Agent-Based Modeling of Physical

Factors That May Control the Growth

of Coccidioides (Valley Fever Fungus)

in Soils

Mark GettingsU.S. Geological Survey, 520 N. Park Ave. Rm 355

Tucson, AZ 85719, USA.(mgetting@usgs.gov)

Frederick S. FisherU.S. Geological Survey, 520 N. Park Ave. Rm 355

Tucson, AZ 85719, USA.(ffisher@email.arizona.edu)

1 Abstract

A model of the spread and survival of the fungus Coccidioides in soil viawind-borne spore transport has been completed using public domain agent-based modeling software. The hypothetical model posits that to successfullyestablish a new site that has never hosted the fungus previously, four factorsmust be simultaneously satisfied. 1) There must be transport of spores froma source site to sites with favorable soil geology, texture, topographic aspect,and lack of biomass competition (favorable ground). 2) There must be sufficientmoisture for fungal growth. 3) The temperature of the surface and soil must befavorable for growth. Finally, 4) the temperature and moisture must remain infavorable ranges for a long enough time interval for the fungus to grow downto depths at which spores will survive subsequent heat, aridity, and ultravioletradiation of the hot, dry climate typical of the Southwest U.S. Using agent-basedmodeling software, a model was built so that the effects of combinations of thesecontrolling factors could be evaluated using realistic temperature, rain and windmodels. The rain probability and amount, temperature annual and diurnalvariation, and wind direction and intensity were based on the weather recordsat Tucson, Arizona for the 107-year period from 1894 to 2001. Favorable groundwas defined using a fractal tree algorithm that emulates a drainage network inaccordance with observations that favorable sites are often adjacent to drainagechannels.

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Numerous model runs produced the following five conclusions. 1) If anyproperty is not isotropic, for example wind direction or narrow paths of rain-storms, parts of the favorable areas will never become colonized no matter howlong the model runs. 2)The spread of new sites is extremely sensitive to mois-ture duration. The amount of wind and temperature after a rain event controlsthe length of time before a site becomes too dry. 3) The distribution of windand rainstorm direction relative to that of the favorable sites is a strong controlon the spread of colonization. East-west winds across an area that has mostlynorth-south favorable sites restricts spread strongly. 4) Soil temperature wasthe least sensitive control in the model, although it does control the ultimatedormancy of a site. And 5), the model results cover the spectrum of completecolonization of all favorable sites from a few source sites to none, one, or twonew sites in three years of model simulation. This implies the probability of newsites depends on the four factors in a Bayesian way. These results indicate thatthe complexity introduced in the model from site favorableness, temperature,moisture, and duration of favorable temperature and moisture conditions pro-duces a set of model distributions that are adequate to explain what is knownof the distributions of real sites. Although the actual distribution of real sitesis unknown, the properties of the set of model distributions cover the knownproperties of real site distributions.

2 Introduction

The fungus Coccidioides is prevalent in the American southwest and in otherareas with similar climatic conditions (Bultman, et al 2005; Fisher et al, 2007)and is a source of the potentially life-threatening disease commonly referred to asValley Fever (Bultman et al, 2004b). This dimorphic fungus spends a portion ofits life cycle in soils (Papagianis, 1980, 1988), and thus the application of geologictechniques for the discovery of spatial and temporal occurrence relationships isa method of investigation that might lead to new means of disease mitigation.Although there are two known species of Coccidioides, Coccidioides immitis andCoccidioides posadasii (Fisher et al, 2002), both are believed to cause humanCoccidioides infections (Fisher et al, 2001; 2002a), and thus for this study, thegenus Coccidioides will be used rather than a species-specific term. The fungushas two forms that can be airborne, the artroconidia that does not survive forlong periods of time and relatively extreme conditions of temperature, humidity,and ultraviolet radiation, and the spherule, which is much more resistant andcan survive long periods of time, especially in the soil (Pappagianis, 1988). Forbrevity in this report, we will refer to the spherules as ”spores”.

The purpose of this work was to evaluate the complex interactions betweentemperature, precipitation, wind and favorable soil properties as possible con-trols on the survival and growth of the fungus Coccidioides in the soil. Models offungal growth have been discussed by Knudsen and Stack (1991) among others,and Bengtsson and Ekere (2001) have developed models useful for evaluating

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the interaction of fungus with soil particles. Several studies have enumeratedpossible factors that might be important in the growth of the fungus (Maddy,1957, 1957a, 1965; Maddy et al, 1961; Lacy and Swatek, 1974, and Papagianis,1980, 1988). Among those factors are soil temperatures, humidity, and composi-tion. The present study was undertaken to model the interactions among thesethree physical factors together with wind-borne transport to evaluate possiblecontrols on the distribution in space and time of sites that might harbor thefungus. Gettings and Fisher (2003) and Gettings et al (2005) have presentedposter versions of this modeling effort.

