Table of Contents
Go Back
System Dynamics applied to epidemics
by Raul Bagni, Roberto Berchi and Pasquale Cariello
Scuola di specializzazione ROSD “La Sapienza” University of Rome
raul.bagni@tiscalinet.it, r.berchi@ tiscalinet.it , pat.car@ tiscalinet.it
Abstract
This paper presents an approach to infectious disease analysis through System
Dynamics methodology, following the early works of Ritchie-Dunham The case study
concerns the Bovine Leukosis Virus (BLV), that exclusively strikes cattle. The infected
animals, exposed to secondary infections, become less productive bringing about an
economic loss. In order to avoid the spread of the infection among dairy farms an
eradication national plan is operative in Italy since 1996, but points of infection are still
being recorded. Hence deeper analyses are required to understand the causes of the
endemic behavior of BLV. Analytical models of epidemic spread have been implemented
since the first decades of the XXth century, but, their practical use is often difficult.
System Dynamics models allow epidemiologists to do a set of what-if analyses, with the
purpose of assessing the system! s behavior under various conditions, and afterwards, to
compare and evaluate the results of alternative sanitary policies.
Keywords: epidemics, health care, public policies, complex nonlinear dynamic
systems.
Introduction
Recently two episodes, concerning epidemics on animal farms, have particularly struck
the European public opinion: the mad cow and foot and mouth disease. The first one
mainly for the tragic consequences that it has had on man. Even if humans are not
directly in danger, since the infective agent is not considered able to “jump species”,
economic damage is however considerable, represented by loss of animals, the negative
effects on meat markets, by resources required to control and eradication policies.
Another animal disease with epidemic characteristics is Enzootic Bovine Leukosis, it
concems a viral pathology, sustained by a retrovirus similar to the human retrovirus of
HIV, that exclusively strikes cattle. The Leukosis is not deadly for most infected
animals, but a small percentage of them develop a malignant form. The infected animals
remain serum positives for life, so they are exposed to secondary infections, due to the
weakening character of the illness, and consequently become less productive bringing
about an economic loss.
In order to avoid the spread of the infection among dairy farms an eradication national
plan is operative in Italy since 1996, in accordance with Directive 64/432/EEC on health
problems affecting intra-Community trade in bovine animals and swine, and the
following Council Directive 97/12/EC.
Notwithstanding the adoption of this plan, with the obligatory elimination of infected
animals, cases of endemic character are still being recorded, especially in dairy farms
situated in Central and Southern Italy. Figures 1 and 2 report the relative percentages of
farms and cows detected as positive in Italy in 1998. Deeper analyses are then required
to understand the causes of the endemic behavior of the Enzootic Bovine Leukosis,
possibly using new approaches.
As already outlined, the main target of sanitary controls is the total eradication of the
disease. To improve the analysis of infection dynamics, with the aim of select the
policies with maximum probability of eradication (in the deterministic model of the
single farm this target can be translated into "minimum time of eradication") and
minimum number of controls, we have built some models of Bovine Leukosis contagion
on a typical dairy farm.
Analytical models of epidemic spread have been implemented since the first decades of
the XX th century, but, their practical use is often difficult, above all for predictive and
quantitative analysis.
A new possible approach is computer simulation, in particular the “System Dynamics”
technique, to solve the problem under exam. This article analyzes the potentialities of
"System Dynamics" simulation technique applied to the epidemiological field.
Developing our models of epidemic simulation we have three goals in mind:
a) to allow the identification of factors and mechanisms of epidemic spread during the
descriptive phase;
b) to allow the sifting of different scenarios in a reasonably rapid way;
c) to allow a qualitative and quantitative evaluation of defense strategy choices;
to implement a valid tool in decisional and control processes for those responsible for
public health.
