vol. 172, no. 6 the american naturalist december 2008 (cid:1) The Role of Heterogeneity in the Persistence and Prevalence of Sin Nombre Virus in Deer Mice F. R. Adler,* C. A. Clay,† and E. M. Lehmer‡ 1.DepartmentofBiologyandDepartmentofMathematics, For an infectious disease to persist, it must have hosts to UniversityofUtah,SaltLakeCity,Utah84112; infect and a sufficiently high rate of infecting those hosts 2.DepartmentofBiology,WestminsterCollege,SaltLakeCity, (Lloyd-Smith etal.2005a).Whenhostsbecomerare,dis- Utah84105; eases are at risk of local extinction because of reduced 3.DepartmentofBiology,FortLewisCollege,Durango,Colorado transmission (Swinton et al. 2002) or stochastic die-off 81301 (Begon et al. 2003; Conlan and Grenfell 2007) or when SubmittedOctober15,2007;AcceptedJuly9,2008; other environmental factors reduce disease persistence ElectronicallypublishedOctober28,2008 within hosts, host contact rates, or probability of trans- mission.Diseasesatlowprevalence,likeotherrarespecies, Onlineenhancements:appendixes. wouldseemmostpronetoextinctionduetosmallchanges in host densities or environmental factors. In a homogeneous population, the equilibrium preva- lencep∗ofadiseasewithoutacquiredimmunityorvertical abstract:Manydiseasespersistatarelativelylowprevalence,seem- transmissionfromparenttooffspringisrelatedtothebasic ingly close to extinction. For a chronic disease in a homogeneous reproductive number R , the number of new infections 0 population,reducingthetransmissionratebyafractionproportional causedbythefirstinfectedindividual(AndersonandMay to the prevalence would be sufficient to eradicate thedisease.This 1992), according to study examines how higherprevalence oftheSinNombrevirusin maledeermice(Peromyscusmaniculatus)mightcontributetodisease persistence.Analyzingdatafromover2,000individualmicecaptured 1 in19sitesover4years,wefoundprevalencesof18.5%inmalesand p∗p1(cid:1) , or 8.8%infemales.Byexaminingrecaptures,wedeterminedthatmales R 0 are more likely to contract the infection because ofhighersuscep- tibility or higher encounter rates. Comparing across 86 sampling 1 R p (1) periods, we found a higher proportion of males when population 0 1(cid:1)p∗ densitieswerelow.Acapture-recaptureanalysisindicatesthatmales livelongerthanfemales.Amathematicalmodelbasedonthemea- sured parameters and population size trajectories suggeststhatthe (app. A in the online edition of the American Naturalist). combinedheterogeneityinencounters,susceptibility,andmortality Inthissimplecase,areductionofR byafractionp∗reduces may buffer the disease from extinctionbyconcentratingdiseasein 0 thesubgroupmostlikelytotransmitthedisease.Thisbufferingeffect R0 below 1 and leads to disease extinction. For example, is not significantly stronger in a fluctuating population, indicating 20%prevalencecorrespondstoR p1.25.Indirectlytrans- 0 that these forms of heterogeneity might not be the key for disease mitteddiseases,R isproportionaltothetransmissionrate 0 persistencethroughhostpopulationbottlenecks. betweenindividuals.Achronicdiseasewith20%prevalence would thus be eradicated by a long-term 20% decrease in Keywords:hantavirus,SinNombrevirus,sex-biasedtransmission. the transmission rate. In rodents, which experience large changesinhostpopulationsizeandpossiblyincontactrates over a range of timescales (Fryxell et al. 1998; Mills et al. * E-mail:[email protected]. 1999), reductions in R0 below 1 could persist for long enough to drive the disease to extinction. † E-mail:[email protected]. For diseases with recovery and acquired immunity, the ‡ E-mail:[email protected]. reductionintransmissionrequiredfordiseaseeradication Am.Nat.2008.Vol.172,pp.855–867.(cid:1)2008byTheUniversityofChicago. is much larger. In measles, which has rapid recovery and 0003-0147/2008/17206-42915$15.00.Allrightsreserved. DOI:10.1086/592405 lifelong immunity, a tiny value of p∗ (far less than 1%) 856 The American Naturalist corresponds to a value of R greater than 10 (Anderson lation size (McCallum et al. 2001). With a mass-action 0 and May 1992; app. A). model, fluctuations inthe hostpopulationhavelargeref- Acquiredimmunitycreatesonetypeofheterogeneityin fects on persistence because contact rates decrease when ahostpopulation.Otherformsofheterogeneity,including the population size becomes small. hostencounterrates,infectiousness,andsusceptibility,can The Sin Nombre virus (SNV) provides a well-studied affecttherelationshipbetweenprevalenceandpersistence. exampleoflong-termpersistenceatrelativelylowprevalence Theory developed in the context of sexually transmitted (summarizedinAdleretal.2008).Thispathogen,primarily diseases has focused on heterogeneity in encounter rates. ofdeermice(Peromyscusmaniculatus),hasreceivedexten- Because these rates appear nonlinearly in the expression sive attention since the first large human outbreak in the for R , a heterogeneous population has a higher value of southwesternUnitedStatesin1993–1994(Schmaljohnand 0 R thanahomogeneouspopulationwiththesameaverage Hjelle 1997;Engelthaleretal.1999).Deermiceremainin- 0 rate of encounters betweenindividuals(Lajmanovichand fected for life (Mills et al. 1999; Kuenzi et al. 2005). There Yorke 1976; Castillo-Chavez