_0 e^{\beta \lt k \gt t}} = 1 \end{equation}
since $i_0 e^{\beta \lt k \gt t} \gt\gt 1$.
If $t = 0$ is put to Eqn \eqref{egn:model-si-normalized-solution} then
\begin{equation} \label{egn:model-si-normalized-solution-t-0} i(0) = \frac{i_0}{1 + i_0 (1 - 1)} = i_0 \end{equation}
will be obtained.
Using MacLaurin series, Eqn \eqref{egn:model-si-normalized-solution} can be written as
\begin{equation} \label{egn:model-si-normalized-solution-t-small-1} i = i_0 + \beta \lt k \gt i_0 (1 - i_0) t + \dots, \end{equation}
which depends on the number of terms. Uncomplete derivation can be found here, a future post.