In this article we will describe the model of simulation that have been developed and we
will also underline the differences between this new approach and the more traditional
ones in the mathematical treatment of infectious diseases.
| This is the more recent publication from Italian Sanitary Authority available from
World Wide Web:
(http://www.sanita.it/alimvet/veterinaria/piani/eradicazione98/leucobov1.html)
Wl o76a 616 ©)
0,45 4 0,76 (1)
0,344 0,45 6)
0152034 (3)
Oo aos
Figure 1- Relative Peroentage of positive farms in Italy
in 1998
0,334 1,47 (3)
19.4 0,33 (4)
Dot ao19 (4)
Doo3a01
Oo a0,03 (6)
Figure 2- Relative Peroatiage of positive cows in
Ttaly in 1998
The mathematical bases of the model
The outstanding impact on public health have given impulse to the development of
representative models of epidemic evolution and propagation (Bailey 1975) (Bartlett
1960) (Hoppensteadt 1975). The most important, from a historical point of view, and for
the number of applications and developed analyses, is the deterministic model (the so-
called SIR model), that depicts the spread of infection in terms of time dependent
differential equation systems.
Deterministic model SIR
The SIR model Kermack and McKendrick 1927) (Kendall 1956) is the point of
reference for mathematical models used to describe epidemic diffusion. We illustrate
here this model, whose differential time dependent equations form the bases of our
simulation models.
The model divides population, into three fundamental blocks:
Ys (susceptible) that is subjects susceptible to contract the infection;
¥ i (infected) the subjects that have already contracted the infection, to the stage of full
development or incubation, so they are vectors of contagion;
Y r (retired) subjects that have died or that have developed immunity to the contagion,
therefore no longer able to infect or to contract infection.
Three differential equations describe time evolution of population P components. We
suppose P constant to simplify.
08 = —isB
Hi =isB -iy
re =iv
AP =r+s +i =oonst
Where the meanings of parameters are:
B(infection-rate or contact-rate) rate of new infected in the unit of time;
y (removal-rate) rate of mortality or immunity in the unit of time.
Therefore we are able to foresee the temporal trend of infection estimating B and y
parameters, and knowing the initial values (so, ip, 0) at time tp. In practice we are more
interested in the derivative of r, that is the “epidemic curve”, given by the new retired
rate. The importance of this curve is due to the format normally available for
epidemiological data: the number of new deceased / recoveries in the last unit of time
(generally day or week).
The famous Kermack and McKendrick's “Threshold Theorem” has derived from these
equations: naming E = Vf the relative removal-rate, no epidemic can spread unless we
had the initial condition xo > @ Defining v = so - pand under some approximations, the
“total size of epidemic”, that is the total number of subjects infected, for t > ois 2v. In
other words:
% =p
kK. =p
In the case of non deadly or non symptomatic disease it is possible to use a simplified
model: the SI. It does not keep track of retired ones and solutions are more simple and
immediate.
Over the years, the model presented has been extended, by the same authors and by
other researchers, to consider other arguments such as: the presence of carriers
(apparently healthy individuals who harbour infection), the spread of illnesses through
vectors (bugs, water, food, etc.), several groups of subjects coexisting (with different
rates of infectivity, mortality and immunity), other states apart from infected /
susceptible (for example, immunity from a previous exposure to contagion), the
contemporary action of more viral agents, incubation periods and so on.
Despite great results reached in the qualitative and quantitative description of the
evolution of infectious diseases, a deterministic model has intrinsic limits, in detail it is
not valid:
y for small populations, in which events are subject to ample statistic fluctuations;
y for non homogeneous populations with different degree of immunity, density,
exposure to contagion factors, etc.;
so its practical use is often difficult, above all for predictive and quantitative analysis.
Other models
Likewise its deterministic counterpart, the epidemic stochastic model can be described
using random variables s(t), i(t) and r(t), that represent respectively the number of
susceptible, infected and retired subjects at time t (with s(t) + i(t) + r(t) =n+awitha
initial infected). In an analogy with the deterministic model we define infection and
removal rates. We define also a time dependent differential-difference equation system,
to determine the probability p (7 that at time tthere are exactly s susceptibles still
uninfected and i infectives in circulation.
