Establish the following identity:
$$\mathbf { E } \left[ \Delta \tilde { R } _ { n } \right] = \rho \mathbf { E } \left[ \tilde { I } _ { n } \right]$$
where each infected person recovers at the end of the day with probability $\rho \in ]0,1[$, independently of others, and $\Delta \tilde{R}_n = \tilde{R}_{n+1} - \tilde{R}_n$.

Show that
$$\mathbf { E } \left[ \Delta \tilde { S } _ { n } \right] = - \mathbf { E } \left[ \tilde { S } _ { n } p \left( \tilde { I } _ { n } \right) \right]$$
then deduce the equation satisfied by $\mathbf { E } \left[ \Delta \tilde { I } _ { n } \right]$.
Here $\Delta \tilde{S}_n = \tilde{S}_{n+1} - \tilde{S}_n$, $\Delta \tilde{I}_n = \tilde{I}_{n+1} - \tilde{I}_n$, $\Delta \tilde{R}_n = \tilde{R}_{n+1} - \tilde{R}_n$, and $\tilde{S}_n + \tilde{I}_n + \tilde{R}_n = M$ for all $n$.

A shopping mall holds a raffle drawing activity with on-site registration. After registration closes, the host places the same number of raffle balls as the number of registrants, of which 10 balls are marked as lucky prizes: 5 balls for a 5000 yuan gift voucher and 5 balls for an 8000 yuan gift voucher. Each ball has an equal probability of being drawn, and balls are not replaced after drawing. Before the drawing, the organizers announce a winning probability of 0.4\% based on the number of prizes and registrants. After the drawing begins, each person draws a ball in order, and each person has only one chance to draw. If among the first 100 participants, exactly 1 person draws a 5000 yuan voucher and no one draws an 8000 yuan voucher, then the expected value of the prize amount that the 101st person can receive is (15－1)(15－2) yuan.