%Carlo C. Maley
%This is a paper on a model of the evolution of virulence in parasites.
%For submission to the Artificial Life IV conference.

% I need to explain the Baveco and Lingeman section more.***
% Check bibliography for things I forgot to \nocite.

\documentstyle[named,alife]{article}
\title{A Model of the Effects of Dispersal Distance on the Evolution of 
Virulence in Parasites}
\author{C. C. Maley \\ 
The Artificial Intelligence Laboratory
\thanks{This research was made possible through the generosity of the
MIT AI Lab.  It was supported by NSF grant 9217041-ASC and ARPA under
the HPCC program.  I would like to thank W.D. Hamilton, L.D. Hurst, S. Rich,
R.C. Berwick, E.S. Johnson, and M.S. Golden for their helpful comments
and support.}
\\ 
NE43-803\\
Massachusetts Institute of Technology \\ cmaley@ai.mit.edu \\
617-253-6567}

\begin{document}


\maketitle


\begin{abstract}
The effects of differing dispersal distances on the evolution of virulence in 
parasites was explored through the use of a configuration model.  Both 
hosts and parasites were individually represented on a two-dimensional 
surface.  Each parasite had a ``gene'' for virulence that determined the 
amount of energy the parasite removed from its host in a single time step.  
When a parasite reproduced, its offspring was placed near to, or far away from 
the parent depending on a dispersal distance parameter.  The 
offspring's virulence was also mutated slightly at reproduction.  Distance of 
dispersal had a small but statistically significant, positive effect on the 
evolution of the parasite's virulence.  That is, those parasites that were forced 
to disperse their offspring further away, evolved a higher level of virulence 
than the parasites that dispersed their offspring locally.
\end{abstract}


\section{Introduction}

There is a growing interest in the application of
evolutionary theory to disease control and treatment \cite{Ewald:1983,Williams:1991}.  At the same time, computer modeling is opening up new
areas of evolutionary theory by allowing us to ask questions that were
previously extremely difficult to test in the lab \cite[for
example]{Koza:1994,Lindgren:1994}.  This paper represents the cross
fertilization of these two research projects in a model of the evolution
of virulence in parasites.

\subsection{The Evolution of Virulence}

Why is the parasite that causes the common cold less deadly than the 
protozoa that causes malaria (genus {\em Plasmodium\/}) or the bacteria 
that cause cholera ({\em Vibrio cholerae\/})?  Are these parasites becoming 
more or less virulent as they coevolve with humans, and why?  These 
questions are neither academic nor intractable.

\subsubsection{The Theory}

I am using the term parasite in the broadest possible understanding.  A 
parasite is any organism that gains its nutrients from other living organisms  and 
usually has a deleterious effect on those organisms \cite{Begon:1990}. It was 
long thought that a parasite's relationship to its host would evolve to become 
less deleterious, or virulent, over time.  This was based on an essentially group 
selectionist, or ``good of the species'' argument.  Parasites that did less damage 
to their hosts would thereby preserve their hosts for longer and thus live 
longer themselves as compared to parasites that were more damaging to their 
hosts.  This kind of argument still persists today despite advances in the 
theory.  Anderson and May have published extensive analyses of the 
coevolution of parasites and hosts.

\begin{quote}
...formal studies make it clear that the coevolutionary trajectory followed by 
any particular host-parasite association will ultimately depend on the way the 
virulence and the projection of transmission stages of the parasite are linked 
together:  depending on the specifics of this linkage, the coevolutionary 
course can be toward essentially zero virulence, or to very high virulence, or 
to some intermediate grade.  \cite{Anderson:1982}
\end{quote}

In other words, there are significant pressures on parasites that might select 
for varying levels of virulence, not just for avirulence or commensalism.  
This can be seen formally in a simple equation for the reproductive rate given 
by May and Anderson (1990).

\begin{equation}
R_{0} = \frac{\lambda (N)}{\alpha + b + v}  	\label{eq:para}
\end{equation}

Here $N$ is the current population size of the hosts and $\lambda(N)$ is 
the rate of successful transmission of the parasite offspring to other hosts as a 
function of the population density of the hosts.  The denominator contains 
the variables that might reduce the pool of hosts available for infection.  
Here, $\alpha$ is the disease induced death rate of the hosts, $b$ is the death 
rate for the hosts from causes other than the disease, and $v$ is the recovery 
rate of the hosts.  After a host has recovered, it is considered to be immune to 
the disease.

