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1 Department of Medicine and Epidemiology, School of Veterinary Medicine, University of California, Davis, California 95616, USA
2 Center for Vectorborne Disease, School of Veterinary Medicine, University of California, Davis, California 95616, USA
3 Department of Population, Health and Reproduction, School of Veterinary Medicine, University of California, Davis, California 95616, USA
4 Department of Landscape Biology, University of California, Davis, California 95616, USA
5 Department of Biological Sciences, California State University, Sacramento, California 95819, USA
6 Corresponding author (email: jefoley{at}ucdavis.edu)
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1LITERATURE CITED |
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infectiveresistantsusceptible (SIRS) model framework to describe transmission of the plague bacterium Yersinia pestis among rodents and fleas in California, 2) to perform sensitivity analysis with 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 parameter values obtained from the literature for Y. pestis in California rodents and fleas. The model PlagueSIRS was implemented in discrete time as a computer simulation incorporating environmental stochasticity and seasonality, by using matrix functions in the computer language R, allowing any number of rodent and flea species to interact through parasitism and disease transmission. Sensitivity analysis indicated that the model was sensitive to flea attack rate, host recovery rate, and rodent host carrying capacity but relatively insensitive to changes in the duration of latent infection in the flea, host and vector competence, flea recovery from infection, and host mortality attributable to plague. Realistic parameters and variable values did allow for the model to predict enzootic plague in some combinations, specifically when rodent species that were susceptible to infection but resistant to morbidity were parasitized by multiple poorly competent flea species, including some that were present year-round. This model could be extended to similar vectorborne disease systems and could be used iteratively with data collection in sylvatic plague studies to better understand plague persistence and emergence in nature. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1LITERATURE CITED |
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At least 18 rodents species and 27 or more flea species are involved in ongoing enzootic plague cycles in the western United States, largely independent of the traditional rat-rat flea cycle (Hubbard, 1968; Smith et al., 1998; Gage and Kosoy, 2005). Most cases of Y. pestis infection, including those in humans and rodents, are acquired through flea bites, but infections can occur through direct exposure to infectious respiratory and oropharyngeal secretions; through predation, particularly by cats; and through scavenging by rodents, such as Onychomys leukogaster, on infected hosts and carcasses (Smith et al., 1998). One theory to account for plague persistence is that unidentified, persistently infected but asymptomatic host species may maintain infection in plague-enzootic communities, whereas lethally susceptible, transiently infected "amplifying hosts" are responsible for most human infections (Pollitzer, 1954). Yet, no persistently infected reservoir hosts have been identified in the western United States. In some situations, fleas may function as de facto reservoirs, because they can retain infection for several months or longer, especially in winter climates (Eskey and Haas, 1940; Gage and Kosoy, 2005). It is likely that the dynamic interactions among hosts and fleas determine whether Y. pestis can be maintained in nature; however, it is not known which critical characteristics of these dynamics permit such maintenance.
There are prospects for enormous ecological complexity in the maintenance of the flea-host-Y. pestis system. Because Y. pestis can spread directly among people in the pneumonic form, can induce fatal disease, and it could be weaponized; it is listed by the US Department of Health and Human Services as a "List A select agent." Additionally, sylvatic plague threatens human health and several endangered and threatened 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 is there insufficient ecologic knowledge to accurately describe the mechanisms for persistence and periodic re-emergence of plague but also there is no established predictive framework for anticipating the possible consequences of intentional release of Y. pestis in regions where the bacteria could persist (Gage and Kosoy, 2005).
Several studies have described modeling approaches to plague dynamics. Some early susceptibleinfectiveresistant (SIR) plague models only dealt with humans and not rodents or fleas, and they did not allow for hosts to recover or for new susceptible individuals to be introduced into the system (Noble, 1974; Raggett, 1982). A recent model of pneumonic plague also focused on humans, and it did not consider enzootic cycles in nonhuman hosts and arthropod vectors (Gani and Leach, 2004). An excellent model of vectorborne plague was developed focusing on Y. pestis infection in humans, R. rattus, and X. cheopis (Keeling and Gilligan, 2000a, b). Although this model recreated many of the worldwide dynamics observed in human plague outbreaks, little insight was offered at the smaller scale of enzootic plague in wild rodent communities in the western United States.