A model of the spread and survival of the fungus Coccidioides in soil viawindborne spore transport has been completed using public domain agent-basedmodeling software (NetLogo, Wilenskey, 1999). The hypothetical model sup-poses that for a successful new site to become established, four factors mustbe simultaneously satisfied. First, there must be transport of spores from anactive source site to locations on ground possessing soil chemistry, texture, to-pographic aspect, and lack of biomass competition such that the fungus canthrive, termed favorable ground in this report. Second, there must be sufficientmoisture for fungal growth. Third, the temperature of the surface and soil mustbe favorable for growth. Fourth, the temperature and moisture must remain infavorable ranges for a long enough time interval for the fungus to grow down todepths at which spores at least will survive subsequent heat, aridity and ultra-violet radiation of the hot, dry season. Using agent-based modeling software,a model was built so that the effects of combinations of the controlling factorscould be evaluated using realistic temperature, precipitation, and wind mod-els. The precipitation probability and amount, temperature annual and diurnalvariation, and wind direction and intensity were based on the weather records atTucson, Arizona, for the 107-year period from 1894 to 2001. Favorable groundwas defined on a rectangular grid as a simple binary relation. Grid cells areeither favorable or unfavorable and were chosen arbitrarily using a fractal treealgorithm that emulates a drainage network.

Maddy (1965) and Maddy and Crecelius (1967) have reported establishmentof a persistent site of Coccidioides for at least 7 years following burial of infectedtissues, and detection of the fungus in previously negative soil 64 m away withinthe 7 year study period. Ajello et al (1965) have recovered Coccidioides fromthe air 3-4 days after a windstorm in the Phoenix, Arizona area. Egeberg andEly (1956) noted that during the dry season, many more positive cultures wereobtained from samples taken at soil depths of 10-30 cm compared to surfacesamples, showing that the fungus survival was much lower in near surface soil.Pappagianis (1988) reviewed considerable evidence suggesting that neither thegrowing fungus nor its spores can survive for long periods in dry, hot soil nor inultraviolet light. Temperatures in excess of 50deg C for periods of 2 weeks killspores but temperatures of 15deg C-37deg C have survival times of 6 monthsor more provided some moisture is present (relative humidities of 10% or more)

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(Stewart and Meyer, 1932; Pappagianis, 1988). Swatek et al (1967), amongothers, have shown that banks of dry stream beds and washes often host thefungus, and this observation is the basis for our use of a drainage network asmodel for the distribution of favorable ground.

The model assumes that an existing site, located in a favorable ground gridcell, has at least viable spores surviving at soil depths of about 20 cm, deepenough to avoid ultraviolet radiation and temperature and moisture extremes.If an existing site does not have proper moisture and temperature for growthto occur, it remains dormant and is termed a ”dormant site”. When sufficientmoisture reaches the spores and temperatures are in a favorable range, thespores begin to grow, and the fungus grows upward toward the surface. Ifthe temperature and moisture conditions remain favorable, the fungus growsto the surface, produces arthrocondidia that can become windborne, and isthen termed a ”sporing” site. Windborne arthroconidia can then be blown toother grid cells, and if they are on favorable ground, the fungus can start togrow in that cell, which is termed ”colonized” or a ”colonized site”. If thetemperature and moisture at the ”colonized site” remain favorable long enoughfor the fungus to grow down to depths of approximately 20 cm, the cell becomesa new (sporing) site. If it dries up and (or) gets too hot, the fungus dies and thecell returns to just being favorable ground. When temperature and moisturebecome unfavorable at a sporing site, it again becomes dormant. Thus, oncea cell becomes a sporing site, it will henceforth always be either dormant orsporing. The extrinsic parameters controlling the model are therefore the timeto grow from depth in the soil to the surface and produce arthroconidia, thetime to grow from the surface down to depth and produce spherules (spores), themaximum number of grid cells an arthroconidia can be blown by the wind, thetemperature above which the fungus cannot survive, and the minimum amountof moisture for growth to or from the surface. Growth time from depth to thesurface and from the surface to depth are assumed equal in this model. Becausea grid cell is either favorable or unfavorable, the model makes no allowance forpartially favorable conditions.

3 Temperature Model

Temperature and precipitation models for this study were based on the 107-year daily record for Tucson, Arizona for the period September 1894 through Au-gust 2001 downloaded from the National Climate Data Center (NOAA, 2007).The data included date, maximum temperature, minimum temperature, pre-cipitation, snowfall, and snow depth. The dataset was processed to eliminaterecords with missing data or bad data (impossible values), estimate the meandaily temperature as the average of the maximum and minimum, snowfall wasconverted to precipitation as water using the factor of 10 in. of snow equalto 1 in. of water, and time values convenient for plotting were added. Traceamounts were entered as 0.01 in. rain to obtain an entirely numerical dataset.