Given this relatively simple equation and the assumption that selection puts 
pressure on the parasites to maximize $R_{0}$, May and Anderson show that 
the virulence of the parasite, $\alpha$, may be expected to evolve toward any 
number of results.  For example, if transmission, $\lambda$, and recovery, 
$v$, are independent of virulence, $\alpha$, then $R_{0}$ is maximized by 
avirulence, $\alpha \rightarrow 0$, a commensal relationship.  However, 
such independence is not the norm.  If we assume that the likelihood of 
transmission, $\lambda$, increases as a linear function of the virulence of 
the parasite, $\alpha$, then $R_{0}$ is maximized by maximum virulence, 
$\alpha \rightarrow \infty$.  This would evolve regardless of the extreme 
likelihood that the parasites would thereby drive their hosts to extinction and 
thus perish themselves.  May and Anderson (1990) conclude, ``What are 
needed are factual data about the functional relationships among $\lambda$, 
$v$ and $\alpha$, derived from the life history details of specific host-parasite 
associations.  In no single instance do we have enough information to 
determine the likely evolutionary trajectory dictated by maximizing $R_{0}$ 
in [equation \ref{eq:para}].''

\subsubsection{The Observations}

In fact, parasite reproduction is generally correlated with disease severity 
\cite{Ewald:1983}.  As Ewald puts it, the question then becomes, ``What 
environmental conditions select for different levels of parasite
reproduction and, hence, for different levels of severity?''  Two
examples help to illustrate the point.  Various species of protozoa in
the genus {\em Plasmodium} cause the disease malaria in human hosts.
These parasites are transmitted to new hosts through the biting of
mosquitoes that transfer small amounts of blood from human to human.
The high levels of virulence in these parasites is hypothesized to be an
adaptation to their mode of dispersal.  By reproducing in great numbers
the parasite achieves two effects.  First, any small amount of blood
taken from the host is likely to contain a few protozoans.  Second, by
causing a sever reaction, the host is immobilized and thus more easily
bitten by mosquitoes.  In contrast, the common cold is transmitted
through social contacts and so a parasite that immobilized its host
would thus be reducing its likelihood of transmission to new hosts.
This may help to explain why the cold is a relatively mild affliction.

Ewald (1983) collected data on a large number of human diseases and found a 
correlation between disease severity and parasites transmitted through biting 
insects.  The cholera epidemic in India in the 1950's and 1960's provides a 
tantalizing evidence of the relationship between mode of parasite 
transmission and virulence.  The bacteria that cause cholera ({\em Vibrio 
cholerae\/}) are generally transmitted through drinking-water supplies.  The 
bacteria cause diarrhea in their hosts.  The diarrhea may contaminate the 
water supplies either through poor sewage disposal or through the washing 
of contaminated sheets.  In any case, the parasite's chances of being 
transmitted are not decreased by immobilizing its host.  In fact, its chances of 
getting into the water supply may even increase if care-takers wash 
contaminated bed-clothes in the water supply of a down-stream community.  

\begin{quote}
As water supplies were purified in cholera-endemic regions in India during 
the 1950's and 1960's, the milder agent of cholera, the El Tor type of {\em 
Vibrio cholerae}, displaced the more dangerous form, classical {\em V. 
cholerae}.  \cite{Ewald:1993}
\end{quote}

Unfortunately, the evidence for the evolution of virulence in relation to the 
parasite's mode of dispersal remains largely observational, and thus only 
correlational \cite{Ewald:1983}.  Ewald (1983) goes on to predict that parasites that 
are highly mobile should cause severe diseases.  

