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THESIS

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This is to certify that the

thesis entitled

Micropopulation Simulation of Disease
Transmission Dynamics: A Non-technical
Model with Applications

presented by
Michael P. Collins

has been accepted towards fulfillment
of the requirements for

Masters degree in Epidemiology

 

 

Date April 17, 1997

 

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1/98 chlRC/DdoDmpGS—ou

MICROPOPULATION SIMULATION OF DISEASE
TRANSMISSION DYNAMICS:
A NON-TECHNICAL MODEL WITH APPLICATIONS
By

Michael P. Collins

A THESIS

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Epidemiology

1997

ABSTRACT
MICROPOPULATION SIMULATION OF DISEASE
TRANSMISSION DYNAMICS:

A NON-TECHNICAL MODEL WITH APPLICATIONS
By

Michael P. Collins

Computer modeling of microparasitic disease transmission is presented
within a general context of modeling as a ubiquitous intellectual procedure.
Justiﬁcation is offered for the introduction of a discrete model (dealing with
hosts as individuals rather than as continuous pools) which is non-technical
enough to be accessible to the general epidemiologist.

The essential elements of a proposed model are explained, and its
potential is explored in four applications: Consequences of host population
size and population structure to the likelihood of acute epidemic spread are
illustratively explored. Within the domain of chronic endemic disease, an
exploration of the source of recurrent cycles of disease incidence is presented,
as is a sensitivity analysis of the consequences of various public health
policies. The presentation concludes with remarks on future directions for
related research, and appendices include the model's Basic program as well as

a number of useful alterations.

To my wife Carolyn, who has endured six years of my vacillation, and a year of my "just
sitting around pushing BUTTONS!"

iii

ACKNOWLEDGMENTS

The author wishes to acknowledge the wisdom and steady enthusiasm of his
primary professor, Dr. Michael Harrison, and of the remainder of his committee: Drs.
Blake Smith, Joseph Gardiner, and Nigel Paneth.

iv

TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................... vi
LIST OF FIGURES ....................................................................................................... vii
INTRODUCTION ........................................................................................................... 1
BACKGROUND - THE COMPARTMENTAL MODEL ................................................ 5
TRANSMISSION PARAMETERS ............................................................................... 17
POPULATION STRUCTURE ....................................................................................... 20
ILLUSTRATIVE EXPLORATIONS
Consequences of Population Size (acute system) ................................................. 24
Consequences of Population Structure (acute system) ......................................... 32
Chronic endemic disease - Endogenous rhythms in a simple system ..................... 43
Endemic Disease - Alternative Public Health Policies .......................................... 64
DISCUSSION .............................................................................................................. 69
APPENDIX A - Program Listing with explanation ........................................................ 73
APPENDIX B - A Guide to Alternative Explorations ..................................................... 87

LIST OF TABLES

Table l ......................................................................................................................... 67

LIST OF FIGURES

Figure l - Caricature of Model Sequences ..................................................................... 13
Figure 2 — Spatial Array of Individuals ............................................................................ 23
Figure 3 - Effect of Population Size on Epidemic Spread ................................................ 28
Figure 4 - Typical Epidemic ........................................................................................... 31
Figure 5 - Random Spacing ............................................................................................ 34
Figure 6 - Strongly Structured Population ...................................................................... 36
Figure 7 - Population Structure and Epidemic Spread ..................................................... 40
Figure 8 - Population Structure and Epidemic Spread II ................................................. 42
Figure 9 - Prevalence of Modelled Infection ................................................................... 47
Figure 10 - Autocorrelation of Prevalence - Time Data ................................................... 49
Figure 11 - Prevalence of Modelled Infection H .............................................................. 51
Figure 12 - Prevalence of Modelled Infection III ............................................................ 53
Figure 13 - Standard Deviations of Prevalence over Time ............................................... 55
Figure 14a - Cyclic Behavior when N = 100 ................................................................... 61
Figure 14b - Cyclic Behavior when N = 200 ................................................................... 63

vii

Introduction

This paper presents a model of disease-host interactions, a
micropopulation simulation, which can be of use to the epidemiologist for
investigatory or heuristic purposes. When a general epidemiologist thinks of
process modeling, particularly of a mathematical ﬂavor, it is probably as an
esoteric activity, involving complex equations, which oversimplify impossibly
complex processes. The overall state of our literature in this area does tend
to reinforce this impression. It is probably not widely enough appreciated and
taught that modeling is a very general activity which we all perform
throughout our daily lives, and which is an integral part of even the most
elementary comparisons which we make in epidemiology. In essence,
modeling involves conceptually stripping down a complex situation into its
essential features. As “the lie that helps us see the truth” (Picasso on Art),
this process may allow insight which is obscured by the details of local
applications of general principles.

Consider the process of modeling a situation of everyday life (can I
pass the car ahead of me?) In general, there are three elements of this and
any situation about which we may or may not have information. There may

be information relating to the components of the process: My car will

 

C6 ’9

accelerate “x” fast, and the oncoming car appears to be approaching at y

2
speed. There may be information about the interaction of the components: If

the oncoming car should reach the same point at the same time that mine
does, the result will be a tragic crash. And there may eventually be
information about the result: Did I make it, or not?

We usually model a situation by stipulating the details of any two of

the three elements in order to gain insight into the third. The highway
example noted above, for instance, is an example of modeling for the sake
of prediction. The result is unknown and in the future, but I will use my
knowledge of the situation’s components and interactions to predict the
result. Ifthe prediction is “likely crash”, then I settle back into line and
wait for a better passing opportunity. Similarly, if we wish to predict the
risk of operative death for an individual, we may know the components of
that risk (the risk factors which apply, with their regression coefﬁcients and
signiﬁcant interactions) and the way in which they interact (a linear logistic
regression model is likely to be applicable), and we use this knowledge to
predict a risk of death. In both cases we are neglecting countless details
which may affect the realization of our model (my engine may fail, this
patient may have an unrecognized risk factor), and our prediction can never

be certain.

3
In epidemiology, on the other hand, we often have knowledge of

results - a “data set” of information on variables we believe to include both
determinants and outcomes. We construct some sort of a model of how these
variables interact. The choice of a linear regression model, for instance,
indicates our belief that the effect of determinant x on outcome y can be
adequately expressed in terms of slope and intercept. From our data and our
assumed model we arrive at conclusions regarding the components involved
in the process, the regression coefﬁcients for the determinants included in the
model.

A third situation ﬁnds us with information in hand on outcomes, and
also convinced that we know a degree of the truth about the components of a
situation. Here the challenge is to ﬁnd a model of the interactive processes
which integrates those two bodies of fact in a satisfying, useful, and
instructive way. It is this third domain of modeling activity within which
much modeling of disease-host interactions falls. We have information on
infectious disease outcomes, such as monthly incidence ﬁgures. We have
information on the components of the process, such as latent and infectious
phase durations, the mode of spread, and the existence or absence of the
immune state. It is understanding of the interactive processes which we

desire. An appropriate model, as a caricature of reality, may provide further

4
insights into the dynamics of the disease within populations, insight which

may have public health signiﬁcance.

The modeling of dynamic processes within epidemiology, such as
parasite-host interactions, has too often been relegated to arcane literature and
to specialized courses within academic departments. We should remember
that the static data which we collect over a short interval of time are often the
results of dynamic processes which ﬁrnctioned before and will continue to
function after our period of observation. In the area of microparasite- host
interactions, there seems to be a need for a heuristically useful model which is
general, and which is straightforward enough that the general epidemiologist
can be comfortable using it. This communication is an attempt to present a
model whose components are familiar and whose workings are transparent,
so that neither expertise in higher mathematics nor in computer programming
are necessary to experience it and modify it. Yet, as will be shown, its
nonlinear and stochastic nature results in complex outputs which appear likely
to be relevant to real situations.

At this point the general epidemiologist might object, “Why should I
want to do any modeling at all?” My answer is that this exercise is always
useful for the clarity of thinking which it promotes; indeed, which it demands.

For instance, one may have a general, half-intuitive idea of how infectious

5

and susceptible hosts interact to produce cyclic waves of disease incidence,
and may be able to construct a fuzzy verbal explanation which would pass the
scrutiny of a superﬁcial listener. But in order to translate this understanding
into a very speciﬁc algorithm which the computer must follow, the
programmer will have to supply very speciﬁc descriptions of relationships
and sequences, an exercise which will leave him with a deeper understanding

of the problem than he had before he began.

Background - The Compartmental Model

Were an epidemiologist to cast about for an example of mathematical
modeling of disease transmission, she would most likely encounter some
variation of the “compartmental model” (1). This approach begins by
conceptually dividing the host population into several sharply deﬁned
categories; for instance “susceptible”, “infected/infectious”, and “immune” or
“removed”. Each category occupies a speciﬁc compartment whose contents
are modeled as a continuous state variable whose value is either a number of
host animals of that category or a proportion of the population. In either case,
the variables can take any value between zero and either the total population
size or 1.0. The dynamic relationships among the compartments are

described through the stipulation of a small number of rate constants. A set

6

of differential or difference equations completes the model, whose behavior
can then be examined through time.

As is true in all modeling activities, simplifying assumptions, in this
case of major signiﬁcance, must be made. Although the compartrnental
model has been extremely useful and has provided major insights in the
understanding of disease transmission dynamics, some of the assumptions are
sufﬁciently troubling that they inhibit the ability of the model to provide
insight into some types of questions, or to apply to some circumstances. Thus
a search for complementary models seems worthwhile. Among the
troublesome assumptions of the compartrnental model are the following:

1. Large population size. This assumption, generally a tacit one, is
implicit in the continuous nature of the state variables. We all know
that there are never, for instance, 37.43 diseased individuals; people
(and other host animals) come in discrete units. If the population is
very large this discrepancy doesn’t really matter; ﬁactions of people
are small proportions of the whole compartment variable. But when
populations are small, a given compartment may be occupied by
two, one, or zero people - and the difference between those
numbers may be of major importance in the fate of the infection

within the population.

7

2. Homogeneous mixing. Common experience tells us that human
societies (and that of other animals as well) contain substantial
structure. We do not in fact have an equal or random chance of
contacting every other member of the population of the city in
which we live, for instance, but rather live very closely with a small
group (such as our families), somewhat less closely with a broader
group (such as a school or work group), and in more random
contact with the remainder of our neighbors. Nevertheless,
compartrnental models generally make the assumption that the
infective individuals are randomly mixing with the susceptibles.

