Rapid High Resolution Genotyping of Francisella tularensis by Whole Genome Sequence Comparison of Annotated Genes (“MLST+”)

The zoonotic disease tularemia is caused by the bacterium Francisella tularensis. This pathogen is considered as a category A select agent with potential to be misused in bioterrorism. Molecular typing based on DNA-sequence like canSNP-typing or MLVA has become the accepted standard for this organism. Due to the organism’s highly clonal nature, the current typing methods have reached their limit of discrimination for classifying closely related subpopulations within the subspecies F. tularensis ssp. holarctica. We introduce a new gene-by-gene approach, MLST⁺, based on whole genome data of 15 sequenced F. tularensis ssp. holarctica strains and apply this approach to investigate an epidemic of lethal tularemia among non-human primates in two animal facilities in Germany. Due to the high resolution of MLST⁺ we are able to demonstrate that three independent clones of this highly infectious pathogen were responsible for these spatially and temporally restricted outbreaks.     

Amblyomma americanum as a Bridging Vector for Human Infection with Francisella tularensis

The γ-proteobacterium Francisella tularensis causes seasonal tick-transmitted tularemia outbreaks in natural rabbit hosts and incidental infections in humans in the south-central United States. Although Dermacentor variabilis is considered a primary vector for F. tularensis, Amblyomma americanum is the most abundant tick species in this endemic region. A systematic study of F. tularensis colonization of A. americanum was undertaken to better understand its potential to serve as an overwintering reservoir for F. tularensis and as a bridging vector for human infections. Colony-reared A. americanum were artificially fed F. tularensis subspecies holarctica strain LVS via glass capillaries and colonization levels determined. Capillary-fed larva and nymph were initially infected with 10⁴ CFU/tick which declined prior to molting for both stages, but rebounded post-molting in nymphs and persisted in 53% at 10³ to 10⁸ CFU/nymph at 168 days post-capillary feeding (longest sampling time in the study). In contrast, only 18% of adults molted from colonized nymphs maintained LVS colonization at 10¹ to 10⁵ CFU/adult at 168 days post-capillary feeding (longest sampling time). For adults, LVS initially colonized the gut and disseminated to salivary glands by 24 h and had an ID₅₀ of <5CFU in mice. Francisella tularensis infected the ovaries of gravid females, but transmission to eggs was infrequent and transovarial transmission to hatched larvae was not observed. The prolonged persistence of F. tularensis in A. americanum nymphs supports A. americanum as an overwintering reservoir for F. tularensis from which seasonal epizootics may originate; however, although the rapid dissemination of F. tularensis from gut to salivary glands in adults A. americanum is compatible with intermittent feeding adult males acting as bridging vectors for incidental F. tularensis infections of humans, acquisition of F. tularensis by adults may be unlikely based on adult feeding preference for larger mammals which are not involved in maintenance of sylvatic tularemia.     

Development of a Multivalent Subunit Vaccine against Tularemia Using Tobacco Mosaic Virus (TMV) Based Delivery System

Francisella tularensis is a facultative intracellular pathogen, and is the causative agent of a fatal human disease known as tularemia. F. tularensis is classified as a Category A Biothreat agent by the CDC based on its use in bioweapon programs by several countries in the past and its potential to be used as an agent of bioterrorism. No licensed vaccine is currently available for prevention of tularemia. In this study, we used a novel approach for development of a multivalent subunit vaccine against tularemia by using an efficient tobacco mosaic virus (TMV) based delivery platform. The multivalent subunit vaccine was formulated to contain a combination of F. tularensis protective antigens: OmpA-like protein (OmpA), chaperone protein DnaK and lipoprotein Tul4 from the highly virulent F. tularensis SchuS4 strain. Two different vaccine formulations and immunization schedules were used. The immunized mice were challenged with lethal (10xLD₁₀₀) doses of F. tularensis LVS on day 28 of the primary immunization and observed daily for morbidity and mortality. Results from this study demonstrate that TMV can be used as a carrier for effective delivery of multiple F. tularensis antigens. TMV-conjugate vaccine formulations are safe and multiple doses can be administered without causing any adverse reactions in immunized mice. Immunization with TMV-conjugated F. tularensis proteins induced a strong humoral immune response and protected mice against respiratory challenges with very high doses of F. tularensis LVS. This study provides a proof-of-concept that TMV can serve as a suitable platform for simultaneous delivery of multiple protective antigens of F. tularensis. Refinement of vaccine formulations coupled with TMV-targeting strategies developed in this study will provide a platform for development of an effective tularemia subunit vaccine as well as a vaccination approach that may broadly be applicable to many other bacterial pathogens.     