The correlation between virulence and parasite mobility fits with a body
of evolutionary theory that analyzes the continuum from purely vertical
transmission, from parent to offspring, to purely horizontal
transmission, i.e., transmission between unrelated hosts.  When a
parasite is transmitted vertically, it maximizes its reproductive
success by maximizing its host's reproductive success.  Thus, the
parasite should evolve to a commensal, if not symbiotic relationship
with the host species.  However, a parasite that is transmitted
horizontally to unrelated hosts has no particular interest in preserving
its current host.  Thus horizontally transmitted parasites are expected
to evolve high levels of virulence \cite{Williams:1991,Herre:1993,Nowak:1991,Hurst:1991}.\footnote{It should be noted that
definitions of virulence vary amongst these authors.  For example, Herre
(1993) operationalizes virulence as a reduction in lifetime reproductive
success of the host, rather than disease induced death rate as per Ewald
(1983) and May and Anderson (1990).}

\subsection{Configuration Models of Parasites}

Differential and difference equations have been used for many years to
model population dynamics \cite{May:1973}.  These models lump
individuals together into homogenous groups that can be represented by a
single number, the number of individuals in that group.  They may break
a population down into a number groups depending on any number of
characteristics, like age, sex, weight, etc. \cite{Caswell:1989}.
However, in each subgroup, the individuals within the groups are still
treated as identical.  In this way, these models may describe the
distribution of individuals in the population.  Caswell and John (1992)
label these distribution models.  In contrast, a relatively new
form of computer model, often called an individual-based model,
explicitly represents each individual in the population as a separate
data structure.  The model keeps track of the configuration of all of
the individuals in the population.  Since the distribution model
is also based on the individuals in the population, this new form of
model may be called a configuration model \cite{Caswell:1992}.

Although configuration models are relatively new to ecological
modeling \cite{Huston:1988}, they have been essential to artificial
life.  Emergent dynamics are often observed through the local
interaction of individuals.  This is the basis of cellular automata
models as well as many of the early models in artificial life
\cite[for example]{Langton:1992,Fontana:1992,Koza:1992,Taylor:1989,Travers:1989,Reynolds:1987}.  Recently, attention has been
focused on the theory behind the construction of configuration
models \cite{Maley:1993,Baveco:1992}.
configuration models open up the possibility of explicitly
representing the interactions between individuals.  On occasion this has
produced evidence that contradicts or extends the analysis of
distribution models \cite{Boerlijst:1992,Collins:1992}.

In the case of a host-parasitoid interaction, a configuration model 
can explicitly model the reproduction and transmission of parasites.  This 
allows an intuitive and direct analogy between the dynamics of the model 
and the dynamics of the biological system.  Baveco and Lingeman (1992) have 
constructed such a model to examine the population dynamics of a host-
parasitoid system in a patchy environment.

Baveco and Lingeman locate the hosts and parasites in a $5 \times 5$ grid of 
patches.  Both hosts and parasites have a three stage life history; eggs, larvae, 
and adults.  Hosts can only be parasitized in their larval stage.  Parasite 
transmission in their system is density-independent.  Similarly, host 
reproduction is density-independent.  Furthermore, the life span of an 
individual is predetermined at birth.  The individual will die after the 
determined number of time steps unless it is a larval host that becomes 
infected by the parasites and so dies before it can reproduce.  Parasites disperse 
from a patch to a neighboring patch when no hosts remain in their patch.

Baveco and Lingeman found that if the reproduction rate of the parasites
was greater than the reproduction rate of the hosts, the parasites
quickly drove the hosts to extinction.  On the other hand, if the reproduction rate of
the parasites was less than the reproduction rate of the host, the
parasite population never caught up to the host population.  This later
case is probably due to the fact that host reproduction was assumed to
be density-independent.  However, in the former case, the extinction of
the hosts was found to be a robust effect for most of their parameter
settings.  Baveco and Lingeman tried increasing the dimensions of the
environment from $5\times5$ up to $10\times10$.  They tentatively
conclude that an environment of $10\times10$ patches is about the
minimum size necessary for the persistence of the host population.