In this study, we extend traditional vector susceptibleinfectiveresistantsusceptible (SIRS) models to create a flexible matrix-based community vector SIRS framework, which was used to investigate plague. There were four specific objectives: 1) to use matrix manipulation within a classical SIRS model framework to describe transmission of Y. pestis among rodents and fleas in California, 2) to perform sensitivity analysis with model parameters and variables to document which values tended to dominate model output, 3) to manipulate model parameters to determine values that would convert transient epizootic infection to enzootic, and 4) to document predicted outcomes of plague with realistic parameter values obtained from the literature for Y. pestis in California rodents and fleas.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1LITERATURE CITED |
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Plague community dynamics were modeled with a matrix version of the basic Kermack-McKendrick SIRS model, with SIRS hosts (Kermack and McKendrick, 1927; Bailey, 1982). This community plague model, designated PlagueSIRS, was implemented in R (The R-Development Core Team, http://www.r-project.org); pseudocode for the sequence of events is given in Appendix 1. Discrete-time differential equations coupled host disease transmission dynamics with those of the flea vectors. Host populations are composed of three groups of individuals: susceptibles of size S, infectious I, and resistant R, with
Flea populations were each separated into susceptibles X, infectious Y, and immunes Z (the latter retained for generality but likely to be rare or lacking in nature):
In a short time interval of 1 day, host individuals move through categories S, I, and R by disease transmission at rate ßSY, recovery at rate 
I, and death at rates dS = dR or dI (mortality attributable to disease, designated in this article as "mad"). Similar rates apply to fleas, except that after fleas acquire Y. pestis infection, there is a substantial latency period during which they are not infective, requiring addition of the vector state E. The rate at which latency shifts to infectivity is 
, and the mean time in latency is 1/. The model dynamics are described by the system of differential equations:
Multiple host-multiple vector system
A plague community contains h host species and v vector species. The community model tracks a vector of host sizes N = [N1...Nh], a vector of S values S = [S1...Sh], and analogous vectors of I, R, X, E, and Y. The community SIRS model is represented below, assuming that host death rates of S and R individuals are similar, X and E flea death rates are similar, and adding the identity matrices (1) required for the matrix multiplication:
The 3h+3v-dimensional, nonlinear system of equations was discretized for computer simulations to obtain the model for the survival of susceptible hosts (Nicholson and Bailey, 1935). If infection is distributed as a Poisson process, then Si 
S* exp(–ßY).
Transmission to new hosts can take place from several possible flea species. Let ßij be the transmission rate from vector j to host i; then, the entire host transmission process is as follows:
ßV is a v*h matrix analogous to ß that gives transmission from the jthhost to the ith vector. To transmit a disease from an infective vector j to a susceptible host i, the vector needs to 1) find a host (proportional to aSY, where a is attack rate), 2) be competent to transmit the disease (with vector competency cvj), and 3) want to use the host (with utilization rate uij). Typical vector-to-host matrix ß entries are as follows:
The host-to-vector matrix ßv has entries of the form:
where cj measures the host competency. The lower limit of disease was set by an extinction threshold of one individual; if either infected flea or host numbers went below the threshold (into fractional individuals), the simulation stopped; upper limits were set by host population upper limits (K). Seasonality was incorporated into the model for flea recovery from infection, which is known to be temperature-dependent (indeed, above 27 C, Y. pestis may not survive or be transmissible in the flea (Pollitzer, 1954), by Fourier transform.
