MODELING PLAGUE PERSISTENCE IN HOST-VECTOR COMMUNITIES IN CALIFORNIA

MODELING PLAGUE PERSISTENCE IN HOST-VECTOR

COMMUNITIES IN CALIFORNIA

Janet E. Foley,1,2,6 Jennifer Zipser,2 Bruno Chomel,3 Evan Girvetz,4 and Patrick Foley5

1 Department of Medicine and Epidemiology, School of Veterinary Medicine, University of California, Davis,California 95616, USA2 Center for Vectorborne Disease, School of Veterinary Medicine, University of California, Davis, California 95616, USA3 Department of Population, Health and Reproduction, School of Veterinary Medicine, University of California, Davis,California 95616, USA4 Department of Landscape Biology, University of California, Davis, California 95616, USA5 Department of Biological Sciences, California State University, Sacramento, California 95819, USA6 Corresponding author (email: [email protected])

ABSTRACT: Plague is an enzootic disease in the western United States, even though long-termpersistent infections do not seem to occur. Enzootic persistence may occur as a function ofdynamic interactions between flea vectors and transiently infected hosts, but the specific levels ofvector competence, host competence, and transmission and recovery rates that would promotepersistence and emergence among wild hosts and vectors are not known. We developeda mathematical model of enzootic plague in the western United States and implemented the modelwith the following objectives: 1) to use matrix manipulation within a classic susceptibleRinfecti-veRresistantRsusceptible (SIRS) model framework to describe transmission of the plaguebacterium Yersinia pestis among rodents and fleas in California, 2) to perform sensitivity analysiswith model parameters and variables to indicate which values tended to dominate model output,and 3) to determine whether enzootic maintenance would be predicted with realistic parametervalues obtained from the literature for Y. pestis in California rodents and fleas. The modelPlagueSIRS was implemented in discrete time as a computer simulation incorporatingenvironmental stochasticity and seasonality, by using matrix functions in the computer languageR, allowing any number of rodent and flea species to interact through parasitism and diseasetransmission. Sensitivity analysis indicated that the model was sensitive to flea attack rate, hostrecovery rate, and rodent host carrying capacity but relatively insensitive to changes in the durationof latent infection in the flea, host and vector competence, flea recovery from infection, and hostmortality attributable to plague. Realistic parameters and variable values did allow for the model topredict enzootic plague in some combinations, specifically when rodent species that weresusceptible to infection but resistant to morbidity were parasitized by multiple poorly competentflea species, including some that were present year-round. This model could be extended to similarvectorborne disease systems and could be used iteratively with data collection in sylvatic plaguestudies to better understand plague persistence and emergence in nature.

Key words: Community modeling, plague, SIRS model, Yersinia pestis.

INTRODUCTION

Yersinia pestis, the agent of plague, isa highly virulent pathogen that persists inpoorly understood ecological foci in thewestern North America, central and South-east Asia, parts of Africa, South America,and the Middle East. Plague was firstintroduced into the New World duringthe Third Pandemic (in the late 19th andearly 20th centuries) via infected rats,Rattus rattus, and rat fleas, Xenopsyllacheopis, through the port of San Francisco,California, USA (Link, 1955). The firsthuman case in Chinatown, San Francisco,in 1900 was followed by epidemics in

California in 1907, 1919, and 1924 (Barnes,1982), with the last pneumonic plagueepidemics in the United States in 1924–25 in Los Angeles (Perry and Fetherston,1997). Simultaneously, Y. pestis becameestablished in native rodents in the SanFrancisco region (Link, 1955), and, as earlyas 1903, massive epizootics were reportedin California ground squirrels (Spermophi-lus beecheyi). The agent of plague was firstisolated in California ground squirrels in1908 in the Berkeley hills in Contra CostaCounty (McCoy, 1908; Wherry, 1908).

At least 18 rodents species and 27 or moreflea species are involved in ongoing enzooticplague cycles in the western United States,

Journal of Wildlife Diseases, 43(3), 2007, pp. 408–424# Wildlife Disease Association 2007

408

largely independent of the traditional rat-ratflea cycle (Hubbard, 1968; Smith et al.,1998; Gage and Kosoy, 2005). Most cases ofY. pestis infection, including those in hu-mans and rodents, are acquired through fleabites, but infections can occur throughdirect exposure to infectious respiratoryand oropharyngeal secretions; through pre-dation, particularly by cats; and throughscavenging by rodents, such as Onychomysleukogaster, on infected hosts and carcasses(Smith et al., 1998). One theory to accountfor plague persistence is that unidentified,persistently infected but asymptomatic hostspecies may maintain infection in plague-enzootic communities, whereas lethallysusceptible, transiently infected ‘‘amplifyinghosts’’ are responsible for most humaninfections (Pollitzer, 1954). Yet, no persis-tently infected reservoir hosts have beenidentified in the western United States. Insome situations, fleas may function as defacto reservoirs, because they can retaininfection for several months or longer,especially in winter climates (Eskey andHaas, 1940; Gage and Kosoy, 2005). It islikely that the dynamic interactions amonghosts and fleas determine whether Y. pestiscan be maintained in nature; however, it isnot known which critical characteristics ofthese dynamics permit such maintenance.

There are prospects for enormous eco-logical complexity in the maintenance of theflea-host-Y. pestis system. Because Y. pestiscan spread directly among people in thepneumonic form, can induce fatal disease,and it could be weaponized; it is listed by theUS Department of Health and HumanServices as a ‘‘List A select agent.’’ Addi-tionally, sylvatic plague threatens humanhealth and several endangered and threat-ened wildlife species, including the black-footed ferret (Mustela nigripes) and black-tailed prairie dog (Cynomys ludovicianus)(Carter, and Gage, 2000; Cully et al., 2000;Biggins and Godbey, 2003). Yet, not only isthere insufficient ecologic knowledge toaccurately describe the mechanisms forpersistence and periodic re-emergence ofplague but also there is no established

predictive framework for anticipating thepossible consequences of intentional releaseof Y. pestis in regions where the bacteriacould persist (Gage and Kosoy, 2005).

