Ira B. Schwartz

Seasonality and period-doubling bifurcations in an epidemic model

The annual incidence rates of some endemic infectious diseases are steady while others fluctuate dramatically, often in a regular cycle. In order to investigate the role of seasonality in driving cycles of recurrent epidemics, we analyze numerically the susceptible/exposed/infective/recovered (SEIR) epidemic model with seasonal transmission. We show that small amplitude periodic solutions exhibit a sequence of period-doubling bifurcations as the amplitude of seasonal variation increases, predicting a transition to chaos of the kind studied in other biological contexts. The epidemiological implication is that the seasonal mechanism generating biennial epidemics may not be able to account for small-amplitude recurrent epidemics of arbitrary periodicity.

Decentralized Environmental Modeling by Mobile Sensor Networks

Cooperating mobile sensors can be used to model environmental functions such as the temperature or salinity of a region of ocean. In this paper, we adopt an optimal filtering approach to fusing local sensor data into a global model of the environment. Our approach is based on the use of proportional-integral (PI) average consensus estimators, whereby information from each mobile sensor diffuses through the communication network. As a result, this approach is scalable and fully decentralized, and allows changing network topologies and anonymous agents to be added and subtracted at any time. We also derive control laws for mobile sensors to move to maximize their sensory information relative to current uncertainties in the model. The approach is demonstrated by simulations including modeling ocean temperature.

Comparison of Culture-Confirmed Erythema Migrans Caused by Borrelia burgdorferi sensu stricto in New York State and by Borrelia afzelii in Slovenia