Host and vector population dynamics
Underlying the disease transmission model is a series of functions that allow for population regulation for each host and vector species, including seasonality and environmental stochasticity. A community consists of h host species, of sizes N1, N2, ..., Nh with species-specific birth rates (bi) given in a diagonal matrix:
so that the number of new individuals per host species is as follows:
Death rates (di) are in an analogous diagonal matrix. A stochastic discrete logistic model in small time units gives host population growth and fluctuation:
where
Ki is the host carrying capacity, bm is the Malthusian birth rate, and 
i(t) ~ N(0,vr) is the environmental stochasticity term at time t, estimated by regression (May, 1974; Foley, 1997). Seasonality of rodent numbers was incorporated into the model by constraining reproduction to those months described by field data and modeling death via a Fourier function of annual periodicity (Keeling and Gilligan, 2000a). Incorporating an approximation, the transform is as follows:
where di(t) is the death rate at time t in radians, 
i is the mean death rate over the year, AHi is the amplitude of the host death rate, and tpeak is the time of peak mortality. A similar function was used for bi. For simplicity, environmental stochasticity was included only in birth and not death, which can be done because the model assumes additivity of b+d+.
Flea populations, for each species V1, V2, ...,Vv, depended numerically on the abundance of rodent hosts. The carrying capacity of a vector species j depends on the abundance of each host species i, together with some measure kij of the vector j’s ability to use host species i as a resource. In the absence of interspecific competition among fleas, the carrying capacity of vector j is given by
The carrying capacity kij offered by an individual host i to vector j depends on 1) uij, the vector’s host utilization rate; 2) qi, host resource quality; and 3) wi, a flea weighting index. The uij ranges from 0 (vector j does not use host i) to 1 (the vector can and will fully use the resources available from this host). qi, the index of the quality of resources a host provides to an average vector, is equivalent to the mean number of fleas of average size on an individual of this host species of average size. wj, the flea weighting index, is related to the size of the vector (or it could be set as some other measure of the amount of resources an individual vector uses). wj has a mean of 1 for a "typical" flea species and is close to 1 for most flea species, but it may rise significantly above 1 for large fleas. Then, the carrying capacity provided by one host to its fleas of species j is shown as follows:
and the whole carrying capacity for vector species j, which depends on the availability of all of the hosts is as follows:
Density-dependent birth of the jth flea is modeled, with the Malthusian parameter bmvj:
Then, flea death is
The model also was implemented with diagonal matrices designated Bv, Dv; seasonality was incorporated for flea birth and death by Fourier transform.
Parameter estimation and sensitivity analysis
Data were maintained in Excel (Microsoft, Redmond, Washington, USA) and analyzed in R. Data on flea distribution, flea host preferences, and plague transmission dynamics were obtained from the literature and are summarized in Table 1
. Data were used as first approximations, to further develop the model, because in many cases, including old literature with very small samples sizes and predictive habitat databases, parameter and variable estimates likely had large confidence intervals. Nevertheless, use of such estimates facilitated investigation to clarify which parameters and variables would require more accurate future estimation.
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calculated as mean time to death, obtained from the literature. Values for rodent carrying capacities, litter sizes, time windows of reproduction, and numbers of litters per year were obtained from the literature (Mayer and Laudenslayer, 1988; Nowak, 1995). Rodent population time series were obtained from the Center for Population Biology Global Population Dynamics Database (Imperial College, London, UK). The bm was calculated as rm–d(S) prorated for the period over which the species reportedly breeds in nature. Assuming a Ricker model of population regulation (Foley, 2000), r(t) was regressed on N(t) to provide estimates of rm and carrying capacity. vr was estimated as the noise component in the regression, although this was considered an overestimate, because it includes both environmental stochasticity as well as sampling variation.