et al. 1989; Adler 1992). In is no vertical transmission from parents to offspring, but contrast,infectiousnessandsusceptibilitybothappearlin- offspringofinfectedmotherscancarryprotectivematernal early,andthevalueofR dependsonlyontheirrespective antibodiesforatime(Boruckietal.2000).Themostlikely 0 means(GalvaniandMay2005),althoughtheycaninteract routeoftransmissionisthroughdirectcontactofdeermice to alter R when their covariance is nonzero (Becker and during aggressive encounters (Bottenet al. 2002).Scarring 0 Marschner 1990). hasbeenusedasameasureofaggressiveinteraction(Botten Otherstudiessuggestthatheterogeneityofinfectionrisk et al. 2002), and several studies have found scarring to be is an important parameter to include in disease models. positivelycorrelatedwithseropositivity(Calisheretal.1999, For example, heterogeneity in contacts in a model of bo- 2007; Douglass et al. 2001). vine tuberculosis (Mycobacterium bovis) in possums ManystudieshaveidentifiedassociationsbetweenSNV (Trichosurus vulpecula) decreases the populationlevelim- infectionandhostcharacteristics.Mostprominently,there pact of the disease and increases the difficulty of disease isawidespreadfindingofhigherprevalenceinmales(Mills elimination(Barlow2000).Similarly,sexuallymaturemale etal.1997;Booneetal.1998;Douglassetal.2001;Calisher yellow-neckedmice(Apodemusflavicollis)withlargebody etal.2005,2007;Allenetal.2006).Arecentstudyidentified sizegarneralargeproportionofticks(Perkinsetal.2003), this pattern inadults but foundnosucheffectinjuveniles and treatment or removal of these hosts would have a and subadults (Calisher et al. 2007), perhaps due to the disproportionate effect on the prevalence of the enceph- presence of maternal antibodies (Borucki et al. 2000). alitis the ticks carry. Several studies have found higher prevalence in indi- Incontrast,studiesthatnormalizethevalueofR rather viduals withhighermass (Millsetal.1997;Escutenaireet 0 thantheunderlyingpropertiesofindividualhostsfindthat al.2000;Douglassetal.2001;Calisheretal.2007).Because heterogeneityininfectiousnesscanincreasetheprobability mass generally increases with age in mice, this is to be of extinction in the initial stages of an epidemic. In this expected in a chronic pathogen. A steeper slope of the case,thefirstinfectedindividualsareunlikelytobetheso- relationship between prevalence and mass in males than called superspreaders that contribute disproportionatelyto in females may indicate more rapid acquisition by older R (Galvani and May 2005; Lloyd-Smith et al. 2005b). males(Calisheretal.2007)butcouldalsobeduetomore 0 Disease persistence is also closely tied to the total host rapid weight gain in females. A recent study directly ex- populationsizeitself(Ostfeldetal.1996).Ifthehoststhat aminedseroconversion(fromseronegativetoseropositive are mostlikelytocarryandtransmittheinfectionarethe status) and found higher rates in adult males in breeding hosts that survive population crashes, the disease is more condition (Douglass et al. 2007). likely to persist through bottlenecks. If those same hosts Thisstudyusesanextensivedatasetcollectedfrommore instead were the hosts most likely to die during periods than 2,000 deer mice captured over the course of 4 years ofdecline,thebottleneckactsmuchlikeatargetedcontrol inthewestdesertofUtahforaninvestigationofthecauses measure and effectively eliminates the disease (Barlow and consequences of higher prevalence in males. We use 2000; Perkins et al. 2003). logistic regression to identify factors associated with sero- For directly transmitted diseases, the interaction be- positivityinthepopulationofanimalsattheirinitialcapture tweenhostheterogeneityandpopulationsizefluctuations andparametricsurvivorshipmodelstostudyseroconversion dependsonhowcontactratesvarywithhostdensity.Mod- in the smaller number of animals that were recaptured in els range from a constant-contact (or frequency-depen- multipleseasons.Inaddition,welookatfactorsleadingto dent) model, where the contact rate is independent of observed patterns of sex-ratiobias, focusing on theroleof population size, to a mass-action (or density-dependent) differential survivorship in the two sexes. model, where the contact rate is proportional to popu- We hypothesize that three types of heterogeneity may Sex Bias in a Virus 857 be important in this disease: higher susceptibility, higher positive status) in mice recaptured in a later season, we encounterrates,andhighersurvivorshipinmales.Wecon- used maximum likelihood to fit models of the form structandparameterizeamathematicalmodeltoexamine theimportanceofthesefactorsfordiseasepersistenceand Pr(mouse i seropositive)p1(cid:1)e(cid:1)[a0(1(cid:2)SajXij)]Ti, (2) prevalence. Specifically, we predict (1) heterogeneity in encounter rates will contribute to disease persistence ina stablepopulation,(2)heterogeneityinmortalitywillmag- where Xij is the value of covariate j in mouse i, aj is the nify this effect in a fluctuating population, and (3) het- coefficient, and Ti is the time between captures of mouse erogeneity in susceptibility will have little effect on prev- i. Covariates considered were scarring, mass z score, sex, alence in either stable