However, as in the case of the deterministic model, the main problem is to apply
formulas to real world.
Other models have been developed that keep track of the geographical distribution of
population components (S,I,R), besides the temporal one: these are “spatial models”.
In other terms the above differential equations describing the spread of infectious
disease are not only dependent on time but also on two-dimensional coordinates (x, y).
So that we analyze the gradients of the functions s(x, y, t), ix, y, t), r(x, y, b.
Another important class of models no longer consider processes that occur in
continuous time but assume that, following the contraction of infection by any
susceptibles, there ensues a latent period of fixed length. The subsequent infectious
period is considered as being contracted to a single point of time. These are the so called
discrete-time or chain-binomial models (Gani 1969) (Gani and Jerwood 1971).
If we also suppose that a susceptible chance of being infected depends only on the
presence of some infectives and not on their actual number, we obtain the Greenwood
model.
If we assume instead, that the chance of infection does depend on the number of
infectives present we obtain the Reed-Frost version.
Because the focus of this paper is the use of simulation models, it’s important to remark
that Monte Carlo techniques have been already used (Elveback et al. 1971) (Elveback
1971) (Ewi et al. 1972) (Gallop 1999) to challenge the complexity of stochastic models
and Markov chain models.
The case study
Enzootic Bovine Leukosis is a viral pathology (BLV) sustained by a retrovirus from the
same family as HIV that exclusively strikes cattle. BLV is not fatal for most infected
animals, although they remain serum-positives for life. Hence, due to the weakening
character of the illness, they are exposed to secondary infections, and consequently
become less productive. Moreover, serum-positive cattle are ineligible for selling.
In order to avoid a great increase in the spread of the infection among farms and to
improve the efficacy of legislation prohibiting the sale of infected animals, rapid
identification and elimination is essential. Therefore, since 1996, an obligatory national
plan for eradication following sanitary controls at frequent intervals has been set up in
Italy. The frequency of the controls depends on the identification of cases of infection.
The adoption of this plan has certainly contributed towards the limitation of the spread
of the infection, but, to date, cases of endemic character are still being recorded,
especially in dairy farms situated in Central and Southem Italy.
This study was developed jointly with ‘Istituto Zooprofilattico Sperimentale delle
Regioni Lazio e Toscana’. The epidemiologists’ aims onto investigate the causes of
endemic behavior of BLV in some farms of Central Italy, and to appraise the
effectiveness of various control measures as alternatives to current ones.
Epidemiological issues
Bovine Leukosis is a weakening disease which occurs over a long period of time. Along
this period the chances of eliminating an infected bovine (retirement or death) from the
farm due to appearance of symptoms are practically negligible; so that the mathematical
reference model do not consider the “retired” in a classical way. Moreover, an infected
animal cannot regain its health, neither can it develop a kind of immunity. Hence
animals that are detected as infected at periodical controls are butchered.
Other aspects of epidemiological interest are the period of latency, of about two weeks,
during which the animal is already infected but cannot transmit the disease to other
animals, and a period of serum-conversion, of about 17 days, in which the infected (and
infectious) animal is not detectable at tests. Therefore, for the simulation of the control
activities, it is necessary to distinguish between infected animals that are detectable and
those that are not.
The situation is even more complex for young calves, in which the simultaneous action
of infective factors (blood and colostrum) and antibodies eventually assumed through
milk, can increase the resistance to the disease for a period of around six months. All
this can be represented in the model with a degree of immunization’ which is diversified
between adults and young bovines, although at an initial stage of simulation we assume
a single degree of immunity in order to simplify the model.
Other important epidemiological subjects are different ways of transmission of the
disease. Basically the means of transmission represented in the model are:
v direct contact between animals of the same department;
¥ simultaneously gynaecological visits in homogeneous groups of animals of the same
sector of the farm;
v transplacental transmission at the moment of parturition;
Y administration of milk of an infected bovine to young calves in their first days of
life;
v injections carried out for the administration of vaccines.