3. Mass action. Those models which assume “strong homogeneous
mixing” assume that the rate at which new cases of disease arise is
proportional to both the number of susceptible and the number of
infective individuals in the population. This has been termed the
principle of mass action. To some extent this model is in accord
with intuition, as increase in the concentration of either group, all
else being equal, should be expected to increase the rate of
formation of new infections. But a simple direct proportion is not
reasonable, for if more and more infectives are added to a constant

number of susceptibles, the new infectives begin to encounter other

8

infectives often enough to dilute their effect on the remaining
susceptibles.

One possible alternative to compartmental models, complementary to
them in that it is largely free of the problematic assumptions listed above, is a
digging model. In this case we model a population of discrete individuals,
each characterized by a collection of such variables as have an effect on the
transmission dynamics which constitute the model. For instance, such
variables might include age, infectious status (susceptible, infectious, or
immune), and, for those infected, the current duration of the illness.

A discrete model is necessarily a stochastic one. In the compartmental
model we deal with large “pools” of susceptibles and infectious individuals,
and we deal with continuous changes in the relative sizes of each of these
pools. This is usually done in a deterministic form, so that for a given set of
parameters the model runs identically every time it is run. But a discrete
model deals with single individuals one at a time. Whereas a compartmental
model might tell us, for instance, that 17% of a class of individuals will
develop infection over a time period, the reality for the single individual will
be a 17% probability that “he” will become infected. The program presented
in this paper utilizes Monte Carlo procedures to enforce this probability. The

computer “rolls the dice” by choosing a random number, and a comparison of

9
this number with the probability of infection results in the status of the

individual either changing from susceptible to infected, or not changing. In
either case the result is an absolute one, and in the latter case the risk of
infection in a later time period will be unaffected by the history of risks in the
past.

The stochastic nature of the discrete model presents a practical
problem to the investigator. A speciﬁed compartmental model can be run, its
result obtained, and the result examined in the security that this is “the result”
of the model. The discrete model, on the other hand, does not behave the
same way each time it is run. Thus no single instance of modeling the
behavior of a system sufﬁces, and one must often instead do multiple runs
with results in the form of frequency among a series of runs. For instance, the
compartmental model might show us that, under speciﬁed conditions, an
epidemic will result and could be expected to show certain characteristics.
The discrete model, on the other hand, might under the same conditions result
in an epidemic in 80% of runs, with a distribution of possibilities as to the
speciﬁcs of those epidemics. This requires considerable computer time, and
patience on the part of the investigator. But it is in accord with our
experience of reality. We probably all sense that there is considerable

uncertainty regarding the details of disease transmission in any community,

10

under all but the most extreme circumstances. And further compensating for
the additional computer time is the fact that working with the discrete model
is simply more fun than with the compartmental ones! The compartmental
model lends itself to analysis by equations (as a scan through the 700+ pages
of Anderson & May will conﬁrm); the discrete model promotes simulations.
At the present time there is no discrete model available to the general
user which seems ideal for the purposes I have envisioned. Discrete models
have been used for very speciﬁc applications (2), are remote from real
biological systems (3), or are available in a form such that the user plugs in
her own constants and watches a simulation happen (4,5). Central to my
thinking is the concept that the workings of the computer program should be
very transparent and available for the user to modify; otherwise the
puriﬁcation of thinking which model construction provides will not happen.

This communication, then, offers such a model.

The Model

The modiﬁable program which I present is written in the classic Basic
programming language; speciﬁcally it uses the QBasic version which is a part
of Microsoft DOS, version 5.0 or later. There are two good reasons for the

choice of this language. Most importantly, the author is not a computer

11

programmer by education or vocation, and writing a program in Basic is as
far as he seemed likely to travel toward programming expertise. But just as
importantly, it is anticipated that the user will be a similar person, and the
Basic language is, after all, “in English”. Its keywords issue commands to the
computer which are straightforward, though potentially very powerful, and
the user will not be discouraged by the program’s language from
understanding and modifying its algorithm. QBasic’s availability as a part of
DOS is another favorable aspect of this language as a medium for a program
designed for widespread use. For users interested in learning more about
programming in QBasic, Reference 6 will be useful.

A listing of the actual QBasic core program itself, along with
explanatory comments, is presented in Appendix A. The reader is
encouraged to peruse it eventually; it is not too obscure for a motivated non-
specialist to follow it. As an overview, the program can be thought of as a
sequential process which takes a set of initial conditions through several
steps, resulting in new conditions, which then are subjected to the same
sequential process, and this repeats indeﬁnitely. The process steps are
explained below, caricatured in Figure 1, and mapped onto the program

listing in Appendix I.

12

Figure 1. Caricature of Model Sequences

This paper focuses upon insights gained by the use of a discrete, stochastic
simulation model of disease-host interaction. The term “discrete” as used here means that
the model treats hosts as individuals rather than as continuous pools of host classes. The
term “stochastic” reﬂects the fact that the transfer of an individual host from one class to
another (e. g., susceptible to infected, infected to dead) is determined by a random number
choice, and is not absolutely determined by the conditions which prevail at the time. This
caricature describes the sequential steps accomplished by the computer model, ﬁrrther
described in the text.

Stage 1 - The program creates the “world” within which the pathogen-host system
will operate. Included are three classes of hosts: Susceptible (S), Infected/infectious (I),
and Immune (R - “resistant”).

Stage 2 - Creates a population of N discrete individual hosts, and establishes their
disease transmission - relevant mutual distances.

Stage 3 - Establishes initial conditions, ordinarily including “seeding” the
population with at least one infective. The little population pictured here begins with 4
susceptibles, 2 infectives, and no immune individuals.

Stage 4 - Determines, for each susceptible, the degree of contact with infectives
happening within the current time interval. For each susceptible, the extent of such
contact depends upon the total number of infectives in the population and also upon the
social distance of each infective from “him”.

Stage 5 (should be thought of as simultaneous with Stage 6) - Determines the fate
of each individual who began the current time interval infected. Some will die, either as a
result of the modeled disease or from old age, and be replaced by newborn susceptibles (it
is assumed that the population size stays the same, so that deaths are immediately balanced
by births). Some will recover, either to a susceptible or (as in this caricature) to an
immune state. Others will simply continue on as infected/infectious to the next time
interval.

Stage 6 - Determines the fate of individuals who began this time interval as
susceptible or immune. Some susceptibles will contract infection, and change to that
state. Some will escape infection, continue in their present state, but age one time period.
And those susceptible or immune individuals who are at the end of their lifespan die, and
are replaced by newborn susceptibles.

Stage 7 - The population, now with new conditions (3 susceptible, 2 infected, and
1 immune), moves back to stage 4, looping through stages 4 through 7 for as many time
intervals as the programmer requests or until there are no more infectives in the
population.

13

© 69 9 Stage 1

 

© © © Stage 2

 

69 © © Stage3 I=2

 

e e e 8...... 63

 

© © @9 Recovery
Death ®© © © Stage 5

(Age 0)

 

®© ©®Infection 9
© © ©©Death Stage 6

(Age 0)

 

© © © Stage7 [=2

 

@ (Back to Stage 4)

Figure 1. Caricature of Model Sequences

14
1. Describe the biological system to be modeled. That is, ﬁx values

for essential constants such as host lifespan, disease duration, disease
transmittance and mortality. Two vital assumptions of the “world” of the
core program are:

a. Every host individual (except for those who die from the effects of
the modeled pathogen) lives exactly L time periods and then dies. This is
“Type I Mortality”, thought to be a reasonable approximation for human
populations in developed countries (1 ).

b. The population size remains constant at N individuals. This means
that upon death, an individual is replaced immediately by a newborn
susceptible.

Neither assumption is absolutely essential, and methods to vary from
them are presented below in the Appendices.

Host individuals are assumed to be of three distinct classes relative to
the modeled disease: Susceptible, infected/infectious, and immune. There
are therefore no “gray areas” such as partially immune individuals or

individuals who are infected but not infectious.

l 5
2. Create a population of N individuals of randomly assigned ages

between zero and L (host lifespan). Establish levels of inter-individual

contact relevant to disease transmission.

3. Fix beginning conditions of infection status, ordinarily including at

least one infective in the population, and begin the simulation.

4. For each susceptible individual, calculate his total contact with

infectives during the time period.

5. Determine the fate of each individual already infected at the
beginning of the current time period (determined by Monte Carlo procedure):

a. Disease-associated death may occur. In this case the individual
is immediately replaced by a newborn susceptible (assumption of constant N).

b. “Natural” death will occur to any individual at the end of
lifespan L. These individuals also are immediately replace by newborn
susceptibles.

0. Others survive. Those who have reached the end of disease

duration recover, either to an immune state or the susceptible state.

16
6. Fate of individuals not infected at onset of this time period.

a. If at the end of lifespan, they “die” and are replaced by newborn
susceptibles.
b. Some susceptibles (determined by Monte Carlo procedure)

contract disease. For each individual, the chance of “catching” disease
depends on level of contact with infectives during that time period. Those
who become infected will contribute next time period to the overall force of
infection within the population.

c. Those susceptibles who escape disease and are not at the end of
their lifespan, and all immunes younger than L, merely have their ages

increased by one.

7. Under the new conditions regarding which individuals are now
susceptible, infected, and immune, move to the next time interval and repeat

steps 4 through 6.

8. If infection dies out completely, or if time interval is the last one

desired, the program quits.

17

Transmission Parameters

As the reader examined the concepts embodied in the algorithm
caricatured above, it was perhaps obvious that the conceptual heart of the
simulation involves the modeling of the contact between susceptible and
infectious individuals. This, in fact, is the unique element of the
microparasite—host dynamic system. Thus a discussion is in order which
focuses on this matter, and in particular the relationship between the
parameters embodied within this model and similar parameters in the better
known compartmental models.

The Bible of disease transmission modeling at this time, expounding
the compartmental models with authority and completeness, is Anderson and
May’s Infectious Diseases of Humans (1). Paraphrasing somewhat the
differential equations which they present as their basic model, we ﬁnd the
expression for the acquisition of disease by susceptibles as:

dX/dt = -AX .

That is, the pool of susceptibles X is decremented (by acquisition of
infection) by the factor lambda, the “force of infection” at that time. Later we

learn that lambda can be

18

expressed in terms of the total number of infectives Y, as:
A=BY.
Thus,

dX/dt =~ -BXY,
and the critical factor B can be understood as the number of new cases of
disease occuring over dt, per susceptible, per infective. It can therefore also
be understood as the probability of disease transmission to a single
susceptible, per infective. Contact between the two is assumed by the
principle of mass action to be directly proportional to the numbers of both
groups. B is “a constant characterizing the infection, and it is not changed by
programmes of immunization or chemotherapy (although it can be altered by
changes in personal or public hygiene, such as greater cleanliness or
sanitation)” (Ref. 1, p. 63).

