Pandemic Dynamics and the Breakdown of Herd Immunity

In this note we discuss the issues involved in attempting to model pandemic dynamics. More specifically, we show how it may be possible to make projections for the ongoing H1N1 pandemic as extrapolated from knowledge of seasonal influenza. We derive first-approximation parameter estimates for the SIR model to describe seasonal influenza, and then explore the implications of the existing classical epidemiological theory for the case of a pandemic virus. In particular, we note the dramatic nonlinear increase in attack rate as a function of the percentage of susceptibles initially present in the population. This has severe consequences for the pandemic, given the general lack of immunity in the global population.


Deriving characteristics of an epidemic from the SIR model
We derive analytic expressions for the attack rate A and for the the duration D of an epidemic described by the SIR model in terms of the parameters γ, R 0 = β γ , R e = S 0 R 0 . Since A, D, and γ are approximately known (see assumptions (a)-(c) in the main text), these expressions enable us to estimate the parameters R e , R 0 , S 0 .

The attack rate
The attack rate in the SIR model is given by the final size formula, which is well-known, see e.g. [1] and recalled for the reader's convenience. From (1),(15), we have so that From (1),(2) we have and integrating both sides, using (15), we have Substituting (6) into (5) we have Taking t → ∞ we have and setting Note that Z is the fraction of susceptibles who get infected during the epidemic. Note that this fraction is determined by R e (we mention that, in the notation of [1], R 0 is what we here call R e ). The attack rate (the fraction of the population who get infected during the epidemic) is Note also that, for given R 0 , the fraction S 0 of susceptibles affects the attack rate A in two ways: first through its effect on R e and hence on Z, and secondly through the relation (9).
We can also write (8) in terms of A, as

Duration of the epidemic
We must give a precise definition of what we mean by the "duration of the epidemic". We define the epidemic period [t 1 , t 2 ] by the following conditions (1) %90 of the cases occur within this period, that is The value D = t 2 − t 1 is called the duration of the epidemic.
There is some arbitrariness in the above definition. The value %90 can of course be replaced by a different fraction. In fact to be general we shall replace 0.9 by α in the derivations below, so that we have The condition I(t 1 ) = I(t 2 ) could be replaced by the condition that the interval [t 1 , t 2 ] is centered at the peak of the epidemic (where the peak can be defined in at least two ways: as the time of the maximum of I(t), or as the time of the maximum of the incidence i(t) = −S (t)). This would not change the duration by much in practice, and is less convenient analytically, and we therefore chose the definition above.

Lemma 1. The duration of an epidemic is given by
where Z is defined by (8).
An important consequence is that the duration of an epidemic depends only on γ and on R e .
Proof. From (7), and using I(t 1 ) = I(t 2 ) and (11) we have (11) and (13) provide us with two equations for S(t 1 ), S(t 2 ) which can be solved to give and using (10) we can rewrite this as Solving (2) for I we have, fixing an arbitrary t 0 , Setting We have Therefore where, using (15), We note that x(t) is independent of the choice of t 0 , which shows that the expression γ log(S(t 0 ))−β(I(t 0 )+S(t 0 )) is independent of t 0 , and in particular we may send t 0 → −∞ and obtain and integrating from t 1 to t 2 we have 1 γ Making the substitution or, using (14), Finally, making the substitution S = S 0 u gives (12).
Using (12) we can calculate the duration of epidemics by numerical evaluation of the integral, with some results given in Table 1.