Epidemic and S-Shaped

["Hayward J (DMath)"]

Dear System Dynamics Community:

I am doing work on epidemics and spread of information, and I am attempting
to understand these models within systems dynamics. However I am having
trouble constructing the causal loop diagram for the simple two
compartmenet epidemic. In terms of stocks and flows this is:

_________________ - ____________
| |______________catching_____+__| |
| Healthy people|______________illness__________|Sick people|
|_______________| | | /|___________|
| + | | + |
|______________>_________| |______<_________|

(View with fixed width font)

catching illness = healthy people X sick people X a constant

The stock sick people follows an S-shaped logistic law and would appear to
be a case of shifting loop dominance. (Mathemtically the equations can be
transormed to a one variable logistic equation). The reinforcing loop on
sick poeple is clear, but where is the balancing loop? Clearly "Sick people"
is limited by the fixed numbers in healthy people who run out thus reducing
the rate "catching illness".

A version of this model appears in Road Maps (D4432)
http://sysdyn.mit.edu
oad-maps/home.html but they have a flow from sick
people to healthy people. However the model is s-shaped without this flow.

Any help in producing a causal loop diagram for this simple situation/model
would be appreciated.

John Hayward
From: "Hayward J (DMath)" <jhayward@glam.ac.uk>

_________________________________________________________________

John Hayward

Division of Mathematics

http://www.maths.glam.ac.uk/maths/

University of Glamorgan, Wales UK

["Michael Bean (Simresource)"]

There is an epidemic model on Simresource that includes causal-loop diagrams
that should meet your requirements.

You can download a copy of this model at:
https://www.simresource.com/_scripts/Mo ... ssed=10166

or go to http://www.simresource.com and search for the term "epidemic"

Michael Bean
From: "Michael Bean (Simresource)" <michael@simresource.com>

["Wayne Wakeland"]

John

The balancing loop is Healthy people to catching illness to Healthy people.
If you have no Healthy people, none can flow out.

When you have interacting loops, loop analysis gets tricky, which is why we
"operationalize" models (in Barry Richmonds terminology) and run them to
help us understand behavior. When you trace a flow diagram, and trace
against a flow, the "sign" of the interaction is negative.

Wayne Wakeland
Adjunct Professor of System Science
Portland State University
wwakeland@uswest.net

[John Sterman]

John Hayward asks about the causal structure of the classic epidemic model
(the so-called SIR (Susceptible-Infectious-Recovered) model, in particular
about the feedback structure of the model.

This class of model, applied to both epidemiology and innovation diffusion,
is treated extensively in chapter 9 of Business Dynamics (my new textbook)
with applications and examples including acute infectious illness such as
measles or chicken pox, as well as mad cow disease (BSE) and HIV/AIDS.
Innovation diffusion examples include a variety of new products from
computers to cable television to VCRs. The cd rom includes a variety of
these models for you to work with.

The reference is:

Business Dynamics: Systems Thinking and Modeling for a Complex World,
Irwin/McGraw-Hill. ISBN 0-07-231135-5.

John Sterman
jsterman@mit.edu

[Scott Rockart]

John,
A causal loop diagram capturing the epidemic stock and flow can be
created with two loops and three variables. One loop connects "healthy
people" and "catching illness." A larger stock of healthy people
contributes to a faster rate of catching illness (positive arrow), and a
faster rate of catching illness depletes the stock of healthy people
(negative arrow). This gives you the missing balancing loop. As you say,
the reinforcing loop is clear. More sick people increase the rate of
catching illnesses which in turn increases the number of sick people.
Looking at your stock and flow diagram you really already have this
diagrammed. A common difficulty in translating from stock and flows to
causal loops is recognizing that a conserved outflow is equivalent to a
negative causal link that runs into the stock. The balancing loop in your
stock and flow shows up with the positive information arrow from the stock
of healthy people to the rate of catching illness AND in the implicit
negative arrow from catching illness to healthy people represented in the
diagram as a conserved outflow.