The clinical presentations of Lyme borreliosis in the United States and Europe seem to differ (1-7). For example, European patients with Lyme borreliosis may develop certain skin manifestations (acrodermatitis chronica atrophicans and borrelial lymphocytoma) that are extremely rare or nonexistent in the United States (6, 7). One explanation of such differences may be the recently appreciated variation in strains of Borrelia species between the two continents (7-9). All isolates from U.S. patients have thus far been members of the genomic group Borrelia burgdorferi sensu stricto (henceforth referred to as B. burgdorferi), whereas European isolates have included two additional genospecies, B. garinii and B. afzelii. Until now, however, no reports of clinical disparities have been based on the systematic study of large numbers of patients with culture-confirmed infection. To investigate possible differences in the manifestations of Lyme borreliosis in Europe and the United States, we compared epidemiologic, demographic, clinical, and laboratory findings in patients from New York State and patients from Slovenia who had microbiologically confirmed erythema migrans caused by B. burgdorferi and B. afzelii, respectively. Methods Patients To isolate Borrelia species from clinical specimens, we recruited patients with a clinical diagnosis of erythema migrans in two separate prospective studies. The Centers for Disease Control and Prevention surveillance definition of erythema migrans (10) was satisfied in almost all cases, although three Slovenian patients with erythema migrans lesions less than 5 cm in diameter were enrolled. Patients from the United States were seen at the Lyme Disease Diagnostic Center, Westchester Medical Center, Valhalla, New York, from June 1991 through August 1995. Slovenian patients were evaluated at the Lyme Borreliosis Outpatient Clinic, Ljubljana, Slovenia, in 1993. All patients were older than 15 years of age and gave informed consent. Patients were included in the present study if Borrelia organisms were recovered from their skin specimens. Several U.S. patients had repeated culture-confirmed episodes of erythema migrans in separate years; for each of these patients, only the first episode was included. Laboratory Methods All isolates were obtained from biopsy or needle aspiration of erythema migrans lesions (11, 12). Cultures were processed as previously reported in modified Barbour-Stoenner-Kelly medium (11) for U.S. patients and modified Kelly-Pettenkofer medium (12) for Slovenian patients. At the U.S. study site, we identified spirochetes as B. burgdorferi by using polymerase chain reaction (PCR) with specific primers directed at the ribosomal RNA genes of Borrelia species [13]. We identified the species of European isolates by using two independent methods: 1) species-specific PCR with different oligonucleotide primer sequences [14] and 2) pulsed-field gel electrophoretic separation of restriction enzyme MluI digestion fragments (15) (the pattern of large restriction fragments produced is diagnostic of the species). Because most Slovenian isolates by far were B. afzelii (12), cases of erythema migrans due to other species (including four cases due to B. burgdorferi and six due to B. garinii) were excluded. Complete blood counts were done, liver function tests were performed, and the erythrocyte sedimentation rate was measured at the time when cultures were obtained. Baseline serum specimens were tested for antibodies to B. burgdorferi. Convalescent-phase specimens were obtained after 1 week to 1 month for U.S. patients and after 2 months for Slovenian patients. In the United States, serum specimens were tested for antibodies to B. burgdorferi with a polyvalent enzyme-linked immunosorbent assay (Whittaker Bioproducts, Inc., Walkersville, Maryland) (11, 16). In Slovenia, the presence of serum IgM and IgG antibodies to B. afzelii was determined separately by immunofluorescent assay without preabsorption (17); a local skin isolate of B. afzelii was used as antigen. Titers of 1:256 or more were considered to indicate positivity. Statistical Analysis Differences in quantitative data were analyzed by using the median test, and differences in qualitative data were analyzed by using the chi-square test (with the Yates correction) or the Fisher exact test. We used Epi-Info 6 for all analyses (Statcalc, version 6.04a, Centers for Disease Control and Prevention, Atlanta, Georgia). All P values were two-tailed. Results Borrelia afzelii was recovered from the skin specimens of 85 Slovenian patients; none of these patients had been reported previously. Borrelia burgdorferi was cultured from 119 U.S. patients; 77 of these patients had been reported previously (11). Of the U.S. isolates, 30 were randomly selected for species identification by PCR, and all 30 were identified as B. burgdorferi. Thus, it is likely (although not certain) that the remainder of the isolates were also B. burgdorferi. Demographic patient data and the characteristics of erythema migrans are summarized in Table 1. All Slovenian patients and 113 of 119 U.S. patients (95%) were white. Slovenian patients were more likely than U.S. patients to recall a tick bite at the erythema migrans site (63.5% compared with 25.2%; P<0.001). Although the size of the erythema migrans lesions was similar in the two groups of patients, the median duration of erythema migrans at presentation was longer (14 days [range, 1 to 206 days] compared with 4 days [range, 1 to 39 days]; P<0.001) and central clearing was more common (68.2% compared with 35.3%; P<0.001) in Slovenian patients than in U.S. patients. Localized pain or burning at the erythema migrans site was more common in U.S. patients (34.3% compared with 18.8%; P=0.02). The occurrence of multiple erythema migrans lesions was greater in the U.S. patients than in the Slovenian patients, but the difference was not significant (13.4% compared with 7.1%; P>0.2). Table 1. Characteristics of Patients with Culture-Confirmed Erythema Migrans Patients in the United States were more likely than those in Slovenia to have systemic symptoms (68.9% compared with 50.6%; P=0.01), including fatigue (54.6% compared with 32.9%; P=0.003), myalgia (44.5% compared with 21.1%; P=0.001), fever or chills (37.8% compared with 8.2%; P<0.001), and stiff neck (38.7% compared with 8.2%; P<0.001) (Table 2). Objective findings on physical examination were also more common in U.S. patients (57.1% compared with 14.1%; P<0.001) and most frequently manifested as regional lymphadenopathy (seen in 28.6% of U.S. patients compared with 8.2% of Slovenian patients; P<0.001) or fever (body temperature 37.8 C) (seen in 15.1% of U.S. patients compared with 1.2% of Slovenian patients; P<0.001). Slovenian patients were more likely to have hepatomegaly (5.9% compared with 0; P=0.01), but U.S. patients more often had at least one abnormal result on a liver function assay (32.2% compared with 18.3%; P=0.04), were more likely to have erythrocyte sedimentation rates two or more times the upper limit of normal (25% compared with 3.7%; P<0.001), were more likely to seroconvert (84.2% compared with 10.9%; P<0.001), and were more often seropositive at presentation (35.3% compared with 22.4%; P=0.07). Table 2. Selected Clinical and Laboratory Findings in Patients with Culture-Confirmed Erythema Migrans Discussion Our findings establish what has long been believed: Some of the clinical and laboratory features of Lyme disease differ in the United States and Europe. The differences seem to result, in part, from a tendency for B. afzelii to be less virulent than B. burgdorferi. Slovenian patients with B. afzelii were significantly less likely than U.S. patients with B. burgdorferi to be systemically ill (Table 2) and to have fever (P<0.001) or regional lymphadenopathy (P<0.001) on physical examination. Although the duration of erythema migrans at presentation was longer in Slovenian patients, lesion size in Slovenian patients was similar to that in U.S. patients; this implies that B. afzelii spreads more slowly in the skin (11). Because central clearing occurs, in part, as a function of time (2), its higher frequency in Slovenian patients (68.2% compared with 35.3%; P<0.001) is probably related to the longer duration of erythema migrans at presentation. Nonetheless, in contrast to what might have been anticipated from earlier reports (1-6, 18), dissemination to multiple cutaneous sites did not occur significantly more often in U.S. patients than in Slovenian patients in our study (P=0.2). Our observations in patients in Westchester County, New York, agree with those of other studies of North American patients with erythema migrans (1, 4, 5), in which reported rates of systemic illness are often greater than 75%. In contrast, reported rates of systemic illness in Europe are often less than 35% (2, 3, 8). In our study, Slovenian patients were much less likely than U.S. patients to be seropositive (26 of 85 [30.6%] compared with 106 of 119 [89.1%]; P<0.001), principally because seroconversion was less common (7 of 64 patients [10.9%] compared with 64 of 76 patients [84.2%]; P<0.001). The seroconversion rate in our U.S. patients, however, was similar to that seen in other recent studies of U.S. patients with erythema migrans (5). Moreover, the contrasting rates of seropositivity at presentation were particularly evident when patients with longstanding erythema migrans (duration 14 days) were compared; 94% of such patients in the United States (15 of 16) but only 28% of such patients in Slovenia (13 of 47) were seropositive (P<0.001). With rare exceptions (19), European studies have reported low rates of seropositivity (20% to 50%) in patients with erythema migrans (3, 18), despite a long duration of illness (often longer than several weeks) (3). However, because convalescent-phase serologic samples from Slovenian patients were obtained at 2 months (rather th