Several studies have described modelingapproaches to plague dynamics. Some earlysusceptibleRinfectiveRresistant (SIR) pla-gue models only dealt with humans and notrodents or fleas, and they did not allow forhosts to recover or for new susceptibleindividuals to be introduced into the system(Noble, 1974; Raggett, 1982). A recentmodel of pneumonic plague also focusedon humans, and it did not consider enzooticcycles in nonhuman hosts and arthropodvectors (Gani and Leach, 2004). An excel-lent model of vectorborne plague wasdeveloped focusing on Y. pestis infection inhumans, R. rattus, and X. cheopis (Keelingand Gilligan, 2000a, b). Although this modelrecreated many of the worldwide dynamicsobserved in human plague outbreaks, littleinsight was offered at the smaller scale ofenzootic plague in wild rodent communitiesin the western United States.

In this study, we extend traditional vectorsusceptibleRinfectiveRresistantRsuscep-tible (SIRS) models to create a flexiblematrix-based community vector SIRSframework, which was used to investigateplague. There were four specific objectives:1) to use matrix manipulation within a clas-sical SIRS model framework to describetransmission of Y. pestis among rodents andfleas in California, 2) to perform sensitivityanalysis with model parameters and vari-ables to document which values tended todominate model output, 3) to manipulatemodel parameters to determine values thatwould convert transient epizootic infectionto enzootic, and 4) to document predictedoutcomes of plague with realistic parametervalues obtained from the literature for Y.pestis in California rodents and fleas.

MATERIALS AND METHODS

Disease model: single host-single vector system

Plague community dynamics were modeledwith a matrix version of the basic Kermack-

FOLEY ET AL.—MODELING PLAGUE PERSISTENCE 409

McKendrick SIRS model, with SIRS hosts(Kermack and McKendrick, 1927; Bailey,1982). This community plague model, desig-nated PlagueSIRS, was implemented in R (TheR-Development Core Team, http://www.r-pro-ject.org); pseudocode for the sequence of eventsis given in Appendix 1. Discrete-time differen-tial equations coupled host disease transmissiondynamics with those of the flea vectors. Hostpopulations are composed of three groups ofindividuals: susceptibles of size S, infectious I,and resistant R, with

N ~ S z I z R: ð1Þ

Flea populations were each separated intosusceptibles X, infectious Y, and immunes Z(the latter retained for generality but likely to berare or lacking in nature):

V ~ X z Y z Z: ð2Þ

In a short time interval of 1 day, host individualsmove through categories S, I, and R by diseasetransmission at rate bSY, recovery at rate cI, anddeath at rates dS 5 dR or dI (mortality attribut-able to disease, designated in this article as‘‘mad’’). Similar rates apply to fleas, except thatafter fleas acquire Y. pestis infection, there isa substantial latency period during which theyare not infective, requiring addition of thevector state E. The rate at which latency shiftsto infectivity is y, and the mean time in latencyis 1/y. The model dynamics are described bythe system of differential equations:

dS

dt~ bN { bSY { dSS

dX

dt~ bV V { bV XI { dX X

dI

dt~ bSY { cI {dI I

dE

dt~ bV XI { yE { dX E

dR

dt~ cI { dRR

dY

dt~ yE { dY Y

ð3Þ

Multiple host-multiple vector system

A plague community contains h host speciesand v vector species. The community modeltracks a vector of host sizes N 5 [N1…Nh],a vector of S values S 5 [S1…Sh], andanalogous vectors of I, R, X, E, and Y. The

community SIRS model is represented below,assuming that host death rates of S and Rindividuals are similar, X and E flea death ratesare similar, and adding the identity matrices(1) required for the matrix multiplication:

dS

dt~ BH N { S1b1 { DSS

dX

dt~ BVV z yV Y { X1bVI { DXX

dI

dt~ S1b1 { yI { dI I

dE

dt~ X1bV I { yE { DXE

dR

dt~ yI { DSR

dY

dt~ yE { yV Y { DY Y :

ð4Þ

The 3h+3v-dimensional, nonlinear system ofequations was discretized for computer simu-lations to obtain the model for the survival ofsusceptible hosts (Nicholson and Bailey,1935). If infection is distributed as a Poissonprocess, then Si rS* exp(–bY).

Transmission to new hosts can take placefrom several possible flea species. Let bij bethe transmission rate from vector j to host i;then, the entire host transmission process is asfollows:

S1bY ~

S1

S2

..

.

S3

26666664

37777775

1 1 . . . 1½ �

b11 b12 b1v

b21 b22 b1n

P

bh1 bh2 bhv

2666664

3777775

Y1

Y2

..

.

Yv

26666664

37777775

~

b11S1Y1zb12S1Y2z:::zb1vS1Yv

b21S2Y1zb22S2Y2z:::zb2vS2Yv

..

.

bh1ShY1zbh2ShY2z:::zbhvShYv

26666664

37777775

ð5Þ

bV is a v*h matrix analogous to b that givestransmission from the jthhost to the ith vector.

410 JOURNAL OF WILDLIFE DISEASES, VOL. 43, NO. 3, JULY 2007

To transmit a disease from an infective vector jto a susceptible host i, the vector needs to 1)find a host (proportional to aSY, where a isattack rate), 2) be competent to transmit thedisease (with vector competency cvj), and 3)want to use the host (with utilization rate uij).Typical vector-to-host matrix b entries are asfollows:

bij ~ acvjuij: ð6Þ

The host-to-vector matrix bv has entries of theform:

bvij ~ acjuji, ð7Þ

where cj measures the host competency. Thelower limit of disease was set by an extinctionthreshold of one individual; if either infectedflea or host numbers went below the threshold(into fractional individuals), the simulationstopped; upper limits were set by host popula-tion upper limits (K). Seasonality was incorpo-rated into the model for flea recovery frominfection, which is known to be temperature-dependent (indeed, above 27 C, Y. pestis maynot survive or be transmissible in the flea(Pollitzer, 1954), by Fourier transform.