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The dataset is available in appendix 1. Figure 1 shows the average daily tem-perature for each day of the year with a least squares cosine fit superimposed.From this fit, the mean annual temperature variation is 38deg F (21.1deg C).Figure 2 shows the daily temperature range plotted as a function of the amountof precipitation. The mean daily temperature range is 29.8deg F (16.6deg C)for these data.

Figure 3 shows hourly mean temperature data for the month of July 2002 and1 February to 12 March 2003 plotted normalized to the maximum and minimum.These data were downloaded from the University of Arizona weather site (2007)and taken to be representative of summer and winter diurnal variations. Figure4 show that the average normalized winter and summer temperatures are nearlyidentical. The least squares fit cosine line shown on Fig. 3 shows that there isassymmetry in the heating and cooling parts of the diurnal cycle; the warmingpart of the cycle is more rapid than predicted by the cosine whereas the coolingis slower. For this study, the cosine function was assumed to be adequate, andthe assymmetry was ignored.

The temperature model used is that of an infinite half-space with a bound-ary condition of a cosine shaped boundary temperature. Two periods are used,one with a 24 hour period, and one with a 365 day period. Rainstorm cool-ing is modeled by adding the solution for a linear temperature variation withtime for the 4-8 days needed to recover from a rainstorm. By the principle ofsuperposition we have

T (x, t) = T0 + Ta(x, t) + Td(x, t) + Tr(x, t) (1)

Where T0, Ta, Td, and Tr are respectively the constant mean temperature, theannual cyclic temperature, the diurnal cyclic temperature, and the temperaturedrop due to a rain event. The mean temperature T0 for the 1894-2001 Tucson,Arizona dataset is 68.65deg F (20.36deg C). Carslaw and Jaeger (1959) give thesolution for temperature at soil depth x and time t in an infinite half-space withharmonic surface temperature as

T (x, t) = Ae−kxcos(ωt− kx− ǫ) (2)

whereT (0, t) = Acos(ωt− ǫ) (3)

andk = (ω/2κ)1/2 (4)

Here, κ is the thermal diffusivity, ω = 2π/T is the frequency with period T ,and ǫ is the phase shift to fit the data relative to clock time. A is an amplitudeconstant. The annual variation is then

Ta(x, t) = Aae−k1xcos(

2πt

365(24)(3600)− k1x− ǫa) (5)

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and the daily variation is

Td(x, t) = Ade−k2xcos(

2πt

(24)(3600)− k2x− ǫd) (6)

k2 = (2π

(2)(24)(3600)κ)1/2k1 = k2/(365)

1/2 (7)

From the 1894-2001 data above, Aa is 19deg F (10.6deg C) and Ad is 14.9degF (8.3deg C), noting the cosine function varies from +1.0 to -1.0.

From the 1894-2001 dataset, Figure 5 shows the average temperature versusthe amount of precipitation on that day, and Figure 6 shows the temperaturedrop the day after precipitation measured as the difference in the average tem-perature the day after precipitation and the day of precipitation. The approx-imate mean drop is about 5deg F (2.8deg C). Figure 7 shows the temperaturefor June and July, 2002, along with the precipitation record. Study of thisand numerous other records shows that there is a temperature drop of at mostapproximately 1/3 of the daily temperature range immediately after the precip-itation event, followed by an approximately linear increase as the ground driesout (Fig. 7). The minimum daily temperature is least affected by the rain, andthe maximum the most. Recovery time varies from about 2 days for events ofless than 3mm precipitation to about 6 days for precipitation greater than 3mm.For these conditions, the temperature distribution for a half-space with initialtemperature zero and surface temperature Tdt increasing with time we obtain(Carslaw and Jaeger, 1959)

Tr(x, t′) = −Td +

Tdt′

D

[

(1 +x2

2κt′)erfc(

x

2(κt′)1/2)−

xe−x2/4κt′

(πκt′)1/2

]

(8)

where D is the duration of the recovery from rain cooling and t′ is the timesince the rain event. The rain cooling term is only evaluated during the recoveryperiod D.

An example of the combined temperature model for a rain event of 6 days isshown in Figure 8 and and its effect in an annual cycle in Figure 9. Althoughthe model is too regular, it nevertheless reproduces the essential features of thetemperature record.