One of my incentives in developing the model presented here has been
my concern that the composite nature of B was not being properly recognized.
There would seem to be three components to the risk that susceptible A will
contract disease from infective B. The ﬁrst is a probability that they will meet
at all. This risk is averaged out in the compartmental model by the term XY,
assuming homogeneous mixing and the principle of mass action. The second

is the probability that a contact will be sufﬁcient to transmit disease. And the

19
third is the probability that, given sufﬁcient contact, a “case” of disease will

result. The latter two risks are both rolled together into the term [3, and this is
unfortunate given the public health differences between the two. The middle
component, that of “sufﬁciency”, can potentially be altered by Public Health
educational measures: Will the infective person wash her hands? Will the
susceptible individual put the pencil, just handled by an ill person, in his
mouth? Will a couple use a condom? The third component of risk (that an
infectious “case” will occur), however, must be assumed to be characteristic
of the microparasite and perhaps the genetically determined defense
mechanisms of the individual host; it is not alterable by Public Health
measures. If we are ever to employ our models to simulate the results of
Public Health policies, those components which can be affected must be
distinguished ﬁ'om those which cannot.

The remainder of this section could be omitted by the less
mathematically driven reader. However, the relationship deserves to be
explored which exists between the parameters of the proposed discrete model
and B and I. discussed above, and to the three components of disease
transmission discussed above. In the ﬁnal stage of calculating transmission,
within the subroutine “Catchit” (see Appendix A), the proposed program

calculates an individual’s probability of infection as the product of two

20

numbers: “Trans”, a constant characteristic of the microparasite, and
“contact”, that individual’s degree of contact with the infectives in the
population. These two factors correspond to the two factors incorporated
into B, discussed above. In fact, if we could establish the mean “contact” for
the time interval (call it contactm), then

B = Trans*contactm, and it = Trans“ contactm*Y.
In the case of the present model, the risk of any sort of contact with a given
infective, and the risk that this contact will be “sufﬁcient”, are both
incorporated into the calculation of “contact”, as both of these factors are
characteristic of the host (and both potentially alterable by Public Health

measures of quarantine or hygiene).

Population Structure

This section deals with a major conceptual issue in taking a population
perspective on disease transmission: How should we most appropriately
model the risk of transmission from any given infective individual to any
given susceptible? As discussed above, the assumption of homogeneous
mixing would rarely be valid for human societies. Humans live at varying
“social distances” from other humans, which would be likely to affect their

probability of contact sufﬁcient to transmit many infections passed by contact

21

with nasal-oral secretions or respiratory droplets. One way to express these
social distances (another will be discussed later) is to assign actual spatial
coordinates to each individual, and then to assume that the inter-individual
contact involving each pair would be inversely proportional to the geographic
distances between their coordinates. This has the advantage of being
portrayable graphically (See Figure 2), with each “family” being a cluster of
nearby individuals, and each “group” a row of such “families”. The core
program utilizes this method.

This method, however, is only one way in which this can be done. It
has a certain appeal, providing a graphic image of the population somewhat
like a little prairie dog village, in which situation the spatial position of
individuals within family units might really correspond to the risk of inter-
individual disease spread. Once the concept of “distance” is understood
through this two-dimensional spatial approach, the distances can be speciﬁed
more directly in ways which would be difﬁcult to portray graphically (2,5).
Alternative forms of the program, then, simply specify a constant “distance”
between individuals within “families”, between individuals of dissimilar
“families” but the same “group”, and between individuals of dissimilar
“groups”. The complexity of structure can be made as intricate as the user

desires.

22

Figure 2. Spatial Array of Individuals

This ﬁgure displays one method of establishing the distances between individuals,
which is to assign each individual a set of two-dimensional coordinates. In this case the
program has clustered a population of 200 individuals into 40 "families" of 5 each. The
tightness of clustering within "families", as well as the total area within which they are
placed, are under the user's control. The situation which results is a spatial array
something like a prairie-dog village, in which it is assumed that the contact of a susceptible
individual with any given infective depends upon their mutual distance. For instance, if it
happened that the only infective in the population was one of the individuals in the lower
left-hand family, one of the other individuals in that same family would be much more
likely to become infected in the next time period than would individuals at the extreme
upper right of the array.

23

 

12 I T l l I

10 r- I. 1 .h r

8 L '1- -._. —
>' 6 — J. I . .g .. d

.-l_ .

4 .. . _ -

2 - ‘ , —

O l l l l l

 

 

 

Figure 2. Spatial Array of Individuals

18

24

In either case, institution of effective Public Health hygienic measures
would be modeled as increases in these distances, to a degree thought
appropriate for intra-family, intra- group, and inter- group interactions under

the new conditions.

Illustrative Explorations

The proposed model is adaptable to the modeling of either acute
epidemic or chronic endemic infections. Two examples of each will be
presented here, to illustrate the use of the model to explore aspects of
parasite-host interactions and perhaps to excite the reader over the fact that

conceptually simple models can have highly interesting results.

Consequences of population size (acute system)

When a single infected individual enters a closed community which
includes individuals susceptible to that infection, an epidemic of the disease
may or may not result. Is the likelihood of epidemic spread affected by the
size of the community? In attempting to give a tentative answer to this
question, one is forced into modeling activity of at least an informal,
conceptual nature. Placing the question into the context of the present model,

one is immediately forced to face a ﬁmdamental question: How do the

25
“social distances” between individuals change when population size

increases?

The spatial caricature of these distances, though obviously only a
beginning toward adequately modeling complex interactions, provides good
conceptual grounds for approaching the issue of social distances. Let us
compare, for instance, two populations which contain, respectively, 100 and
400 individuals. If we place both populations within the same spatial area,
we are in effect saying that we believe that as populations grow, the
maximum distances separating them (with respect to the possibility of disease
transmission) do not increase. This might be valid for a vector-transmitted
infection which moves by water or highly motile insects, but probably is not
true for directly acquired infections. If, on the other hand, we maintain the
same spacing between individuals, thus making the spatial area containing
them proportional to the population size, we are implying that even in a group
of 100, each individual meets about as many others over one time interval as
he/ she can, and that added individuals result in greater maximum inter-
individual distances than was true within the smaller population.

Neither assumption seems likely to be completely true in other than
extreme conditions. The latter is more interesting, however, for it isolates the

factor of numbers alone from the factor of population density. For the

26

purposes of the comparison illustrated here, then, inter-individual spacing was
held constant and thus the total area within which the population was
contained was allowed to increase proportionate to the population size.
When an acute, inﬂuenza-like illness is to be modeled, we think of the
time steps within the program as one day, and set the host lifespan, in days,
accordingly. Thus no host individual will die of “other causes”, and all
mortality will be disease-associated. The comparison depicted is the chance
of epidemic spread within groups of 50, 100, 200, and 400, in which the host
lifespan is 25550 “days” (70 years x 365 days), the disease duration is 4
days, and recovery is to an immune state. Each trial begins with a single
infective at the periphery of the population. 100 trials were run for each
population size, for four different levels of disease transmittance, and the
number of those trials which resulted in epidemic spread determined
(Deﬁnition: at least 20 cases or 20 time intervals involved. In practice there
is no difﬁculty determining epidemic runs; the results are always obvious
epidemics or non-epidemics). The results are depicted in Figure 3. Though
this ﬁgure shows the number of epidemics without associated conﬁdence
intervals, the trend is very obvious: For a range of parasite transmittances,

the likelihood of epidemic spread increases with population size.

27

Figure 3. Effect of Population Size on Epidemic Spread

Plotted is the number of computer runs resulting in epidemic spread of disease, out
of 100 trials, for four population sizes (N = 50, 100, 200, and 400) and for increasing
levels of disease transmittance. Mutual individual distances were established as a spatial
array, as in Figure 1. Distances between adjacent individuals and between families were
held constant as population size was altered; thus, the total " social area" containing the
population increased as population size increased. A general trend favoring epidemic
spread with increasing population size is evident.

28

 

 

100 r l r I
ooo
o”
1"
’z
I
80 " 1’
’z
I” cg
"
[I r ﬂﬂﬂﬂ
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E 1’ I
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:8 /
I
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I” aaa
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0 5---». "i’ r

 

 

D EPID400
0 EP|D200
A EPID100
O EPIDEO

0.006 0007 0.008 0.009 0.010 0.011 0.012

Transmlttance

Figure 3. Effect of Population Size on Epidemic Spread

29
Is this result what would intuitively be expected? Again the exercise of

doing formal modeling forces us to face a question very clearly. Since social
spacing between individuals is identical in our model whether populations are
large or small, why should in infection spread more certainly in a larger
group? If spread through the population were largely maintained by
transmission to closely-related individuals, an analogy with holding a match
to a piece of paper would seem to hold. The ﬂame either does or does not
catch on the corner to which the ﬂame is held; it is of no consequence
whether the remainder of the page is large or small.

The proposed model allows us to look more closely at the dynamics of
disease spread, as we can print out a grid depicting the infectious state of all
individuals at each time interval. When this is done for runs which do result
in epidemics, the mode of spread is typically like that shown in Figure 4: One
or more “distant” individuals happen to become infected, and the infection
spreads from more than one front through the whole population. Thus the
analogy to a ﬂame is not valid, and this model would suggest that epidemic
spread might often substantially depend upon early stochastic transmission to
socially distant individuals, who then bring the infection into new

“neighborhoods”.

30

Figure 4. Typical Epidemic

Pictured are "snapshots" from the early stages of a typical epidemic,
showing the common indirect pattern of spread. The population of 100 individuals
are represented here as 0's for susceptible hosts, 1’s for infected/infective hosts,
and 9's for immune hosts. Families are arranged in two-dimensional coordinates
similar to Figure 2, but are represented in this display by segments of 5 on each
line. Thus, there are four families on each line. The three ﬁgures preceding each
matrix of individuals are the number of time increments passed, number of
infectives in the population, and the proportion infected. The original infective is
seen in time 0 to be at the periphery of the population. By time 3, a new case has
developed in the family of the original infective, but also in another distant family.
By time 5 the original infective is immune, but individuals from several other
families are infected. By time 8 infection is widespread , and the population is
certain to experience a general epidemic. Observation of this mechanism of
epidemic spread, in which multiple fronts develop early, helps to clarify the
association seen in Figure 3 between population size and the likelihood of epidemic
spread.