Host and vector population dynamics

Underlying the disease transmission modelis a series of functions that allow for popula-tion regulation for each host and vectorspecies, including seasonality and environmen-tal stochasticity. A community consists of hhost species, of sizes N1, N2, …, Nh withspecies-specific birth rates (bi) given in a di-agonal matrix:

BH~

b1 0 0

0 b2 0

P

0 0 bh

26664

37775, ð8Þ

so that the number of new individuals per hostspecies is as follows:

BHN ~

b1 0 0

0 b2 0

P

0 0 bh

26664

37775

N1

N2

Nh

26664

37775~

b1N1

b2N2

bhNh

26664

37775:

Death rates (di) are in an analogous diagonalmatrix. A stochastic discrete logistic model insmall time units gives host population growthand fluctuation:

N(tz1) ~ N(t)(1 z bi(t) { di(t)), ð10Þ

where

bi(t) ~ bm 1 {N(t)

K

� �z ei(t), ð11Þ

Ki is the host carrying capacity, bm is theMalthusian birth rate, and ei(t) , N(0,vr) is theenvironmental stochasticity term at time t,estimated by regression (May, 1974; Foley,1997). Seasonality of rodent numbers wasincorporated into the model by constrainingreproduction to those months described byfield data and modeling death via a Fourierfunction of annual periodicity (Keeling andGilligan, 2000a). Incorporating an approxima-tion, the transform is as follows:

di(t) ~ AHi cos (t { tpeak) z �ddi ð12Þ

where di(t) is the death rate at time t inradians, d̄i is the mean death rate over theyear, AHi is the amplitude of the host deathrate, and tpeak is the time of peak mortality. Asimilar function was used for bi. For simplicity,environmental stochasticity was included onlyin birth and not death, which can be donebecause the model assumes additivity ofb+d+e.

Flea populations, for each species V1, V2,…,Vv, depended numerically on the abun-dance of rodent hosts. The carrying capacity ofa vector species j depends on the abundance ofeach host species i, together with somemeasure kij of the vector j’s ability to use hostspecies i as a resource. In the absence ofinterspecific competition among fleas, thecarrying capacity of vector j is given by

Kvj ~X

ikijNi: ð13Þ

The carrying capacity kij offered by anindividual host i to vector j depends on 1)uij, the vector’s host utilization rate; 2) qi, hostresource quality; and 3) wi, a flea weightingindex. The uij ranges from 0 (vector j doesnot use host i) to 1 (the vector can and willfully use the resources available from thishost). qi, the index of the quality of resourcesa host provides to an average vector, isequivalent to the mean number of fleas ofaverage size on an individual of this hostspecies of average size. wj, the flea weightingindex, is related to the size of the vector (or itcould be set as some other measure of theamount of resources an individual vectoruses). wj has a mean of 1 for a ‘‘typical’’flea species and is close to 1 for most fleaspecies, but it may rise significantly above 1 forlarge fleas. Then, the carrying capacity pro-

(9)


vided by one host to its fleas of species j isshown as follows:

kij ~ uijqiwj, ð14Þ

and the whole carrying capacity for vectorspecies j, which depends on the availability ofall of the hosts is as follows:

Kvj ~X

ikijNi ~

XiuijqiwjNi: ð15Þ

Density-dependent birth of the jth flea ismodeled, with the Malthusian parameter bmvj:

bj(t) ~ bmvj(1 { Vj=Kvj): ð16Þ

Then, flea death is

dj(t) ~ Avj cos (t { tpeak) z dj: ð17Þ

The model also was implemented with di-agonal matrices designated Bv, Dv; seasonalitywas incorporated for flea birth and death byFourier transform.

Parameter estimation and sensitivity analysis

Data were maintained in Excel (Microsoft,Redmond, Washington, USA) and analyzed inR. Data on flea distribution, flea host prefer-ences, and plague transmission dynamics wereobtained from the literature and are summarizedin Table 1. Data were used as first approxima-tions, to further develop the model, because inmany cases, including old literature with verysmall samples sizes and predictive habitatdatabases, parameter and variable estimateslikely had large confidence intervals. Neverthe-less, use of such estimates facilitated investiga-tion to clarify which parameters and variableswould require more accurate future estimation.

Life history and habitat association data forrodents were obtained primarily from theCalifornia Department of Fish and GameWildlife Habitat Relationships database(Mayer and Laudenslayer, 1988). Death ratesfor fleas and rodents were calculated approx-imately assuming that death times wereexponentially distributed with the parameterl calculated as mean time to death, obtainedfrom the literature. Values for rodent carryingcapacities, litter sizes, time windows of re-production, and numbers of litters per yearwere obtained from the literature (Mayer andLaudenslayer, 1988; Nowak, 1995). Rodentpopulation time series were obtained fromthe Center for Population Biology GlobalPopulation Dynamics Database (Imperial Col-lege, London, UK). The bm was calculated asrm–d(S) prorated for the period over which thespecies reportedly breeds in nature. Assuming

a Ricker model of population regulation(Foley, 2000), r(t) was regressed on N(t) toprovide estimates of rm and carrying capacity.vr was estimated as the noise component in theregression, although this was considered anoverestimate, because it includes both envi-ronmental stochasticity as well as samplingvariation.

The parameters q and w were estimatedfrom reports of the mean flea numbers ofrodents (Davis et al., 2002), with w scaled toa maximum value of 3 and a mean of 1. Thematrix u was constructed from field data givingthe fraction of all flea fauna on each hostrepresented by each flea species. c (rates ofrecovery for fleas and rodents) and y (rate ofmovement from latent to active infection infleas) were estimated as the mean time fromthe first point at which there was evidence ofinfection to the first time the flea was infectiveto rodents (y) or the flea or rodent recovered(c). Vector and host competence were aver-aged across all published reports of experi-mental Y. pestis infection in that species. If novalues were reported, the estimate was usedfor the most phylogenetically proximal speciesfor which data were available.

