--start---

epi
2/11/98

remarks: it's been interesting watching my kids 
go through social studies classes here in the US, because US history 
seems to start in the 1600s when jamestown was founded, but there were 
spanish people here then, and other indigenous peoples as well. you can't 
help but notice in areas of the midwest, there are alot of mounds called 
"spanish forts" that were built by native americans a long time ago. 
Europeans who arrived in 17th century couldn't believe that native 
americans had built these things, so they attributed them to the 
spaniards. some monk in the 15th century decided the indians were 
descended from the lost tribes of israel. many christians used this as an 
argument against slaughtering them. some jesuit priest propounded the 
heretical view that the native americans had arrived from asia via a land 
bridge. no one believed him. people liked the lost tribes of israel idea. 
they couldn't ask the people who built the mounds, because most of them 
were dead because of smallpox and other dzs. b/w 1492-1600 about 30 
million people died - twice as many as during the "black death" in 
Europe.

epidemic disease still and has always exerted a tremendous 
effect on world history. you can read books about this by some people. 
"rats, lice, and history," "the coming plague" etc. infectious disease is 
always a threat. 

he put up a diagram of the X,Y,Z graph solved by a 
computer. the computer solutions are great, but the point of the model is 
to generate rules about the behavior of disease. you can customize a 
model for a particular disease to see how it will behave.  there are 
better ways to generate these rules than using a computer solution.

last 
hour we were left with the "basic reproduction ratio", denoted Ro. you 
can derive a formula for the Ro from your model - then you have a really 
powerful tool, because if you know the formula and numerical value, you 
can tell alot about the disease. why is it so important?

if Ro > 1, an 
endemic infection can be established, or an epidemic can occur
if Ro < 1, 
an endemic can not be established, an epidemic can't occur

although this 
may sound high tech, there is a concrete biological reference. the Ro is 
just the number of secondary cases arising from the first infectious  
case introduced into a naive population. obviously, it must be >1 for the 
infection to spread. if you know the formula that gives you the number 
Ro, you can see what you need to adjust to reduce Ro to a number below 1. 
the formula helps you figure out control options, and explains how much 
you have to adjust things. 

steps in model building:

1. draw flow chart

2. write diff equation for each compartment
3. analyze model behavior

** 
these three are the basis for 2 exam questions. ***

you have to know how 
to do this.

once you have this, you have a few choices:

a) solve 
equations by hand - usually impossible
b) examine equilibrium behavior - 
see where endpoint might be. this means setting all derivatives equal to 
zero and figure out what must be true for that to be equal to zero, and 
what's common at that point. for the epidemic, we found the endpoint was 
y=0, no infectious individuals left. we couldn't say anything about x or 
z. 
c) use computer to solve equations - rapid
d) derive a formula for Ro 
- more powerful than (c) - helps us plan and design controls.


The basic 
reproduction ratio, Ro, is the average number of secondary infections 
produced when one infectious individual is introduced into a population 
where every host is susceptible. **know this by heart** this is going to 
be on the test **

Ro > 1, epidemic will occur
Ro < 1, no epidemic will 
occur (endemic will not be established.

**this part is also very 
important!

deriving the formula for Ro:

ask what must be true for dY/dt 
to be positive?
why is that important? if dY/dt isn't positive, then Y 
won't increase. with an epidemic, Y increases. so when will it increase? 
when dY/dt is positive.
the density of Y will increase, provided dY/dt is 
positive. dY/dt is positive when BXY is greater than dY (deltaY, that 
is). remember, dY/dt = BXY - dY
note to self: dY on the right side of the 
equation is *delta* Y not the same dY as in dY/dt

density of Y increases 
when BXY > dY
notice that Ys cancel. so, BX > d 
we're interested in 
whether or not an epidemic will occur. at the beginning, X = N, so an 
epidemic will occur when BN > d (that's [beta * total population] is 
greater than delta) N = total population density of susceptible 
individuals

math people aren't satisfied with that. they rewrite this to 
be:
for an epidemic to occur:

Ro = BN/d > 1

an epidemic is more likely 
to occur if...
-the density of susceptible hosts (N) is high. (this is 
why we house large groups in large facilities, to avoid a very high 
density)
-the transmission parameter (B) is large (this is the 
probability that contact with infected animal will lead to infection - 
so, could change animal behavior, nutritional status, etc, to try to 
change susceptibility)
-the recovery rate (d) (that's delta) is low - the 
longer it takes to recover, the lower the recovery rate, so the more 
chance of an epidemic you have. 

if density of the susceptible hosts is 
too low, an epidemic will not occur:

Ro = BN/d > 1

Nt > d/B

Nt is the 
critical minimum threshold density, below which an epidemic will not 
occur. if you make N > d/B, you risk an epidemic (i think?)

we can 
prevent epidemics if we vaccinate enough hosts. 