31

01.01

0
0
0
0
0

000000000000000000
0
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O

33.03

10000
00000
00000
00000
00000
00000
00000
00000
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00000
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56.06

10000
00000
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00001
00000
00000
00000
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00000
00000
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00000
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90000
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00009
00000
00100
10100
101101
00000
00000
10000
90000
00001
01101
00000
00000
00000
90000
10100
90000

Figure 4. Typical Epidemic

32
The concept of a threshold population necessary to maintain an

endemic disease has been known for some time, based on theoretical (7) and
to some extent empirical (8, 9) observations. But the extension of this
concept to the possibility of acute epidemic spread, and the demonstration of

the mechanism behind this effect, is, to the author’s knowledge, new.

Consequences of Population Structure (acute system)

As mentioned above, the assumption of homogeneous mixing is not a
realistic one for human populations, nor for those of many other host
organisms as well. We do not ﬁnd individuals within human societies mixing
randomly like molecules boiling in a pot, or like ants milling within a swarm.
Humans are gathered into family units for many of the most intimate activities
of daily life, and for many other activities there is structure of more varied
and complex types. The proposed model is capable of generating a random
disposition of individuals; a typical result is shown in Figure 5. But in the
form reprinted in Appendix A, the program models the most elementary step
in structuring of populations, the clustering of individuals within families of
uniform size. The tightness of clustering within families, always within the
same overall area, is controlled by manipulation of factors within the

program; moderate and tight clustering are depicted in Figures 2 and 6.

33

Figure 5. Random spacing

Individuals can be randomly placed within the same area as that utilized for the
clustered population in Figure 2. Although there is no structure in this population, it is not
the same as homogenous mixing, in which all individuals are assumed to contact every
other individual equally, as if they were percoloating through a pot or were elements of a
seething anthill. In the spatial array pictured here, obviously individuals are much closer
to some neighbors than they are to others.

34

 

12 I

10-

 

 

 

Figure 5. Random Spacing

15

18

35

Figure 6. Strongly structured population

In contrast to the randomly placed population pictured in Figure 5, and the
moderately structured one of Figure 2, this population has its individuals very tightly
clustered into groups. In interpreting this difference, recall that the inter-individual
distances are to be thought of as the reciprocal of the likelihood of contact between two
individuals of such type that would be sufﬁcient to pass an infection of the type modelled.
Thus if the relevant activity for this disease were sharing objects with one another, the
population of this ﬁgure would be thought of as engaging in such sharing very intensely
with a small number of others, but being more reluctant to do so outside the group than is
true of the populations of Figures 2 and 5. The fact that the overall area is identical in all
three cases makes the maximum distances similar for all. Thus the overall "object sharing
in the three populations is similar, and the degree of sharing between the most distant of
individuals is similar, but the ﬁerceness of loyalty to small groups differs markedly. The
text poses the question of how such differences might affect the likelihood of epidemic
spread of a pathogen.

36

 

 

 

 

Figure 6. Strongly structured population

12 I I r r
10— r r e s '. d
8- I a I I 2 _
>6- g r .. n r 4
4r 4 I. .. . -. ..
2L 0 r. c r - 4

0 I l r 1
0 8 6 12 15 18

 

37
Recalling that the distances between individuals in this model represent

the possibility of contact sufﬁcient for disease transmission, the aggregation
of a few individuals into tight clusters models increasing intimacy within the
family regarding such contact as might transmit the infection of interest.

When the overall area is maintained as constant, such clustering is inevitably

associated by an increase in the distances separating individuals from unhke
families.

Will the development of such structure, all else being equal, facilitate
the spread of an epidemic over the population, or inhibit this spread? On one
hand, since the individuals of each family are (relative to the homogeneous or
random condition) isolated from others, it might be more likely than infection
would be conﬁned within a family. On the other hand, if whole families are
more certain to become quickly infected, the overall force of infection would
be increased. Might there be an optimum degree of clustering for epidemic
control?

The results of my own exploration of this question provide some
insight, not only into the resolution of the question itself, but also into the way
the model’s results should be interpreted. Using values of host and parasite
characteristics identical to the population size study described above, and

populations of size 200, 100 trials were run at each of 8 levels of “tightness”

38

of clustering, from that illustrated in Fig. 6 to near-random spacing. The
result, shown in Figure 7, would seem suggestive of a minimum near “clus” =
1.6, a structure not visually very different from random spacing.

However, considering the stochastic nature of the program, the results
should be considered like observational data from any natural population.

The proportion of runs which result in epidemics is a proportion subject to the
same variance expected from any other proportion. With N=100 in each
case, the standard errors for each proportion are too large to come to any
conclusion regarding an optimum from the presented data. Realizing this,
200 more trials were run at the critical clustering levels, resulting in the data
of Figure 8. Now the conclusion would have to be that, given the
assumptions and constants of this particular exploration, the likelihood of
epidemic spread is a monotonically increasing function of the tightness of
family clustering.

The consequences of population structure to the threshold level of
immunization necessary to eliminate a disease (10, 11), and to the possibility
of epidemic spread (2), have been explored previously. However, the present
model offers complete ﬂexibility in the speciﬁcs of the structure to be
explored, and a mode of deﬁnition of this structure which is likely to be

reasonably clear to the average potential user.

 

39

Figure 7. Population Structure and Epidemic Spread

The number of runs out of 100 trials resulting in epidemic spread within a
population of 200, as a function of degree of structure. Higher values of "clus” indicate
looser clustering of families (Figure 1 was produced with "clus" = 1.0, Figure 6 with
"clus" = 0.2). There appears to be a possible minimum at moderate to loose levels of
clustering. However, the stochastic nature of the model requires the user to consider the
standard error for each proportion, which would be quite large when each data point
comes from a sample of only 100 trials.

% Epldemlcs

4o

 

100 . . I
90 - r

80 - \

7o — \

60 — \

50 — ‘--*\

 

 

 

0.0 0.6 1 .2 1 .8

Clusterlng factor

Figure 7. Population Structure and Epidemic Spread

2.4

41

Figure 8. Population Structure and Epidemic Spread [1

Now 200 more trials have been completed at critical levels of structure, and the
results are reported.with standard errors. Now it would appear that the likelihood of
epidemic spread is a monotonically decreasing function of population structure. When
small, intensely inter-acting groups form, it becomes nearly certain that if one member
becomes infected, the whole group does so also. This increases the force of infection
within the whole population more than enough to compensate for the fact that the contact
between members of unlike groups is decreased.

% Epldemlcs

42

 

100 I I I

80 " \\

\
l
\

60 — \

.0- H,
4° ' 11“} “PM.

30— l

20 l l l

 

 

 

0.0 0.6 1 .2 1 .8

Clusterlng factor

Figure 8. Population Structure and Epidemic Spread H

43

Chronic endemic disease - Endogenous rhythms in a simple system

In the two explorations discussed above, the infection modeled was an
acute one which could not possibly be maintained in populations as small as
those stipulated here. Each computer run ﬁnishes quickly, as the number of
infectives drops to zero either without resulting in epidemic spread, or rapidly
travels throughout the population and then dies out from lack of available
susceptibles (though rarely having affected all individuals). When the
duration of illness is much longer, however, relative to the lifetime of host
individuals, we have a chronic disease which may persist for almost an
indeﬁnite number of time intervals. One would probably intuitively conclude
that the level of infection within the population (the proportion actively
infected at any one time) would either (1) fall to zero and die out, (2) rise
eventually to uniform infection, or (3) stabilize at some intermediate level
with minimal ﬂuctuation, in the absence of perturbation. In fact, both in
nature and in the results of this model, it may be that none of these is the
case!

My introductory comments made the point that modeling activities
frequently take place in a situation in which we have prior information both
about outcomes of the system, and about the separate components of the

system. The modeling effort, I stated, then becomes one of supplying

44

assumed interactions of these components, simulating the process with the
goal of comparing the model results with the reality of known data. With
regard to chronic endemic disease, one aspect of the incidence of many is that
they have a cyclic nature. The annual epidemics of inﬂuenza are well-known,
for instance, and cycles of varying length were seen in pre-immunization data
as well for measles, rubella, and mumps. The cause of this rhythmicity is not
well understood, and it is not duplicated by compartmental models, which
tend to show dampened cycles following the epidemic phase of a new disease
(1). Often a search for an explanation for the rhythmicity has led observers to
postulate some form of external “pumping” of the system (like regular re-
introduction of the infection through migration, or the annual coming together
of schoolchildren at the beginning of the academic year) rather than a
conﬁdence that the source could be found within the dynamics of the system
itself (Reviewed in Ref. 1). Thus the cyclic nature of disease incidence data
is an aspect of the reality of the system which has not heretofore been well
matched by available models.

For this exploration I simulated an infection using the modiﬁed version
of the program which directly stipulates inter-individual distances, and with
constants of N = 100, L (lifespan) = 20, and D (duration of disease) = 4.

Mortality was stipulated to be 20% per year, and an immune state was

 

45

assumed to exist. The “Seed” subroutine introduced a single infective as in
the acute simulations described above. The result, in the form of a
prevalence/time plot, is as shown in Figure 9. After an initial epidemic and
subsequent trough, the system settles into a varied pattern which is quite
“noisy”, yet seems suggestive of holding an underlying rhythmicity.
Autocorrelation analysis (Figure 10) strongly suggests that a cycle is indeed
present, at a frequency of 23 or 24 time intervals. The simulation was carried
out to 5000 intervals, amounting to 250 lifetimes of the host and 1250
lifetimes of the disease duration. Figure 11 and 12 compare intervals 4901-
5000 with intervals 101-200. They are qualitatively similar, and
autocorrelation analysis of the last portions show the same cycle
characteristics as do the ﬁrst portions. The variance of the prevalence
ﬁgures, displayed in Figure 13, shows no tendency to decrease with the
passage of time. In other words, there appears no be a cyclicity inherent
within this relatively simple system of only 100 individuals which requires no
external pumping and does not decay with time.

In his book At Home in the Universe (12), Stuart Kauffman discusses
what he calls “order for free”. This is the tendency of dynamic networks,
comprised of interconnected elements, to exhibit spontaneous order under the

proper conditions. The N=100 network of individual hosts created by the

46

Figure 9. Prevalence of Modelled Infection

Shows prevalence vs. time for an infection whose parameters are discussed in the
text, in a population of N = 100. Shown are the ﬁrst 100 time intervals. As a continuous
model would predict, the initial epidemic spread through the population is followed later
by a dampened secondary rise after susceptibles are replenished by birth. Though a
"noisy” curve, there is a suggestion of periodic behavior, with a possible cycle length of 24
or 25 time intervals.