Diversity of Francisella Species in Environmental Samples from Martha’s Vineyard, Massachusetts

We determined whether Francisella spp. are present in water, sediment, and soil from an active tularemia natural focus on Martha’s Vineyard, Massachusetts, during a multiyear outbreak of pneumonic tularemia. Environmental samples were tested by polymerase chain reaction (PCR) targeting Francisella species 16S rRNA gene and succinate dehydrogenase A (sdhA) sequences; evidence of the agent of tularemia was sought by amplification of Francisella tularensis-specific sequences for the insertion element ISFTu2, 17-kDa protein gene tul4, and the 43-kDa outer membrane protein gene fopA. Evidence of F. tularensis subsp. tularensis, the causative agent of the human infections in this outbreak, was not detected from environmental samples despite its active transmission among ticks and animals in the sampling site. Francisella philomiragia was frequently detected from a brackish-water pond using Francisella species PCR targets, and subsequently F. philomiragia was isolated from an individual brackish-water sample. Distinct Francisella sp. sequences that are closely related to F. tularensis and Francisella novicida were detected from samples collected from the brackish-water pond. We conclude that diverse Francisella spp. are present in the environment where human cases of pneumonic tularemia occur.     

Prevalence of Francisella tularensis in brown hare (Lepus europaeus) populations in Lower Saxony, Germany

Francisella tularensis is the aetiological agent of tularemia. Hares, rabbits, and small rodents are the main hosts, but humans can be infected and develop severe clinical symptoms. In Germany, tularemia in humans was a rare disease during the last four decades, but since 2005, this zoonosis seems to be re-emerging. However, only very little is known about the prevalence in the host populations. Therefore, in a study performed from 2006 to 2009, we investigated 2,121 brown hares (Lepus europaeus) and 41 European rabbits (Oryctolagus cuniculus) located in Lower Saxony, Germany for the occurrence of this zoonotic bacterium by PCR and bacterial culture. F. tularensis subsp. holarctica was detected in an average of 1.1% of these animals. Two hot spots were found in northern Lower Saxony indicating outbreaks of tularemia even in hares. This study demonstrates the occurrence of F. tularensis subsp. holarctica within the hare population in Germany. Hunters, medical practitioners, and public health professionals should be aware of the risk which could come from this zoonotic agent especially in the hot spot areas.     

Targeting of a Fixed Bacterial Immunogen to Fc Receptors Reverses the Anti-Inflammatory Properties of the Gram-Negative Bacterium,Francisella tularensis,during the Early Stages of Infection

Production of pro-inflammatory cytokines by innate immune cells at the early stages of bacterial infection is important for host protection against the pathogen. Many intracellular bacteria, including Francisella tularensis, the agent of tularemia, utilize the anti-inflammatory cytokine IL-10, to evade the host immune response. It is well established that IL-10 has the ability to inhibit robust antigen presentation by dendritic cells and macrophages, thus suppressing the generation of protective immunity. The pathogenesis of F. tularensis is not fully understood, and research has failed to develop an effective vaccine to this date. In the current study, we hypothesized that F. tularensis polarizes antigen presenting cells during the early stages of infection towards an anti-inflammatory status characterized by increased synthesis of IL-10 and decreased production of IL-12p70 and TNF-α in an IFN-ɣ-dependent fashion. In addition, F. tularensis drives an alternative activation of alveolar macrophages within the first 48 hours post-infection, thus allowing the bacterium to avoid protective immunity. Furthermore, we demonstrate that targeting inactivated F. tularensis (iFt) to Fcγ receptors (FcɣRs) via intranasal immunization with mAb-iFt complexes, a proven vaccine strategy in our laboratories, reverses the anti-inflammatory effects of the bacterium on macrophages by down-regulating production of IL-10. More specifically, we observed that targeting of iFt to FcγRs enhances the classical activation of macrophages not only within the respiratory mucosa, but also systemically, at the early stages of infection. These results provide important insight for further understanding the protective immune mechanisms generated when targeting immunogens to Fc receptors.     