From the evidence reported by experts it is clear that the course of an epidemic of BLV
depends on the interaction between the cows and the logistic aspects of the farm.
The operative ways dairy cows are managed on the farms are strictly related to their
physical conditions, in particular to their ability to yield milk and to their possible
pregnant state. These aspects reflect the organisation of the dairy farm that is structured
in several sectors which are physically separated, each containing a homogeneous group
of animals.
In our simulation models, we have considered a typical medium - large farm from the
centre of Italy with a population of a few thousand cows and seven physical sectors
which bovines pass through depending on specific events like birth, growth, pregnancy,
delivery, nursing, etc. During their lives, usually about five and a half years, the cows
go into calf areas immediately after their birth and subsequently they complete the
production cycle (insemination, pregnancy, delivery and nursing) three times on
average.
The controls
The Italian control policy for bovine Leukosis consists, at the moment, of periodic tests
conducted on the animals by the Veterinary Services, in correspondence with the
national plan for eradication of the disease. Such controls are limited to the cows of ages
older than twelve months.
2 fee is the resistance to also contract the infection following exposure to infective
‘actors.
The dairy farms that do not exhibit cases of Leukosis acquire a sanitary warranty, in
order to be able to sell animals. On the other hand, the presence of infected animals
determines the carrying out of controls at closer intervals. The farms get the sanitary
certification again after two negative consecutive controls, the second of which is
performed after a period of quarantine.
In the case of positive controls the farm must eliminate all the infected cows within one
month. According to empirical evidence and to simplify the structure of the simulation,
we assumed that all the infected animals are butchered at the end of the period allowed
for this operation and hence the latter test is carried out two months after the former.
Simulation
As an altemative to methods used so far to define epidemic descriptive / predictive
models, we have developed a new approach through computer simulation, using
“System Dynamics” paradigms (Forrester 1961).
Building an epidemic model using System Dynamics archetype, as in Hemorrhaging
Dengue Fever study (Ritchie-Dunham 1995a) Ritchie-Dunham 1995b), we focus on
time dependent differential equations systems and their iterative solutions. The
population characteristics are aggregate variables and the model describes the change of
these variables as a whole system.
The main sources of information were interviews with the subject-matter experts,
evidence taken from scientific literature conceming estimates of the infectivity ratios of
the various ways of transmission and a first series of real data acquired from a medium-
large Italian dairy farm.
We used the commercial product “Vensim” to implement our models.
Some details of model
Here we resume and list the fundamental characteristics of the deterministic model,
whose picture we can see in Figure 3.
vY We have three stock variables: susceptible, infected and detectable subjects, and
transitions between the three stocks are adjusted according to the SI contagion
model basis idea.
v Only susceptibles and detectables are measurable variables. The infected is a
variable that is fundamental, influences other variables and obviously affects the
model behavior. However, the observer will never know its value as already stated
in the discussion on serum-conversion period in "Epidemiological issues".
v The life cycle of dairy cows passes through seven physical sectors and nine logic
states. The mean times of permanence in the various sectors are determined by
physical characteristics of cows (lifetime, gestation time and so on).
v The overall number of cows tends to be stable in the neighborhood of the firm’s
target. This is because we are examining a farm whose production is milk so that the
company would like to manage cows as far as possible at the upper bound limit
established by logistic considerations. Retired animals are replaced as soon as
possible, any surplus in numbers is drained by selling calves in excess, all the males
bom on the farm are sold within about eleven days of birth.
The sanitary controls are scheduled on an annual basis if the farm has the sanitary
warranty. If an infected animal is detected, the farm will lose the sanitary warranty and,
at the same time, controls will be scheduled at closer intervals until two consecutive
controls, the second performed after a period of quarantine, give negative results.