Sensitivity analysis was performed to de-termine which parameters and variables weremost responsible for driving observed dynam-ics, by allowing each parameter, one at a time,to range across very broad possible values forthat parameter, as shown in Table 2. All otherparameters were fixed corresponding to opti-mal estimates from literature reviewed asmean values for each parameter. The analysiswas run with parameters for a three-host, two-flea community consisting of Peromyscusmaniculatus, woodrats (Neotoma fuscipes),and S. beecheyi and the fleas Aetheca wagneriand Oropsylla montana (Table 3). The follow-ing outputs were used as indicators of modelresults: the mean expected duration of in-fection in the community (.100 simulationruns) and the expected maximum number ofinfected hosts. By using output from thisexploration, critical values of parameters thatwould convert epizootic infection to enzooticwere determined. Default values were usedfor all parameter values were except one, asdescribed for sensitivity analysis, whereas thesimulation was run until a cutoff for eachparticular value was found (if at all) where theinfection was predicted to be enzootic (de-fined as infected hosts or fleas for $400 days).

To document whether enzootic mainte-nance would be predicted with realisticparameter values for Y. pestis in Californiarodents and fleas, the model was run withparameter values from the well-studied pla-


TABLE 1. Parameters and variables used in the model PlagueSIRS for simulating plague dynamics.

Abbreviation Definition Estimate Source

H No. of host species in thecommunity

Davis et al., 2002

V No. of vector species inthe community

Davis et al., 2002

K Vector of host-carryingcapacities

Tables 3, 4, 7 Nowak, 1995; Wildlife and HabitatData Analysis Branch, 2004

kij Composite carryingcapacity parameter

Calculated in modelfrom u, q, and w

N, rmi, rmvj Species-specific hostpopulation size,Malthusian parameterfor hosts and fleas

Calculated in model

bm Vector of hostmaximum birth rates

Tables 3, 4, 7 Linsdale, 1946; Linsdale and Tevis,1951; Longanecker andBurroughs, 1952; Rutledge et al.,1979; National Center forEcological Analysis andSynthesis, 2004; Wildlife andHabitat Data Analysis Branch,2004

Mean no. oflitters/yr


Peak date of birthand window


bmv Vector of flea meandaily hatch rates

Daily50.016 Davis et al., 2002

Peak date,amplitude

Amplitude and peakdate of flea hatches

Amplitude50.016;Table 6

vr Vector of environmentalstochasticity affectinghosts


vrv Vector of environmentalstochasticity affectingfleas

0.01 None

a Attack rate 0.001 None



comp Vector of hostcompetence

0.75 all McCoy, 1908, 1909, 1911a, b; Mc-Coy and Smith, 1910; McCoyand Chapin, 1912; Eskey andHaas,1940; Holdenreid andQuan, 1956; Marchette et al.,1962a, b; Quan and Kartman,1962; Quan et al., 1985

compv Vector of flea vectorcompetence

Tables 6, 8 Eskey and Haas, 1940; Burroughs,1944, 1947; Wheeler and Dou-glas, 1945; Holdenreid, 1952;Kartman and Prince, 1956;Kartman, et al., 1958

u Matrix of relativeutilization preferences(scaled 0–1) of fleaspecies for host species

Table 5 Davis et al., 2002

q Vector of quality ofhosts: expectednumber of fleas on hostindividuals of eachspecies

Tables 3, 4, 7 Davis et al., 2002

w Vector of flea speciesweighting indices

Tables 6, 8 Davis et al., 2002

Y Vector of rates at whichlatently infected fleasbecome infective

Tables 6, 8 Eskey and Haas, 1940; Burroughs,1944, 1947; Wheeler and Dou-glas, 1945; Holdenreid, 1952;Kartman and Prince, 1956;Kartman et al., 1958

c Vector of rates ofrecovery for each hostspecies

Tables 3, 4, 7 McCoy, 1908, 1909, 1911a, b; Mc-Coy and Smith, 1910; McCoyand Chapin, 1912; Eskey andHaas, 1940; Holdenreid andQuan, 1956; Marchette et al.,1962a, b; Quan and Kartman,1962; Quan et al., 1985

cv, mean, peakdate, amplitude

Vector of rates ofrecovery for eachvector species

Peak date 7/1;mean51/5 for all;Tables 6, 8

Eskey and Haas, 1940; Burroughs,1944, 1947; Wheeler and Dou-glas, 1945; Holdenreid, 1952;Kartman and Prince, 1956;Kartman et al., Quan, 1958

dS5dR, mean,peak date,amplitude

Vector of death ratesof S and R hosts

Peak date 1/1 for all;amp50.1;Tables 3, 4, 7

McCoy, 1908, 1909, 1911a, b; Mc-Coy and Smith, 1910; McCoyand Chapin, 1912; Eskey andHaas, 1940; Holdenreid andQuan, 1956; Marchette et al.,1962a, b; Quan and Kartman,1962; Quan et al., 1985

dI5‘‘mad’’ Vector of death ratesof I hosts

Tables 3, 4, 7 Linsdale, 1946; Linsdale and Tevis,1951; Rutledge et al., 1979

dX, mean, peakdate, amplitude

Vector of death ratesof X vectors

Tables 6, 8 Eskey and Haas, 1940; Burroughs,1944, 1947; Wheeler and Dou-glas, 1945; Holdenreid, 1952;Kartman and Prince, 1956;Kartman et al., 1958

TABLE 1. Continued.


gue-enzootic site at Chuchupate Campground(CG) in Ventura County (Tables 4–6). Chu-chupate CG is at 1,890 m on Frazier Moun-tain in pine (Pinus spp.)-oak (Quercus spp.)forest. A previous study in Chuchupate CGhas documented 20 flea species and 10 rodentspecies potentially involved in the ecology ofplague (Davis et al., 2002).