Empirical Estimation of R 0 for Unknown Transmission Functions: The Case of Chronic Wasting Disease in Alberta

We consider the problem of estimating the basic reproduction number R ₀ from data on prevalence dynamics at the beginning of a disease outbreak. We derive discrete and continuous time models, some coefficients of which are to be fitted from data. We show that prevalence of the disease is sufficient to determine R ₀. We apply this method to chronic wasting disease spread in Alberta determining a range of possible R ₀ and their sensitivity to the probability of deer annual survival.     

Mechanisms of Heme Utilization by Francisella tularensis

Francisella tularensis is a highly virulent facultative intracellular pathogen causing the severe disease tularemia in mammals. As for other bacteria, iron is essential for its growth but very few mechanisms for iron acquisition have been identified. Here, we analyzed if and how F. tularensis can utilize heme, a major source of iron in vivo. This is by no means obvious since the bacterium lacks components of traditional heme-uptake systems. We show that SCHU S4, the prototypic strain of subspecies tularensis, grew in vitro with heme as the sole iron source. By screening a SCHU S4 transposon insertion library, 16 genes were identified as important to efficiently utilize heme, two of which were required to avoid heme toxicity. None of the identified genes appeared to encode components of a potential heme-uptake apparatus. Analysis of SCHU S4 deletion mutants revealed that each of the components FeoB, the siderophore system, and FupA, contributed to the heme-dependent growth. In the case of the former two systems, iron acquisition was impaired, whereas the absence of FupA did not affect iron uptake but led to abnormally high binding of iron to macromolecules. Overall, the present study demonstrates that heme supports growth of F. tularensis and that the requirements for the utilization are highly complex and to some extent novel.     

Identification of Francisella tularensis Live Vaccine Strain CuZn Superoxide Dismutase as Critical for Resistance to Extracellularly Generated Reactive Oxygen Species

Francisella tularensis is an intracellular pathogen whose survival is in part dependent on its ability to resist the microbicidal activity of host-generated reactive oxygen species (ROS) and reactive nitrogen species (RNS). In numerous bacterial pathogens, CuZn-containing superoxide dismutases (SodC) are important virulence factors, localizing to the periplasm to offer protection from host-derived superoxide radicals (O₂⁻). In the present study, mutants of F. tularensis live vaccine strain (LVS) deficient in superoxide dismutases (SODs) were used to examine their role in defense against ROS/RNS-mediated microbicidal activity of infected macrophages. An in-frame deletion F. tularensis mutant of sodC (ΔsodC) and a F. tularensis ΔsodC mutant with attenuated Fe-superoxide dismutase (sodB) gene expression (sodB ΔsodC) were constructed and evaluated for susceptibility to ROS and RNS in gamma interferon (IFN-γ)-activated macrophages and a mouse model of respiratory tularemia. The F. tularensis ΔsodC and sodB ΔsodC mutants showed attenuated intramacrophage survival in IFN-γ-activated macrophages compared to the wild-type F. tularensis LVS. Transcomplementing the sodC gene in the ΔsodC mutant or inhibiting the IFN-γ-dependent production of O₂⁻ or nitric oxide (NO) enhanced intramacrophage survival of the sod mutants. The ΔsodC and sodB ΔsodC mutants were also significantly attenuated for virulence in intranasally challenged C57BL/6 mice compared to the wild-type F. tularensis LVS. As observed for macrophages, the virulence of the ΔsodC mutant was restored in ifn-γ^−/−, inos^−/−, and phox^−/− mice, indicating that SodC is required for resisting host-generated ROS. To conclude, this study demonstrates that SodB and SodC act to confer protection against host-derived oxidants and contribute to intramacrophage survival and virulence of F. tularensis in mice.Francisella tularensis is considered a potential biological threat due to its extreme infectivity, ease of artificial dissemination via aerosols, and substantial capacity to cause illness and death. A hallmark of all F. tularensis subspecies is their ability to survive and replicate within macrophages (18) and other cell types (6, 11, 25, 28). While recent work has furthered our understanding of F. tularensis virulence mechanisms, little is known with respect to its ability to resist the microbicidal production of reactive oxygen species (ROS) or reactive nitrogen species (RNS).Superoxide dismutases (SODs) are metalloproteins that are classified according to their coordinating active site metals. SODs catalyze the dismutation of the highly reactive superoxide (O₂⁻) anion to hydrogen peroxide (H₂O₂) and O₂ (26). The dismutation of O₂⁻ prevents accumulation of microbicidal ROS and RNS in infected macrophages. Three major categories of SODs have been identified in bacteria and include Mn-, Fe-, and CuZn-containing SODs (SodA, SodB, and SodC, respectively) and are required for aerobic survival (27). The F. tularensis genome encodes SodB (FTL_1791) and SodC (FTL_0380). In several intracellular bacterial pathogens, SodC is an important virulence factor, and its localization to the periplasmic space protects bacteria from host-derived O₂⁻ and NO radicals (8, 9, 21, 32). Moreover, many virulent bacteria possess two copies of the sodC gene (4). The evolutionary maintenance of an extra sodC gene copy suggests that it serves some essential function in survival (4). As an intracellular pathogen, F. tularensis is exposed to ROS and RNS generated by inflammatory cells during the macrophage activation process, which suggests that SODs may play an important role in its intracellular survival and pathogenesis. We have demonstrated that decreases in SodB activity render F. tularensis sensitive to ROS and attenuate virulence in mice (2). However, the contribution of F. tularensis SodC in virulence and intramacrophage survival has not been defined. In this study we have constructed a F. tularensis sodC mutant (ΔsodC) and a F. tularensis sodBC double mutant (sodB ΔsodC) and determined that SodC in conjunction with SodB primarily protects the pathogen from host-derived ROS and is required for intramacrophage survival and virulence of F. tularensis in mice.     

