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title: "Time Discounted Infection and Susceptibility"
author: "George G. Vega Yon"
date: "November 11, 2015"
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In Myers (2000), susceptibility and infection is defined for a given time period and as a constant throughout the network--so only varies on $t$. In order to include effects from previous/coming time periods, it adds up through the \emph{severity} of the rioting, which in our case would be strength of tie, hence a dichotomous variable, whenever the event occurred a week within $t$, furthermore, he then introduces a discount factor in order to account for decay of the influence of the event. Finally, he obtains

$$
V_{(t)} = \sum_{a\in \mathbf{A}(t)} \frac{S_{(a)}m_{T(a), T\leq t-T(a)}}{t- T(a)}
$$

where $\mathbf{A}(t)$ is the set of all riots that occurred by time $t$, $S_{(a)}$ is the severity of the riot $a$, $T(a)$ is the time period by when the riot $a$ accurred and $m$ is an indicator function.

In order to include this notion in our equations, I modify these by also adding whether a link existed between $i$ and $j$ at the corresponding time period. Furthermore, in a more general way, the time windown is now a function of the number of time periods to include, $K$, this way, instead of looking at time periods $t$ and $t+1$ for infection, we look at the time range between $t$ and $t + K$.

# Infectiousness

Following the paper's notation, a more generalized formula for infectiousness is

$$\label{eq:infect-dec}
\left(
\sum_{k=1}^K\sum_{j\neq i} \frac{x_{ji(t+k-1)}z_{j(t+k)}}{k}
\right)\left(
\sum_{k=1}^K\sum_{j\neq i} \frac{x_{ji(t+k-1)}z_{j([t+k;T])}}{k}
\right)^{-1}
$$

\noindent Where $\frac{1}{k}$ would be the equivalent of $\frac{1}{t - T(a)}$ in mayers. Alternatively, we can include a discount factor as follows

$$\label{eq:infect-exp}
\left(
\sum_{k=1}^K\sum_{j\neq i} \frac{x_{ji(t+k-1)}z_{j(t+k)}}{(1+r)^{k-1}}
\right)\left(
\sum_{k=1}^K\sum_{j\neq i} \frac{x_{ji(t+k-1)}z_{j([t+k;T])}}{(1+r)^{k-1}}
\right)^{-1}
$$

Observe that when $K=1$, this formula turns out to be the same as the paper.


# Susceptibility

Likewise, a more generalized formula of susceptibility is

$$\label{eq:suscept-dec}
\left(
\sum_{k=1}^K\sum_{j\neq i} \frac{x_{ij(t-k+1)}z_{j(t-k)}}{k}
\right)\left(
\sum_{k=1}^K\sum_{j\neq i} \frac{x_{ij(t-k+1)}z_{j([1;t-k])}}{k}
\right)^{-1}
$$

\noindent Which can also may include an alternative discount factor

$$\label{eq:suscept-exp}
\left(
\sum_{k=1}^K\sum_{j\neq i} \frac{x_{ij(t-k+1)}z_{j(t-k)}}{(1+r)^{k-1}}
\right)\left(
\sum_{k=1}^K\sum_{j\neq i} \frac{x_{ij(t-k+1)}z_{j([1;t-k])}}{(1+r)^{k-1}}
\right)^{-1}
$$

Also equal to the original equation when $K=1$. Furthermore, the resulting statistic will lie between 0 and 1, been the later whenever $i$ acquired the innovation lastly and right after $j$ acquired it, been $j$ its only alter.

(PENDING: Normalization of the stats)