RESULTS

A vector-SIRS–based simulation modelwas created to investigate multiple inter-acting rodent hosts and flea vectors ofplague in the western United States.Sensitivity analysis performed for a three-host, two-flea plague community indicatedthat the expected duration of infection inthat community and the maximum num-ber of infected hosts expected in thatepizootic were most sensitive to hostcarrying capacity, attack rate, and hostand flea recovery rates (Table 2). Increas-ing host carrying capacity led to increasesboth in host and flea numbers, resulting inprolonged duration of infection. Thispattern was saturated at high levels ofcarrying capacity (Figs. 1, 2) and increas-ing K did not induce permanent enzooticinfection at any level. Changing a also didnot induce enzootic disease. Very lowattack rates did not allow for any diseasemaintenance, whereas increasing a re-sulted in an increased epizootic durationup to a plateau value, with a considerableconcurrent increase in maximum infectedhost number. Modifying cH, host recovery,had an expectedly important effect. As cH

was decreased, the duration of infectionpersistence and maximum number ofinfected animals both increased until, atlow cH, enzootic infection was maintainedin the simulation. Although not apparent

over most values used for cV, very lowflea recovery rates also had the capacityto convert the epizootic infection toenzootic.

Parameters to which model outcomewas relatively insensitive included y, u, w,host competence unless a host was com-pletely incapable of transmitting disease,vector competence, mortality attributableto disease (except that at high rates ofmortality, there was no enzootic disease),and q. Thus, sensitivity analysis clarifiedwhich parameters had the greatest in-dividual impacts on model output andsuggested that changes in cH and cV weremost likely to convert transient epizooticsto long persistence-time enzootics.

Parameter values supporting a plague-enzootic community were explored usingpublished parameter values for the Chu-chupate CG community, allowing a (forwhich no good estimates were available) tovary in order to observe expected changesin plague persistence. With a universalattack rate (all fleas on all hosts) of 0.01bites/host/day, plague was enzootic in thecommunity (Fig. 3). Seasonality in rodentand flea numbers was apparent, as wereepizootic pulses of plague primarily oc-curring only in fleas (with only smallnumbers of infected hosts). The principalrodent and flea species contributing topersistence were woodrats, Californiaground squirrels, and chipmunks (Tamiasspp.), with the fleas A. wagneri and M.telchinus. The features of these hostscontributing to their important roles inplague epizootics included relative resis-tance to the development of fatal plague(except for ground squirrels) and theirability to support more flea species,


dY (fleas)5‘‘mad’’ Vector of death ratesof Y vectors.

Tables 6, 8 Eskey and Haas, 1940; Burroughs,1944, 1947; Wheeler andDouglas, 1945; Holdenreid,1952; Kartman and Prince, 1956;Kartman et al., 1958

TABLE 1. Continued.


particularly the highly abundant M. telchi-nus. Prolonged duration of infection withinindividuals did not occur and was nota contributing factor to enzootic plague in

the community. When a was increased to0.1 bites/host/day, plague remained enzo-otic but apparent stable cycles of infectionwere disrupted. Reduction of a to 0.009 led

TABLE 2. Results of analysis of the sensitivity of the model PlagueSIRS, measured as changes in expectedepizootic duration and maximum infected host number (MIHN), as a function of changes in parameter values.Critical values are the values of each parameter (if any) that permits expected epizootic infection to becomepersistent in the community.

Parameter Level ValueMean epizootic

durationChange in epizootic

duration (%)Mean maximuminfected host no.

Change inMIHN (%)

K Low 15 75 –25 80 –20Default 150 100 100High 1,500 350 250 10 –90Critical None

a Low 0.005 0 –100 0 –100Default 0.05 100 100High 0.5 100 NC 150 50Critical None

comp Low 0.1 0 –100 0 –100Default 1 80 80High NAc NA NA NA NACritical NA

compv Low 0.1 130 –48 40 12.5Default 1 250 35High NA NA NA NA NACritical NA

u Low 0.08 0 –100 0 NADefault 0.8 10 80High 1 100 NCc 80 NCCritical None

q Low 1 0 –100 0 –100Default 10 50 8High 100 50 NC 20 150Critical None

w Low 0.5 160 NC 30 –25Default 1 160 40High 3 260 62 50 25Critical None

Y Low 0.01 250 NC 150 NCDefault 0.1 250 150High 1 250 NC 150 NCCritical None

cH Low 0.001 persistent NA 120 71Default 0.01 80 70High 0.1 10 –87 5 –93Critical 0.05a

cV Low 0.1 250 NC 150 NCDefault 0.52 250 150High 1 250 NC 150 NCCritical 0.01b

a The critical value, c50.05 induced persistence inconsistently (60% of the runs), compared with no persistent enzooticfor higher values of c.

b The critical value, cV50.01 induced persistence inconsistently (10% of the runs), compared with no persistent enzooticfor higher values of cV.

c NA indicates results not applicable, NC indicates no change.


to a single epizootic of approximately 200-day duration.

DISCUSSION

We present development and explora-tion of a flexible, matrix-based simulationof a vector-SIRS model for highly complexrodent and flea communities supportingplague in western North America. Unlikemost vector-SIRS analytical models, theapproach used here has the advantage ofoffering a framework for inclusion ofdemographic data for hosts and fleas,expansibility to virtually any size commu-nity, and the ability to manage verycomplex communities without require-

ments for oversimplification. Explorationof the model indicated that the mainfeatures of epizootics, duration and mag-nitude (i.e., infected host number), weredriven by only a few key parameters,a finding that should guide data collectionand understanding of natural epizootics inthe future. Moreover, simulation of a largeplague community documented emergentproperties of this ecological system, name-ly, that indefinite plague persistence waspredicted in the highly complex system,even when simplified, three host-two fleasystems with very similar parameter valueswere predicted to go extinct.