Live Attenuated Francisella novicida Vaccine Protects against Francisella tularensis Pulmonary Challenge in Rats and Non-human Primates

Francisella tularensis causes the disease tularemia. Human pulmonary exposure to the most virulent form, F. tularensis subsp. tularensis (Ftt), leads to high morbidity and mortality, resulting in this bacterium being classified as a potential biothreat agent. However, a closely-related species, F. novicida, is avirulent in healthy humans. No tularemia vaccine is currently approved for human use. We demonstrate that a single dose vaccine of a live attenuated F. novicida strain (Fn iglD) protects against subsequent pulmonary challenge with Ftt using two different animal models, Fischer 344 rats and cynomolgus macaques (NHP). The Fn iglD vaccine showed protective efficacy in rats, as did a Ftt iglD vaccine, suggesting no disadvantage to utilizing the low human virulent Francisella species to induce protective immunity. Comparison of specific antibody profiles in vaccinated rat and NHP sera by proteome array identified a core set of immunodominant antigens in vaccinated animals. This is the first report of a defined live attenuated vaccine that demonstrates efficacy against pulmonary tularemia in a NHP, and indicates that the low human virulence F. novicida functions as an effective tularemia vaccine platform.     

Bayesian Monitoring of Emerging Infectious Diseases

We define data analyses to monitor a change in R, the average number of secondary cases caused by a typical infected individual. The input dataset consists of incident cases partitioned into outbreaks, each initiated from a single index case. We split the input dataset into two successive subsets, to evaluate two successive R values, according to the Bayesian paradigm. We used the Bayes factor between the model with two different R values and that with a single R value to justify that the change in R is statistically significant. We validated our approach using simulated data, generated using known R. In particular, we found that claiming two distinct R values may depend significantly on the number of outbreaks. We then reanalyzed data previously studied by Jansen et al. [Jansen et al. Science 301 (5634), 804], concerning the effective reproduction number for measles in the UK, during 1995–2002. Our analyses showed that the 1995–2002 dataset should be divided into two separate subsets for the periods 1995–1998 and 1999–2002. In contrast, Jansen et al. take this splitting point as input of their analysis. Our estimated effective reproduction numbers R are in good agreement with those found by Jansen et al. In conclusion, our methodology for detecting temporal changes in R using outbreak-size data worked satisfactorily with both simulated and real-world data. The methodology may be used for updating R in real time, as surveillance outbreak data become available.     

Pneumonic tularemia in rabbits resembles the human disease as illustrated by radiographic and hematological changes after infection

Background

Pneumonic tularemia is caused by inhalation of the gram negative bacterium, Francisella tularensis. Because of concerns that tularemia could be used as a bioterrorism agent, vaccines and therapeutics are urgently needed. Animal models of pneumonic tularemia with a pathophysiology similar to the human disease are needed to evaluate the efficacy of these potential medical countermeasures.