The parameters q and w were estimated from reports of the mean flea numbers of rodents (Davis et al., 2002), with w scaled to a maximum value of 3 and a mean of 1. The matrix u was constructed from field data giving the fraction of all flea fauna on each host represented by each flea species. (rates of recovery for fleas and rodents) and (rate of movement from latent to active infection in fleas) were estimated as the mean time from the first point at which there was evidence of infection to the first time the flea was infective to rodents () or the flea or rodent recovered (). Vector and host competence were averaged across all published reports of experimental Y. pestis infection in that species. If no values were reported, the estimate was used for the most phylogenetically proximal species for which data were available.
Sensitivity analysis was performed to determine which parameters and variables were most responsible for driving observed dynamics, by allowing each parameter, one at a time, to range across very broad possible values for that parameter, as shown in Table 2. All other parameters were fixed corresponding to optimal estimates from literature reviewed as mean values for each parameter. The analysis was run with parameters for a three-host, two-flea community consisting of Peromyscus maniculatus, woodrats (Neotoma fuscipes), and S. beecheyi and the fleas Aetheca wagneri and Oropsylla montana (Table 3). The following outputs were used as indicators of model results: the mean expected duration of infection in the community (>100 simulation runs) and the expected maximum number of infected hosts. By using output from this exploration, critical values of parameters that would convert epizootic infection to enzootic were determined. Default values were used for all parameter values were except one, as described for sensitivity analysis, whereas the simulation was run until a cutoff for each particular value was found (if at all) where the infection was predicted to be enzootic (defined as infected hosts or fleas for 
400 days).
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–6). Chuchupate CG is at 1,890 m on Frazier Mountain in pine (Pinus spp.)-oak (Quercus spp.) forest. A previous study in Chuchupate CG has documented 20 flea species and 10 rodent species potentially involved in the ecology of plague (Davis et al., 2002).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1LITERATURE CITED |
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). Increasing host carrying capacity led to increases both in host and flea numbers, resulting in prolonged duration of infection. This pattern was saturated at high levels of carrying capacity (Figs. 1, 2) and increasing K did not induce permanent enzootic infection at any level. Changing a also did not induce enzootic disease. Very low attack rates did not allow for any disease maintenance, whereas increasing a resulted in an increased epizootic duration up to a plateau value, with a considerable concurrent increase in maximum infected host number. Modifying H, host recovery, had an expectedly important effect. As H was decreased, the duration of infection persistence and maximum number of infected animals both increased until, at low H, enzootic infection was maintained in the simulation. Although not apparent over most values used for V, very low flea recovery rates also had the capacity to convert the epizootic infection to enzootic.
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, u, w, host competence unless a host was completely incapable of transmitting disease, vector competence, mortality attributable to disease (except that at high rates of mortality, there was no enzootic disease), and q. Thus, sensitivity analysis clarified which parameters had the greatest individual impacts on model output and suggested that changes in H and V were most likely to convert transient epizootics to long persistence-time enzootics.
Parameter values supporting a plague-enzootic community were explored using published parameter values for the Chuchupate CG community, allowing a (for which no good estimates were available) to vary in order to observe expected changes in plague persistence. With a universal attack rate (all fleas on all hosts) of 0.01 bites/host/day, plague was enzootic in the community (Fig. 3). Seasonality in rodent and flea numbers was apparent, as were epizootic pulses of plague primarily occurring only in fleas (with only small numbers of infected hosts). The principal rodent and flea species contributing to persistence were woodrats, California ground squirrels, and chipmunks (Tamias spp.), with the fleas A. wagneri and M. telchinus. The features of these hosts contributing to their important roles in plague epizootics included relative resistance to the development of fatal plague (except for ground squirrels) and their ability to support more flea species, particularly the highly abundant M. telchinus. Prolonged duration of infection within individuals did not occur and was not a contributing factor to enzootic plague in the community. When a was increased to 0.1 bites/host/day, plague remained enzootic but apparent stable cycles of infection were disrupted. Reduction of a to 0.009 led to a single epizootic of approximately 200-day duration.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1LITERATURE CITED |
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The three most important parameter values required to predict epizootic duration and magnitude were H, V, and a, the attack rate of fleas on hosts. Reduction in H and V not surprisingly increased the duration of epizootics or induced enzootic disease by essentially increasing reservoir potential in the classical sense of some of the rodents. This interesting finding could help explain some of the differences in the risk of plague in diverse communities where rodent hosts have different H levels. There are estimates of H available in the literature for many California rodents and some fleas. Flea recovery rates in particular can be highly variable because those flea species with a narrow esophagus and proventriculus are more susceptible to blockage with a mat of bacteria and inflammatory debris (and thus more likely to regurgitate infectious material into the host while attempting to feed). Other flea species have greater ability to enzymatically destroy the block and less susceptible anatomic configurations (Perry and Fetherston, 1997). Undoubtedly, more accurate estimates from experimental infection would be valuable for both rodent and flea recovery rates.