47

 

 

 

 

 

 

 

 

 

80 I I I I
70 - l -
60 " "
so _. \I -
iii
‘2 4° " d
O l
30 — "
20 —
10 " ‘
O I I I r l
0 20 40 60 80 100

TIME

Figure 9. Prevalence of Modelled Infection

48

Figure 10. Autocorrelation of Prevalence - Time Data

From the same system depicted in Figure 9, this ﬁgure depicts the autocorrelation
of the time series of prevalence data. Pictured is the correlation between data points at
increasing lag intervals ﬁ'om each other. The resulting curve is strongly conﬁrmatory of
the presence of periodicity within the series, with a cycle interval of 24 time intervals.

Autocorrelatlon Coefflclent

49

 

1.0 I T I I

l
r.
0.5 — I .,' ‘ ~ _.
.5 '-
'. .' '- 4’ '5
'-. . 3 "-,
'. :' '1 . ‘.
". .' ' a' 'I
0.0 — '1‘ ‘ 1. .. ' —I
' ' f: 1r
‘3 j v'.
a a ‘ l,
.' 2";
-O.5 r _ l." ‘
”I. an.

 

l

 

 

-1'0 1 l l 1
0 1O 20 30 40

Lag Interval

Figure 10. Autocorrelation of Prevalence - Time Data

50

60

50

Figure 11. Prevalence of Modelled Infection 11

Depicted are intervals 100 - 200 of the same model run as that of Figure 9. By
now the initial high variance of the prevalence has been dampened, and the system has
settled down to an endemic state. Yet there remains a suggestion of persistent cycles at
the same period length as earlier.

51

 

80 l I I I

70-

60-

50-

4o...

CASES

 

30-

20-

10-

 

o I I l l

 

 

 

100 120 140 160 180
TIME

Figure 11. Prevalence of Modelled Infection 11

200

52

Figure 12. Prevalence of Modelled Infection 1]]

Shown here are intervals 4900 - 5000 from the same run as Figures 9 and 11. By
this time the system has completed time equal to 250 host lifetimes, or 1250 times the
duration of the infection. The qualitative similarity to the pattern of Figure 11 is obvious,
with cycles apparently persisting in the absence of any perturbation.

53

80 I I I I

 

70-

60-

50-

40...

CASES

30 -
20

10r-

 

O I l I I

 

 

4900 4920 4940 4960 4980
TIME ‘

Figure 12. Prevalence of Modelled Infection III

5000

54

Figure 13. Standard Deviations of Prevalence over Time

This ﬁgure plots the standard deviations of 200-time-interval segments, showing
no tendency to decrease with time after the initial dampening of early high variance. This
conﬁrms the impression gained from the autocorrelation analyses and visual inspection of
comparisons like Figure 11 vs. Figure 12: The system exhibits persistent cycles which do
not decay with time, and require no external perturbations or other "pumping" forces to
continue the cycles.

Standard Devlatlon

12

10

55

 

 

I I

 

 

15 20

Interval

Figure 13. Standard Deviations of Prevalence over Time

25

56
proposed model, depending on the perspective with which one looks at it, can

seem vastly complex or quite simple. From the standpoint of the host, since
there are three possible states for each individual (susceptible, infected, and
immune), the total number of possible states of the system is 3 ‘00, or about 5
x 1047. This is certainly a hyperastronomic number. Yet from the standpoint
of the parasite perhaps only the total number of infected hosts matters, and
there are only 101 possible states of the system relative to this quantity. In
spite of the great complexity-yet-simplicity of the system, the result is a
regular “pulse” of infection whose intensity does not decay and whose period
does not lose its organization. This is indeed “order for free”.

It is important to recall that the periodicity exhibited by this model is
not that of a natural organism, whose characteristics have passed the ﬁlter of
natural selection. It is a fact that the incidence of many natural infections
(and other predator-prey relationships) show cyclicity, but we do not know
whether this fact is “good” or “bad” from the viewpoint of the parasite or the
host. It may be a character available to natural selection, or it may be purely
an epiphenomenon of other characteristics.

What the present results do imply is that there may be a cyclicity which
arises naturally in host-parasite relationships of the type modeled here. The

period and intensity of the cycles are dependent in part on characteristics of

57
the parasite’s eﬂ’ect on the host - transmissibility, mortality and morbidity,

duration, and the stimulation of immunity. These characteristics are probably
inter-related in terms of constraints on possible parasite genotypes. For
instance, a mutant strain possessed of increased transmissibility might
inevitably also exhibit increased mortality, and thus result in a new cycle
period, should it become dominant. Thus cycle attributes would be a
character potentially visible to natural selection, whose action could be
primarily on this character rather than the infection characteristics which
produce it.

Why might nature act directly on the cycle characteristics of an
infection? One possibility is to tune it to match a cycle present within the
host population, or to some multiple of such a cycle. Thus we could imagine
that if the host-parasite dynamics of measles now produce a natural two-year
cycle, this could be the result of selection’s “tuning” the cycle to a multiple of
the annual tendency of humans to aggregate more closely during the cold
months of the year.

What is the mechanism by which the stochastic model produces
persistent cycles? As stated, the tendency of a host-pathogen system to
exhibit regular oscillations subsequent to a major perturbation has been

known for some time, and is a prediction which follows from the

58
deterministic SIR compartmental model (for discussion see Ref. 1, Chapter 6)

as long as a regular birth rate (which replenishes the stock of susceptibles) is
assumed. However, this model predicts steadily damped cycles in the
absence of further perturbation, as each wave of disease incidence becomes
less intense than the one which preceded it. Eventually an equilibrium is
reached, in which the proportion of susceptibles is approximately equal to the
reciprocal of R0, the basic reproductive rate of the pathogen within this
system. Perturbations like immigration of infectives or institution of
immunization can result in the initiation of a new set of cycles. The present
model, however, seems to show cycles, in the absence of any external
perturbation, which do not decay . Why should this be so?

In essence, it is because every step of a discrete, stochastic model
results in a perturbation. To see this, note that if a continuous, deterministic
model were to indicate, at a speciﬁc time, that 6.3 of every hundred
susceptibles should contract infection within the current time period, this
cannot be done precisely in a discrete population of 100. Furthermore, since
the model is stochastic, the number of new infectives it produces will not
necessarily even be 6 or 7, but could be substantially higher or lower. As the
number is, at any rate, not that which is “expected” (in a statistical sense),

this is a perturbation. Ifwe compare this cyclic behavior to that of a

59
pendulum, the pendulum in the stochastic model is constantly being bumped

in one direction or another. It never achieves an equilibrium. Though its
swing is somewhat irregular due to all of the stochastic buffeting it receives, it
swings with an intrinsic period which can be approximated (l) by the formula
T = 21r[AD]'/’, in which A is the average age of contacting disease, and D is
disease duration.

This eﬁ’ect is most prominently seen in smaller populations, as
stochastic effects will tend to average out in larger populations. Even the
difference between population size of 100 vs. 200 is signiﬁcant, as shown in
Figures 14a and 14b. Therefore one might be skeptical of this effect (which
Andrson and May call demographic stochasticity) as causative of the cycles
which are seen in data from very large populations . Yet it may be that
hmnan (or other) populations ﬁmction largely as aggregates of small,
semi-independent groups, which might be small and independent enough to
maintain cycles on the basis of demographic stochasticity .

Ifthat assumption is accepted, then the issue is a different one.
Presumably transmittance would vary among all the small groups. This
would affect A, the average age of disease, which would be lower in the high
transmittance groups. Thus each group would cycle with its own period, and

the resulting “noise” would not exhibit large scale cyclicity. Perhaps,

60

Figure 14a. Cyclic Behavior when N = 100

Depicted is the prevalence - time series of an infection in which mortality is zero,
with other parameters the same as those used previously. Cyclic behavior of the model is
even more obvious when the additional stochasticity of disease-associated mortality is not
signiﬁcant. Average age of contracting disease in this system is 2.49. Estimated period
length can be calculated as T = 21t(AD)‘/’ , with A = the average age of contracting
disease, and D = duration of disease. In this case T = 19.8, very compatible with that
observed in the "empiric data" of the model output.

6l

 

80 I I I I
70 " ‘

60- "

I " i.

30-

 

CASES

 

 

 

 

 

 

 

 

20‘

10I -

0 1 I I I
100 120 140 160 180 200

TIME

 

 

 

 

 

Figure 14a. Cyclic Behavior when N = 100

62

Figure 14b. Cyclic Behavior when N = 200

Here everything is identical to Figure 143 except that population size is increased
to 200. Cycles of the same period persist, but are a less prominent feature of the system
than is true in the smaller population. As the text describes, the cause of the cycles in the
stochastic model is the fact that each time interval actually creates a perturbation, as the
number of new infectives is never exactly that which a continuous model would have
predicted. This is less true as population size increases and the stochastic nature of the
model becomes smoothed by the larger size of the population. Stochastic outputs from a
larger population are closer to that which a continuous model would have predicted.

63

 

80 I I I

70— {I

60

 

50-

 

40..

CASES

30_

20-

10-

 

o I I I

 

 

 

100 120 140 160
TIME

Figure 14b. Cyclic Behavior when N = 200

180

200

64

Endemic Disease - Alternative Public Health Policies

In dealing with the effects of possible Public Health policies upon
endemic disease, the compartmental model is generally used to predict effects
on the equilibrium situation. The discussion above illustrates that in dealing
with a stochastic discrete model of disease within a small population, there is
no equilibrium. Therefore, rather than dealing with equilibrium values of
incidence and prevalence before and after institution of the measures, we
must run the stochastic model over a substantial period of time under both
sets of conditions, and determine mean values for these parameters.

The exploration of this section deals with a system involving the
following constants as a baseline: Population size = 100, host lifetime = 20,
duration of infectiousness = 4, mortality = zero, and all inter-individual
distances = 4 units (homogeneous mixing is assumed). Recovery is to an
immune state. Under these baseline conditions the population experiences an
average of 5.02 cases per time period, a mean prevalence of infection equal to
.258, and an average age of contacting disease equal to 2.26 time intervals.
The average proportion of susceptibles is .106. Estimating R0, the basic
reproductive rate of the infection, as the reciprocal of the proportion
susceptible (1), this parameter would be approximately 9.5. This number is

to be thought of as the number of new infectives produced over time D (the

65

duration of infectiousness) by one infective when all of “his” contacts are
with uninfected susceptibles.