or fluctuating populations. and season. For a given model, the maximum likelihood was found using the “optim” function in R. Covariates were included in the final model if they increased thelog Methods likelihood by more than 2.0. Confidence limits were es- timated by computing where the log likelihooddecreased Field Data by 2.0 (Edwards 1972; Hilborn and Mangel 1997) using the “uniroot” function in R. Deer mice were nondestructively sampledfrom19differ- We used logistic regression (“glm” in R with the bi- entsitesneartheWestTinticMountainsintheGreatBasin nomial family; Hosmer and Lemeshow 2000) to analyze Desert of central Utah (Juab and Utah counties; Lehmer thefactorsassociatedwithseropositivityintheinitialcap- et al. 2007). A total of 2,032 deer mice were captured at tures. We first tested the effects of season,year,sex,scars, leastonce,with166recapturedinasubsequentseasonand reproductive status, mass z score, and population size in 1,658 recaptured within the same season (a totalof 3,856 a univariate analysis. We then included those in a full captures). Captures took place over 19 sites using a 3.14- model without interactions and used backward selection ha trapping web (Mills et al. 1999) during each spring (with a Pp.1 cutoff) to remove insignificant terms. We (May and June) and fall (August–October) in the years tested for interactions between male sex and the other 2002–2005. Each site was trapped for 3–7 nights during covariates using forward selection (adding terms with a season. Mouse densities per hectare were computed as P!.05).Wealsousedlogisticregressiontotestforfactors the number captured divided by 3.14. Not all sites were associated with the proportion of males. In each case, we trappedineachoftheeightpossibleseasonandyearcom- report the results on the final multivariate model. binations, leading to a total of 86 such combinations. Toestimatetheprobabilityofcaptureandapparentsur- Animals were identified to species, weighed and sexed, vivorship, we extracted a series of events for each mouse and given uniquely numbered ear tags (Lehmer et al. starting on its date of first capture, using the Cormack- 2007). Mice had reproductivestatusdeterminedbyvisual Jolly-Sebermodelwithaconstantsurvivalprobabilityper estimationofexternalreproductivecondition(recordedas unit time (Brownie et al. 1986; Nichols 2005). For each positive for lactating or perforate females and scrotal subsequent trapping night at that site, the mouse was re- males;Bottenetal.2002;Calisheretal.2005)andscarring corded as either trapped or nottrapped.Foreachmouse, (present or absent) recorded by visual examination. On the log likelihood of a given record is their first capture in any given season, mice were tested forSNVexposurewithenzyme-linkedimmunosorbentas- says for antibodies (immunoglobulin G) against SNV in ln(L)p(cid:1)d(t (cid:1)t)(cid:2)cln(p)(cid:2)mln(1(cid:1)p) l f blood in a laboratory (biosafety level 3) at the University (cid:1) [ ] of Nevada, Reno(Otteson etal. 1996). Tocorrectfordif- (cid:2) ln1(cid:1)p (1(cid:1)p)j(cid:1)1e(cid:1)d(tj(cid:1)tl), ferencesintechnique,season,andyear,amasszscorewas tj1tl computed as the number of standarddeviationsfromthe mean mass for that sex, season, and year. Juveniles were wheredistheestimatedprobabilityofdisappearancedur- not excluded because mass z score, often used to identify ing 1 day, p is the probability of capture conditional on young animals, was included in the analyses. presence, t is the last day that mouse was seen, t is the l f first, c is the number of times captured between the first andlastnights,misthenumberoftimesmissedbetween Statistics the first and last nights, and t is the jth date after t for j l All statistical analyses were done in R, version 2.4.0 (R trapping at that site. The total log likelihood was found DevelopmentCoreTeam2007).Toanalyzetheprobability by summing over all mice. There were 2,039 mouse/date of seroconversion (transition from seronegative to sero- combinationswhenamouseknowntobealivecouldhave 858 The American Naturalist been recaptured and was not, and there were 15,917 aged13.2%acrosssites(fig.1B)andwasuncorrelatedwith mouse/date combinations after the last sighting. the current population density but was significantly cor- To test for effects of sex and seropositivity, modelswere related with the density 1 year earlier. builtwithdifferentparameters(valuesofdandp)foreach subgroup,parametersthatwereoptimizedwiththe“optim” function in R, and improvement in model fit was tested Sex-Biased Prevalence and Population Sex-Ratio Bias with the likelihood ratio test (Hilborn and Mangel 1997). Ofthe2,157initialcapturesinanyseasonwitharecorded We tested for the effects of mass z score by setting dp sex, 81 out of 915 females (8.8%) and 230 out of 1,242 d (cid:2)dz and testing whether d differs significantlyfrom0. 