Working with Daniel Park on generalized exponentials (in theory), but we’ve gotten deeply into a bunch of moment stuff related to my thesis. On the bus today, while I was missing the stop, I started thinking yet again about terminology.

I think I want to try writing \(Y_o(v; D)\) for Type, order, variable, Domain.

My first type is total, \(T\). Specifically, \(T_o(v; D) = \int v(a)^o D(a) da\).

Examples:

  • \(T_0(.;S)\equiv T(S)\) is the total number of susceptibles (\(v\) doesn’t affect anything when \(o=0\)).

  • \(T_1(\sigma; I)\) is the total amount of susceptibility to infection in the infectious population

SI example

I want to integrate: \(\dot S(a) = -\Lambda \sigma(a) S(a)\)

\(a\) is an abstract “aspect variable” that indexes the population.

to get:

\[\dot T_0(\sigma; S) \equiv \dot S = -\Lambda T_1(\sigma; S)\]

More generally, I can multiply both sides by \(\sigma^i\) and integrate to get:

\[\dot T_i(\sigma; S) \equiv \dot S = -\Lambda T_{i+1}(\sigma; S)\]

Now define:

\(M_i(.) = T_i(.)/T_0(.)\).

Then: \(\dot M_i = \frac{\dot T_i}{T_0} - M_i\frac{\dot T_0}{T_0}\).

Thus: \(\dot M_i(\sigma; S) = -\Lambda \left(M_{i+1}(\sigma; S) - M_i(\sigma; S) M_1(\sigma; S)\right)\).

In particular, we define \(M\equiv M_1\) and suppress the arguments to write:

\[\dot S = -\Lambda MS.\] \[\dot M = -\Lambda(M_2 - M^2).\]

The Dwyer-Dushoff assumption is that \(M_2 = (1+\kappa) M^2\), with κ constant. This closes the chain, and we can then integrate to find:

\(M = \hat M S^\kappa\). \(\hat M\) is the value of \(M\) when everyone is susceptible; observed values are lower.

We can define \(κ = κ_2\) and extend this idea to \(\kappa_i = \frac{M_iM_{i-2}}{M_{i-1}^2} - 1.\)

[[Equation chain for the κs]]. Can we get some understanding from the equation for κ itself?

SIS model

[[Can we extend the above to cover this case?]]