Principal Findings

Rabbits exposed to aerosols containing Francisella tularensis strain SCHU S4 developed a rapidly progressive fatal pneumonic disease. Clinical signs became evident on the third day after exposure with development of a fever (>40.5°C) and a sharp decline in both food and water intake. Blood samples collected on day 4 found lymphopenia and a decrease in platelet counts coupled with elevations in erythrocyte sedimentation rate, alanine aminotransferase, cholesterol, granulocytes and monocytes. Radiographs demonstrated the development of pneumonia and abnormalities of intestinal gas consistent with ileus. On average, rabbits were moribund 5.1 days after exposure; no rabbits survived exposure at any dose (190–54,000 cfu). Gross evaluation of tissues taken at necropsy showed evidence of pathology in the lungs, spleen, liver, kidney and intestines. Bacterial counts confirmed bacterial dissemination from the lungs to the liver and spleen.

Conclusions/Significance

The pathophysiology of pneumonic tularemia in rabbits resembles what has been reported for humans. Rabbits therefore are a relevant model of the human disease caused by type A strains of F. tularensis.     

Description of two new plasmids isolated from Francisella philomiragia strains and construction of shuttle vectors for the study of Francisella tularensis

Francisella tularensis is the causative agent of tularemia, a zoonotic disease often transmitted to humans by infected animals. The lack of useful specific genetic tools has long hampered the study of F. tularensis subspecies. We identified and characterized two new plasmids, pF242 and pF243, isolated from Francisella philomiragia strains ATCC 25016 and ATCC 25017, respectively. Sequence analysis revealed that pF242 and pF243 are closely related to pC194 and pFNL10 plasmids, respectively. Two generations of pF242- and pF243-based shuttle vectors, harboring several antibiotic resistance markers, were developed. We used the first generation to compare transformation efficiencies in two virulent F. tularensis subspecies. We found that electroporation was more efficient than cryotransformation: almost all vectors tested were successfully introduced by electroporation into Francisella strains with a high level of efficiency. The second generation of shuttle vectors, containing a multiple cloning site and/or gfp gene downstream of Francisella groES promotor, was used for GFP production in F. tularensis. The development of new shuttle vectors offers new perspectives in the genetic manipulation of F. tularensis, helping to elucidate the mechanisms underlying its virulence.     

Prior Inoculation with Type B Strains of Francisella tularensis Provides Partial Protection against Virulent Type A Strains in Cottontail Rabbits

Francisella tularensis is a highly virulent bacterium that is capable of causing severe disease (tularemia) in a wide range of species. This organism is characterized into two distinct subspecies: tularensis (type A) and holarctica (type B) which vary in several crucial ways, with some type A strains having been found to be considerably more virulent in humans and laboratory animals. Cottontail rabbits have been widely implicated as a reservoir species for this subspecies; however, experimental inoculation in our laboratory revealed type A organisms to be highly virulent, resulting in 100% mortality following challenge with 50–100 organisms. Inoculation of cottontail rabbits with the same number of organisms from type B strains of bacteria was found to be rarely lethal and to result in a robust humoral immune response. The objective of this study was to characterize the protection afforded by a prior challenge with type B strains against a later inoculation with a type A strain in North American cottontail rabbits (Sylvilagus spp). Previous infection with a type B strain of organism was found to lengthen survival time and in some cases prevent death following inoculation with a type A2 strain of F. tularensis. In contrast, inoculation of a type A1b strain was uniformly lethal in cottontail rabbits irrespective of a prior type B inoculation. These findings provide important insight about the role cottontail rabbits may play in environmental maintenance and transmission of this organism.     

Estimation of the Basic Reproductive Number and Mean Serial Interval of a Novel Pathogen in a Small,Well-Observed Discrete Population

Background

Accurately assessing the transmissibility and serial interval of a novel human pathogen is public health priority so that the timing and required strength of interventions may be determined. Recent theoretical work has focused on making best use of data from the initial exponential phase of growth of incidence in large populations.

Methods

We measured generational transmissibility by the basic reproductive number R₀ and the serial interval by its mean T_g. First, we constructed a simulation algorithm for case data arising from a small population of known size with R₀ and T_g also known. We then developed an inferential model for the likelihood of these case data as a function of R₀ and T_g. The model was designed to capture a) any signal of the serial interval distribution in the initial stochastic phase b) the growth rate of the exponential phase and c) the unique combination of R₀ and T_g that generates a specific shape of peak incidence when the susceptible portion of a small population is depleted.