0 1 1 males (18.5%) were seropositive (x2p39.1, P!.0001). Overall prevalence was 14.4% in individuals. Of the 86 combinationsofsite,season,andyear,69hadatleastone Empirical Results seropositive individual, meaning that the infection was The populations at the 19 sites fluctuatedaroundamean absent in 17, or nearly 20%, of samples. Of these 69, 54 of 7.98 mice/ha (fig. 1A). The infection prevalence aver- had male prevalence greater than female prevalence, one Figure 1: Summary of empirical data. A, Trajectoriesof population densityatthe19sites(graylines)andtheaveragepopulationdensity(solid black line) over the course of the study. B, Trajectories of prevalence at the 19 sites (gray lines) and the average prevalence(solid blackline). C, Relationshipofmaleandfemaleprevalenceatthefullsetof86measurements(triangles),themeasurementsaggregatedbysite(circles)anddata fromthe 23sites reported by Calisher etal. (2007;squares).Linearregressionlinesareincludedforillustration.Maleandfemaleprevalenceare significantlycorrelatedwiththerawdata(Kendall’stp0.26,P!.01)andtheCalisherdata(Kendall’stp0.63,P!.0001)butnotfortheaggregated data(Kendall’stp0.25,Pp.14).D,Relationshipoftheproportionofmaleswiththetotalmousedensityforthefullsetof86measurements (triangles), the measurements aggregated by site (circles) and data from the 23 sites reported byCalisheretal. (2007;squares).Linearregression lines are included for illustration. The population sex ratio and density are significantlycorrelated withtherawdata(Kendall’stp(cid:1)0.20,P! .01)andtheCalisherdata(Kendall’stp(cid:1)0.39,P!.01)butonlymarginallysowiththeaggregateddata(Kendall’stp(cid:1)0.32,Pp.06). Sex Bias in a Virus 859 had equal prevalence, and 14 had male prevalence lower Table 2: Factors associated with seropositivity thanfemaleprevalence(fig.1C).Therelationshipbetween Factor Coefficient SE P male and female prevalence is similar to that in a recent Intercept (cid:1)3.86514 .37667 !.0001 studyofthesametwospecies(Calisheretal.2007),where Male sex 1.03009 .25889 !.0001 reportedresultspooledmeasurementsinsitesoverseveral Scars .63560 .15489 !.0001 seasons.Formoredirectcomparison,figure1Calsoshows Mass z score .25162 .13066 .054 male and female prevalence after combiningoverseasons Spring 1.19153 .27306 !.0001 and years, giving 19 data points of which 17 show male- Spring: male sex (cid:1).69476 .32381 .03 biased prevalence. z score: male sex .55329 .16040 !.0001 Year 2003 .08414 .33246 .8 The fraction 0.568 of males, based on the 1,146 males Year 2004 .92934 .32436 !.0001 and871femalescapturedatleastonce,differssignificantly Year 2005 .92505 .32175 !.0001 from0.5(x2p37.49,P!.0001).However,thisdifference isdueatleastinparttothehigherprobabilityoftrapping males(seebelow).Theobservedfractionofmalesdepends ropositive,aswereindividualswithscars.Althoughmales on the population density, with more male-biased popu- are more likely to have scars, both factors are significant lations in sites with lower density (Kendall’s tp0.20, in the multiple regression, while their interaction is not. P!.01; fig. 1D). Again, the recently published data of Seropositivity was higher in spring and higher in the last Calisheretal.(2007)showasimilar,ifweaker,trendthan 2 years of the study (2004 and 2005), after population found in our data aggregated by site. sizes had recovered from the effects of the 2002 drought. The roughly 1-year delay between increased population sizeandincreasedprevalenceisconsistentwiththeresults Sex-Biased Prevalence foundatothersites(Adleretal.2008).Malesexinteracted withseason,creatingasmaller(butstillpositive)increase We looked in two ways for factors leading tohighermale inmaleseropositivityinthespring.Maleswithhighermass prevalence:(1)usingmaximumlikelihoodtoidentifyfac- hada higher probabilityofseropositivity.Thecoefficients torsassociatedwithseroconversioninmicethatwerecap- and significance of the covariates were robust to removal turedinmultipleseasonsand(2)usinglogisticregression of year from the model. Reproductive status and current to identify factors associated with seropositivity for each population density had no significant effect on the prob- mouseatitsinitialcapture.Ofthe166individualsrecorded ability of seropositivity. as recaptured in a different season, 123 wereseronegative We looked at characteristics of the site in the previous ontheirinitialcaptureandhadconsistentlyrecordeddata seasontoattempttoidentifythesourceormodeoftrans- on sex, mass, and scars. Of these, 22 out of 82 males mission. Starting from the model in table 2, we usedfor- seroconverted,asdidfiveoutof41females.Usingapara- ward selection to test for effects of the densities of males metricsurvivalmodel(eq.[2]),maximumlikelihoodiden- and females, the densities of infected males and females, tifiedtwofactorsassociatedwithseroconversion,malesex, the summed mass z scores of all males and females, and and mass z score (table 1). The interactionbetweenthese the summed mass z scores of infected males andfemales. factors was not significant. Averaging over all mice gives These analyses excluded 336 individuals lacking data for an estimated mean rate of seroconversion of 0.00170/day the previous season (for a total of 1,696). formalesand0.00072/dayforfemales,witharatioof2.37. All variables with the exception of the summed mass z We tested the effects of season (spring or fall), year score of infected females were significant when added to (2002, 2003, 2004, or 2005 recordedasfactors),sex,scar- the model in table 2. However, backward selection re- ring, reproductive