4 Precipitation Model

Precipitation (including snow converted to rain) for the daily record 1894-2001 is shown as a function of time in Figure 10. At this scale, the distributionis rather uniform with no strong trends, implying that for this study the precip-titaion can be modelled on the basis of day of the year with no need for a long

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term trend. Figure 11 shows the same data plotted as a function of day of theyear. The winter wet season (approximately days 320 to 110) and the summermonsoon (approximately days 180 to 260) are quite obvious. The spring dryseason from days 110 to 180 is well delineated, but the fall dry season (days260 to 320) is less obvious in the record. The frequency distribution of precipi-tiation amplitude is shown in Figure 12 and shows a smooth exponential-likedecay. The probability of precipitation of any amplitude versus day of the yearcalculated from the data of Fig. 11 is shown in Figure 13. Note this patternis similar to that of fractional brownian motion with persistence (Feder, 1988),a clear demonstration of the fractal or multifractal nature of the likelihood ofprecipitation (Schertzer et al, 2002).

Figure 14 shows the model probability functions for precipitation as a functionof amplitude of precipitation for the two wet and two dry seasons derived fromthe 1894-2001 dataset. Histogram versions of these functions, together with theprobability of precipitation (Fig. 13) were used in the model to determine theoccurrence of precipitation and, if nonzero, its amplitude.

5 Wind and Favorable Ground Model

Figure 15 is a record of wind direction and speed for the period 24 October- 20 November, 2003, at the University of Arizona, Tucson, Arizona. From therecords reviewed, Fig. 15 is fairly typical for the Tucson area for the last decade.Note that there is generally an approximately 90 degree band for a given day, andthat speed is most frequently 0-5 m/sec range, and least frequently 15-20 m/sec.Thus, similar distributions were used in the model to select wind direction andspeed. Within the model one can choose a preferred wind direction and rangeto model trade wind type behavior, or isotropic wind directions can be chosen.Within the selected range, the wind direction is chosen from a uniform randomdistribution.

Favorable ground (as defined above) in the model was chosen arbitrarily us-ing a fractal L-system tree of four iterations (Wilensky, 1999). This models adrainage network, corresponding to the observation that sites known to produceCoccidioides often occur adjacent to local watercourses (Swatek et al, 1967). Atthis stage of modeling, we only distinguish favorable ground in a binary sense,that is, a patch is either favorable or not favorable. An important extension ofthe model will be to incorporate a favorableness function so that patches thatare only partially favorable will allow the fungus to grow only at a slower rate.Bultman et al (2004, 2004a, 2005) have created a mosaic of favorable groundbased on fuzzy adaptive maps combining criteria known to control the incidenceof Coccidioides. Fisher et al (2007) have given a careful review of factors thatmay control favorable ground and conclude that temperature range and finesandy or silty soils are important factors at known sites of Coccidioides. These

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results were not available until well after this model study (2003), but they willbe incorporated in any future modeling.

6 Agent-Based Model

The model was developed using NetLogo software, a publicly available agent-based modeling software package from Northwestern University (Wilensky, 1999).This software combines simple and flexible coding to keep track of agent setswith the ability to do fairly complex calculations. The Coccidioides ecosystemmodel is fairly complex and involves several criteria for all agents. The modelextent is usually a 100x100 cell grid with east-west measured by columns androws increasing north-south. Thus, north is the top of the grid as in a nor-mal map view in the screenshots of the following figures. Moisture content ofa grid cell of the ground (”patch” in NetLogo) is variable for every grid cell.Moisture content is modeled by keeping track of the amount of rain on eachgrid cell and calculating the heat flow from the grid cell. The moisture contentof a grid cell is thus increased by rain on the cell and decreased by heat flowcausing evaporation. All grid cells have fixed soil thermal properties. A uniformtemperature distribution for all grid cells was adopted because the inclusion ofthe temperature drop equations defined above made the model run unaccept-ably slow. In this model, the grid cells are the immobile agent set of favorableground (yellow, Figure 16) or unfavorable ground (black) with the same dailyand annual temperature variations. The model keeps track of the temperatureat the surface, 1 cm and 20 cm depths. Heat flow is calculated using the thermalgradient defined by the surface and 1 cm depth temperatures and subject to aboundary condition of 3 times the surface conductive heat flux to simulate theeffect of forced convection at the soil surface. Mobile agent sets (”turtles” inNetLogo) are wind and precipitation. Wind can have a preferred azimuth andrange of azimuths to simulate trade winds. Precipitation in the model occurs asa rainstorm with an azimuth and a specified width. Rainstorm azimuths wereisotropic in this model. Wind occurrence, direction, and intensity are chosenrandomly from the observed probability distributions for the Tucson, Arizonaarea, as are precipitation occurrence and amplitude. The parameters that areset by the user for a model run are as follows together with typical values. Thenumber of existing dormant sites on favorable ground is typically 20. The max-imum temperature for long term survival at 20 cm depth in the soil is usuallychosen as 40deg C. The maximum number of grid cells a spore (arthroconidia)can jump with maximum wind velocity is chosen, usually in the range of 1 to5 grid cells. The minimum moisture content for growth is chosen, usually inthe range of 0.2 to 0.5 in (0.5 to 1.2 cm). The minimum time for growth tothe surface from 20 cm or growth down from the surface to 20 cm (they areassumed equal in this simple model) is specified in days, usually 14-30 daysbased on observations at some sites (Pappagianis, 1988). Finally the preferredazimuth for wind direction blown to and the angular range about that directionare specified. Typical values are 0, for winds from south to north and a 90deg

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angle gives winds from southeast to southwest. An angle range of 360deg givesisotropic wind directions. The width in grid cells of a rainstorm swath is alsospecified; 15 grid cells is a typical width. Finally, the fade rate for rainstormswaths on the map display is specified so that older rainstorm paths vanish;otherwise, the map display gets unacceptably cluttered.