Findings

Extensive repeat simulation and parameter estimation revealed no bias in univariate estimates of either R₀ and T_g. We were also able to simultaneously estimate both R₀ and T_g. However, accurate final estimates could be obtained only much later in the outbreak. In particular, estimates of T_g were considerably less accurate in the bivariate case until the peak of incidence had passed.

Conclusions

The basic reproductive number and mean serial interval can be estimated simultaneously in real time during an outbreak of an emerging pathogen. Repeated application of these methods to small scale outbreaks at the start of an epidemic would permit accurate estimates of key parameters.     

The Transmission and Antibiotic Resistance Variation in a Multiple Drug Resistance Clade of Vibrio cholerae Circulating in Multiple Countries in Asia

Vibrio cholerae has caused massive outbreaks and even trans-continental epidemics. In 2008 and 2010, at least 3 remarkable cholera outbreaks occurred in Hainan, Anhui and Jiangsu provinces of China. To address the possible transmissions and the relationships to the 7^th pandemic strains of those 3 outbreaks, we sequenced the whole genomes of the outbreak isolates and compared with the global isolates from the 7^th pandemic. The three outbreaks in this study were caused by a cluster of V. cholerae in clade 3.B which is parallel to the clade 3.C that was transmitted from Nepal to Haiti and caused an outbreak in 2010. Pan-genome analysis provided additional evolution information on the mobile element and acquired multiple antibiotic resistance genes. We suggested that clade 3.B should be monitored because the multiple antibiotic resistant characteristics of this clade and the ‘amplifier’ function of China in the global transmission of current Cholera pandemic. We also show that dedicated whole genome sequencing analysis provided more information than the previous techniques and should be applied in the disease surveillance networks.     

Nonrandom Distribution of Vector Ticks (Dermacentor variabilis) Infected by Francisella tularensis

The island of Martha''s Vineyard, Massachusetts, is the site of a sustained outbreak of tularemia due to Francisella tularensis tularensis. Dog ticks, Dermacentor variabilis, appear to be critical in the perpetuation of the agent there. Tularemia has long been characterized as an agent of natural focality, stably persisting in characteristic sites of transmission, but this suggestion has never been rigorously tested. Accordingly, we sought to identify a natural focus of transmission of the agent of tularemia by mapping the distribution of PCR-positive ticks. From 2004 to 2007, questing D. variabilis were collected from 85 individual waypoints along a 1.5 km transect in a field site on Martha''s Vineyard. The positions of PCR-positive ticks were then mapped using ArcGIS. Cluster analysis identified an area approximately 290 meters in diameter, 9 waypoints, that was significantly more likely to yield PCR-positive ticks (relative risk 3.3, P = 0.001) than the rest of the field site. Genotyping of F. tularensis using variable number tandem repeat (VNTR) analysis on PCR-positive ticks yielded 13 different haplotypes, the vast majority of which was one dominant haplotype. Positive ticks collected in the cluster were 3.4 times (relative risk = 3.4, P<0.0001) more likely to have an uncommon haplotype than those collected elsewhere from the transect. We conclude that we have identified a microfocus where the agent of tularemia stably perpetuates and that this area is where genetic diversity is generated.     

Physiological analysis on oscillatory behavior of glucose-insulin regulation by model with delays

Despite temporally forced transmission driving many infectious diseases, analytical insight into its role when combined with stochastic disease processes and non-linear transmission has received little attention. During disease outbreaks, however, the absence of saturation effects early on in well-mixed populations mean that epidemic models may be linearised and we can calculate outbreak properties, including the effects of temporal forcing on fade-out, disease emergence and system dynamics, via analysis of the associated master equations. The approach is illustrated for the unforced and forced SIR and SEIR epidemic models. We demonstrate that in unforced models, initial conditions (and any uncertainty therein) play a stronger role in driving outbreak properties than the basic reproduction number R₀, while the same properties are highly sensitive to small amplitude temporal forcing, particularly when R₀ is small. Although illustrated for the SIR and SEIR models, the master equation framework may be applied to more realistic models, although analytical intractability scales rapidly with increasing system dimensionality. One application of these methods is obtaining a better understanding of the rate at which vector-borne and waterborne infectious diseases invade new regions given variability in environmental drivers, a particularly important question when addressing potential shifts in the global distribution and intensity of infectious diseases under climate change.