All dairy cows detected during scheduled controls are butchered in a single day, the unit
time of our model, so that the retired is a pulse function, whose schedule depends on
actual sanitary warranty and the previous number of detectable subjects, and whose
amplitude depends on the previous number of detectable subjects.
Researchers can set up simulated sanitary controls to build different scenarios (see Fig.
4), varying: the initial control time, the delay between two controls when the farm has
the sanitary warranty, the delay between two controls when the farm is in quarantine
and the delay after the quarantine. There is also a "control type" variable to set: no
controls, controls on animals older than twelve months, controls on all animals.
The stock array variables have nine rows: Cages, Growth, Calves, pregnant Calves,
Dries, Delivery room, Feeding, Production, pregnant Cows. These are a mix between
physical sectors of farm, and logical conditions of Cows; anyway they are fundamental
factors for epidemic spread.
They are connected by feedback to the flows of the newly infected that depend on intra-
sector infection rates (contact and iatrogenic) and on the infected subjects within the
same sectors, in the same way as they depend on inter-sector (contact and colostrum)
infection rates and on infected subjects from other sectors. Of course, flows and
feedback are of array type too.
There is also another continuous flow for the same stock (an auto loop) from one sector
to the subsequent one, its size depends on stock size and from the inverse of its mean
time of permanence.
The stability of the total number of cows on the farm, is guaranteed by making a
comparison with the desired farm target that is also the initial population. Differences
are balanced by an input flow to represent animals bought or sold, this way retired
animals are replaced as soon as possible.
“Popidition™
by coutrol>
‘Populaiee?
otal bed
nein
sretredoy
v_
waranty
rw rarcepeble>
Fig. 4- the sanitary controls
The model produces two types of output. The first one is a “control output”, to verify
the correctness of model. It sketches for the simulated dairy farm:
the number of susceptible cows, for each sector;
the number of infected cows, for each sector;
the number of detectable cows, for each sector;
the total sum of infected cows and of detectable, the sum of cows that are able to
transmit the disease.
The second one is an output to depict the dynamic of the system, not only in terms of
time development but in terms of stability too. So we have:
e)
)
the classical epidemic curve, i.e. the total sum of new infected, detectable,
infectious, for each time step (a very important curve in epidemic analyses), in
other words the derivatives;
the epidemic orbit projections (two graphs): these are the projections on two-
dimensional surfaces of the infection represented as a point in multidimensional
phase space, in other words it is a snapshot of the dynamic behaviour of the
whole system so that the stability or instability of the system may be seen at a
glance;
the number of retired subjects due to scheduled controls.
We show, as an example, the results of two runs:
v in figures 5 and 6 we can see the epidemic evolution when we have an external
shock of 4 infected animals in "pregnant calves" sector at day 30, and no controls
v in figures 7 and 8 we can see the epidemic evolution for the same shock but actual
veterinary controls are applied.
We have designed a first stochastic counterpart of this model too, applying the Poisson
distribution to flows with deterministic estimation as expected value. The stochastic
model is actually under development and evaluation.