Baveco and Lingeman were not trying to model the evolution of virulence.  
In fact, they were not trying to model evolution at all.  They took a classical 
 distribution model \cite{Nicholson:1935} of an ecological 
interaction between hosts and parasites and examined the dynamics of the 
system when all the individuals were explicitly represented.  However, a 
similar model might be constructed where the parasites might have a ``gene'' 
for virulence.  Given the possibility of mutation at the time of reproduction, 
the parasites might evolve varying levels of virulence over time in response 
to the selective pressures in the model.



\section{A Model of the Evolution of Virulence}

Ewald (1983), May and Anderson (1982 and 1990) all predict that virulence 
will evolve in relation to the mode of transmission of the parasite.  However, 
mode of transmission is not an easily quantifiable characteristic.  The
problem can be simplified by focusing on the distance over which the
parasites disperses.  Whether it 
is transmission through biting insects, contaminated water supplies, or 
through sneezing, the dispersal of the parasites can be quantified by the 
distance of dispersal from the infected host.  The question can now be 
asked, what is the effect of the distance of dispersal on the evolution of 
virulence in parasites?

\subsection{The Hypothesis}

Hamilton (1992)\nocite{Hamilton:1992} suggested that we should expect parasites that disperse over 
short distances to evolve to be less virulent than parasites that disperse over 
long distances.  This follows from the continuum from vertical to horizontal 
transmission.  Parasites that reproduce locally will be likely to share their host 
with their offspring.  Thus they damage their own reproductive fitness by 
damaging their host.  On the other hand, a parasite that disperses over long 
distances is less likely to share its host with its offspring, and so can exploit its 
host with near impunity.

\subsection{General Structure}

The hypothesis as it stands requires a system of only four significant 
components: host organisms, parasite organisms, spatial structure in the 
environment so that the parasites can disperse to new hosts, and a parasite 
gene for level of virulence that can evolve over time.  It is necessary to use a configuration model because the spatial relationships between 
organisms directly affects the dynamics of the model.  Because the
hypothesis does not involve the immune response of the host, host immune
systems were excluded from the model.

I chose to use an energy flow model \cite{Dewdney:1989,Rizki:1986,Taylor:1989}.  Each discrete time step, a quantified amount of energy 
flows from the environment into the host organisms.  At the same time, 
parasites are removing energy from the hosts to support the loss of energy in 
the parasites due to their metabolism.  Additional energy is used by both hosts 
and parasites when they reproduce and that energy is given to the newborn 
organisms.

Hosts and parasites interact in a two-dimensional environment, or grid.  The 
edges of the grid are connected to form a toroidal surface.  Each patch in the 
grid can hold a single host.  Each host can hold a virtually unlimited number 
of parasites while it still lives.\footnote{The maximum number of parasites 
allowed to infest a single host was set to 500.  Since the actual number of 
parasites infesting a host tended to stay below 10, this should not have 
influenced the dynamics of the model.}  Death occurs when an organism's 
energy 
level drops to zero.

Hosts are relatively uninteresting.  They only have one attribute:
their energy level.  Hosts cannot move.  They can reproduce when their 
energy reaches a certain threshold.  However, the newborn only survives if 
there is an empty patch in the grid adjacent\footnote{Every patch has eight 
adjacent patches corresponding to the eight points of the compass.  Since the 
edges of the grid wrap around, the surface actually forms a torus with no 
edge effects.} to the parent.  Since the hosts have no ``genes'' they cannot 
evolve.

Parasites have two attributes: energy and virulence.  The virulence
attribute is the number of energy units the parasite takes from its
host each time step.  Like hosts, parasites reproduce when their
energy level reaches a certain threshold.  However, they have genes
for virulence, so the virulence attribute may mutate (e.g., $\pm 6$) in the 
newborn parasite.  Furthermore, there is a parameter in 
the model, dispersal distance, that restricts where the newborn 
parasite may be randomly placed in the environment.