Nv = density of 
susceptible hosts after vaccination.

if Nv < Nt there will be no 
epidemic.

realize that vaccine failure can occur, so it's common to 
engage in vaccination program and still end up with some susceptible 
hosts (Nv). but as long as the number of susceptible hosts is below the 
Nt critical minimum threshold density, there will be no epidemic.

if an 
epidemic already started, the density of Y the infectious individuals 
will continue to rise as long as BX > d 
we can shorten the course of the 
epidemic by vaccinating the remaining susceptibles, to reduce X, or by 
increasing the recovery rate d by treating infectious animals or by 
removing infectious animals from the population via culling or 
quarantine. these both essentially increase the recovery rate.

why does 
removing infectious animals effectively increase recovery rate? we're 
measuring avg time an animal is infectious in the population. if we see 
the animals and manually remove them, then the time they are infectious 
in the population is reduced - so we've increased the recovery rate. if 
they're infectious 6 days normally, but we remove them on the 2nd day, 
then the recovery rate is increased.

if we let the epidemic run its 
course, will every susceptible eventually get infected? will all the Xs 
go to Y and then Z? depends on Ro, the basic reproduction ratio. if Ro = 
1 then there is no epidemic. 

attack rate: the proportion of the 
original group at risk who get the infection. if AR = 1, every animal 
gets the disease. as Ro increases, AR increases, until you get to about 4 
= Ro, where every animal gets infected, pretty much. Ro < 4, not every 
animal gets infected - due to chance events. 

end of epidemic model 
lecture
start endemic model lecture:

remember, typically, a great many 
endemics show recurrent epidemic behavior. measles is classic in this 
sense. before vaccinations, epidemics would occur q 2-3 yrs. rabies 
epidemics occur about every 4 yrs. 

flow chart for simple endemic model:


--u---> |X|---u-->
         |
         |BY
        \|/
         V
        
|Y|--u-->
         |
         |d
        \|/
         V
        |z| 
--u-->

assumptions: birth rate (u) equals death rate. infection doesn't 
cause mortality. 

three equations for this model:

dX/dt = uN - BXY - uX       
note: uN is total birth rate, uX is mortality
dY/dt = BXY - dY - uY
dZ/dt 
= dY - uZ

dN/dt = dX/dt + dY/dt + dZ/dt = 0

the simple endemic model 
takes account of births and deaths; because births equal deaths, 
population density remains unchanged. 

again, there is no analytical 
solution to these equations. just notice that if you solve them with a 
computer, you see characteristic oscillations of the number of 
susceptible, immune, and recovered populations. these are damped 
oscillations - they get smaller with time. 

there's a lag b/w infectious 
phase and time for animals to reach minimum critical density.

damped 
oscillations decrease in amplitude over time.

some endemic infections 
also show damped oscillations, at least for some time period. there are 
some endemic infections that show this type of behavior. but, some other 
endemic infections show sustained oscillations whose amplitude doesn't 
change over time. 

in the simple model the oscillations rae like a 
pendulum that slowly stops swinging. but sometimes it doesn't stop 
swinging. what forces the sustained oscillations? changes in the 
transmission constant, B. could be seasonal, annual, or longer term 
periodic changes in the transmission constant - birth rate, nutritional 
status change, bunches of kids reaching high density in school every 
september, etc. 

---break---

basic reproduction ratio Ro is still 
defined the same way in the simple endemic model. it's always the same, 
in whatever model you write. however, the formula differs. now, to work 
out Ro, what makes Y increase? dY/dt must be positive. so that means that 
BXY must be greater than dY + uY, or in other words

density of Y 
increases if BXY > (d+u)Y
the Ys cancel so: BX > d+u
we want to know if 
infection will be established. at the very start, X=N
an endemic 
infection becomes established if BN > d+u

remember - u in this equation 
means death rate, not birth rate. also, note: i'm typing u but i mean mu.


an endemic infection will establish itself when BN > d+u. in other words:


Ro = BN/d+u > 1

processes that increase the value of Ro will tend to 
promote the establishment of endemic infections. evolution in its cunning 
way has come up with many processes that increase Ro, making 
establishment of infection more certain. some of these things:

long 
infectious periods
asymptomatic carriers 
recrudescent infections
long 
lived free living stages (things that persist in environment)
vertical 
transmission
sexual transmission

these are all mechanisms that ensure 
that the infection persists in the population. sexual transmission is the 
best way to ensure transmission in a low density population. now, how 
many of these apply to aids? four of them. possibly five, if you consider 
maybe recrudescence. this is why it's so hard to handle, 
epidemiologically. it's going to be really hard to get rid of it. in 
animal populations, STDs are easy to eliminate through management - but 
not in wild animals or humans. you can't get rid of them. 

once an 
infection is established, when the first infectious individual has passed 
it on, we now thing in terms of the "effective reproduction ratio" Re, 
instead of Ro

Re = BX/d+u >1

the only difference is we're using X 
instead of N. Ro is simply Re at time zero. as infection progresses, X 
changes - generally it declines, then increases, then declines. this Re, 
therefore, changes all the time, whereas Ro is a constant that's defined 
for a particular population. but it's useful to consider Re, to figure 
out what to do after endemic ifnection is established.

processes that 
decrease Re will tend to increase the probability of fade out
these 
include:

-increasing the recovery rate via test and removal, cull or 
quarantine, or treatment.

-decreasing density of susceptibles (value of 
X) by culling irrespective of infection status, or by vaccination. this 
is done a lot.

-altering susceptibility by changing nutritional status, 
taking advantage of age resistance, hoping for an acquired immune 
response - eg, in case of Rubella you will deliberately expose a child 
hoping it won't get it when she's pregnant, later (or something).


controlling Re: vaccination and culling reduce X. testing and removal 
will increase delta. changing nutritional and immune status will reduce 
beta, and culling will increase u. 

control or eradication of infectious 
diseases: actual programs that are used.
example: in NSW australia, they 
had a brucella eradication program. 

vaccination is a strategy used to 
control (reduce prevalence) and eradicate  (reduce prevalence to zero) 
infxs dz. from the point of view of the ecological epidemiologist, there 
are two types of vaccines:
those that reduce morbidity andmortality w/o 
preventing infxn and shedding (eg equine influenza vaccine, mareks 
vaccine)
those that provide complete protection against establishment of 
infection.

when vaccines do not protect against infection but reduce 
morbidity and mortality:

B is reduced - susceptibility is reduced, takes 
larger insult for infection to occur
d is increased - animals aren't sick 
as long, and they shed pathogen for a shorter time. 
therefore, even 
these vaccines which protect against m&m without protecting against 
infection, they can reduce Re to below 1 and eradicate the disease. under 
small herd densities, this can work. 

if goal of vaccination is 
eradication, then it is more efficient to use a vaccine that provides 
complete protection against infection. 

if you use a vaccine that 
provides complete protection against infection, it is often not necessary 
to vaccinate every animal in the population in order to eradicate the 
disease. this is b/c all you're doing is reducing X to below the minimum 
threshold density or whatever it's called. the infection will disappear 
if you do this. it might take a while, but it will work. people call this 
"herd immunity" and use it as an excuse not to vaccinate their kids. 
unvaccinated members of a population derive protection by reason of the 
association with vaccinated animals. 

you can use the basic reproduction 
ratio to calculate the fraction of animals we have to vaccinte to 
eradicate an infection:

assuming we're using a vaccine that provides 
complete protection:

p = 1 - 1/Ro

p = proportion of population you must 
vaccinate to get rid of disease. 

measles has Ro of about 10. so you 
have to vaccinate 90% of people to get rid of it. 

Ro of smallpox was 
about 3 - only had to vaccinate 2/3.
Ro of malaria is 33. uh....

if 
you're dealing with an endemic infection, we can estimate Ro as follows:

assuming no huge age differences in probability of infection:

Ro = 1 + 
L/A

L = avg life expectancy of animals in the population in absence of 
dz
A = avg age at infection

so in africa, avg age of infection w/measles 
is about 2. avg lifespan is 30. so Ro is about 16 for measles in africa. 
hard to eradicate.

in europe, A = 7 or 8 and L is about 70, so Ro is 10 
or 11.

consequences of vaccination before total eradication:

1. 
interval b/w recurrent epidemics gets longer
2. amplitude of recurrent 
epidemics decreases
3. finally, avg age of infection increases. this is 
bad. there's a big vaccination program to eradicate rubella. now - we 
reduce incidence by vaccinating. incidence is 1/age of onset, so as 
incidence decreases, age of onset increases. so for rubella, we're 
increasing the amount of women who get the disease while of childbearing 
age. so you must understand this is going to happen while you work toward 
eradication. 

eradication strategies:

eradication is hard. it takes a 
lot of effort. a lot of effort. it's not only hard to do, it's hard to 
know when you're done. people argued for years over smallpox, if it was 
gone or not. once you're done, once you've eradicated it

note: 
eradication means disease is gone from face of earth. elimination means 
it is gone from a particular population. if you eradicate a disease from 
a particular pig unit, you can never never stop preventing it in that 
population. you have to implement your strategy, get Ro below 1, and keep 
that below 1 for long enough for the disease to totally disappear from 
that population, which can take years or decades, and then once you've 
done that, you want to make sure it doesn't come back in from somewhere 
else, set up rules, set up monitoring system, this all costs money of 
course. disease eradication is very hard. 

test and removal: objective 
is to reduce the average duration of the infectious period.

brucella 
ovis - eradicated from falkland islands. started in 1980, finished 1993. 
there were only 8000 rams on the islands. it took 13 years and 65,266 
serological tests and surveillance testing continues!

now, there are 
about 8000 rams in PA, and PA isn't even a big sheep state. 
think about 
the costs involved. 

mass culling: objective is to reduce the density of 
the susceptible hosts to below the critical minimum threshold density. 
often used against rabies and hardly ever works. effort required is 
enourmous. works best when used in areas with natural barriers like 
mountains or rivers.

in alberta, they did a big mass culling program - 
they stayed rabies free for 10 years and then it came back, because they 
didn't keep up the surveillance program

depopulation: objective is to 
contain the infection by killing all the hosts in the affected population 
and thus prevent spread to other populations. also have to dispose of 
carcasses w/o spreading dz. premises where animals were kept are subject 
to long period of cleaning and disinfection. this is common w/poultry, 
swine, other intensive production systems. used to control swine fever - 
kill every pig in a 3 mi radius, set up surveillance in 10 mi radius. 
this is removal of entire infectious population to avoid spread. 
frequently used, very effective.

combined strategies: some are more 
efficient or cheaper in the end phase of  a program. consider the 
brucellosis eradication - started out with vaccination, then vaccination 
and test/remove, and then at the end, depopulation. why? it's more 
efficient and cheaper to change strategy as prevalence declines - for 
cost, resource availability, time availability, and acceptance. early on, 
they won't depopulate - every farm was infected. when only a few infected 
farms remain, depopulation was well accepted by the majority of farmers 
(except the infected farms, of course).

there is new material in the 
practice problems, btw. you can't blow them off. you have to know all the 
stuff in the problems for the exam.

---practice problems---

1. during 
an epidemic episode, investigators recognized the following types of 
hosts: susceptible (X), infectious (Y) and immune (Z). investigators 
measured the following parameters: transmission constant (0.001), 
infectious period (4 days), duration of immunity (10 wks)

a) draw flow 
chart, b) write equations:
____
 X			dX/dt = -BYX + gZ
____ <-|
 |     |
 
|BY   |
\|/    |
 V     | g = 1/70/animal/day
____   |
 Y	   |	dY/dt = 
+BY(X) - d(Y)
____   |
 |     |
 | d   |
\|/    |
 V     |
____   |
 Z	
---		dZ/dt = +dY - gZ
____

c) derive basic reproductive ratio, Ro:


deriving the formula for Ro:

ask what must be true for dY/dt to be 
positive?
if dY/dt isn't positive, then Y won't increase. In an epidemic, 
Y always increases. so when will it increase? when dY/dt is positive.
the 
density of Y will increase, provided dY/dt is positive. dY/dt is positive 
when BXY is greater than dY (deltaY, that is). remember, dY/dt = BXY - dY

note to self: dY on the right side of the equation is *delta* Y not the 
same dY as in dY/dt

density of Y increases when BXY > dY
notice that Ys 
cancel. so, BX > d 
we're interested in whether or not an epidemic will 
occur. at the beginning, X = N, so an epidemic will occur when BN > d 
(that's [beta * total population] is greater than delta) N = total 
population density of susceptible individuals

Ro = BN/d **this is the 
answer!

math people aren't satisfied with that. they rewrite this to be:

for an epidemic to occur:

Ro = BN/d > 1

d) derive expression for 
critical minimum threshold density:

Nt > d/B

Nt = d/B ** this is the 
answer

Nt = d/B = .025/.001 = 250 animals/unit area

e) calculate each 
parameter:
B = 0.001/day
d = 0.25/animal/day

f) calculate minimum 
threshold density: Nt

Nt = d/B

0.25/0.001 = 250 animals per area

I 
think this means that you must have at least 250 animals in order for the 
epidemic to occur???**??