The three most important parametervalues required to predict epizootic dura-

TABLE 3. Parameter values for hypothetical three host-two flea system for sensitivity analysis of theplague simulation.

Host species K/ha bm vr q dS/day dI5mad cLittersize

Birthwindow (mo)

No.litters/yr

N. fuscipes 17.5 0.02 0.4 11.3 0.0042 0.3 0.001 2–3 2–7 1–5P. maniculatus 17.5 0.02 0.18 10.9 0.006 0.001 0.06 3 3–8 2–4S. beecheyi 45 0.01 0.26 25 0.0014 0.17 0.001 6–7 5–10 1

Flea speciesPeak

birth moPeakdeath compv w y dX dY5mad cv

A. wagneri 6 12 0.6 1 0.02 0.01 0.05 0.02O. montana 5 8 0.6 3 0.012 0.015 0.08 (Wheeler

and Douglas,1945)

0.02 (Wheelerand Douglas,1945)

TABLE 4. Parameter values used in the model PlagueSIRS for rodents in Chuchupate Campground, VenturaCounty, California.

Species K/ha bm vr q dS/day dI5mad cLittersize

Birthwindow

(mo)No.

litters/yr

Chaetodipus californicus 7.5 0.009 0.05 10 0.003 0.01 0.06 3 3–8 2Dipodomys agilis 7.5 0.03 0.05 10.7 0.003 0.01 0.09 3 1–7 1–2Microtus californicus 250 0.005 1 12.7 0.0025 0.04 0.06 4 1–12 2–5Neotoma fuscipes 17.5 0.02 0.4 11.3 0.0042 0.3 0.001 2–3 2–7 1–5Peromyscus boylii 17.5 0.03 0.18 10.7 0.006 0.04 0.06 3 4–8 1–4P. maniculatus 17.5 0.02 0.18 10.9 0.006 0.001 0.06 3 3–8 2–4P. truei 17.5 0.03 0.18 10.6 0.006 0.13 0.01 3 5–9 2Reithrodontomys megalotis 37.5 0.04 0.49 11 0.012 0.2 0.001 2–4 4–10 14S. beecheyi 45 0.01 0.26 25 0.0014 0.17 0.001 6–7 5–10 1T. merriami 12.5 0.01 0.2a 11.8 0.0014 0.185 0.007 4 1–6 1

a Extrapolated results, poor data availability in literature.


tion and magnitude were cH, cV, and a,the attack rate of fleas on hosts. Reductionin cH and cV not surprisingly increased theduration of epizootics or induced enzooticdisease by essentially increasing reservoirpotential in the classical sense of some ofthe rodents. This interesting finding couldhelp explain some of the differences in therisk of plague in diverse communitieswhere rodent hosts have different cH

levels. There are estimates of cH availablein the literature for many Californiarodents and some fleas. Flea recoveryrates in particular can be highly variablebecause those flea species with a narrowesophagus and proventriculus are moresusceptible to blockage with a mat ofbacteria and inflammatory debris (andthus more likely to regurgitate infectiousmaterial into the host while attempting tofeed). Other flea species have greaterability to enzymatically destroy the blockand less susceptible anatomic configura-

tions (Perry and Fetherston, 1997). Un-doubtedly, more accurate estimates fromexperimental infection would be valuablefor both rodent and flea recovery rates.

Attack rate, for which very little in-formation is available from Californiasystems, was also an important parameterin determining durations of epizootics.This finding is comparable with resultsfrom an earlier vector-SIRS model ofhumans, rats, and fleas that was sensitiveto flea search efficiency (Keeling andGilligan, 2000a). Other vectorborne dis-ease systems also are highly sensitive toattack rate, including the pioneering mod-els of Ross (1928) and MacDonald (1957)showing the nonlinear effect of attack rateof mosquitoes transmitting malaria to hu-mans. The estimation of this parameterfrom field data likely represents a criticallyimportant target for future research.

The earlier vector-SIRS plague modelof Keeling was also particularly sensitive to

TABLE 5. ‘‘u matrix’’, containing values for flea utilization of various rodent species in ChuchupateCampground, Ventura County, California (Davis et al., 2002).

Rodent

Flea

Aethecawagneri

Anomiopsfalsical

A.nudatus

Atyph.e.l.

Atyph.m.m.

Catal.luski

Eumol.e.

eumolpiE.

fornacisHoplops.anomalus

Hystrich.o.

linsdaleiMalaraeustelchinum

Dipodomys 0.000 0.000 0.004 0.000 0.000 0.053 0.000 0.000 0.000 0.000 0.000Microtus 0.005 0.000 0.000 0.000 0.000 0.000 0.000 0.009 0.000 0.071 0.667Neotoma 0.118 0.913 0.961 0.846 0.680 0.263 0.000 0.026 0.020 0.286 0.140Peromyscus boylii 0.093 0.000 0.004 0.000 0.000 0.105 0.000 0.000 0.000 0.143 0.032P. manic 0.760 0.087 0.019 0.154 0.280 0.526 0.000 0.000 0.000 0.500 0.129P. truei 0.015 0.000 0.000 0.000 0.040 0.000 0.000 0.000 0.000 0.000 0.000Reithrodonomys 0.005 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.022Spermophilus 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.980 0.000 0.000Tamias 0.005 0.000 0.012 0.000 0.000 0.053 1.000 0.966 0.000 0.000 0.011

Megarth.divisus

Meringiscummingi

Opisodasysnesiotus

Orchopeassexdentatus

Oropsyllamontana

Peromysco.h. adelpha

Rhadinop.s. sectilis

Thrassissp.