The model starts with a given number of dormant sites that will becomeactive sources of spores when they receive enough moisture and have the correcttemperature for a growth period corresponding to growth from spores beneath20 cm in the soil up to the surface. When this occurs for a dormant site (shownas brown in the model) it becomes a ”sporing” site (shown as red in the model).A site becomes ”colonized” when the wind blows from a sporing site to a siteon favorable ground, and there is at least a minimum amount of moisture inthe ground at that site and the temperature must be below the maximum forgrowth of the fungus. The colonized site will continue to grow until it runsout of moisture or the temperature becomes too warm for growth. If the sitehas grown long enough for the fungus to penetrate the soil to 20 cm or deeper,the colonized site becomes a ”sporing” site, that is, a site that will reactivateand after a growth interval produce spores at the surface. A colonized site dies(goes back to yellow) if it does not have enough moisture or gets too hot beforethe growth interval to become a permanent site passes. A sporing site becomesdormant when there is too little moisture or the temperature gets too high,killing the fungus down to 20 cm in the soil. The code for the netlogo model isattached as appendix 2.

7 Model Results

Models were initially run for about 1 year (Fig. 16), but we soon realizedmore information was gained by running the model for multiple years, so mostmodels were run for 10 to 20 years. A twenty-year model simulation takes about10 hours on a 1.5 GHz PC. Nearly 100 models have been run.

Output consists of: 1) an image (jpeg format) of the Graphical User Interfaceshowing the parameter values, the state of the favorable ground, and plots overmodel time of the numbers of dormant sites, active (sporing) sites, colonizedsites, precipitation, and temperature; and 2) a comma-delimited file of the plotsthat defines all parameter setting and contains the numerical data used to gen-erate the plots. The present netlogo software only allows runs up to about 23years because of memory limitations.

Fig. 16 shows an approximately one year long simulation with winds blowingfrom the south in an 89deg wide band (used in most simulations), that is, inrandom directions from 135deg through north to 224deg azimuth. Note thatthe branches of the favorable ground ”tree” aligned approximately north-southexperienced the largest number of new sites. Branches perpendicular to the

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prevailing wind experienced only little growth of number of sites. Parametersin this simulation were set so that the fungus survival rates would be high, thatis a high soil temperature that kills the fungus, a relatively small amount of soilmoisture for survival, and a fairly high growth rate to form a site. This results inan order of magnitude increase in the number of sporing or dormant sites from20 to about 200 within a year (note the red ”sporing” and brown ”dormant”lines on the plot in Fig. 16), much higher than the rate at which the evidencesuggests the fungus spreads (Pappagianis, 1988). Figure 17 shows a three yearsimulation with more stringent temperature and moisture survival conditions,and winds prevailing from the west, resulting in only about a five-fold increasein sites in three years. Figure 18 shows a 17 year simultion with similar tempera-ture and moisture survival conditions and westerly winds again. The number ofsporing or dormant sites increases from 20 to about 175 over the entire period,but mainly only in four events when moisture, favorable ground, temperature,and wind combined optimally. Figure 19 shows a 9 year simulation with fa-vorable conditions: high temperature and low moisture for survival and windfrom the south maximizing favorable ground availability to wind-borne colo-nization. The total number of sporing or dormant sites increases approximatelyas a logarithmic function of time.

Figure 20 shows a 20 year simulation with fairly restrictive temperature andmoisture conditions for fungus survival and in addition a wind direction re-stricted to a 60deg wide band centered on south. This case exhibits again the”staircase” behavior of the increase in number of sporing or dormant sites, withlong periods of no change in total number of sites. Figure 21 shows a similarcase but with less restrictive temperature and moisture conditions and windin a 90deg band from the east. In this case, two events at about 1.7 and 2.7years dominate the growth of sporing or dormant sites, with a slow ”staircase”growth for the rest of the period. Note also that the growth is mainly in aneast-west direction (brown, green and red cells of the favorable ground tree inFig.21), illustrating the relation between the geometry of the favorable groundrelative to existing sites and the wind direction, since it is the vector for growth.Finally, Figure 22 shows another 20 year simulation with somewhat less restric-tive survival parameters than Fig. 20, but only a 60deg range of winds from thesouth. The total number of new sporing or dormant sites is about the same,but the growth pattern is much more ”linear” because of the occurrence of morefavorable wind, rain and temperature events, emphasizing the stochastic natureof these occurrences.