status, mass z score, and population moved all but two: the summed mass z score of infected density, using the 1,877 individuals (out of 2,032) with males (coefficient 0.05847, SE 0.02853, Pp.04) and the complete data at their initial capture. Logistic regression summed mass z score of all females (coefficient 0.04242, identified several factors and interactions associated with SE 0.01708, Pp.013). Inclusion of these variables pro- seropositivity (table 2). Males were more likely to be se- duced only small changes in the coefficients of the basic model (app. B in the online editionoftheAmericanNat- Table 1: Factors associated with seroconversion uralist). The effect of summed infected malemasszscore Factor Coefficient Lower CL Upper CL indicates that transmission may occur largely from heavy Baseline rate .001100 .0000736 .00171 infectedmales.Theeffectofsummedfemalemasszscore Male sex 9.1314 4.2269 18.080 could be due to a higher probability of subsequentinfec- Mass z score 6.6051 2.518 11.924 tionbylargeterritorialfemales.However,duetothehigh Note:CLpconfidencelimit. degree of collinearity among these variables, these results 860 The American Naturalist mustbetreatedwithcaution.Bootstrappinganalysisfound Table 4: Analysis of recapture data that these terms were retained in a majority of samples, Factor Coefficient Lower CL Upper CL althoughothertermswereoftenretainedinthefinalmodel Male catchability .345 .327 .362 (app. B). Female catchability .306 .285 .326 Uninfected male apparent mortality .0110 .0100 .0122 Population Sex Ratio Bias Uninfected female apparent mortality .0144 .0128 .0164 Weusedlogisticregressiononallinitialcapturestoidentify Disease-induced factors that correlate with a higher proportion of males mortality 1.276 1.047 1.611 (table3).Thenegativecoefficientfordensitysupportsthe Note:CLpconfidencelimit. finding that males are more common when population density is low, and the negative coefficient for spring in- probability of capture among individuals and seasons dicates that males are less common in spring. (Nichols2005),theseresultsshouldbetreatedashypoth- The correlation of the population sex ratio bias with eses subject to improved estimation using more flexible density could be due to at leastthree causes:alterationof methods, such as Bayesian models that can treat individ- birthsexratioasafunctionofconditions,differentialmale uals as random effects (Clark 2007; Zheng et al. 2007). and female survivorship, or density- and sex-dependent dispersal. We have insufficient data on juvenile mice to test the birth sex ratio, and we cannot effectively distin- guishdispersalfrommortality(ofthe166micerecaptured, onlyfourcanbeunequivocallyidentifiedashavingmoved Model and Parameter Estimates between sites). We used a parametric survivorship mark-recapture Model Derivation model to examine the effects of genderandseropositivity on the probability of recapture (“Methods”). Males are slightlymorelikelytobecapturedthanfemales,indepen- The data analysis identified three possible forms of het- dentoftheirinfectionstatus(table4).Becausewedidnot erogeneity: (1) males have lower apparent mortality, (2) implement a full Jolly-Seber model due to the complex males are more likely to become infected (higher suscep- structure of this data set, we cannot accurately estimate tibility), and (3) males may have higher encounter rates. the true fraction of males in the population. However, a Our data cannot distinguish the last two possibilities, so substantial portion oftheobservedsex-ratiobiasisprob- we study them and their interaction with mortality sep- ably due to this difference. arately. The effects of mass z score and season will be Overall male apparent mortality is about 20% lower modeled in future work. than that of females. Interpreting apparent mortality as Our model is a two-sex version of the SI model of true mortality, we estimate a mean life span of 90 days susceptible and infected individuals (Anderson and May for uninfected males and 69 days for uninfected females. 1992;DiekmannandHeesterbeek2000)trackingtheden- Inaddition,wefindthattheapparentmortalityofinfected sities of susceptible males and females (S andS,respec- m f individuals is approximately 28% higher than thatofun- tively) and infected males and females (I and I, respec- m f infected individuals for both sexes, with no support for tively;Allenetal.2006).Forconvenience,wealsousethe different effects of infection on the two sexes. We found male and female prevalences p andp andthe totalpop- m f noeffectofmasszscoreonsurvivorshipineithersex,but ulations N and N (variables and parameters are sum- m f we cannot rule out that some other covariate that is cor- marized in table 5): related with infection underlies the observed difference. The estimated values correspond to a mean life span of dS 71daysforinfectedmalesand54daysforinfectedfemales. mp(cid:1)v2j g p S (cid:1)v vj g pS dt m m mm m m m f m mf f m Becauseourmodeldoesnotincorporatedifferencesinthe (cid:1)d S (cid:2)bN, (3) m m f Table3:Factorsassociatedwithahighproportionofmales dS Factor Coefficient SE P fp(cid:1)v vjg p S (cid:1)v2jg pS Intercept .765285 .104378 !.0001 dt m f f fm m f