The Influence of Social Network Graph Structure on Disease Dynamics in a Simulated Environment Page: 20
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slightly higher at 4.97. This higher value of Ro0 may be attributed to the close living quarters on
the ships resulting in more frequent contacts.
2.3.2. Deriving Ro0 Mathematically
The fundamental equations of the SIR model can be used to derive Ro0 mathematically. The
established differential equations below represent the movement from susceptible to infected to
removed. The constant a is a probability that describes the likelihood of disease transfer. The
constant 7 is the removal rate, which is the reciprocal of the average number of days in the Infected
state. The three SIR differential equations are defined as follows:
(1) AS = -aStlt
(2) I = a StIt - It
(3) AR = 7It
The SIR equations correspond directly to Figure 2.4. The negative sign in Equation (1) in-
dicates that as the disease spreads, the number of susceptibles decline. Likewise, the number
of removed individuals, Equation (3), increases. The number of infected individuals initially in-
creases and then decreases following a bell-shaped curve. Equation (2) provides the basis for the
calculation of Ro. If the rate of infection is faster than the rate of removal (A2I > 0), for some time,
t, an epidemic occurs. Factoring TIt from Equation 2, the change in infected individuals over time
(4) AI = 7It (acSt 1)
It is now evident that if cs > 1, the number of infected individuals will increase. The mathe-
matical definition of R0 is taken directly from Equation 4. Because R0 is measured at the beginning
of the outbreak (t = 0), the definition is as follows:
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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/m1/30/: accessed May 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; .