2. a vet was investigating a viral infxn 
endemic in a cattery. X, Y, Z, and C (recrudescent infections) were 
present. average time spent in Z before returning to C was 10 days. 
average time that recrudescent animals shed virus was 2 days. B = 0.002 
and time spent infectious was 5 days. birth and death rates were constant 
at 0.05/animal/day and cat population was constant at 200. no cats died 
of this infection.

a) draw flow chart:

--u---> |X|---u-->
         |
         
|BY
        \|/
         V
        |Y|--u-->
         |
         |d
        
\|/
         V
    |-> |Z| --u-->
	|	 |
   r|	 |q
	|	\|/
    |	 V
	
----|C| --u-->

assumptions: birth rate (u) equals death rate. infection 
doesn't cause mortality. 

b) equations for this model:

dX/dt = uN - BXY 
- uX       note: uN is total birth rate
dY/dt = BXY - dY - uY
dZ/dt = dY 
+ rC - qZ - uZ
dC/dt = qZ - rC - uC

dN/dt = dX/dt + dY/dt + dZ/dt + 
dC/dt = 0

c) basic reproduction ratio:

Ro = BN/d+u 

d) derive 
expression for critical minimum threshold density:

Nt = d+u/B 

e) 
calculate parameters:

B = 0.002 /day (given)
d = 1/5 = 0.2 /animal/day
q 
= 1/70 = 0.1/animal/day
r = 1/2 = 0.5/animal/day
u = 0.05/animal/day

f) 
calculate minimum threshold density and basic reproduction ratio for this 
cattery:

Ro = BN/d+u 
   = .002 * 200/.2+.05 = 1.6
Nt = d+u/B = .25/.002 
=125

g) a very effective vaccine is available for this virus but vaccine 
failure is a common problem. what proportion of cats must be vaccinated 
to eradicate this disease from the cattery?

p = 1 - 1/Ro

p = proportion 
of population you must vaccinate to get rid of disease.

p = 1 - 1/1.6 = 
.375 -->37.5% 

h) could this problem be reduced if there were fewer cats 
in the cattery?
yes, if he reduced the number of cats below the minimum 
threshold density of 125, there would be no endemic infection (after a 
while).

i) if the only solution were depopulation, could the cattery 
owner immediately repopulate? 

no, he'd have to disinfect and allow time 
to ensure the absence of the virus.

j) if the cattery owner eradicated 
the disease but insisted this was to be a one time expense, what steps 
could be taken to ensure virus didn't become reestablished?

- maintain 
population below minimum threshold density
- implement a surveillance 
program
- implement vaccination program
- other? ???

3. how does a 
vaccine that protects against M&M but not against infection and shedding 
lead to eradication?

when vaccines do not protect against infection but 
reduce morbidity and mortality:

B is reduced - susceptibility is 
reduced, takes larger insult for infection to occur
d is increased - 
animals aren't sick as long, and they shed pathogen for a shorter time. 

therefore, even these vaccines which protect against m&m without 
protecting against infection, they can reduce Re to below 1 and eradicate 
the disease. under small herd densities, this can work. 

4. why do 
bacterial infections acquired in a hospital have a higher fatality rate 
than infection acquired outside?

population density is so high, and 
handling of patients occurs so often, that an etiological agent can have 
high virulence without risk of killing the patient before spreading to 
another host.

5. why do eradication programs use a mix of strategies, 
and change the mix as the program progresses?

combined strategies: some 
are more efficient or cheaper in the end phase of  a program. consider 
the brucellosis eradication - started out with vaccination, then 
vaccination and test/remove, and then at the end, depopulation. why? it's 
more efficient and cheaper to change strategy as prevalence declines - 
for cost, resource availability, time availability, and acceptance. early 
on, they won't depopulate - every farm was infected. when only a few 
infected farms remain, depopulation was well accepted by the majority of 
farmers (except the infected farms, of course).


---end---