f f ff f f Spring (cid:1).327917 .088343 !.001 (cid:1)dS (cid:2)bN, (4) Density (cid:1).028343 .007564 !.001 f f f Sex Bias in a Virus 861 Table 5: Variables and parameters in the model Symbol Description N , N Densities of male and female mice m f I , I Densities of male and female infected mice m f S , S Densities of male and female susceptible mice m f p , p Prevalence of infection in male and female mice m f d , d Daily mortality probability for susceptible males and females m f k Proportional increase in mortality due to infection j , j Susceptibility of males and females m f b Production of female offspring per females per day v , v Encounter parameters for males and females m f g , g , … Density dependence of contacts between males, males and females, etc. mm fm N Normalizing population density for the mass-action model d dI not sensitive to the order chosen or the details of mpv2j g p S (cid:2)v vj g pS dt m m mm m m m f m mf f m interpolation. We solve for the birth rate b(t) that produces a given (cid:1)kd I , (5) m m trajectory of the total population N(t). In the absence of disease-induced mortality (where kp1), the equations dI fpv vjg p S (cid:2)v2jg pS forthetotalpopulationsN andN arelinear,andwecan dt m f f fm m f f f ff f f m f solve for b(t) directly (Adler et al. 2008). When k(1, (cid:1)kdI . (6) thismethoddoesnotwork(becausethediseaseaffectsthe f f population dynamics). We instead adjusted the birth rate Weassumeproportionatemixingbetweenthesexes(Cas- in each interval so that the population size matches the tillo-Chavez et al. 1989) as described by the encounter given population time series. parametersv andv,potentiallydifferentmaleandfemale The differential equations are deterministic, and they m f susceptibilities of j andj, a per-femalebirthrateof2b, give trajectories of population size of infected and unin- m f and a sex ratio of 0.5 at birth. The uninfected mortality fectedmalesandfemalesinaregionof3.14ha.Thepop- rates for males and females are d and d, increased by a ulations studied here are embedded ina muchlargerand m f factor of k in infected individuals. relatively homogeneous population throughout the west The terms g , g , g , and g describe the density de- desert of Utah and are thus samples rather than isolated mm mf fm ff pendence of encounters. For simplicity, we compare two siteswiththeirowndynamics(KeelingandGrenfell2002). extreme cases (McCallum et al. 2001). In the constant- To compare with empirical data, we sampled from the contact model (or frequency-dependentmodel),eachgis results in a way to mimic the field observations at each equal to 1. In the mass-action (or density-dependent) ofthe86timescorrespondingtopopulationschosenfrom model,thecontactrateisproportionaltopopulationden- the field data (excluding the interpolating values). First, sity, and g pg p2N /N and g pg p2N/N , we chose a Poisson-distributed number of males with mm fm m d mf ff f d where N normalizes the models to match when thetotal mean given by the model output for N . We chose a d m population is equal to N with an even sex ratio. Thus, Poisson-distributednumberoffemaleswithmeangivenN d f the rate at which males are infected by females, for ex- butreducedbyafactorestimatingthelowerprobabilityof ample, is proportional to the product of the number of catchingafemaleoveranaverageofthreenightsofsampling infected females and the number of susceptible males. (afactorof0.926p{1(cid:1)[1(cid:1)0.306]3}/{1(cid:1)[1(cid:1)0.345]3}). To model a fluctuating population, we created a pop- The numbers of infected males and females were chosen ulation trajectory using the 86 measured population sizes as binomial random variables with probabilities given by (thenumbersmeasuredineach3.14-hasite,shownasper- the model output of p and p. m f hectaredensitiesinfig.1A).Weembeddedthesevaluesin As an alternative to comparing samples from the sim- a single time series by linearly interpolating, withlognor- ulation with the empirical data, we treated the measured mally distributed noise (with meanp1 and SDp0.8), maleandfemaleprevalencesineachsiteasrandomeffects between the last value in one site and the first value in and used empirical Bayes methods to estimate the un- thenext,creatingonaverageninenewdatapointsbetween derlying distribution from which these prevalences were each pair. The new time series has a total of 245 points. drawn (Cassella 1985; Clark 2007). If these values closely Sitesareconcatenatedinarandomorder,andresultswere match the deterministic results of the simulation, the re- 862 The American Naturalist sults are consistent with heterogeneity created solely by population fluctuations because the model has no other form of heterogeneity among sites. Model Parameterization Themortalityratesareestimatedfromthemodelintable 4 and simplified to d p0.0145 and d p0.8d. We use f m f kp1.276astheestimateoftheincreaseinmortalitydue to the disease. The results on seroconversion (table 1) suggest that males are roughly twice as likely to become infected. We thus set either v p2v (heterogeneity in m f encounters) or j p2j (heterogeneity in susceptibility). m f We compute the average values of the mortality