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epidemiology
2/9/98

note: the midterms are graded but 
can't be returned yet, because some people are still taking them, up til 
5 pm. however, tomorrow, we should get our exams back with a distribution 
curve. great. i'm *thrilled* by this news.

some guy once said "a donkey 
is a horse that has been translated into Dutch."

models of disease are 
similar. they're just aids. they aren't going to absolutely predict an 
answer for you. regard them as simply a tool, an aid to thought. we have 
a demo all ready to go, but Chuck didn't bring in a projector, 
apparently.

so. The Epidemic Model: aka the Simple Epidemic Model:

we 
always follow the same steps when we construct a model. 

1. construct a 
flow chart - most important, clarfies your thinking. the model consists 
of three boxes, or compartments, so you draw those:

____
 X
____
 |
 |BY

\|/
 V
____
 Y
____
 |
 | d
\|/
 V
____
 Z
____

now, you divide your 
animal population up by disease status, such that X = animals as yet 
uninfected, and susceptible. X = animals at risk. Y = infectious animals. 
Z = recovered and immune, noninfectious. X also stands for density of 
susceptible animals, Y for density of infectious animals, and Z for the 
density of recovered immune animals. Not the NUMBER of animals, but the 
density of them. 

the arrow moving from Y --> Z we call this delta (d). 
d = recovery rate. Just like incidence, d is an instantaneous per capita 
rate. the units of d are per animal per time --> d = #/animal/time. when 
you have a parameter representing the rate at which animals are lost from 
the box (Y) for whatever reason, we can estimate the numerical value of 
that simply - it's the reciprocal of the length of time animals are in Y. 
so if an animal is infectious for ten days, the recovery rate d = 
0.1/animal/day.

rate of movement from X --> Y is not a single parameter. 
it depends on several things, eg how often a susceptible X animal 
encounters an infectious Y animal. that will depend on the density of Y - 
the more Y there are, the more often an X will meet one. so the rate of 
movement is also somehow proportional to Y. so, if this rate is 
proportional to Y you have to add in a constant. so this rate is called 
beta Y (BY) where beta (B) is the transmission constant. what is B? many 
things. it conceals many subtle relationships. it has something to do 
with the probability that an encounter b/w X and Y will result in an 
infection. because, you can have some encounters that do not result in 
infection. the probability that it will lead to infection is governed 
somehow by B. 

so those are all the parameters we need for our model. 
X,Y and Z are the population variables, and B and d are the parameters. 
that's all step one. now, we must convert that into a series of 
equations.

why was step 1 the most important step? well, does it look 
accurate to you? are you only going to have X, Y, and Z animals? no. you 
can have infected, noninfectious animals, you can have recovered 
infectious animals. you could have animals die instead of recovering. 
this model excludes any infection-attributable deaths. this models only a 
benign infection. it also excludes a category of infected/noninfectious 
animals. it also excludes the possibility of a recovered animal becoming 
reinfected. you would need an arrow from Z to X for that. but this model 
assumes that infection confers lifelong immunity. these aren't criticisms 
of the model, just reminders of what to think about. that's why you do 
this first, so you remember to think about these things. we could have 
carriers that are immune but later recrudesce. there are a lot of things 
that aren't on this flow chart. when we make the chart, we have to think 
about the dz really analytically. Ok. onward...

2. construct the 
equations. this will be on the exam. ** we'll have to make a flow chart, 
and draw the equations ** it's simple. we'll get a lesson now on how to 
do it.