Best,

Scott
From: Scott Rockart <srockart@MIT.EDU>

==========================================================================
Scott Rockart
Sloan School of Management
Massachusetts Institute of Technology
50 Memorial Drive, E52-511
Cambridge, MA 02142 USA
Tel (617) 253-4023 Fax (617) 253-2660
==========================================================================

["William J. Costello"]

Dr. Hayward,

In its most basic form the causal loop for an epidemic does have a
reinforcing loop as you suggest but also has a balancing loop that would
include the "infected population" and "susceptible population" feeding
into the "rate of new infections", which would then
influence the "infected population". Unfortunately I am working away from
home today and
cannot include graphics in this response.

This caual relationship assumes a finite population of susceptible
individuals, with no migration, no cures or deaths, and an equal
probability of contact among both compartments of the population.

Excellent resources:
"Exploring S-Shaped growth" (D-4476), which I belive is part of Roadmaps
and
"Study Notes in System Dynamics", by Michael Goodman ISBN 0-262-57051-3
which might be available through Pegasus (www.pegasuscom.com)

The study of the dynamics of epidemics has always provided me with a fine
avenue for introducing children (as young as 7-8 years) to SD in schools.

regards,
Will Costello
Systems Mentor, Waters Foundation Project
Chittenden South School District
Hinesburg, VT 05461
From: "William J. Costello" <costello@panther.middlebury.edu>

[L J Wilkinson]

hi

i think the situation is a little more complicated than you describe.

the rate of catching illness is dependent on the number of sick people, the
number of healthy ones, the probability of transmission of the disease and
the number of social contacts made (where the disease can be transmitted).

enclosed is a stella sd of a student example i use when teaching sd
example is about spread of infectious disease
[Hosts note: Attachements do not go through on the mailing list but I am sure you
can email Larry directly if you would like a copy]

larry wilkinson

______________________________________________________________
Larry J Wilkinson Room 321
Newcastle Business School Northumberland Building
University of Northumbria
Newcastle upon Tyne 0191 2274374
NE1 8ST
l.j.wilkinson@unn.ac.uk
______________________________________________________________

["Roberto Vacca"]

Dear John Hayward
epidemics are very often modeled very exactly with Volterra-Lotka equations
The number x of people who get sick varies proportionally to the product of
itself times the number of people who are not sick and are not immune. Call
N the total number at the end of the epidemics:

dx/dt = k x(N-x)

The solution is

x = N/(1 + EXP(a + bt))

I have developed SW for determining N, a, b based on time series.

best

Roberto Vacca
Director of Research ISIS (Institute of Systems Integration Studies), Rome
Italy
From: "Roberto Vacca" <mc4634@mclink.it>

[Mohammad@mail.yazd.co.ir]

John,

A rigorous method for producing causal loop diagrams and locating
dominant feedback loops in simulation models is developed in my doctoral
thesis (1997) at the University at Albany.

Based on this method, the reinforcing loop that is mainly responsible in
creating the reinforcing growth in Sick People is:

Sick People --> probability of contact with sick people --> catching
illness --> Sick People

At the same time, the above feedback loop is the major cause of
reinforcing decline in Healthy People.

The balancing feedback loop that creates a balancing decline in the
behavior of Healthy People and a balancing growth in the behavior of
Sick People is either:

Healthy People --> catching illness --> Healthy People
or
Sick People --> recovering --> Sick People

depending on the duration of illness relative to population interactions
and probability of catching illness. A very small duration of illness
can cause the latter feedback to become dominant instead.

The story is that the reinforcing feedback loop drains Healthy People
through the flow of catching illness until they run out. The balancing
loop then comes play to control the the outflow of Healthy People which
happens to be the inflow of Sick People. When the duration of illness
is quite small, Sick people get well immediately. A larger flow of
recovery can lead to a slower decline of Healthy People and, at the same
time, a slower growth of sick people.

