**Lecture 32 (19-Apr-13)**

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The spread of infectious diseases (continued).

Last time, I introduced a simple difference (discrete) equation model for the spread of an infectious disease.

which has the following equilibrium value (Eqn 32.1 (= Eqn 31.2)

Eqn 32.2 (= Eqn 31.8)

What does **equilibrium** mean here? It means that there will be **no change** in the infection level --
that is the rate of addition of new infections will be balanced by the recovery rate
(a population growth analog would be d*N*/d*t* = 0 when births = deaths) .

We also solved for some aspects of the graph of *I*_{t} as
a function of time. We know its *Y*-intercept, given by:

{and its slope given byY-intercept}Eqn 32.3 (= Eqn 31.4)m = (1 - g - r) {slope}Eqn 32.4 (= Eqn 31.5)

Let's say we have an empirically observed plot of
*I*_{t}
values over time. We have only **two** equations (Eqn
32.3
= Eqn 31.4 and Eqn 32.4 = 31.5) for **three** unknowns, *L*, *g*,
and *r -- *too few equations to solve for the unknowns. We can,
however,
use some tricks, based on other knowledge about constraints on the
system,
to obtain estimates of these unknown parameters from the data.

First, let's find the equilibrium solution from
the
observed data. Remember from Eqns 32.3 and 32.4
that *gL* and (1 -
*g* - *r*) are the intercept and slope
respectively. But the equilibrium [Eqn 32.2] is just the intercept (*gL*)
divided by 1 minus the slope [1 - (1 -* g* - *r*) = *g* + *r*], so we
can
calculate the equilibrium value

If we calculate the slope from ordinary least squares regression (that is, by analyzing the numerical values of the data points statistically) or simply by using a cruder rise-over-run approach on a graph, we'll then be able to calculate the equilibrium level of infection. Next let's look at reasonable bounds onEqn 32.5

From Eqn 32.3

soEqn 32.6(= Eqn 32.3 = Eqn 31.4)

But we know that 0 and 1.0 are the bounds on the possible values ofEqn 32.7

Look just at theEqn 32.8

g > bEqn 32.9

That is, *g* (the infection rate per unit time) will be greater than the *Y*-intercept of our empirical plot. We now have a **lower** bound for *g*. Let's
use similar logic to calculate an **upper** bound. Remember from Eqn 32.4 that the slope is given by

which can be rearranged to solve form = (1-g-r){slope}Eqn 32.10(= Eqn 32.4 = Eqn 31.5)

As forr = 1-g-mEqn 32.11

Now just move the0 < 1-g-m < 1Eqn 32.12

g < 1-mEqn 32.13

That is, we now know that the upper limit for *g* is given by 1 minus the slope (*m*) of our empirical plot.

Combine Eqns 32.9 and 32.13
together
for the upper and lower bounds on *g*

Let's make the reasonable assumption that the possible estimates ofb < g < 1-mEqn 32.14

where the "hat" over the variable denotes the fact that it is anEqn 32.15

and substituting into Eqn 32.7 we getEqn 32.16

Let's say that our observed data had an intercept,Eqn 32.17

[Note: all the above should be Eqn 32.18 not 31.18] We have used knowledge of constraints on the system to calculate equilibrium values and parameter estimates from relatively little information. From the observed infection data we have been able to estimate

1) the rate of spread (g),

2) the rate of recovery (r), and

3) the upper level of proportion infectable (L).

Having an estimate of any one of those three previously unknown parameters could be very useful in managing a population subject to
infectious disease. Estimating them directly would be very difficult. Doing the empirical plot of *I*_{t+1} against
*I*_{t} is much more feasible, and is all we really need.

References:

Huckfeldt,
R.R., C.W. Kohfeld, and T.W. Likens. 1993. Dynamic Modeling: An Introduction.

Quant. Applic. Soc. Sciences No. 27.
Sage University Press, Thousand Oaks, CA

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