For each time step, the model executes four steps.  1.~The energy levels
in the hosts are updated in parallel.  2.~The energy levels in the
parasites are updated in parallel.  3.~Newborn parasites are placed in
the environment.  4.~Newborn hosts are placed in the environment.

Energy flows in the following way:  First, each host gains a fixed
amount of energy, {\em food\/}, each time step.  Second, each host loses
energy equal to the sum of the virulences of the parasites infesting
it.  Third, each parasite gains its {\em virulence\/} worth of energy units
each time step, as long as its host is still alive, {\em host energy\/} $> 0$.
Fourth, each parasite loses a fixed amount of energy, {\em cost of living\/} 
each time step.  In addition to this basic energy flow, an organism loses
a fixed amount of energy, {\em cost of reproduction\/}, each time it 
reproduces  The newborn organism starts off with that amount of energy.

\subsection{Parameters}

Although the basic structure of the model has only four components, there 
are a host of details that must be specified in order to model the life histories 
and interactions of the organisms.  These details can be divided into three 
categories:  fundamental parameters, parameters that can be derived from 
previously defined parameters, and the initial conditions in the model at the 
beginning of a simulation.  [In all cases the default settings will be put in 
brackets.]

{\small
\begin{table*}[t]
\centering
\begin{tabular}{|l|l|l|}	\hline
{\bf Parameter} & {\bf Description} & {\bf Default} \\ \hline \hline

{\em energy} & Energy units accumulated by an organism & variable\\
{\em virulence} & Amount of energy a parasite removes from its host & variable\\ \hline

{\em dispersal distance\/} & Radius of dispersal for parasite offspring & 1-15 
\\
{\em reproduction rate\/} & Uninfested host generation time & 2 \\
{\em host cost of reproduction\/} & Energy penalty for host reproduction & 
1000 \\
{\em size\/}	& Size scale between hosts and parasites	& 5 \\ 
\hline

{\em food\/} & Energy given to a host each time step & 500 \\
{\em cost of living for parasites\/} & Energy taken from a parasite each time 
step & 100 \\
{\em host reproduction threshold\/} & Amount of energy triggering host 
reproduction & 2000 \\
{\em parasite cost of reproduction} & Energy penalty for parasite 
reproduction & 200 \\
{\em parasite reproduction threshold} & Amount of energy triggering 
parasite reproduction & 400 \\
{\em mutation factor\/} & Scales the range of mutation in virulence & 8 \\
{\em carrying capacity of the hosts\/} & Maximum number of parasites in a 
host & 500 \\ \hline
\end{tabular}

\caption{The variables, fundamental parameters, and derived parameters that 
define the model.}
\end{table*}
}

\subsubsection{Fundamental Parameters}

The fundamental parameters in the model are the bottom line.  They are 
used to derive reasonable quantities for many other parameters in the model, 
and so define basic choices about the scale of the model.  There are only four 
such fundamental parameters: {\em dispersal, reproduction rate, host cost of 
reproduction\/} , and {\em size\/} .

{\em Dispersal distance\/} is the independent variable in the experiment.  
The 
{\em dispersal distance\/} represents the maximum distance between a 
parasite and 
the placement of its offspring.  {\em Dispersal distance\/} defines the size of 
the 
square, with the parental parasite's patch in the center, in which the offspring 
parasite is randomly placed.  The probability of placement for a parasite was 
the same for every patch within that square.

{\em Reproduction rate\/} is the number of time steps it would take an 
uninfested host to build up enough energy to reproduce.  In other words, it is 
a sort of generation time.  {\em Reproduction rate\/} determines the 
granularity of 
time in the model.  [2]

{\em Host cost of reproduction\/} is the number of energy units transferred 
from a 
host parent to its offspring at birth. This determines the basic energy scale or 
granularity in the model.  The larger the number the finer the grain.  [1000]

Finally, {\em size\/} determines the relative scale between the hosts and 
parasites.  
It is roughly the number of parasites that can fit in a host.  The larger the 
number, the smaller the parasites.  [5]



\subsubsection{Derived Parameters}

Most of the parameters in the model are derived from {\em size, host cost of 
reproduction\/}, and {\em reproduction rate\/}.  They generally determine 
the 
specifics of energy flow through the model.

