The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier
1998 ◽
Vol 35
(3)
◽
pp. 671-682
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Keyword(s):
The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a / n1/3, m ≈ bn1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K / n2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t2/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.
1998 ◽
Vol 35
(03)
◽
pp. 671-682
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2013 ◽
Vol 31
(4)
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pp. 695-707
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Keyword(s):
1987 ◽
Vol 1
(1)
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pp. 69-74
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1977 ◽
Vol 14
(04)
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pp. 850-856
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1996 ◽
Vol 33
(01)
◽
pp. 164-175
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Keyword(s):
1983 ◽
Vol 20
(01)
◽
pp. 197-201
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Keyword(s):