For analysis, output images are cataloged on a website in a table with theimage name linked. Thus one can scan the table for desired parameter combi-nations and see the image instantly. Using several browsers at once allows quickvisual comparisons and enables detailed study of the model runs in a convenientway.

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When all four conditions are satisfied, an increase in the number of sites isobserved. Each of these events represents a case of self-organized criticality(SOC) (Jensen, 1998). The frequency of SOCs depends upon the values of thecontrolling parameters (principally minimummoisture, number of days required,and maximum tolerable temperature) relative to the climate (precipitation andtemperature) of the model.

The model achieves climatic variability very similar to that observed in theclimate dataset for Tucson for the period 1894-2001. There are dry years, wetyears, and multi-year wet and dry periods. This is likely due to the use ofprobability distributions from the observed data for precipitation occurrence andamplitude. It is unlikely that the actual numbers of sporing or dormant sites orgrowth rates produced by the model are correct because of the simplified natureof the model. However, the relative increases and decreases from different modelruns provide insight into the controls of the possible spread of new dormant sitesby the mechanisms of this model. Known infectious sites are relatively scarce(Bultman et al, 2005; Fisher et al, 2007) whereas the model predicts quite a largenumber of sites. Jammalamadaka et al (2007), using more realistic favorableground models and growth rates, report a much reduced rate of increase of sites.

The model SOC events tend to occur in early spring and post-monsoon peri-ods, similar to the observed time of maximal occurrence of Valley Fever (Pap-pagianis, 1988). This is not built into the model in any way that we are awareof, and represents an encouraging result. We believe this behavior is due to in-creased probability of simultaneous satisfaction of the four criteria during thesetwo periods. Figure 23 shows a plot of the total number of sporing or dormantsites for most of the model runs. Note the ”Devil Staircase” appearance of eachof the curves for each model run, suggesting fractal behavior as a generalizedmultiplicative process (Feder, 1988). There is an appearance of clustering attimes longer than 10 years in Fig. 23, suggesting that bifurcation (Turcotte,1997) may be occurring but there is an insufficient number of model runs toestablish this.

8 Conclusions

Any anisotropy (distribution of favorable ground, prevailing wind direction,etc.) reduces the rate of formation of new dormant or sporing sites. More-over, the anisotropy causes certain parts of the area to never be subject to newsite formation. This is a robust conclusion shown by every long term modelcomputed.

Models generally exhibit Devil Staircase behavior, that is, they persist witha fixed number of dormant or sporing sites for several years, and then a SOCevent occurs and more dormant or sporing sites are produced.

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The rate of increase of number of sites (dormant or sporing) is sensitive tomoisture duration and maximum temperature for survival.

The correspondence between the pattern or distribution of favorable groundand wind and rain direction is a strong control on the rate of increase of sites.

A mechanism for limiting the spread of new dormant or sporing sites must ex-ist; otherwise all favorable ground would be a site after a few decades. We modelthis by the maximum temperature value, which is a proxy for the combinationof high temperature and, possibly ultraviolet radiation.

Future modeling will include different patterns of favorable ground based onfuzzy logic models of known factors (Bultman et al, 2005), continuous ratherthan binary growth rates on favorable ground, and using actual temperatureand precipitation records for each year rather than probability distributions.Jammalamadaka et al (2005 and 2007) have reported results of efforts to im-plement versions of the model incorporating some of these improvements on acomputer cluster in order to speed up the model runs sufficiently for parametervariation studies. Ultimately, we will test whether Bayesian probability modelscan be used to set limits on the rate of new dormant or sporing sites, that is,whether the product of the prior probabilities for wind, precipitation, favorableground and duration can predict rates of new site formation. Based on themodel results presented here, we believe the Bayesian probability approach willdefine maximum rates of site formation.

9 Acknowledgments

The work described here was carried out under the medical geology task ofthe the Complex Systems Applied to Basin Margins of the American SouthwestProject, Mineral Resources Program, U. S. Geological Survey. This manuscripthas benefited greatly from discussions and technical reviews by M. Bultman andF. Gray, U.S. Geological Survey. Publication of this report has been approvedby the Director of the U. S. Geological Survey.

Note Added in Proof Subsequent to completion of this report R. Jammala-madaka completed his dissertation, Jammalamadaka (2008), which includes fur-ther modelling of the scenario presented here. His modeling included continu-ously variable favorable ground rather than binary as used in this report, anda diffusion-controlled model of soil moisture calibrated by observations of soilwetting and drying in the Tucson, Arizona area.