Now we might imagine three possible Public Health approaches
designed to reduce the impact of this disease on the population. One
approach would be some sort of hygienic measure which would increase the
effective distances between individuals. A second would be immunization,
which reduces the production rate of new susceptibles by converting them
directly to immune status. And a third would be a program either of antibiotic
therapy or of quarantine, which would reduce the period of time over which
an individual case was infectious to his neighbors. The results of each are not
qualitatively different from that which would be predicted by a deterministic
analysis of a compartmental model, but there is satisfaction to be gained by
seeing the predictions realized in the more realistic simulations done through
the present model.

The hygienic approach was modeled by increasing inter-individual
distances from 4 to 15 units. This results in some striking changes, in that the
average age of contracting disease increases to 6.67 , and the proportion of
susceptibles increases to .444 (estimated R0 = 2.3). Yet incidence only

decreases to 4.41 cases/time period, and prevalence to .212.

66

For the immunization model inter-individual distances are returned to 4
units. Now the assumption is that 40% of newborns are immediately
immunized, or that this is done as soon as maternal antibodies decline. Thus
the program declares 40% of the newborns as immune, and these remain
immune throughout their lifespan. Now the age of disease is 3.50, with the
proportion susceptible averaging .122. Since R0 is conceptualized to be the
number of cases produced by an infective surrounded by a sea of
susceptibles, this really should not change appreciably from baseline values.
Indeed, it is estimated here as 8.2. But now incidence is decreased to 3.01
cases/time period, and prevalence to .144.

We model the institution of antibiotic therapy or quarantine as a
reduction in disease duration from 4 to 2 time periods. All other constants are
those of the baseline. This single alteration results in a reduction of average
prevalence to .150 but an incidence only minimally reduced to 4.50 cases/
time period. Age of disease is now 3.72, and a susceptible proportion of .198
yields an R0 of 5.1. In order to compare the basic reproductive rate in
absolute time, however, when D varies we must compare Ro/D. The basic
reproductive rate per time period is not appreciably different from baseline.

For a tabular summary of all of these effects, see Table l.

67

 

 

Table 1
Model Prevalence Incidence Age at Infect. Est. R9 Est. R._,/_D
Baseline .258 5.02 2.26 9.5 2.4
Hygiene .212 4.41 6.67 2.3 0.6
Immunization .144 3.01 3.50 8.2 2.1
Quarantine, Rx .150 4.50 3.72 5.1 2.5

Each of these approaches can be seen to have salutary effects for the
population, but each does so by different mechanisms which produce diverse
results. A look at the results of the hygienic approach recalls speculations
which have been made regarding the effects of increasing hygiene on
poliomyelitis in pre-immunization days . Transmission-relevant inter-
individual distances are increased, and thus the pathogen’s ability to spread
quickly through the population (its R0) declines. Thus the age at infection
increases and the proportion immune (who are recovered cases from earlier
times) declines. Yet unless these measures can completely eliminate the
disease, the number of cases per time interval and the prevalence of infection
will be affected only minimally. The infection takes longer to get to each
host, but gets there eventually. Thus the value of this approach may depend
largely on age-dependent effects of the disease. As may have been the case
with polio, it is even possible for such a program to be deleterious from a

population perspective, if disease is more serious in older individuals.

68

Immunization works by reducing the rate of production of the
infection’s limiting resource, new susceptible hosts. To a micro-organism the
production rate of new susceptible hosts is as important as rabbits to wolves
or nitrogen for grasses. It is the resource which determines the supportable
“biomass” of infection, measured as the prevalence of infected individuals
within the population. The theoretical ability to spread (R0) remains high, but
the production rate of susceptibles can no longer sustain the same number of
infectives. The overall promrtion of susceptible individuals in the population
is not actually appreciably different after immunization (1), as the proportion
of infectives has decreased. Since the force of infection is low, the incidence
of new infections is reduced as well. Most of those who are immune became
so as a result of immunization rather than of recovery from disease.

Either quarantine or antibiotic therapy can reduce the ability of each
case to infect others, by reducing the period of infectiousness. Therefore any
given individual will, on the average, be older before contacting the infection.
Each case lasts a smaller proportion of the lifespan, and the overall
prevalence of infectious individuals thus is reduced. Yet the birth rate of new
susceptibles is unchanged, and therefore the incidence of new cases may not
be appreciably affected. Thus the value of such a program may depend upon

what is desired. If overall prevalence is to be targeted, this approach may be

69

more useful than if it is the incidence of new cases which is to be focused
upon.

Again, many of the conclusions related to these three potential Public
Health programs have been stated before, based on theory or on data. The
proposed model has great heuristic value in this area, however, due to its
realistic nature and its ability to produce epidemiologic “ ta” which is

usefully predictive.

Discussion

The foregoing examples suggest a rich variety of theoretical problems
which can be approached using this stochastic, discrete model. Yet those
who experience this model of disease-host interactions may be disappointed
in its apparent lack of “real-world” applicability. There are no baseline
values of “contact” now established for any real population, nor values of
“transmittance” for any real disease, in units which could be plugged in to this
model. If one wished to explore the consequences of some public health
action to the risk of epidemic spread of polio, for instance, the only recourse
would be to toy with the constants in the model until one obtained results
similar to those of a known epidemic in a population of similar size. And

then, if one wished to model the likely effect of regular handwashing, there

70

exists no data upon which to judge to what degree this action would increase
“distance” and decrease “contact”, in the terms of this model. It would seem,
then, that this model might be more useﬁrl in the exploration of general
principles of disease-host interaction than in making real decisions regarding
speciﬁc infectious diseases.

This criticism is valid, but perhaps inevitable given the current state of
our understanding related to population structure as it relates to disease
transmission, and our level of quantization of pathogen transmittance.
Perhaps an analogy is in order: At one time the meaning of temperature was
unclear, and the effect of temperature as a measurable character of objects
was not conceptually separate from other aspects affecting the human
perception of heat, such as air movement, or conduction of heat by objects
which were touched. Yet people did conceptual modeling, at least in
primitive ways, based on temperature as they understood it. They could
expect water, or animals, or materials to behave differently depending upon
their temperatures. Now, however, we have a speciﬁc deﬁnition of
temperature, as the energy of the random motion of molecules. We have
ways of measuring temperature, usually not directly but by measuring an
effect of temperature; thus, we observe and measure the progressive

expansion of a colunm of mercury as it is heated.

71
Before temperature could be deﬁned it had to be conceptually

separated from the other factors which affected human perception of heat and
cold. Similarly, if we ever are to quantify the factors affecting disease
transmission, we must deﬁne the components of the system in a useful way.
As discussed above, the present model offers a separation of those factors
characteristic of the host (likelihood of contact, plus likelihood that the
contact will be sufﬁcient, both incorporated into the program parameter
“contact”) from those characteristic of the pathogen (“Trans”, the likelihood
of acquiring disease given sufﬁcient contact). This appears potentially more

productive of further insight than does the BXY term of the classic SIR

model, in which the critical term B includes both host and pathogen
charactensucs.

Thinking in the terms of the present model could facilitate the future
development of units of measurement of population distances and of
pathogen transmittance. As is the case with temperature, the units might be
based on an indirect measure rather than an observation of the central
deﬁning quality itself. Thus we might conceive of something like a “measles
unit” of disease transmission, measuring the ability of an organism to infect
household contacts compared to a that of the measles virus. In the area of

population contact, perhaps a “Peoria unit” of inter-individual contact. This

72
could be based on the attack rate within a larger group compared to that of

household contacts. In the meantime, models such as that which this paper
proposes will promote a healthy clarity of thinking in discussions of infectious
disease, a portion of epidemiology which now seems likely to remain

prominent.

APPENDICES

 

APPENDIX A

73

Appendix A - Program Listing with explanation

In the display of the program which follows, actual components of the
program are in bold print, whereas my comments which follow each section
are normally printed.

DECLARE SUB Mix (infectl, coordI, contactI, N)
DECLARE SUB Seed (L, Mort, id!( ), coord!( ), infect!( ), duration!( ),

DgEgIBARE SUB Rebirth (a, agel, infectl, durationl, contactl, coord!)
DECLARE SUB Catchit (k, Trans, contactl, infectl, agel, D, durationl,
gOQMON SHARED year, N, F, G, L, D, Mort, Trans, id!( ), coord!( ),
infect!( ), duration!( ), age!( ), contact!( ), k

QBasic requires that if the program includes any subroutines, these
must be declared initially along with a list of parameters which must be
passed from the main program to or from the subroutine. The main program
is printed below followed by the subroutines in the order in which they are
called. In addition to declaring the four subroutines, the above section lists
those parameters which are common to all components of the program.
CLS

This command clears the output screen in preparation for receiving
output from the program. If the program is directing its output to some ﬁle

which existed previously, this command would have the effect of erasing this

pre-existing ﬁle.

 

74

year=0
N=200
F=5

G=5
L=20
D=4
Mort=.2
Trans=.013

The preceding is a list of constants which can all be changed by the user. N
is the population size, L the lifetime of the host, D the duration of the illness
(both in arbitrary units of time). If no recovery is to be allowed, D can be
made to equal L, and individuals will always die before recovering. Mort is
the probability of disease-associated death per time interval. Trans, which
will be discussed ﬁuther later, is a constant stipulating the transmissibility of
the microparasite. F and G are factors which are used later to divide up the
population into social groups (N needs to be evenly divisible by G‘F).
OPTION BASE 1

DIM id!(N)

DIM coord!(N, 2)

DIM infect!(N)

DIM age!(N)

DIM contact!(N)

DIM duration!(N)

The DIMension command reserves N places in the computer’s memory

for each of several variables. These variables will contain all that the

 

75
program needs to know about each individual in order to run the simulation.

The BASE 1 command says that the ﬁrst observation of each variable is
number 1, not number 0.
RANDOMIZE TIMER

At several points within its algorithm, the program must obtain a
random number. In order to make this choice always truly random, the
computer is asked to base its number choice on the exact time its internal
clock shows at the moment of each choice.