rate, susceptibility,andencounterratesastheaverageweighted by the fractions of males and females in the population at equilibrium. The fraction of males will approach d/(d (cid:2)d). We choose the mean encounter rate so that f m f the mean prevalence across sites matches the measured average of 13%. Mean susceptibility is fixed at 1, and the mean death rate is fixed at 0.0129 (the measured mean). ThenormalizingdensityN issettothemeanpopulation d density of 8.0 mice/ha observed in this study. Comparison of Model with Data The models with constant contact, whether with higher encounterparameterorhighersusceptibilityinmales,ac- curatelypredicttheratioofmaletofemaleprevalenceand capture the distribution of male and female prevalence (fig. 2). The distribution of samples from the simulation and the empirical data match well (P1.3 for both males Figure2:Smootheddistributionofprevalencesformales(A)andfemales and females using the Kolmogorov-Smirnovtest“ks.test” (B) across 86 measurements using the raw data (solid black line), the inR).ThedistributionofempiricalBayesestimatesofthe deterministicsimulationwithheterogeneityinencountersandmortality (dashedgrayline),samplesfromthedeterministicsimulation(solidgray prevalenceandthedeterministicsimulationresultslargely line),andempiricalBayesestimatesbasedonthemeasuredvalues(dashed overlap, sharing 88% of area for males and 81% of area black line). Parameter values are kp1.276, v p0.0615, v p0.123, f m for females. This concordance indicates that the simula- v¯p0.0956,j pj p1,d p0.0145,andd p0.0116,assumingcon- f m f m tions, which include only population fluctuation as a stantcontact.Resultswithheterogeneityinsusceptibilityandmortality source of variation, capture the extent of the observed aresimilar. variation without appealing to other forms of heteroge- suggests thatsexratio variationhasonlyasmalleffecton neity. With mass action, the simulation predicts a large the dynamics of prevalence. proportionofsiteswithunrealisticallyhighandlowprev- alence (results not shown). The model produces a significantly lower fraction of Effects of Heterogeneity on Prevalence maleswhenthepopulationdensityishigher,butitiswith a slope significantly less than that in the data (fig. 3A). Weusethesimulationstostudyhowheterogeneityaffects Samplesfromthesimulationgenerallyshownosignificant the prevalence of the disease by comparing mean preva- relationship. The empirical Bayes estimates of the distri- lence over a range of parameters with and without the butionoftheproportionofmalesacrosssiteshaveawider threeformsofheterogeneity(encounter,susceptibility,and distribution than those in the simulation, indicating that mortality). Figure 4A uses a baseline with heterogeneity another source of heterogeneity, probably seasonality(ta- inencountersandmortalitybutnoneinsusceptibility,and ble 3), plays an important role (fig. 3B). That the simu- figure 4B uses a baseline with heterogeneity in suscepti- lationnonethelesscloselyfitstheempiricalprevalencedata bility and mortality but none in encounters. Removing Sex Bias in a Virus 863 dashedgrayarrowsinfig.4).Inparticular,weexpectedto see the largest effect of combined heterogeneity in en- counters and mortality in the presence of fluctuations. Their removal leads to a 35% reduction in overall prev- Figure3:A,Relationshipoftheproportionofmaleswithmousedensity intherawdata(triangles)andthesimulation(circles).Parametervalues as in figure 2; linear regression lines are included for illustration. The values are significantly correlated in the simulation results (Kendall’s tp(cid:1)0.22, P!.01). B, Smoothed distributionoftheproportionmale usingthesameparametervaluesandsymbolsasinfigure2. Figure 4: Effects of removing different factors on the mean male and female prevalence generated by the simulation while maintaining the heterogeneityin encounterrates(whilekeepingthemean meanvalue.Numbersbeloworabovethelettersgivetheoverallprev- rate the same) lowers both male and overall prevalence alence,andthelargedotindicatestheempiricalresult.Thethindotted lineisthelineofequalmaleandfemaleprevalence,achievedonlyinthe (cf. points B and M in fig. 4A). However, removing het- absenceofheterogeneity.A,Fluctuationsandheterogeneityinencounters erogeneity in mortality (point E in fig. 4A and point S in andmortality(pointB),fluctuationsandheterogeneityinmortality(M), fig. 4B) or in susceptibility (point M in fig. 4B) leads to fluctuationsandheterogeneityinencounters(E),fluctuationsonly(N), overallincreasesinprevalence.Theprevalencewithneither constantpopulationwithheterogeneityinencountersandmortality(C), form of heterogeneity is similar to that with mortality constantpopulationwithheterogeneityinmortality(grayM),constant populationwithheterogeneityinencounters(grayE),andconstantpop- heterogeneityonly(pointNinfig.4).Removingthefluc- ulationwithnoheterogeneity(grayN).Parametervaluesasinfigure2. tuations in population size leads to an increase in prev- B,Fluctuationsandheterogeneityinsusceptibilityandmortality(point alence (point C in fig. 4). B), fluctuations and heterogeneity in mortality (M), fluctuations and We can directly solve for the equilibria in the absence heterogeneityinsusceptibility(S),fluctuationsonly(N),constantpop- of