Talk:Mathematical modelling in epidemiology
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What about the probability of a mutation that circumvents the vaccine? The more the disease is exposed to the vaccine, the greater the probability that it will suffer a mutation that permits it to survive the vaccine. This is over-simplified, but on point.
Over a sufficiently long period of time, this probability becomes greater if not inevitable. The factors are, simplistically:
- the mutability of the virus,
- the capacity for the vaccine to handle variations.
In reality, there are a large but discrete number of viable mutations, M={M0, M1, ... Mn}, and a discrete set of mutations that the vaccine protects against V={V0, V1, ... Vm}. The difference M-V={E1, E2, ... Ek}=E, is the set of mutations to which the vaccine is ineffective to a significant degree.
The probability of a circumvention is the sum of probabilities all of Ei, for i=1..k, occuring, which is a function of time (E(t)). It is however, not singly a function of time. It is also a function of opportunity. If the virus can spread without mutation, as M0, the probability of E(t) spreading broadly is decreased because M0 is predominant in the hosts. Unless Ei has a selective advantage (faster transmission, greater infectious rate, etc), it will likely remain curtailed by the presence of the dominant strain.
The situation is different in the presence of a vaccine. If the vaccine reduces transmission of the original strain, M0, then the opportunity arises for a circumventing mutation, Ei to spread. Presupposing that the vaccine is ineffective for Ei at X percent of the time, the probability of Ei spreading is the probability of Ei occuring (over time, asymptotically approaches 1) Ei(t), times the ineffectiveness (X): Ei(t) * X. As t approaches infinity, the probability of spreading to a given host goes to X.
Thus, on the one hand, a vaccine can increase the probability of an immune mutation occuring. This is in line with expectations.
In the contrapositive, a vaccine-less disease such as AIDS will decrease in mortality over time, as longer lifespans translate into greater transmission rates. In effect, diseases tend toward an equilibrium with their hosts. Vaccines (and cures, such as antibiotics) increase the probability of mutations that circumvent the treatment or prophylactic. On the other hand, prolonged exposure to a disease weakens it; variants of the disease with lower short-term mortality increase transmission rates.
All of these elements directly affect the mathematics of epidemology, and are relevant to this article, I believe.
[edit] Referencing
Hi there. Nice article, but it desperately needs some referencing in there. Not least for the great gods of epidemiological modelling, Bob May and Roy Anderson. Otherwise, it's a good intro to the subject. --Plumbago 17:22, 9 March 2006 (UTC)
[edit] Partial immunity
How is the maths changed when instead of a proportion q of the population being 100% immune, 100% of the population is q immune? Cyberia
