Page 116 - ITU Journal, Future and evolving technologies - Volume 1 (2020), Issue 1, Inaugural issue

ITU Journal on Future and Evolving Technologies, Volume 1 (2020), Issue 1

individual is not infected by the virus at time , and { ( )} as follows,

=1
( ) = 0 if individual is infected by the virus but can-

not spread the virus, i.e., is in incubation, ( ) = +1 ℙ{ ( ) = 1 ∣ ℱ }

if individual is infected by the virus and can spread (15)
∑
the virus, and finally ( ) = −2 if they have recovered = 1 ( )≠0 ( =−1,0,1 1 ( )= ⋅ →1 ),

from the virus or have been isolated already. We call
( ) = 0 “inactive” and ( ) = 1 active. ℙ{ ( ) = 1 ∣ ℱ }

The rest of the discussion in this section makes the fol- = 1 ( )≠0 ( ∑ 1 ( )= ⋅ →−1 ), (16)
lowing simplifying assumptions: =−1,0,1
where whether ( ) = 0 or not is determined by the
• States ( ) and decisions ( ) do not change tracing or testing algorithms.

within a slot. Clearly, ( ) = { ( )} is a Markov process, and if

∈
we are provided with the values of (0), (1), (2), …,

• Upon becoming an active spreader, an individual then our goal is to find the most likely values of ( ).
can spread the disease only after the current time- In order to do this, we might use Markov chain Monte
slot ends. This might seem to be restrictive, but is Carlo (MCMC) algorithms such as Gibbs Sampling.
justifiable since our modeling procedure already in- Readers may refer to [27] for a review of MCMC al-
troduces “noise” due to erroneous tracing and test- gorithms.
ing.
4.1.3 Graph Embedding
• The spreading probability, denoted as , is a con-

stant that is independent of other parameters such Computing the infection probabilities of individuals di-
as the values of the states, the number of days one rectly will be computationally cumbersome, and we can
has been infected, etc. utilize graph embedding [5] techniques in order to find
suspicious infected individuals. These techniques map

• After becoming an inactive infected person, at each the nodes of a graph to points in ℝ , where is a natural
time slot the individual becomes active with a prob- number. If the graph embedding algorithm is properly

ability equal to 0,1 . chosen, then if two points are close in the space ℝ then
they are also close in the susceptibility graph, so that the
probability that the virus spreads from one individual to
• For an active infected person, for every time slot, the other is high. Note that in the graph each node may
this individual has a constant probability to get re- have up to | | edges, but in the embedded graph, each
moved. We use 1,−2 to denote this probability. node only has coordinates. Since the number of edges
might be much more than , performing computations
• For an individual at state ( ) = , it has a with the embedded coordinates is much more efficient

constant probability → to be tested to be state than directly working with the original graph.
( ) = .

5. SIMULATIONS
Let ℱ be the filtration generated by ( ( ), ( ), ∈

, ≤ ). With the above assumptions in place, we can In this section, we use a simulation to indicate the ne-
write the “dynamics” or transition probabilities govern- cessities of contact tracing and building a contact graph.
ing ( ) = { ( ))} as follows, Contact tracing is an essential technique for finding po-
=1

tential infectious people. A commonly employed naive
contact tracing technique is to trace and test only those
ℙ{ ( + 1) = 2 ∣ ℱ } who have had contact with a confirmed positive person.

= 1 ( )=−2 + 1 ( )=1 ⋅ 1,−2 , (11) We call this simple and intuitively appealing contact
ℙ{ ( + 1) = 1 ∣ ℱ } tracing policy as Policy 1. However, this method may

= 1 ( )=1 (1 − 1,−2 ) + 1 ( )=0 ⋅ 0,1 , (12) not be optimal, especially under circumstances when a
ℙ{ ( + 1) = 0 ∣ ℱ } sizeable proportion of the population is infected. To
see why this might be the case, consider the scenario

= 1 ( )=0 ⋅ (1 − 0,1 ) when two people are waiting to get tested. The first
+ 1 ( )=−1 (1 − ∏ (1 − ( ))), (13) person had a close contact with a confirmed infected
person, while the second person did not have any such

∶( , )∈ℰ, ( )=1 close contact with a confirmed infected person; but did
ℙ{ ( + 1) = −1 ∣ ℱ } closely contact 500+ untested people (for example this

= 1 − ℙ{ ( + 1) = 1, 0, or − 2 ∣ ℱ }, (14) person works in a supermarket). Policy 1 will suggest to

us to test the first person; however, when a significant
and the dynamics or transition probabilities of ( ) = proportion of the population (e.g., 1%) are positive, in
96 © International Telecommunication Union, 2020