Energy enters the cycle as {\em food\/} .  {\em Food\/} is the amount of 
energy given to a 
host each time step.  $food =$ {\em host cost of reproduction / reproduction 
rate\/}.  
[500]

Similarly, energy leaves the cycle in the {\em cost of living for parasites.\/}   
This is 
the amount of energy removed from every parasite each time step.  {\em cost 
of 
living for parasites = food / size\/}.  [100]

Hosts automatically reproduce when their energy level reaches the {\em host 
reproduction threshold\/} .  At that point, the host loses the {\em host cost of 
reproduction\/} amount of energy.  {\em Host reproduction threshold\/} $= 
2 \times$ 
{\em host cost of reproduction\/}.  [2000]

Parasites also lose energy when they reproduce equal to the {\em parasite cost 
of 
reproduction = host cost of reproduction / size\/}.  [200]

Like hosts, parasites automatically reproduce when their energy level rises 
above the {\em parasite reproduction threshold = host reproduction 
threshold / size\/} .  However, unlike hosts, parasites might be so virulent 
that their energy level jumps far above the {\em parasite reproduction 
threshold\/} .  Parasites are forced to reproduce multiple times until the 
repeated deductions of the {\em parasite cost of reproduction\/} lowers the 
parasite's energy level below the {\em parasite reproduction threshold.\/}  In 
this way, highly virulent parasites reproduce more quickly than less virulent 
parasites.  [400]

The {\em mutation factor\/} scales the range of mutation of the virulence 
gene in 
the parasites.  [8]  It is not strictly a derived parameter since it does not 
formally depend on other parameters.  However, it has been set such that the 
actual range of mutation ($\frac{food}{2 \times mutation \: factor \times 
size}$ [= 
6]) in the virulence gene is small compared to the amount of energy 
entering a host, {\em food.}

Finally, for ease of programming, there is a {\em carrying capacity of the 
hosts\/} 
parameter that puts a strict limit on the number of parasites that may infest 
the host.  This was set high enough that in actuality it never affected the 
dynamics of the model.  [500]


\subsubsection{Initial Conditions}

The only pieces that remain to describe are the initial settings of the
characteristics in the organisms at the start of a simulation.  Hosts
only possess a single characteristic, energy, while parasites possess
both energy and virulence characteristics.  Hosts start with an energy
that is determined randomly in the range of 0 to {\em host reproduction
threshold.\/} However, any newborn host starts out life with {\em host
cost of reproduction\/} worth of energy units.  Similarly, the
parasite's initial energy was set randomly between 0 and the {\em
parasite reproduction threshold,\/} and a newborn parasite starts life
with {\em parasite cost of reproduction\/} energy units.  The parasites
initially placed in the model have virulences that are determined
randomly in the range from the {\em cost of living for parasites\/} to
$3 \times${\em cost of living for parasites.\/} [100 - 300] \footnote{If
a parasite has a {\em virulence} less than the {\em cost of living for
parasites\/} then it will inevitably die before reproducing.}


\subsection{Quirks and Caveats}

This model of a host-parasitoid system is extremely simplified.  The principle 
of parsimony suggests that the bare minimum of detail should be included so 
as to test the hypothesis.  Beyond this, there were a number of decisions that 
had to be made more or less arbitrarily.  Usually ease of model construction 
prevailed over the choices.  



\subsubsection{Major Dynamics Left Out}

The most glaring exemption is the lack of resistance genes in the hosts.  This 
removes the entire dynamic of coevolution.  It also removes the possibility 
for sexual reproduction in hosts.   Hamilton, Axelrod, and Tanese (1990) 
\nocite{Hamilton:1990}argue that sexual 
reproduction may have evolved as an adaptive response to parasitism.

