Suppose you are studying the spread of a rare disease among the set of people pi
ID: 3120684 • Letter: S
Question
Suppose you are studying the spread of a rare disease among the set of people pictured in the below image
The contacts among these people are as depicted in the network in the above image, with a time interval on each edge showing when the period of contect occured. We assume that the period of observation runs from time 0 to time 20.
a.) Suppose that s is the only individual who had the disease at time 0. Which nodes could potentially have acquired the disease by the end of the observation period, at time 20?
b.) Suppose that you find, in fact, that all nodes have the disease at time 20. You're faily certain that the disease couldn't have been introduced into this group from other sources, and so you suspect instead that a value you're using as the start or end of one of the time intervals is incorrect. Can you find a single number designating the start or end of one of the time intervals, that you could change so that in the resulting network, it's possible for the disease to have flowed from s to every other node?
[7,12] W [1,3] 10,16] (5,9] 114,18] [12,16]Explanation / Answer
a) u, v and w. Notice that any interaction of x,y and z with s will have to go through v. So x, y and z can contract disease only if they interact with v after v catching the disease. v interacts with u between time 1 and 3. But u interacts with s only in [4,8] thus u cannot communicate disease to v. w can possibly interact with v after interacting with s thereby transmitting disease to v. This can happen only in the interval [10,16]. But v's interaction with y is in an interval prior to this [5,9] so it cannot pass disease to y and hence not to x and z as well.
b) Change [4,8] on the edge s-u to [1,8]. This results in the possibility that catches the disease within the interval [1,3] and transmits it to v. Now v in [5,9] passes it to y and y subsequently to x and z. w anyway directly interacts with s.
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