____
 X			dX/dt = -BY (X)
____
 |
 |BY
\|/
 V
____
 Y			dY/dt = 
+BY(X) - d(Y)
____
 |
 | d
\|/
 V
____
 Z			dZ/dt = +dY
____

there is an 
equation for each compartment. for each compartment:

dX/dt = rate of 
change in the density of X with respect to time

dY/dt =  rate of change 
in the density of Y wrt time

dZ/dt = rate of change in density of Z wrt 
time

to the right of the = sign goes the sum of all the losses and 
gains. with an epidemic, you have losses (-) from X, because you start 
with more animals in X, then they move to Y. you multiply the thing next 
to the arrow and multiply it by the box it refers to. see above. gains 
are arrows going IN to the box, as in Y. so the loss from X is the gain 
to Y, so now it's +BY; the losses from Y are -dY (note: this is delta Y, 
not the dY from the derivative.). anyway, and so forth.

now, can you 
prove that population density isn't changing with this model? let N = 
total population density = X + Y + Z

well, we're saying dN/dt = 0. so, 
dX/dt + dY/dt + dZ/dt = 0

-BYX + BYX - dY + dY = 0. 

so, N is constant 
in this model. this is built in to the model. this is a technique we'll 
use later on in the endemic model. any questions?

ok. 

3. we've got the 
equations and flow charts, what assumption do we make? with tremendous 
hubris and amazing arrogance, we assume that the behavior of this model 
is identical to that of the epidemic we're trying to study. so we assume 
that by studying the behavior of the equations, we're studying the 
behavior of the epidemic we're interested in, w/o starting an epidemic 
ourselves. just by looking at our equations, we can say a lot about 
what's going to happen. it would be nice to be able to solve the 
equations, but we can't. they aren't solvable. you can't solve for X, Y 
and Z. for almost every realistic epidemiological model, it's impossible 
to solve the equations. so why do we write them in the first place? b/c 
there are other ways to crack this nut. for example, we can see what the 
final values will be for X, Y, and Z. we do this by saying that the final 
values are those attained at equilibrium - when there is no further 
change in density. what's true at equilibrium? well, the derivatives must 
be zero. there is no change in the densities, therefore it's obviously 
going to be zero. how do you make these derivatives equal zero?

dX/dt: 
for -BYX to be zero, Y or X must be zero.
dY/dt: BYX - dY is zero if Y = 
zero, or BYX = dY
dZ/dt: dY is zero if Y = zero

the only common thing is 
if Y = zero. so we know, at the end of the epidemic, there are no 
infectious individuals left - that's how we define the end of the 
epidemic. however, it doesn't say, necessarily, that X will equal zero. 
this simple model doesn't require X to equal zero, we don't know what X 
is. we also don't know, and can never know, the final value of Z.

the 
next thing we can do is say - let's just graph this sucker on the 
computer. 
these are called "numerical simulations"

you plot density on 
the y axis vs time on the x axis, and you get a curve that starts high, 
and ends low - a declining sigmoidal curve - for susceptibles. the 
infectious will increase and then decrease, from and to zero. recovered 
individuals will increase from zero and plateau.

the point? we're trying 
to generate rules about model behavior that we can apply to real 
epidemics. so you have to run the model over and over again. so it's a 
bad way of generating rules. numerical simulations are ok as a first 
look, but not a good way of investigating model behavior. a better way is 
to derive a formula for the "basic reproduction ratio".

this is called 
other things, too - basic reproductive rate, basic reproduction number, 
etc. you'll see it in terms of a simple formula. the meaning? the number 
of secondary cases that are produced by that first infectious individual 
in a population that is wholly susceptible.

if you introduce one 
infectious chicken into a flock of susceptible chickens, and it infects 3 
chickens before it recovers, then the basic reproduction ratio is 3. 


how do we derive a formula for the basic reproduction ratio? ** you need 
to know for the exam ***

look at the equation for Y: dY/dt = BYX - dY


what must be true for Y to be increasing? dY/dt must be positive. this 
defines an epidemic - animals are getting sick. so, BYX must be a bigger 
number than dY.

for Y to increase, dY/dt > 0 is required. therefore BYX 
> dY is required. 
therefore, BXY/dY > 1 is required. then, the Y 
cancels, so BX/d > 1. we're still not done, though. we're interested in 
the # of new cases when that first individual goes in there. so, in a 
large population when the first infectious animal goes in there is nearly 
N. so, make it BN/d > 1. this is the number of new secondary cases 
produced by that first infectious individual. 

BRR  = BN/d
now, if we 
can make BN/d less than 1, we can stop the epidemic.  that's the point of 
calculating it. 

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