We lack data on the relationship between parasite reproduction and 
virulence in general \cite{May:1990,Ewald:1983}, or $\lambda$ 
and $\alpha$ in equation \ref{eq:para}.  Therefore I have let the relationship 
be linear as a first approximation.  Specifically, a parasite will produce $\lceil 
\frac{energy - parasite \: reproduction \: threshold}{parasite \: cost \: of \: 
reproduction} 
\rceil$ number of offspring in one time step.  Furthermore, the model does 
not contain any sense of host recovery or immunity ($v$ in equation 
\ref{eq:para}).

On a more physical level, there is no dissipation of energy during transfer 
from environment to host, from host to parasite, and from parent to 
offspring.  Specifically, parasites gain exactly the same amount of energy as 
their host loses in a single time step.

No distinction was made in the model between microparasites and 
macroparasites.  Nor is there any representation of life histories of the 
parasites and host beyond the dispersal distance.  Finally, there is no 
interspecific competition between different parasites species within the hosts.  
The model only simulates intraspecific competition amongst the parasites.



\subsubsection{Additional Details is in the Model}

When a host dies all of the parasites infesting it die.  However, when a host 
reproduces, half of the parasites infesting the parent are transmitted to its 
offspring.  This allows parasites to be transmitted vertically.  

There are a number of quirks surrounding reproduction.  If a host tries to 
reproduce when there are no empty adjacent patches, the host will lose the 
{\em host cost of reproduction\/} amount of energy, but no new host will be 
created, and so the parental host will retain all of its parasites.  Similarly, 
there are a few rare cases where two parasites will try to place offspring in the 
same position of a single host, thus causing a ``collision'' since the model 
operates in parallel.  In this case, the more virulent offspring will win out and 
the other parasite offspring is killed.  In an even more unlikely case, a parasite 
will try to place an offspring into a host that already has the {\em carrying 
capacity 
of the hosts} number of parasites infesting that host.  Again, the parasite's 
offspring is killed under these conditions.


\section{The Experiment}

The model was constructed in the language {\em C*\/} and was run on a
Thinking Machines CM5 parallel computer.  Once the model had been
constructed, the experiment was straight-forward.  What is the
relationship between distance of dispersal and virulence in parasites?
The independent variable was {\em dispersal distance\/} and the
dependent variable was the {\em virulence\/} in the population of
parasites after a specified amount of time.

\subsection{Method}

Each run of the model began with a host in every patch of the $30 \times
30$ environment.  One of the hosts was also infected by a single
parasite.  The model began by simulating the invasion of the host
population by the parasite.  Initial runs indicated that the
oscillations of the host and parasite population sizes were so violent
as to crash in many cases when the {\em size\/} parameter was too large,
so {\em size\/} was set at five for all of the runs.

The virulences of the parasites were sampled after a specified number of
time steps.  In order to determine an appropriate number of time steps,
the model was run a few times under different dispersal conditions for
extremely long periods of time (e.g., 10,000 time steps) and the average
virulence was observed over time.  Virulence seemed to level out after
about 4,000 time steps in most cases, so the sampling time was set to be
5,000 time steps.  At the end of this period of time, the virulence for
each parasite in the population was saved in a file for later analysis.

The independent variable, {\em dispersal distance\/}, ranged from 1 to
15.  If the {\em dispersal distance\/} was 1, parasites could only
transmit their offspring to the nearest neighbor hosts.  In contrast,
with a {\em dispersal distance\/} of 15, a parasite could transmit its
offspring anywhere in the environment.  The model was run under a {\em
dispersal distance\/} of 1, 2, 4, 8, and 15.  For each of these
possible settings the model was run five times using different random
number generator seeds.  In total, data was collected for 25 runs of the
model.



\subsection{Results}

All of the runs of the models produced oscillations in the host and
parasite populations that were stable enough that neither population
crashed in the sampled interval.  However, the resulting sizes of the
parasite populations varied from about 125 to 250 at the 5,000 time step
mark.

