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1. Estimating the Length of a Chain of Infection: Vaccinia
From class, you know that for an infectious disease (such as HIV), the reproductive
rate of infection, denoted by Ro, is the expected number of secondary infections
transmitted directly by a single infected person early in the epidemic (when the
fraction of the population infected is negligible). Suppose a single infectious per-
son is introduced to a huge population (so we will take it as infinite in size) of
otherwise uninfected (i.e. susceptible) persons. The infection in question can be
characterized by R0 as discussed above. In class, we focused our attention on cases
where R0 > 1. Here we investigate what happens when R0 < 1 (so the expected
number of infections generated by this new infected person is less than one).
(a) Let T be the expected total number of infections generated over all time
starting with this single infected person (that is, T is 1 (for the initially infected
person) plus the sum of the number of infections directly transmitted by the
initially infected person, plus the number of infections generated by each of these
secondary infections, plus the sum of all tertiary infections, etc. etc. etc.). Use
the repetition method to determine a simple expression for T in terms of R0.
(b) A concern raised during the Great Smallpox Debates of 2002 was that
the vaccinia virus used in smallpox vaccination could itself be transmitted from
a recently vaccinated person to others, with possibly problematic complications
resulting. Indeed, an alarming presentation was made at an important national
meeting suggesting that 20% of all smallpox vaccine complications stemmed from
transmitted vaccinia infections. However, a disagreeable Yale professor suggested
that these same data implied that 80% of all complications were not due to vaccinia
transmission (and hence due to direct vaccination), implying that the ratio of
complications due to vaccinia transmission to complications unrelated to vaccinia
transmission equals 20%/80% or 0.25. Using the result from (a) above, what
is the implied expected total number of vaccine complications over all time per
direct vaccine complication (that is, the sum of the initial direct complication
plus all future transmitted vaccine complications resulting from the original direct
complication)? And, given a death rate of 1 per million from vaccination (due
to direct complications), how should this death rate be adjusted to account for
deaths that could result from complications due to transmitted vaccinia?

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