Graph for cows Graph for infected
so Foo TT Todd ddd 200 [| Ild)jd)odiJ)).i.10001
400 100 |
0 0 | ASS
0 390 780 1170 1560 1950 2340 0 390 780 1170 1560 1950 2340
Time (Day) Time (Day)
a
ES infect Calves] = Caest on
cowsiprogn Calves}: Curent cow aoe
tows Dalvery rom Care ee info Daivey ao]: Comer
Th fnfeced{Peedng) Cure cor
ae oe rr LA ee
‘cows{pregn Cows} : Curent, "cow infected{pregn Cows) : Curent. —————— cow
Gieph for ceanle Epidemic evolution
ne 4,000 [ TdT) )od)iii....1100 1
400 |
2,000 |
0
0 390 780 1170 1560 1950 2340 |
Time (Day) 0
eT 0 390 780 1170 1560 1950 2340
dletectaelGrowth} : Curent’ cow Time (Day)
detectable(Calves) : Curent
detectablelpregn Calves} : Current
dtectable(Dries) : Curent
detectable[Delivery room] : Curent. ——————— cow
detectable{Feeding}: Curent, —— cow
detectable[Production}: Curent, cow
detectablefpregn Cows}: Curent, ——— cow
total infected : Curent. —————__
tot detectable : Current. ————__
grand total infected : Curent
Fig. 5 - first group of graphs, 4 infected at time 30, without controls
Epidemic Curve
Epidemic Orbit
4,000
2,000
0
9 632 1255 1877 2500
total susceptible
grand total infected : Current
20
10
0
0 390 780 1170 1560 1950 2340
Time (Day)
total new infotnd : Curent
total nov detectable : Curent
epidemic curve : Curent
2nd Epidemic Orbit Plane
20
10
0
-3 -2 Al 0 al
total new susceptible
epidemic curve : Current
G raph for retiredby antrols
0.2
0
0 390 780 1170 1560 1950 2340
Time (Day)
retired by contos{Cages) : Curent
retired by controls{Growth} : Curent
tied by controls{Calves} : Curent
retired by controls{pregn Calves) : C
‘tied by controls{Dries] : Curent,
ried by controls{Delivery room) : Curent
tired by controls{Feeding] : Cument
retired by controls|Production] : Current
retired by controls{pregn Cows] : Curent
Fig. 6 - second group of graphs, 4 infected at time 30, without controls
Graph for cows Graph for infected
800 4
400 2
0 0
0 363 726 1088 1451 0 363 726 1088 1451
Time (Day) Time (Day)
cows{Cages) : Curent cow ete) Cue, on
rao $e
aac = —$<$<$<$<$<———" an infectedprogn Caves: Cumen ox
cows{Dries} : Current cow
infected{Dries} - Curent
Festi eal a =
SS 7 re
SS —_— _—_______.@ Ts
Graph for detectable Epidemic evolution
4 20
10
0 363 726 1088 ~—«1451
Time (Day) 0
feign Vine aay 0 363726 10881451
ae. & Time (Day)
Aetectable{Calves) : Current
‘detectable{pregn Calves} : Current
detectabl[Dries) : Curent
detectable{Delivery room : Current. cow
detectable|Feeding] : Cument. cay
detectable|Production} ; Current, cow
detectablelpregn Covi}: Current, cow
total infected : Current
tot detectable : Current. ————_
grand total infected:: Curent. —————————_
Fig. 7 - first group of graphs, 4 infected at time 30, with "standard" controls
Epidemic Curve
0.2
0.1
0
0 363 726 1088 1451
Time (Day)
‘total new infected : Current
‘total new detectable : Current
epidemic curve : Curent
2nd Epidemic Orbit Plane
0.2
total new susceptible
epidemic curve : Current
Epidemic Orbit
20
10
0
2483 «2486 «62489 «2492 ©2495 «2498
total susceptible
grand total infected : Curent: —————————
Graph for retired by antrols
4
|
|
0 | re Fa La
0 363 726 1088 1451
Time (Day)
retired by contros{Cages) : Cuent; —$_————
retired by contol{Growth) : Curent
relied by contols{Calves}: Curent
retired by contol{preqn Calves) : Curent
refired by contol[Dries}: Cunent
refed by contos{Delivery room] : Cument
relied by contol Feeding} : Curent
relied by contols{Production) : Curent ——————
relied by contols{pregn Cows) : Curent
Fig. 8 - second group of graphs, 4 infected at time 30, with "standard" controls
Conclusions
First results
The constructed model allows the user to regularly monitor the simulation by means of
a series of graphs, as well as to customize experiments by changing characteristic
quantities of the system at the initialization step.