Dipodomys 0.000 0.964 0.000 0.000 0.000 0.000 0.063 1.000Microtus 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000Neotoma 0.933 0.000 0.000 0.997 0.024 0.089 0.375 0.000P. boylii 0.004 0.000 0.333 0.000 0.000 0.311 0.063 0.000P. manic. 0.064 0.036 0.667 0.003 0.005 0.578 0.438 0.000P. truei 0.000 0.000 0.000 0.000 0.000 0.022 0.063 0.000Reithrodontomys 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000Spermophilus 0.000 0.000 0.000 0.000 0.967 0.000 0.000 0.000Tamias 0.000 0.000 0.000 0.000 0.004 0.000 0.000 0.000


changes in the carrying capacity of fleasper rat, the rat’s reproductive rate, and therat’s carrying capacity. The model de-veloped in this present study also issensitive to rodent K: increasing K byorders of magnitude increased the ex-pected epizootic duration but did not, in

the three host-two flea system, induceenzootic persistence. This feature may berelated to the particular collection of otherparameter values used for that system. Aprevious study in natural plague foci inKazakhstan documented an abundancethreshold in great gerbils, Rhombomys

TABLE 6. Parameter values used in the model PlagueSIRS for fleas in Chuchupate Campground, VenturaCounty, California.

SpeciesPeak

birth moPeakdeath compv w y dX dY5mad cv

Aetheca wagneri 6 12 0.6 1 0.02a 0.01a 0.05a 0.02a

Anomiopsyllusfalsicalifornicus

4.5 10.5 0.1 0.5 0.02a 0.01a 0.05a 0.02a

A. nudatus 3.5 9.5 0.3 1 0.02a 0.01a 0.05a 0.02a

Athyphloceras echislongipalpis

12 6 0.3 1 0.02a 0.01a 0.1 (Eskeyand Haas,1940)

0.02 (Eskeyand Haas,1940)

A. m. multidentatus 10 4 0.3 1 0.02a 0.01a 0.1 (Eskeyand Haas,1940)

0.02 (Eskeyand Haas,1940)

Catallagia luski 3 9 0.3 1 0.02a 0.01a 0.05a 0.02a

Eumolpianus eumolpieumolpi

7 1 0.6 0.5 0.02a 0.01a 0.05a 0.02a

E. fornacis 6 12 0.6 0.5 0.02a 0.01a 0.05a 0.02a

Hoplopsyllusanomalus

3 12 0.4 3 0.03 0.01a 0.07 (WheelerandDouglas,1945)

0.06 (WheelerandDouglas,1945)

Hystrichopsyllaoccidentalislinsdalei

3–7 11 0.6 1 0.02a 0.01a 0.05a 0.02a

Malaraeus telchinus 10 4 0.1 0.5 0.105 0.019 0.0038(Burroughs,1944)

0.0099(Burroughs,1944)

Megarthroglossusdivisus

1 7 0.3 3 0.02a 0.01a 0.05a 0.02a

Meringis cummingi 6.5 12.5 0.3 1 0.02a 0.01a 0.05a 0.02a

Opisodasys nesiotus 7.5 1.5 0.6 1 0.25 0.035 0.076(Burroughs,1944)


Orchopeassexdentatus

7 1 0.6 1 0.18 0.006 0.08(Burroughs,1944)


Oropsylla montana 5 8 0.6 3 0.012 0.015 0.08 (WheelerandDouglas,1945)

0.02 (WheelerandDouglas,1945)

Peromyscoscyllahesperomysadelpha

6 12 0.6 1 0.02a 0.01a 0.05a 0.02a

Rhadinopsylla s.sectilis

3 8 0.4 1 0.02a 0.01a 0.05a 0.02a

Thrassis aridis 10 4 0.6 1 0.02a 0.01a 0.05a 0.02a

a Extrapolated results, poor data availability in literature.


opimus, above which plague was persis-tent (Davis et al., 2004). Such a thresholdcould apply in California as well, althoughthe abundance threshold refers to theactual numbers of animals in the commu-nity, whereas the carrying capacity refersto a potential with actual populations oftencycling around K. Other important differ-ences are that the plague communities inCalifornia are highly complex, comparedwith systems dominated by a single reser-voir such as a rat or great gerbil. Althoughaccurate estimates of K are difficult toobtain and values are site-specific, someattempt should be made to acquire

estimates with reasonable accuracy, al-though other, possibly more easily esti-mated, parameters such as c and a may bemore critical.

The structure of PlagueSIRS has ad-vantages of flexibility and expansibility,while focusing on short- and long-termdynamics associated with disease trans-mission in communities. This model ad-vantage is in contrast to some invasionanalysis models, which focus on epidemicthresholds such as R0.1. The matrixmethod (Diekmann et al., 1990) couldoffer some insight into the communitysystem, but two difficulties would need tobe overcome. First, in reality, Y. pestistransmission dynamics must include ex-tensive variability in recovery and trans-mission parameters to reflect the diversebiology of the hosts and vectors. Becausethe infection time varies across rodent andflea species, the K matrix would requireconsiderable adjustment before it couldcalculate the diverse collection of I values

FIGURE 1. Expected dependence of duration ofa plague epizootic on K, the carrying capacity of themost abundant rodent host. The data were obtainedfrom the model PlagueSIRS with a three rodent-twoflea plague community (parameter values givenin Table 2).

FIGURE 2. Example of model output from Pla-gueSIRS for a three rodent-two flea plague commu-nity with low K. I 5 number of infected rodents; Y 5

number of infected fleas.

FIGURE 3. Sample model output of simulationPlagueSIRS by using parameter values estimated inChuchupate CG, Ventura County, California, with10 rodent species and 20 flea species, with attack rate5 0.005. I 5 number of infected rodents; Y 5

number of infected fleas; N 5 number of rodentstotal; V 5 total fleas. First and second panels: solidpanel 5 N. fuscipes; dotted line 5 Tamias spp.;dashed line 5 S. beecheyi. Third panel: dotted line 5

M. telchinus; solid line 5 A. wagneri.


appropriately. Second, to investigate per-sistence, the K matrix would need to bemade frequency-dependent.