Fluctuating epidemics on adaptive networks

A model for epidemics on an adaptive network is considered. Nodes follow a susceptible-infective-recovered-susceptible pattern. Connections are rewired to break links from noninfected nodes to infected nodes and are reformed to connect to other noninfected nodes, as the nodes that are not infected try to avoid the infection. Monte Carlo simulation and numerical solution of a mean field model are employed. The introduction of rewiring affects both the network structure and the epidemic dynamics. Degree distributions are altered, and the average distance from a node to the nearest infective increases. The rewiring leads to regions of bistability where either an endemic or a disease-free steady state can exist. Fluctuations around the endemic state and the lifetime of the endemic state are considered. The fluctuations are found to exhibit power law behavior.

Impact of Genotypic Variation of Borrelia burgdorferi Sensu Stricto on Kinetics of Dissemination and Severity of Disease in C3H/HeJ Mice

ABSTRACT Various genotypes of Borrelia burgdorferi sensu stricto have been previously identified among a large collection of isolates cultured from patients with Lyme disease in the United States. Furthermore, association of specific genotypes with hematogenous dissemination early in the disease course has been observed. The present study assessed kinetics of spirochete dissemination and disease severity in C3H/HeJ mice infected with two different genotypes ofB. burgdorferi. Spirochete load in plasma and ear and other tissue samples of infected mice was measured by quantitative PCR, and these data were compared to those obtained by culture and histopathologic analysis. In mice infected with isolate BL206 (a type 1 strain), the peak number of spirochetes was observed in plasma between day 4 and 7, in heart and ear tissue on day 14, and in joints on day 28 postinoculation. There was a correlation between the peak number of spirochetes in plasma on day 4 or 7 and that in ear biopsy and joint specimens on day 14. By contrast, spirochete burdens in plasma of mice infected with isolate B356 (a type 3 strain) were 16- and 5-fold lower than those of BL206-infected mice on days 7 and 14 of infection, respectively. Similarly, approximately 6- and 13-fold fewer spirochetes were detected in the heart tissues of B356-infected mice compared to BL206-infected mice. Histopathologically, severe arthritis and aortitis were noted only in mice infected with isolate BL206. Spirochete dissemination and disease severity vary significantly in mice infected with distinct genotypes of B. burgdorferi, suggesting that genotypic differences in the infecting spirochetes play a key role in the pathogenesis and development of clinical disease.

Multiple stable recurrent outbreaks and predictability in seasonally forced nonlinear epidemic models

A seasonally forced nonlinear SEIR epidemic model is used to simulate small and large amplitude periodic outbreaks. The model is shown to exhibit bistable behavior for a fixed set of parameters. Basins of attraction for each recurrent outbreak are computed, and it is shown that the basins of two coexisting stable outbreaks are intertwined in a complicated manner. The effect of such a basin structure is shown to result in an obstruction in predicting asymptotically the type of outbreak given an uncertainty in the initial population of susceptibles and infectives.