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[] Fisher, F.S., M.W. Bultman, S.M. Johnson, D. Pappagianis, and E. Zaborsky(2007), Coccidioides niches and habitat parameters in the SouthwesternUnited States, a matter of scale, Ann. N.Y. Acad. Sci. 1111, 47-72.

[] Fisher, M.C., G.L. Koenig, T.J. White, G. San-Blas, R. Negroni, A.I.Gutierez, B. Wanke, and J.W. Taylor (2001), Biogeographic range expansioninto South America by Coccidioides immitis mirrors New World patterns ofhuman migration, Proc. Nat. Acad. Sci. USA 98(8), National Academy ofSciences, Washington, D.C.

[] Fisher, M.C., G.L. Koenig, T.J. White, and J.W. Taylor (2002), Molecularand phenotypic description of Coccidioides posadasii sp. nov., previously rec-ognized as the non- California population of Coccidioides immitis, Mycologia94(1) 73-84.

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[] Fisher, M.C., B. Rannala, V. Chaturvedi, and J.W. Taylor (2002a), Diseasesurveillance in recombining pathogens: Multilocus genotypes identify sourcesof human Coccidioides infections, Proc. Nat. Acad. Sci. USA 99 9067-9071.

[] Gettings, M.E., and F.S. Fisher, (2003), Agent-Based Modeling of PhysicalFactors That May Control the Growth of Coccidioides immitis (Valley FeverFungus) in Soils: Eos Trans. AGU, 84(46), Fall Meet. Suppl., Abstract B21B-0713.

[] Gettings, M. E., F.S. Fisher, and M.W. Bultman (2005), Agent-Based Model-ing of Physical Factors That May Control the Growth of Coccidioides (ValleyFever Fungus) in Soils: 1st All-USGS Modeling Conference, 15-17 November,2005, Port Angeles, WA.

[] Jammalamadaka, R. (2008),Multilevel Methodology for Simulation of Spatio-Temporal Systems with Heterogeneous Activity: Application to Spread ofValley Fever Fungus: PhD dissertation, Electrical and Computer EngineeringDept., University of Arizona.

[] Jammalamadaka, R., J. Nutaro, M. Gettings, B. Zeigler (2005), DEVS Re-Implementation of an Agent-Based Valley Fever Model, 2005 DEVS Integra-tive Modeling and Simulation Symposium, 2005 Spring Simulation Multicon-ference, SpringSim’05, San Diego.

[] Jammalamadaka, R., M. Gettings, M. Bultman,B. Zeigler, and MingZhang (2007), Complex System Simulation: DEVS Implementation ofthe valley fever model: 2007 DEVS Integrative Modeling and Sim-ulation Symposium (DEVS’07) Spring Simulation Multiconference2007 (SpringSim’07), March 25 - 29, 2007, Norfolk, Virginia, USA,http://www.scs.org/confernc/springsim/springsim07/finalProgram/finalProgram.htm

[] Jensen, H.J. (1998), Self-Organized Criticality, Emergent Complex Behaviorin Physical and Biological Systems, 153p., Cambidge Univ. Press.

[] Knudsen, G., and J. Stack (1991), Modeling growth and dispersal of fungiin natural environments, in Handbook of Applied Mycology, vol. 1: Soils andPlants, Eds. D.K. Arora, B. Rai, K. Mukerji, and G. Knudsen, 625-645, Mar-cel Dekker, Inc., New York.

[] Lacy, G. H. and F.E. Swatek (1974), Soil ecology of Coccidioides immitis atAmerindian middens in California, Appl. Microbiol. 27, 379-388.

[] Maddy, K. T. (1957), Ecological factors of the geographic distribution ofCoccidioides immitis, J. Am. Vet. Med. Assoc.130, 475-476.

[] Maddy, K.T. (1957a), Ecological factors possibly relating to the geographicdistribution of Coccidioides immitis, Proceedings of the Symposium on Coc-cidioidomycosis. U.S. Public Health Service, Pub. no. 575, 144-157, CDC,Atlanta, GA.

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[] Maddy, K.T. (1965), Observations on Coccidioides immitis found growingnaturally in soil, Arizona Medicine 22, 281-288.

[] Maddy, K.T., and G.T. Crecelius (1967), Establishment of Coccidioides im-mitis in negative soil following burial of infected animals and animal tissues,in Coccidioidomycosis, L. Ajello, Ed., University of Arizona Press, Tucson,Arizona, 309-312.

[] Maddy, K.T., H.G. Crecelius, and R.G. Cornell (1961), Where can coccid-ioidomycosis be acquired in Arizona, Ariz. Med. 18, 184-194.

[] NOAA (2007), National Climatic Data Center, National Oceansand Atmospheres Administration, U.S. Department of Commerce,URL:http//www.ncdc.noaa.gov, data downloaded October 18, 2001.