Hope it helps.
From: Mohammad@mail.yazd.co.ir

Mohammad Mojtahedzadeh
Assistant Professor,
Department of Management and Systems,
Sharif Industrial University, Tehran
Iran

[George Richardson]

>I am doing work on epidemics and spread of information, and I am attempting
>to understand these models within systems dynamics. However I am having
>trouble constructing the causal loop diagram for the simple two
>compartmenet epidemic.

You might look at the note called "Problems with Causal Loop Diagrams
Revisited" in the Systems Thinker, vol. 7, issue 10 (1996-97) and/or
the slightly longer version in the System Dynamics Review, vol. 13,
issue 3 (1997).

They provide a version of the causal loop diagram I think you are
looking for, and perhaps the explanation for why you might be having
a tussle with this.

...George
--

-------------------------------------------------------------------------
George P. Richardson G.P.Richardson@Albany.edu
Chair, Dept. of Public Administration and Policy 518-442-5258
Rockefeller College of Public Affairs and Policy 518-442-5298
University at Albany, Albany, NY 12222 http://www.albany.edu/~gpr
-------------------------------------------------------------------------

[Richard Karash]

John Hayward asks about the causal loop diagram ("CLD") for the
S-Shaped epidemic model, and questions the diagram in the Road Maps
document D-4432. I note good answers from others here.

I, too, had a problem with the Road Maps diagram. I think, a best,
the Road Maps diagram doesnt emphasize the important relationships.
At worst, its misleading. (Overall, Ive found the Road Maps series
to be very, very good.)

Ive put the diagram from Road Maps D-4432 on the web at

http://www.learning-org.com/graphics/SickCLD.gif

My version of the Causal Loop diagram for epidemic is at

http://www.learning-org.com/graphics
evSickCLD.gif

My explanation is: New Infections are created when healthy and sick
people meet; thus new infections rise when the product Sick * Healthy
rises. New infections add to the number Sick, creating even more New
Infections, a reinforcing loop.

What slows the growth of Sick? Two things...
a) The more Sick, the more Recovering each month, and each recovery
reduces the number of Sick.
b) When Sick increases, this reduces the number of Healthy and
thereby the rate of New Infections.
~~~~~

I wondered, is there a lesson to be gained from finding an
ineffective causal loop diagram in Road Maps? I think that when we
focus on stock/flow and simulations (these ARE better), we can easily
gloss over the construction of good causal loop diagrams (they seem
SO easy) and miss opportunities to create diagrams and explanations
that will have punch.

In spite of their restrictions (see George Richardsons excellent
"Problems with Causal Loop Diagrams"), some of us find CLDs an
effective tool, especially for communicating.

Hope this is helpful.

-=- Rick Karash

--

Richard Karash ("Rick") | <http://world.std.com/~rkarash>
Speaker, Facilitator, Trainer | mailto:Richard@Karash.com
"Towards learning organizations" | Host for Learning-Org Discussion
(617)227-0106, fax (617)523-3839 | <http://www.learning-org.com>

["Hayward J (DMath)"]

I would like to say thank you to all who responded to my query concerning
the simple two compartment epidemic model and its S-shaped behaviour. The
discussion was a few weeks ago, but various email and system crashes have
prevented me replying earlier.

I cant remember which replies prompted the solution but the solution lay in
keeping to the simple two compartment model.

The model was missing one vital converter: The probability of an ill person
making contact with a well person. Assuming homogenous mixing this is given
by:

probability of catching illness = Well/(Well+Sick)

As the number of Ill people goes up, this probability goes down. This is the
negative link. To see this keep all variables constant apart from Sick, as
suggested by one of the replies.

Catching Illness = Probability of Catching Illness X Sick (X
a constant)

Thus the balancing loop is: Sick --> probability --> catching illness -->
Sick
The reinforcing loop is: Sick --> Catching Illness --> Sick

This gives the S-shaped shifting loop dominance.

In the classical three compartment general epidemic model the balancing loop
through the removed category reduces the upper limit of the S-shaped curve
to something less than the whole population.

I hope this has added some understanding to the situation.

John Hayward
From: "Hayward J (DMath)" <jhayward@glam.ac.uk>

University of Glamorgan

Wales UK