Attack rate, for which very little information is available from California systems, was also an important parameter in determining durations of epizootics. This finding is comparable with results from an earlier vector-SIRS model of humans, rats, and fleas that was sensitive to flea search efficiency (Keeling and Gilligan, 2000a). Other vectorborne disease systems also are highly sensitive to attack rate, including the pioneering models of Ross (1928) and MacDonald (1957) showing the nonlinear effect of attack rate of mosquitoes transmitting malaria to humans. The estimation of this parameter from field data likely represents a critically important target for future research.
The earlier vector-SIRS plague model of Keeling was also particularly sensitive to changes in the carrying capacity of fleas per rat, the rat’s reproductive rate, and the rat’s carrying capacity. The model developed in this present study also is sensitive to rodent K: increasing K by orders of magnitude increased the expected epizootic duration but did not, in the three host-two flea system, induce enzootic persistence. This feature may be related to the particular collection of other parameter values used for that system. A previous study in natural plague foci in Kazakhstan documented an abundance threshold in great gerbils, Rhombomys opimus, above which plague was persistent (Davis et al., 2004). Such a threshold could apply in California as well, although the abundance threshold refers to the actual numbers of animals in the community, whereas the carrying capacity refers to a potential with actual populations often cycling around K. Other important differences are that the plague communities in California are highly complex, compared with systems dominated by a single reservoir such as a rat or great gerbil. Although accurate estimates of K are difficult to obtain and values are site-specific, some attempt should be made to acquire estimates with reasonable accuracy, although other, possibly more easily estimated, parameters such as and a may be more critical.
The structure of PlagueSIRS has advantages of flexibility and expansibility, while focusing on short- and long-term dynamics associated with disease transmission in communities. This model advantage is in contrast to some invasion analysis models, which focus on epidemic thresholds such as R0>1. The matrix method (Diekmann et al., 1990) could offer some insight into the community system, but two difficulties would need to be overcome. First, in reality, Y. pestis transmission dynamics must include extensive variability in recovery and transmission parameters to reflect the diverse biology of the hosts and vectors. Because the infection time varies across rodent and flea species, the K matrix would require considerable adjustment before it could calculate the diverse collection of I values appropriately. Second, to investigate persistence, the K matrix would need to be made frequency-dependent.
PlagueSIRS has the additional advantages over many vector-SIRS models in realistically representing underlying host and flea population dynamics in the model. This advantage is important because the disease dynamics being studied often occurs over distinctly long periods, a problem that is avoided in some models focusing only on rapid epidemics or disease emergence. In PlagueSIRS, rodent population growth was bounded by a carrying capacity, whereas flea density was given as a function of flea use of specific hosts, abundance of fleas on hosts, attack rates, and seasonal constraints; both rodent and flea dynamics were modified by environmental stochasticity. The earlier model of Keeling used simple flea searching efficiency and an arbitrary flea carrying capacity not explicitly tied to rodent density. Analysis of the model shown here clarifies why population regulation needs to be included in the model, and future model validation with data from field studies should allow for any necessary modifications to the functions predicting host and flea numbers to be incorporated into PlagueSIRS.