The most evident improvement in the use of the simulation is the availability of
conspicuous quantities of artificial data, hardly ever obtainable in the real world,
because of the elevated costs of polls or the rarity of phenomenon. This has allowed the
experts to do a preliminary “what-if” analysis with the purpose of assessing the
system’s behavior under various conditions and evaluating which altemative policies to
adopt.
Thanks to model results, the “Zooprofilattico” Institute researchers have hypothesized
that disease may be endemic because of the failure of the current methods of control in
the identification of the very young infected calves. These can start a new process of
the spread of infection between one control and the following one.
The values of parameters have been inferred by current literature, from the experience
of researchers and from some data results picked up from the field.
Using optimization and “goal seeking”, offered by the Vensim product, we were able to
conduct a first verification of the hypothesized infection rates. A calibration model was
developed to match real data collected on field. The first results of parameter values
matches very well that available in veterinary literature obtained by experiments.
Other more precise hypotheses, and their validations, will however, only be possible
using a model whose parameters are well calibrated. So it becomes necessary to receive
more data from the field and extend the model with other necessary elements to fit
phenomenon fully.
Acknowledgements
The authors are grateful to Ferrari Giancarlo, Sala Marcello and Scavia Gaia of the
Epidemiological observatory of the “Istituto Zooprofilattico Sperimentale delle Regioni
Lazio e Toscana ” for their invaluable contributions and for the precious discussions.
References
Bailey, N.TJ., 1975, The Mathematical Theory of Infectious Diseases and its
Applications II ed, Griffin & Co.
Banks, J., 1998, Handbook of Simulation: Principles, Methodology, Advances,
Applications, and Practice, John Wiley & Sons.
Bartlett, M. S., 1960, Stochastic population models in ecology and epidemiology,
Methuen, London,
Elveback, L. , 1971, “Simulation of stochastic discrete-time epidemic models for two
agents”, Adv. Appl. Prob., 3, pp. 226-228.
Elveback, L., Ackerman, E., Gatewood, L., Fox, J.P. , 1971, “Stochastic two-agents
epidemic simulation models for to community of families’, Am J. Epidem, 93, pp. 267-
280.
Ewy, W., Ackerman, E., Gatewood, L., Elveback, L., Fox, J.P., 1972, “To generalized
stochastic model for simulation of epidemics in to heterogeneus population (model
You)”, Comp. Biol. Med., 2, pp. 48-58.
Forrester, J.W., 1961, Industrial Dynamics, The MIT press.
Gallop, R.J., 1999, Modeling General Epidemics: SIR MODEL, Proceedings of the 12th
Annual NorthEast SAS" Users Group Conference.
Gani, J., 1969, “A chain binomial study of inoculation in epidemics”, Bull. LS. L,
43(2), pp. 203-204.
Gani, J., Jerwood, D., 1971, “Markov chain methods in chain binomial epidemic
models”, Biometrics, 27, pp. 591-604.
Hoppensteadt F., 1975, Mathematical Theories of Populations: Demographic, Genetics
and Epidemics. SIAM Regional Conference Series in Applied Mathematics 20.
Kermack, W.O., McKendrick, A.G., 1927, “Contributions to the mathematical theory of
epidemics”, Proc. Roy. Soc., To 15, (Part I).
Kendall, D.G., 1956, "Deterministic and stochastic epidemics in closed populations”,
Proc. Third Berkeley Symp. Math. Statist. & Prob., 4, pp. 149-165.
Law, A.M., and Kelton, W.D., 2000, Simulation Modeling and Analysis, Third Edition.
McGraw-Hill.
Ritchie-Dunham, J.L., 1995a, "Application of Systems Thinking in the Mexican Health
Sector: An Epidemiological Case Study of Decision Policies in Combating
Hemorrhaging Dengue", Third International Decision Sciences Institute Meeting,
Puebla, Mexico.
Ritchie-Dunham, J.L., 1995b, “Evaluating Policies / Strategies to Combat Epidemics
with Systems Thinking”, from World Wide Web: www.sdsg.com/new/appbntrf.htm
Back to the Top