The following section deals with a major conceptual issue in taking a
population perspective on disease transmission: How should we most
appropriately model the risk of transmission from any given infective
individual to any given susceptible? As discussed above, the assumption of
homogeneous mixing would rarely be valid for human societies. Humans live
at varying “social distances” from other humans, which would be likely to
affect their probability of contact sufﬁcient to transmit many infections passed
by contact with nasal-oral secretions or respiratory droplets. One way to
express these social distances (another will be discussed later) is to assign
actual spatial coordinates to each individual, and then to assume that the
inter-individual contact involving each pair would be inversely proportional to

the geographic distances between their coordinates. This has the advantage

76
of being portrayable graphically (See Figure 2), with each “family” being a

cluster of nearby individuals, and each “group” a row of such “families”.

[The above corresponds to section 1 from the overview above]

The following section, then, sets up the initial conditions for each
individual, creating a population of N individuals grouped into F families and
G groups, assigning them two-dimensional social "positions" based on the
size of the F (family) and G (group) axes. It assigns everyone an age, and
everyone begins with a lifespan equal to L.

FORi=2TO(2 *G) STEP2
FORj=2TO(2 * F) STEP2
FORy= l TO(N/(G * F))
id = id + 1
id!(id) = id
coord!(id!, l) = i + (RND - .5)
coord!(id!, 2) = j + (RND - .5)
age!(id!) = INT(RND * L)
duration!(id!) = 0
contact!(id!) = 0
infect(id!) = 0
NEXTy
NEXTj
NEXT i

[This concludes Section 2]

77
Now the original group is "seeded" with at least one infective, as the

main program calls the Seed subroutine. The reader may review that module
now (it is printed below) or later.
CALL Seed(L, Mort, id!( ), coord!( ), infect!( ), duration!( ), age!( ))
Count = 0
FOR 111 = 1 TO N

*[Count = Count + infect!(m)]
NEXT m
Proportion = Count / N
PRINT year; Count; Proportion
‘FOR row = 1 TO (N l 20)
‘ FOR element = 1 TO 20
‘ PRINT infect!(20 * (row - l) + element);
‘ NEXT element
‘PRINT
‘NEXT row

The initial conditions are now in place, and the simulation is poised to

begin. The section above is the ﬁrst to direct the production of any kind of
output. Those details of the initial conditions chosen by the user will be
directed to the screen or wherever else the program is eventually directed to
print. The program assumes that the user will want lines of output, one for
each time period, which include the output variables of interest to him, and
that this should begin with a line which states the baseline conditions. As

reproduced above, this would include the number of time intervals passed, the

78
number of actively infectious individuals in the population, and the proportion

of those individuals in the population.

The status of each individual relative to the infection is reﬂected in a
single integer which is carried as the variable “infection”. Susceptible
individuals are designated as 0, infectious individuals as l, and immune as 9.
Thus the last six lines above would produce a grid of these integers, in lines
of length 20, representing the infectious status of each individual in the
population. Those lines are currently preceded by apostrophes, which tells
the computer to disregard them and treat them as remarks. The user who
wishes to “see” the spread of her simulated epidemic, and experience the
discrete nature of the model, needs only to delete the apostrophes before
executing the program.

[This concludes Section 3]

FOR year = 1 TO 60
CALL Mix(infect!, coordl, contactl, N)

The FOR statement above (which is coupled with the “NEXT year”
statement near the end of the main program) swings the computer into a loop
of calculations which it will repeat for as many time intervals as we designate
- 60 as printed above (I couldn’t call the variable “time” because that word is
a reserved one within Basic, so I chose “year”, though the user can imagine

the intervals to be whatever actual time is most appropriate). The ﬁrst step is

79

to calculate the degree of contact each susceptible individual in the population
has, over the current time interval, with each infective. This portion of the
program requires the bulk of the computer time of the whole program, and is
accomplished in the subroutine called Mix.
[This concludes Section 4]
FOR k = 1 TO N
[F infect!(k) = 1 THEN infect!(k) = 1
IF infect!(k) = 9 THEN infect!(k) = 9
IF infect!(k) = 0 THEN CALL Catchit(k, Trans, contactl,
infectl, agel, D, durationI, Mort)
NEXT k
In the above lines, the program examines each individual, allowing
everyone except the susceptibles to pass. The susceptibles, however, are
passed to the Catchit subprogram. Here their degree of contact with
infectives plus the transmissibility of the infection are translated into a
probability of contracting disease. A random number is obtained, and if this
number is within that probability range, the individual’s infectious status is
changed to “1 For the duration of the infection, that individual will

contribute to the risk of disease in subsequent time periods for susceptibles in

the population, and will be subject to disease-associated mortality, if any.

80

FOR a = 1 TO N

IF infect!(a) = 1 THEN chance = RND

IF infect!(a) = 1 AND chance < Mort THEN CALL Rebirth(a,
agel, infectl, span!, contactl, coord!)

IF infect!(a) = 1 THEN duration(a) = duration(a) - 1

IF infect!(a) = 1 AND duration!(a) = 0 THEN infect!(a) = 0

IF age!(a) = L THEN CALL Rebirth(a, agel, infecﬂ, durationl,
contacti, coord!)

age!(a) = age!(a) + 1
NEXT a

[The above accomplishes the functions of Sections 5 and 6, which
should be thought of as occuning simultaneously]

The program is now evaluating the population at the end of time period
1, and now knows which individuals are infected - those who began the
interval infected and those who just became infected during the interval.
Each time period (“year”), those infected may die of disease with chance
equal to the variable Mort. The ﬁrst two [F . . . THEN lines above
accomplish this grim reaping.

Those who have reached the end of their lifespan must also die (a
“natural” death, unassociated with the disease process being simulated). This
is Type I Mortality, the assumption that every individual lives to exactly age
L, miless eliminated earlier by disease-induced death. It is perfectly possible

to have the program assume Type H Mortality, which would stipulate a

81
constant yearly probability of death. The program commands which model

this condition are discussed below in Appendix B.

For those who don’t die, their age is incremented by a year. For those
infected individuals who survive, their disease duration is reduced by one
(you win some, you lose somel). In the fourth IF line above, those infecteds
who have reached duration = 0 have their infectious status altered either to 0
(indicating renewed susceptibility, as printed above) or to 9 (indicating they’ll
now be immune, if the user has altered the line appropriately).

A major assumption of this program is wrapped up in the name of the
subroutine which deals with death: It is called Rebirth. The program assumes
a constant population size (which many compartmental models do also, but
are not forced to do). Population size can remain constant only if the birth
rate equals the death rate, whatever that may be. This program accomplishes
that with maximum efﬁciency, producing an immediate birth of a susceptible
individual aged 0, into the same “social neighborhood” as the immediately
departed ancestor. The assumption of constant population size is not
absolutely necessary; one way to vary from this is discussed below.

Count = 0
FOR m = 1 TO N
IF infect!(m) = 1 THEN Count = Count + 1

NEXT In

[F Count = 0 THEN END

82

[The above accomplishes Section 8]

The current time interval has ended and all individuals’ variables have
been altered as chance and the simulated microparasite have dictated. The
program now counts the number of active “cases” in preparation for data
printing. If the infection happens to have died out completely there is no
point in continuing, so the program ends early.

Proportion = Count / N

PRINT year; Count; Proportion

‘FOR row = 1 TO (N / 20)

‘ FOR element = 1 TO 20

‘ PRINT infect!(20 * (row - l) + element);
‘ NEXT element

‘PRINT

‘NEXT row

NEXT year

[With the line above, the program accomplishes Section 7]
END

The program prints a line of data reﬂecting the conditions at the end of
the current time interval, and should do so identically to the baseline printing

above. The semicolons between variable names, incidentally, tell the

computer to print them all on the same line. The absence of a semicolon

83
following the last variable to be printed results in the next time interval’s data

being printed on a new line.

The “NEXT year” statement then directs the computer back to begin
the next time interval’s calculations, based now on the new conditions of each
individual. If the current time interval happens to be the last requested, the

program quits after printing. This is the end of the main program.

SUB Seed (L, Mort, id!( ), coord!( ), infect!( ), duration!( ), age!( ))
'infect!(l) = l
'age!(l) = INT(L * .3)
'duration!(l) = D
FORs=(2*N)/(G* F)TONSTEP(2 * N)/(G * F)
infect!(s) = 1
age!(s) = INT(L * .3)
duration!(s) = D
NEXT s

END SUB

This subroutine, the ﬁrst which the program encounters, is given N
individuals with their respective ages and coordinates, all designated as
susceptible. It changes the status of either one of these individuals (if the ﬁrst
set of lines is executed) or of one per family (the second set of program lines)
to infected/infectious. To be sure these individuals endure for a signiﬁcant

time, their ages are set at 30% of L. Thus the scenario begins, by the user’s

84
choice, either as though infection has just entered the population, or as though

it has existed, perhaps endemically, for some time.
SUB Mix (infectl, coordl, contactl, N)

FOR b = 1 TO N
contact = 0
FOR h = 1 TO N
C = 0
Z = 0
IF infect!(b) = 1 THEN EXIT FOR
IF infect!(b) = 9 THEN EXIT FOR
IF infect!(h) = 0 THEN C = 0
IF infect!(b) = 9 THEN C = 0
'Line below gives Euclidian distances
'IF infect!(b) = 0 AND infect!(b) = 1 THEN C =
((ABS(coord!(b, l) - coord!(h, 1))) A 2 + (ABS(coord!(b, 2) - coord!(h,
2») " 2) " -5
'Line below gives Manhattan distances
IF infect!(b) = 0 AND infect!(h) = 1 THEN C = (ABS(coord!(b,
l) - coord!(h, 1)) + (ABS(coord!(b, 2) - coord!(h, 2))))
IFC>0THENZ=1IC
contact = contact + Z
NEXT h
contact!(b) = contact
NEXT b

END SUB

The Mix subroutine takes the population as it exists at the begimring of
the time interval and examines each individual in tIun to see if it is
susceptible. For each susceptible, it ﬁnds every infective in the population
and determines the distance of that infective ﬁom this susceptible. The

measure of distance used can either be the as-the-crow-ﬂies Euclidian

85
distance, which uses much computer time, or the right-angle-tums-only

Manhattan or City Block distance, which uses about half as much. Whatever
the distance measure calculated, the contact with that individual is calculated
as the inverse of that distance. And the total potentially infectious contact
calculated for each susceptible is the sum of the calculated contacts with each

infective.