It was very interesting how the model represented the enzootic plague community at Chuchupate CG, by using parameter values from the literature. Based on serologic testing of rodents and flea collections, vector control biologists had concluded that plague was enzootic in this region and that three host-flea complexes were important in the local plague ecology, although how these complexes could function as true reservoirs was unclear. These different complexes were 1) California ground squirrel (S. beecheyi) and O. montana/H. anomalous fleas, 2) Merriam’s chipmunk (Tamias merriami) and Eumolpianus spp. fleas, and 3) dusky-footed woodrats (N. fuscipes) and Orchopeas sexdentatus and Anomiopsyllus spp. fleas. Ground squirrels were generally infested with only two host-specific fleas, whereas other hosts, including deer mice and woodrats, harbored various flea species that they shared with other rodent species. The model recreated enzootic plague with seasonal outbreaks, corresponding to published data (Davis et al., 2002). Hosts with relatively low mortality, including woodrats and chipmunks, were critically important in maintaining enzootic plague, although neither would likely be considered typical reservoirs given the short duration of infection in any individual. Deer mice and voles have been reported previously 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 flea species were A. wagneri and M. telchinus, the former reported in Chuchupate CG to infest seven different rodent species, especially deer mice (Davis et al., 2002). It was reported that A. wagneri is a competent, but not excellent plague vector (Eskey and Haas, 1940). The flea M. telchinus is found year-round and feeds on multiple hosts, although with a preference for voles. A pool of this flea species collected from a vole was reported positive for Y. pestis in July 1997 in the Chuchupate CG (Davis et al., 2002). Despite that experimental inoculations indicated that M. telchinum is a poorly competent vector (Burroughs, 1947), reducing vector competency in the model did not seriously diminish the role of this flea species in modeled epizootics, probably because numbers of this flea are commonly very high. Several flea species that had been thought to be very important in plague ecology played very minor roles in our model, including O. montana and H. anomalus, probably because they fed on rodents that were so susceptible to disease that they died. Therefore, the host-flea couple could not support anything more than a very transient epizootic. Nevertheless, O. montana is important in amplifying plague and particularly in transmitting it to humans, because this flea aggressively seeks new hosts, including humans, when its preferred host dies (Douglas and Wheeler, 1943; Holdenreid, 1952; Quan et al., 1960; Nelson, 1980; Barnes, 1982; Gage and Kosoy, 2005).
It was interesting that enzootic plague was predicted at Chuchupate, even with similar parameter values to a hypothetical three host-two flea system where plague died out. One possible explanation is that the variability across rodent and flea species in infection dynamics created heterogeneity that is, in some ways, analogous to spatial heterogeneity facilitating longer disease persistence. Specifically, the inclusion of stochasticity into an earlier plague metapopulation model was accompanied by rapid local extinction, which was remedied by the inclusion of spatial structure (Keeling and Gilligan, 2000a, b). Extending the present simulation modeling approach to spatially structured systems would be a logical, although challenging, exercise.
To summarize, a predictive model of plague in complex rodent-flea communities in California clarified key interactions driving plague local extinction or persistence. The described model represents an innovative, highly tractable method of managing ecological complexity typical of communities of multiple hosts and vectors such as occurs in plague but also other vectorborne disease systems such as West Nile virus, granulocytic anaplasmosis, and others. This model also helps establish a framework for ongoing data collection that will allow the model to be refined and ultimately more realistic. This model should be helpful in ongoing surveillance activities and in the event of apparently increased activity that could indicate intentional release.
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APPENDIX 1 |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1LITERATURE CITED |
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Main loop (repeat once a day for tmax days)
Step 1:
V 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 and X, E, and Y fleas by using changes in step 2.
Step 4: Append new S, I, R, X, E, and Y values to history data frame.
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ACKNOWLEDGMENTS |
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LITERATURE CITED |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX 1LITERATURE CITED |
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Received for publication 22 April 2005.
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