[] Pappagianis, D. (1980), Epidemiology of coccidioidomycosis. In Coccid-ioidomycosis D.A. Stevens, Ed. Plenum Medical Book Company, New York.63-85.

[] Pappagianis, D. (1988), Epidemiology of coccidioidomycosis. In Current top-ics in Medical Mycology, V2 Springer Verlag, New York, 199-238.

[] Schertzer D., S. Lovejoy, and P. Hubert (2002), An Introduction to StochasticMultifractal Fields, in Mathematical Problems in Environmental Science andEngineering, A. Ern and L. Weiping, editors, Series in Contemporary AppliedMathematics, pp. 106-179, Higher Education Press, Beijing.

[] Stewart, R.A., and K.F. Meyer (1932), Isolation of Coccidioides immitis(Stiles) from the soil, Proc. Soc. Exp. Biol. Med. V 29, 937-938.

[] Swatek, F.E., D.T. Omieczynski, and O.A. Plunkett (1967), Coccidioidesimmitis in California, in Coccidioidomycosis, L. Ajello, Ed., University ofArizona Press, Tucson, Arizona, 255-264.

[] Turcotte, D.L. (1997), Fractals and Chaos in Geology and Geophysics, Seconded., 398p., Cambridge Univ. Press.

[] University of Arizona (2007), Atmospheric Sciences, weather page,URL:http://www.atmo.arizona.edu/, data downloaded October 6, 2003.

[] Wilensky, U. (1999), NetLogo. http://ccl.northwestern.edu/netlogo/. Centerfor Connected Learning and Computer-Based Modeling, Northwestern Uni-versity. Evanston, IL.

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Figure 1: Average daily temperature for day of the year, 1894-2001, TucsonArizona, with cosine fit.

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Figure 2: Daily temperature range versus amount of precipitation, Tucson Ari-zona, 1894-2001.

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Figure 3: Diurnal variation of temperature, summer (pluses, July, 2002) andwinter (diamonds, February-March, 2003), Tucson, Arizona. Green line is leastsquares cosine fit. Temperature range normalized to 1.0

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Figure 4: Average diurnal temperature variation, summer and winter , Tucson,Arizona. Temperature variation normalized to 1.0

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Figure 5: Average daily temperature versus amount of precipitation, Tucson,Arizona, 1894-2001.

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Figure 6: Temperature drop the day after precipitation, Tucson, Arizona, 1894-2001.

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Figure 7: Plot illustrating temperature drop and approximately linear recoveryafter precipitation, Tucson, Arizona, 2002.

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Figure 8: Temperature model of temperature and range drop due to precipita-tion.

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Figure 9: Illustration of one rain event temperature drop at approximate timeof 7e+06 in the temperature model for one year.

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Figure 10: Precipitation versus year, Tucson, Arizona, 1894-2001

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Figure 11: Precipitation versus day of the year, Tucson, Arizona 1894-2001.Notethe two wet and two dry periods.

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Figure 12: Histogram showing frequency of precipitation amount, Tucson, Ari-zona, 1894-2001

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Figure 13: Probability of precipitation, versus day of the year, Tucson, Arizona,1894-2001. This pattern is similar to fractional brownian motion with persis-tence (Feder, 1988), which demonstrates the fractal or multifractal nature ofthe likelihood of precipitation.

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Figure 14: Model probability functions for the two wet and two dry seasons.

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Figure 15: Wind direction and speed for the period 24 October - 20 November,2003, Tucson, Arizona.

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Appendix 1- Temperature and precipitation datafor Tucson, Arizona, 1894-2001

Please see supplementary file 1894 2001TUCAZ.dat.

Appendix 2- Netlogo program code

Please see supplementary file coccicosystem.nlogo.

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Figure 16: A typical simulation of a little more than one year resulting in about240 dormant or sporing sites from the original 20. The red-shaded stripes showthe paths of recent rainstorms, and the red spots on the yellow ”tree” of favorableground are ”sporing” sites while the brown are ”dormant” sites. Below are plotsof the variables temperature (green), rain (blue), colonized (green), sporing(red) and dormant (brown) sites, and time in black in 10 day cycles. See textfor explanation.

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Figure 17: A three year simulation. See text for discussion.

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Figure 18: A 17 year simulation. Note ”staircase” form of growth of number ofdormant sites. See text for discussion.

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Figure 19: A 9 year simulation. Note logarithmic form of growth of number ofdormant sites. See text for discussion.

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Figure 20: A 20 year simulation. Note growth along predominant wind direction.See text for discussion.

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Figure 21: A 20 year simulation. Note abrupt growth in the number of sites inthe first four years. See text for discussion.

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Figure 22: A 20 year simulation. Note long periods of little growth in thenumber of sites. See text for discussion.

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Figure 23: Plot of total number of sites capable of ”sporing” for several models.Note suggestion of clustering of number of sites at about 10 yr and later.

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