SUB Catchit (k, Trans, contactl, infectl, agel, D, durationl, Mort)
RANDOMIZE TIMER

odds = Trans * contact!(k)

r01] = RND

IF roII >= odds THEN infect!(k) = 0

IF r01] < odds THEN infect!(k) = 1

IF roII < odds THEN duration!(k) = D + 1

END SUB

The Catchit subroutine is as close to operating like a deity as a
computer gets. Each susceptible in the population is separately referred to it,
along with its contact number for this time interval, just calculated by Mix.
This number is multiplied by the constant “Trans”, which is characteristic of
the microparasite, to produce a probability (“odds”) of contracting infection.
The routine “rolls” the celestial dice in the form of obtaining a random

number. Ifthis number (between 0 and l) is within the bounds of “odds”

86
(which is large if “contact” and/or “Trans” are large) then the individual’s

status is changed to infected/infectious, signiﬁed by the integer l. The
duration of disease is placed at D + 1 because this number will be

decremented by the later sections of the main program.

SUB Rebirth (a, agel, infectl, durationl, contactl, coord!)
RANDOMIZE TIMER

age!(a) = -1

infect!(a) = 0

duration!(a) = 0

contact!(a) = 0

coord!(a, 1) = CINT(coord!(a, 1)) + (RND - .5)

coord!(a, 2) = CINT(coord!(a, 2)) + (RND - .5)

END SUB

The Rebirth subroutine is also deity-like. Here each individual is sent
who has been marked for death in the main program, either by disease or by
reaching the end of the allotted lifespan L. In either case this routine,
obedient to the assumption of constant population size, creates a new
susceptible newborn whose spatial coordinates are close to those of the
recent deceased. The variable “age” must be set at -1 , so that the later

portions of the main program will advance it to 0 as the time interval ends.

APPENDIX B

87
Appendix B - A Guide to Alternative Explorations

Alternative Printing

The reader will note that the program prints its output in two places,
once printing the conditions at the onset of the simulation (immediately after
“seeding” with infectives), and again printing the conditions at the end of
each time interval. Usually the user will want both of these program sections
to direct identical printing. As reproduced here, the program prints only time
and the prevalence of infection, expressed both as numbers of cases and as a
proportion of the population. Any variable carried in the computer’ 5 memory
can be printed, however, and need only be added to the list of those already
there, separated by semi-colons. The user should plan the gathering of data
by the program as she would plan data collection in a ﬁeld study, being
certain what variables are needed and for what reason prior to running the
pro gram. Fortunately, the penalty for poor planning in a computer simulation
is less stringent than running a multi-year epidemiological study and then
realizing that the wrong data had been collected.

Sometimes the printing of variable values can be useful for reasons of
program debugging. If one wanted to pick out one individual and follow the

value of “age” (to be certain that the program was handling it as one wished),

88

the line

PRINT year; Count; Proportion; age(49)
would add a column for the age of the 49‘11 individual to be included in each
data line.

As the limiting resource for the parasite is always the pool of available
susceptibles, the user may at times wish to print the number or proportion of
these as well as the infected group. Note the form of the lines just above the
ﬁnal print section, as the program counts the number of infectives in a
variable called “Count”. One can add another variable of a different name to

count the susceptibles, who will be those for whom “infect” = 0.

Vertical Transmission

It is in the Rebirth subroutine that new babies are born into the
population. As reproduced here, all newborns are born susceptible. But it is
possible to alter this portion to build in some vertical (congenital)
transmission of disease. Suppose one wishes to build in a 20% probability
that a baby born of an infected mother will be born infected. The probability
that the “infant’” 5 mother will be infected is the proportion of infected

individuals within the population.

89
At the time that the Rebirth subroutine operates, the value of

“Proportion” is the proportion of infectives at the end of the last time interval,
applicable to the birth which is happening in the present time interval. Within
the Rebirth subroutine, then, replace the two lines which stipulate that
infection and duration both equal zero, with the following:

Chance = Proportion * 0.2

Luck = RND

IF Luck >= Chance THEN infect(a) = 0 AND duration(a) = 0

IF Luck < Chance THEN infect(a) = 1 and duration(a) = D.

However, one may wish to stipulate a different value for the duration of

congenitally acquired disease than for horizontally acquired disease.

Age-dependent transmission or mortality

As printed here, the original program includes the parasite—dependent
transmission probability (“Trans”) and the disease-associated mortality rate
(“Mort”) as constants which apply equally to all members of the population.
Either of these can be made age-dependent.

Altering the transmission rate would be done in the Catchit subroutine.
If one rate is to apply to children, another to adults, precede the application of

the transmission rate by something like:

90

IF age(a) <= 5 THEN Trans = 0.15
IF age(a) > 5 THEN Trans = 0.10 .

If transmissibility is to be a continuous function of age, then the
preceding lines could be:

Trans = (age(a))“.5 *0.04 ,
and there would no longer be a need to declare a value of Trans among the
constants.

Mortality is applied within the main program, and could be made age-
dependent with exactly the same type of commands as suggested above for
transmission rates. Just be certain that the added lines are within the FOR. .
.NEXT loop which examines each individual in turn, so that the mortality will

be separately determined for each individual.

Type H Mortality

As discussed above along with the presentation of the program, the
original model assumes Type 1 Mortality. Under this assumption every host
individual who escapes disease-associated death dies at exactly the same age
L. It is also possible to assume Type II Mortality, which stipulates a constant
rate of death, which in a discrete model would be a constant probability of

death. The model modiﬁed for Type II Mortality would utilize lines like:

91

FOR a = 1 TO N
deathchance = RND
IF deathchance < deathrate THEN CALL Rebirth(a, agel,
infectl, durationl, contactl, coord!)
NEXTa
In this case the constant “deathrate” would be approximately l/L,

where L is the desired mean lifespan. More precisely, calculate this rate as

1.0 minus the Lth root 0f0.5.

Sex-dependent transmission or mortality

We move now into potential alterations which I will describe
conceptually, but for which I will not supply all speciﬁc program lines.

In the program with all of its modiﬁcations discussed so far, there are
individuals of three kinds: Susceptible, infected/infectious, and immune,
designated by values of the infect! variable of 0, 1, and 9. The population
could be ﬁnther subdivided into two classes, each of which could include
individuals of all three types. The classes could be thought of as the two
sexes, or two races, or risk groups, or any other divisions important to the
concept being modeled. Susceptibles of the two classes might be represented

as 0 and 6, infecteds as 1 and 7, and immunes as 9 and 8.

92
The distinction between classes would be meaningful if mortality or

transmission were dependent upon these classiﬁcations. Such distinctions
would be made through lines such as:
IF infect(a) = 6 THEN Mort = . . . , or

IF infect(b) = 6 AND infect(h) = 1 then contact = . . .

Existence of a Latent Phase

Adding a latent phase to the three infection types already in the model
would add a fourth value to the infect! variable. The latent phase might be
signiﬁed, for instance, by the value of 4. A new constant will be needed to
hold the duration of the latent phase; perhaps this would be “E”. The Catchit
routine could then be modiﬁed to change the infection state of newly infected
individuals not from 0 to l, but 0 to 4. The variable “duration” would be
changed to E + 1.

With the passage of each time interval thereafter, the value of duration
would be decremented by one. Add a line in that portion of the main program
which sends the susceptibles to the Catchit routine which will activate those
in the latent phase whose duration has reached 0:

IF infect(k) = 4 AND duration(k) = 0 THEN infect(k) = 1 and duration(k)
= D + l.

93
Duration will be decreased to D by the last phases of the program, and

the individual will contribute to the force of infection for the next D time

intervals.

Assumption of constant population size - An alternative

Among the various assumptions and oversimpliﬁcations which are a
part of this model and its described modiﬁcations, probably the one which is
most likely to seem troublesome is the assumption of constant population
size. Though a usual assumption of most models of disease-host association,
its presence in this more realistic discrete model seems more starkly
oversimpliﬁed. The immediate “rebirth” of each individual at death is not
realistic. Furthermore, if strains of varying transmissibility/mortality
combinations are to be compared, a strain incorporating greater
transmissibility should probably “pay” with higher mortality, and this cannot
be done if rebirth is instantaneous.

This problem can at least partially be overcome by building in a lag
time between death (or only disease-associated death) and rebirth. A possible
modiﬁcation of the program could include a “purgatory” state as a fourth

class of individual. Such a designation would indicate that the site was

94
temporarily unoccupied. More picturesquely, we could think of the individual

at that site as dead and not yet reborn.

The modiﬁed program includes a new constant P which stipulates the
number of time intervals between disease-associated death and rebirth. There
is a new subroutine Purgatory to which individuals are referred by the main
program upon such death. This routine reassigns them to their new state, and
sets a new individual variable “purg” to the value of P, as follows.

SUB Purgatory (P, a, agel, infectl, purgl, contact!)

age!(a) = -1

infect!(a) = 5

purg!(a) = P + 1

contact!(a) = 0

END SUB

The variable “purg” is then decremented at each time interval by the
main program, until P steps have passed and the individual can be referred to
Rebirth. That portion of the main program reads:

FOR a = 1 TO N

IF infect!(a) = 1 THEN chance = RND

IF infect!(a) = 1 AND chance < Mort THEN CALL
Purgatory(P, a, agel, infecti, purgl, contact!)

IF infect!(a) = 1 THEN duration(a) = duration(a) - 1

IF infect!(a) = 1 AND duration!(a) = 0 THEN infect!(a) =

IF infect!(a) = 5 THEN purg!(a) = purg!(a) - 1

IF infect!(a) = 5 AND purg!(a) = 0 THEN CALL
Rebirth(a, agel, infectI, durationl, contactl, coord!)

95
IF age!(a) = L - 1 THEN CALL Rebirth(a, agel, infectl,

durationl, contactl, coord!)
IF age!(a) = L THEN CALL Rebirth(a, agel, infectl,
durationl, contactl, coord!)
age!(a) = age!(a) + 1
NEXT a
Another approach, probably more biologically well-founded, is to base
the population’s ability to produce newborns on the total number of
individuals of reproductive age within the population. One can deﬁne a
portion of the host lifespan which is assumed to be reproductively active.
The program is asked to total the number of individuals within that age range
at each time interval. Given assumptions relative to the sex ratio and the
duration of pregnancy in time units of the program, one can restrict the
number of newborns at each time interval to some fraction of the number of
reproductive adults. This approach is written in to some of the model variants
which I offer to supply on disc.
Note
The author would be happy to provide a disc containing the

fundamental model, along with several of its modiﬁcations, upon request.

BIBLIOGRAPHY

96

BIBLIOGRAPHY

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10. Anderson R and May R, Spatial, temporal, and genetic heterogeneity in
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