fluctuations (app. Cin the onlineeditionoftheAmer- ulationwithheterogeneityinsusceptibilityandmortality(C),constant populationwithheterogeneityinmortality(grayM),constantpopulation icanNaturalist).Contrarytoourprediction,removinghet- with heterogeneity in susceptibility (gray S), and constant population erogeneityhasnearlyidenticaleffectsonprevalenceinthe withnoheterogeneity(grayN).Parametervaluesasinfigure2,except presence or absence of fluctuations (cf. black arrows and v pv pv¯p0.0996andj p0.6429,j p1.286. f m f m 864 The American Naturalist alencewhetherornotfluctuationsareincluded(cf.points B and N in fig. 4A). How Heterogeneity Changes Equilibrium Prevalence and R 0 Thisstudyaddressesanapparentparadox:ifapopulation were homogeneous, chronic diseases with low prevalence would have values of the basic reproductive number R 0 dangerously close to the extinction threshold at R p1. 0 Wehypothesizethatheterogeneityassociatesalargervalue ofR withaparticularprevalencethanisgivenbyequation 0 (1). In a heterogeneous population, computing R is fa- 0 cilitated by finding the next-generation operator, which tabulatesthenumberofmaleinfectionscreatedbyamale, the number of male infections created by a female, and Figure 5: Effectsofheterogeneityontherelationshipbetweenequilib- soforth.ThebasicreproductivenumberR istheleading rium mean prevalence and R, comparing cases with noheterogeneity 0 0 eigenvalueofthismatrix(Diekmannetal.1990).Detailed (solidcurve),heterogeneityinsusceptibilityandmortality(dottedcurve), andheterogeneityinencountersandmortality(dashedcurve).Thesolid calculationsaregiveninappendixDintheonlineedition dot plotstheR associatedwithheterogeneityinencountersandmor- of the American Naturalist. 0 tality,usingthebaselineparametervaluesfromfigure1againsttheprev- Figure 5 compares the case with no heterogeneity with alencefoundinthesimulationwiththeseparametervaluesandfluctu- variousformsofheterogeneity.Asalways,thepopulation- ations. The open dot plots the R associated with heterogeneity in 0 wide average encounter parameter, susceptibility, and susceptibility and mortality using the baseline parameter values from figure 2 against the prevalence found in the simulationwiththesepa- death rate are normalized to ensure that observed effects rametervaluesandfluctuations. are due to heterogeneity itself. We see that a particular prevalence is associated with a larger R in the presence 0 of heterogeneity and that this effect is enhanced when in males and females. This disease-induced mortality fluctuationsareincluded.ThecomputationofR doesnot would require the disease to have substantially higher 0 explicitly include fluctuations, and an extension of the transmissioninordertosurvive,butithaslittleinteraction theory in periodically forced populations might be nec- with heterogeneity. These results contrast with an earlier essary (Bacaer and Guernaoui 2006). Although the ab- findingthatantibody-positivePeromyscusboyliihadhigher solute effects are not large, the R associated withaprev- apparent survival (Abbott et al. 1999). More complete 0 alence of 0.13 changes from 1.15 in the absence of all analysis of this complex data set is required to verify this heterogeneity to 1.22 with fluctuations and heterogeneity result, in particular to deal with heterogeneities among in encounters and mortality. This value is 50% farther individuals and differences among sites and seasons. from the extinction threshold. We used these data to parameterize a system of differ- ential equations tracking infected and susceptible males and females. A mass-action model fit the distribution of Discussion prevalences poorly, and we focus on a constant-contact We hypothesized that SNV persists at low prevalence in model(McCallumetal.2001;Adleretal.2008),although deer mouse populations due in part to various forms of therealitycouldbeamorecomplexcontactprocess(Ryder heterogeneity and focused here on heterogeneity between et al. 2007). In combination, the estimated higher male the sexes. Our data analysis identifiedthree formsofhet- susceptibility or encounter rates along with lower male erogeneity in this population. The higher prevalence and mortality are sufficient to explain higher male prevalence seroconversion rate in males indicates roughly doubled and part of the bias in the sex ratio. The simulations susceptibility or encounter rates. Without more detailed capturemuchoftheobservedvariationinprevalence(fig. timeseries,weareunabletodistinguishbetweenthesetwo 2), arguing that this variation might be largely the result mechanisms. A bias in the population sex ratio at times ofpopulationsizechangesratherthanintrinsicdifferences of low density is consistent with our direct estimate of a among sites and years, as found in other wildlife diseases 20% lower mortality rate in males. (Smith 2006). However, the simulations capture only a Ourpreliminarycapture-recaptureanalysisfoundasig- portion of the observed sex ratio bias (fig. 3), implying nificant increase of 28% in the mortality rate in infected thatanotherfactor,quitepossiblyseasonality,needstobe animalsandcouldnotdistinguishtheextentofthiseffect included (Altizer et al. 2006).