# The Influence of Social Network Graph Structure on Disease Dynamics in a Simulated Environment Page: 47

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3.1%. For k > 14 and p > 60%, the percent infected rises to 76% with a standard deviation of

2.4%.

With a population of size 500, a mean of 70% of the population becomes infected over all

averaged simulations. The lowest average of total infections is 6.58, or 1% of the population which

occurs when k = 2 and p = 0. The highest average is 84% at k = 100 and p = 80%. Similar to

the results with a population of size 30, Figure 3.8 (a) reveals that simulations with neighborhood

sizes of k = 2 to k = 10 and p = 0 result in a very small proportion of the population becoming

infected, ranging from 1% to 7%. There is a sharp incline as k and p increase. In fact, for all

values of k > 4 and p > 30% the average remains above 70%, with a mean of 77% and a standard

deviation of 3.1%. Figure 3.8 (b) illustrates that, with the exception of k = 20 and p = 0 at 38%,

the proportion of the population infected remains relatively consistent. For these values, exception

noted, the mean is 77% with a standard deviation of 2.9%.

Similar results are observed in both the small and large graph simulations. When the neigh-

borhood size is restricted and the probability of random contacts is low, the proportion of the

population that is infected is greatly reduced. Moderate increases in either one or both of these

parameters greatly increases this proportion. In the small graph simulations, approximately 65%

or more of the population is infected for all values of k when p > 50%; for all values of p when

k > 10; and when k > 6 and p > 30%. In the large graph simulations, approximately 65% or more

of the population is infected for all values of k when p > 40%; for all values of p when k > 40;

and when k > 6 and p > 20%. This implies that small-world graphs are very conducive to the

spread of disease, even with relatively small values for k and p. It should be noted, however, that

the parameters selected for these experiments generate a high probability of producing an epidemic

and no preventative measures are taken during any simulations.

3.5. Summary

Two groups of experiments were presented and discussed in this Chapter. The first involves a

series of simulated outbreaks on a population of size 30. This small population size was purposely

chosen to allow visual inspection of the resulting contact and outbreak graphs. The second involves47

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### Reference the current page of this Dissertation.

Johnson, Tina V. The Influence of Social Network Graph Structure on Disease Dynamics in a Simulated Environment, dissertation, December 2010; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc33173/m1/57/: accessed May 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; .