Recognition of Borrelia burgdorferi, the Lyme Disease Spirochete, by TLR7 and TLR9 Induces a Type I IFN Response by Human Immune Cells

Borrelia burgdorferi is the spirochetal agent of Lyme disease, a multisystemic disorder characterized by inflammation. Using global transcriptional profiling, we characterized the response of human PBMCs exposed to B. burgdorferi in an ex vivo coculture system. The expression profiles induced by B. burgdorferi were marked by the intense up-regulation of IFN-responsive transcripts and transcripts involved in the JAK/STAT signaling pathway. Transcript levels of IFN-α, IFN-β, and IRF7, and protein concentrations of IFN-α, were significantly elevated relative to those in unstimulated PBMCs. The induction of IFN-α was completely dependent upon phagocytosis of B. burgdorferi. Addition of a soluble type I IFN receptor, B18R, did not abolish the induction of IFN-inducible genes, indicating that B. burgdorferi directly elicits enhanced expression of these genes independently of type I IFN feedback signaling. Inhibitors of either TLR7 or TLR9 significantly reduced B. burgdorferi-stimulated IFN-α protein expression and transcription of IFN-induced genes. Simultaneous inhibition of both TLR7 and TLR9 completely abrogated IFN-α induction. The IFN-α-producing populations in PBMCs were identified as plasmacytoid dendritic and CD14+CD11c+ cells. These results reveal a TLR7/9-dependent signaling pathway used by human PBMCs to initiate a type I IFN response to the extracellular bacterium B. burgdorferi.

A unified prediction of computer virus spread in connected networks

We derive two models of viral epidemiology on connected networks and compare results to simulations. The differential equation model easily predicts the expected long term behavior by defining a boundary between survival and extinction regions. The discrete Markov model captures the short term behavior dependent on initial conditions, providing extinction probabilities and the fluctuations around the expected behavior. These analysis techniques provide new insight on the persistence of computer viruses and what strategies should be devised for their control.

Enhanced vaccine control of epidemics in adaptive networks

We study vaccine control for disease spread on an adaptive network modeling disease avoidance behavior. Control is implemented by adding Poisson-distributed vaccination of susceptibles. We show that vaccine control is much more effective in adaptive networks than in static networks due to feedback interaction between the adaptive network rewiring and the vaccine application. When compared to extinction rates in static social networks, we find that the amount of vaccine resources required to sustain similar rates of extinction are as much as two orders of magnitude lower in adaptive networks.

Differentiation of Reinfection from Relapse in Recurrent Lyme Disease

BACKGROUND Erythema migrans is the most common manifestation of Lyme disease. Recurrences are not uncommon, and although they are usually attributed to reinfection rather than relapse of the original infection, this remains somewhat controversial. We used molecular typing of Borrelia burgdorferi isolates obtained from patients with culture-confirmed episodes of erythema migrans to distinguish between relapse and reinfection. METHODS We determined the genotype of the gene encoding outer-surface protein C (ospC) of B. burgdorferi strains detected in cultures of skin or blood specimens obtained from patients with consecutive episodes of erythema migrans. After polymerase-chain-reaction amplification, ospC genotyping was performed by means of reverse line-blot analysis or DNA sequencing of the nearly full-length gene. Most strains were further analyzed by determining the genotype according to the 16S-23S ribosomal RNA intergenic spacer type, multilocus sequence typing, or both. Patients received standard courses of antibiotics for erythema migrans. RESULTS B. burgdorferi isolates obtained from 17 patients who received a diagnosis of erythema migrans between 1991 and 2011 and who had 22 paired episodes of this lesion (initial and second episodes) were available for testing. The ospC genotype was found to be different at each initial and second episode. Apparently identical genotypes were identified on more than one occasion in only one patient, at the first and third episodes, 5 years apart, but different genotypes were identified at the second and fourth episodes. CONCLUSIONS None of the 22 paired consecutive episodes of erythema migrans were associated with the same strain of B. burgdorferi on culture. Our data show that repeat episodes of erythema migrans in appropriately treated patients were due to reinfection and not relapse. (Funded by the National Institutes of Health and the William and Sylvia Silberstein Foundation.).