PlagueSIRS has the additional advan-tages over many vector-SIRS models inrealistically representing underlying hostand flea population dynamics in themodel. This advantage is important be-cause the disease dynamics being studiedoften occurs over distinctly long periods,a problem that is avoided in some modelsfocusing only on rapid epidemics ordisease emergence. In PlagueSIRS, rodentpopulation growth was bounded by a car-rying capacity, whereas flea density wasgiven as a function of flea use of specifichosts, abundance of fleas on hosts, attackrates, and seasonal constraints; both ro-dent and flea dynamics were modified byenvironmental stochasticity. The earliermodel of Keeling used simple flea search-ing efficiency and an arbitrary flea carry-ing capacity not explicitly tied to rodentdensity. Analysis of the model shown hereclarifies why population regulation needsto be included in the model, and futuremodel validation with data from fieldstudies should allow for any necessarymodifications to the functions predictinghost and flea numbers to be incorporatedinto PlagueSIRS.

It was very interesting how the modelrepresented the enzootic plague commu-nity at Chuchupate CG, by using param-eter values from the literature. Based onserologic testing of rodents and fleacollections, vector control biologists hadconcluded that plague was enzootic in thisregion and that three host-flea complexeswere important in the local plague ecolo-gy, although how these complexes couldfunction as true reservoirs was unclear.These different complexes were 1) Cali-fornia ground squirrel (S. beecheyi) and O.montana/H. anomalous fleas, 2) Merriam’schipmunk (Tamias merriami) and Eumol-pianus spp. fleas, and 3) dusky-footedwoodrats (N. fuscipes) and Orchopeassexdentatus and Anomiopsyllus spp. fleas.Ground squirrels were generally infested

with only two host-specific fleas, whereasother hosts, including deer mice andwoodrats, harbored various flea speciesthat they shared with other rodent species.The model recreated enzootic plague withseasonal outbreaks, corresponding to pub-lished data (Davis et al., 2002). Hosts withrelatively low mortality, including woo-drats and chipmunks, were critically im-portant in maintaining enzootic plague,although neither would likely be consid-ered typical reservoirs given the shortduration of infection in any individual.Deer mice and voles have been reportedpreviously to be reservoirs for plague(Miles et al., 1957; Goldenberg et al.,1964; Nelson, 1980; Larson et al., 1996).In our model, the most important fleaspecies were A. wagneri and M. telchinus,the former reported in Chuchupate CG toinfest seven different rodent species,especially deer mice (Davis et al., 2002).It was reported that A. wagneri isa competent, but not excellent plaguevector (Eskey and Haas, 1940). The fleaM. telchinus is found year-round and feedson multiple hosts, although with a prefer-ence for voles. A pool of this flea speciescollected from a vole was reported positivefor Y. pestis in July 1997 in the Chuchu-pate CG (Davis et al., 2002). Despite thatexperimental inoculations indicated thatM. telchinum is a poorly competent vector(Burroughs, 1947), reducing vector com-petency in the model did not seriouslydiminish the role of this flea species inmodeled epizootics, probably becausenumbers of this flea are commonly veryhigh. Several flea species that had beenthought to be very important in plagueecology played very minor roles in ourmodel, including O. montana and H.anomalus, probably because they fed onrodents that were so susceptible to diseasethat they died. Therefore, the host-fleacouple could not support anything morethan a very transient epizootic. Neverthe-less, O. montana is important in amplify-ing plague and particularly in transmittingit to humans, because this flea aggressively


seeks new hosts, including humans, whenits preferred host dies (Douglas andWheeler, 1943; Holdenreid, 1952; Quanet al., 1960; Nelson, 1980; Barnes, 1982;Gage and Kosoy, 2005).

It was interesting that enzootic plaguewas predicted at Chuchupate, even withsimilar parameter values to a hypotheticalthree host-two flea system where plaguedied out. One possible explanation is thatthe variability across rodent and fleaspecies in infection dynamics createdheterogeneity that is, in some ways,analogous to spatial heterogeneity facili-tating longer disease persistence. Specifi-cally, the inclusion of stochasticity into anearlier plague metapopulation model wasaccompanied by rapid local extinction,which was remedied by the inclusion ofspatial structure (Keeling and Gilligan,2000a, b). Extending the present simula-tion modeling approach to spatially struc-tured systems would be a logical, althoughchallenging, exercise.

To summarize, a predictive model ofplague in complex rodent-flea communi-ties in California clarified key interactionsdriving plague local extinction or persis-tence. The described model represents aninnovative, highly tractable method ofmanaging ecological complexity typical ofcommunities of multiple hosts and vectorssuch as occurs in plague but also othervectorborne disease systems such as WestNile virus, granulocytic anaplasmosis, andothers. This model also helps establisha framework for ongoing data collectionthat will allow the model to be refined andultimately more realistic. This modelshould be helpful in ongoing surveillanceactivities and in the event of apparentlyincreased activity that could indicate in-tentional release.

ACKNOWLEDGMENTS

The authors thank N. Drazenovich and R.Kasten for input and technical expertise.Funding for this project was provided by theUniversity of California Davis Center forVectorborne Diseases.

APPENDIX 1

Pseudocode for main simulation loop of Plague-SIRS

Main loop (repeat once a day for tmax days)Step 1:

1. Determine season2. Adjust rodent birth and death rates to season3. Adjust rodent birth and death rates to density

dependence4. Calculate environmental stochasticity in host

and flea birth rates5. Adjust cV to season.

Step 2: Calculate changes in disease states (i.e.,transmissions, recoveries, births, and deaths).

Step 3: Adjust numbers in S, I, and R rodents andX, E, and Y fleas by using changes in step 2.

Step 4: Append new S, I, R, X, E, and Y values tohistory data frame.

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Received for publication 22 April 2005.


MODELING PLAGUE PERSISTENCE IN HOST-VECTOR COMMUNITIES IN CALIFORNIA

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