An analysis of the variance in the resulting populations indicated that
dispersal had a statistically significant effect on the evolved
virulence of the parasites ($p < 0.0001$), as did the seed for the
random number generator ($p < 0.05$).  The line of best fit for the
$virulence = 594.23 + 0.91 dispersal - 0.11 seed$.

\section{Discussion}

Why is the parasite that causes the common cold less deadly than the
protozoa that causes malaria (genus {\em Plasmodium\/}) or the bacteria
that cause cholera ({\em Vibrio cholerae\/})?  Perhaps the differences
in virulence depend in part on the differences in the dispersal
mechanisms of the parasites.  The model gives some evidence that the
distance over which a parasite transmits its offspring has a significant
effect on the parasite's evolution of virulence.  However, according to
the model, this effect is relatively small.  This is probably due to the
fact that selection pressure against a high virulence is relatively
small compared to the advantage of rapid reproduction in the short
dispersal distance conditions.  High virulence boosts a parasite's
reproductive success primarily by allowing the parasite to
reproduce quickly.  Only on a secondary level does it reduces the
resources in the area upon which the parasite's offspring must depend.
Even if a parasite kills off its host, the parasite's offspring could
have been transmitted to any of that host's eight neighbors.  Thus, a
high virulence only removes a fraction of the resources available to its
offspring.

It is likely that the details of the life history of a real biological
parasite would have a much stronger effect on its evolution of virulence
than just its dispersal distance.  For example, many parasites of humans
are transmitted through the close physical contact of medical workers with their
patients.  The medical workers carry the parasites from
patient to patient.  Even though
dispersal distance is relatively short in this case, it is to the parasite's 
reproductive advantage to have a severe effect on its human host, so that the 
host is taken to the hospital.

One oddity of the model ought to be addressed.  It was observed that a
high value for {\em size\/} (e.g., 100) caused the populations to crash
for most settings of the {\em dispersal distance\/} parameter.  That is,
when many parasites could infect a host, the populations tended to
crash.  As {\em size} was adjusted downwards, the populations began to
enter stable oscillations.  This is probably due to the resultant
infection rate in the population of hosts.  Once a host has been
infected, its days are numbered.  The parasites begin draining off
energy making it less likely that the host will be able to reproduce.
Even if it does manage to reproduce, half of its parasites are
transmitted to the offspring.  Thus, if there are many parasites in a
host, the only introduction of uninfected hosts must come from the
reproduction of other uninfected hosts.  Meanwhile, parasite virulence
levels evolve to the point of draining all of the host's food in one
time step.  A large {\em size\/} parameter means that parasites can
reproduce relatively easily.  Thus a single infected host can be the
staging ground for many new infections in a single time step.  In this
way, a parasite population can sweep through the host population until
all the hosts are infected.  Given even moderate levels of virulence,
this spells disaster for both species.

In contrast, when the {\em size\/} parameter is small, only a few
parasites can infect a single host.  The result is that there is some
chance that no parasite will be transmitted to an infected host's
offspring.  More importantly, a single infected host can only be used as
a staging ground for the transmission of a few new parasites each time
step.  Thus, infection rates in the host population are lower.  This maintains a
resource of uninfected hosts producing uninfected offspring.  The high
levels if infection in the host population is probably the effect of two
dynamics.  First, it is
relatively easy for a parasite to transmit its young to a new host.  In
fact, the survival of a parasite during transmission is almost 100 percent.
Second, the hosts lack an immune system which might also serve to
reduce infection rates in the host population. 

The model is highly simplified.  In the biological world, a parasite's
mode of transmission is not fixed, and may evolve along with virulence.
It is also clear that there are many complex interactions between real
parasites and the host's immune system, as well as between different
parasite species, not to mention other factors \cite{Begon:1990}.  These
effects are likely to be dramatic compared to the effect of dispersal
distance on virulence.  {\samepage However, it was the hope that by excluding these
factors, and thus considering all things to be equal, it may be possible
 to begin to pick apart the complexities of the evolution of virulence in
parasites.}



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