Modeling Relapsing Disease Dynamics in a Host-Vector Community

Tammi L. Johnson; Erin L. Landguth; Emily F. Stone

doi:10.1371/journal.pntd.0004428

Abstract

Vector-borne diseases represent a threat to human and wildlife populations and mathematical models provide a means to understand and control epidemics involved in complex host-vector systems. The disease model studied here is a host-vector system with a relapsing class of host individuals, used to investigate tick-borne relapsing fever (TBRF). Equilibrium analysis is performed for models with increasing numbers of relapses and multiple hosts and the disease reproduction number, R₀, is generalized to establish relationships with parameters that would result in the elimination of the disease. We show that host relapses in a single competent host-vector system is needed to maintain an endemic state. We show that the addition of an incompetent second host with no relapses increases the number of relapses needed for maintaining the pathogen in the first competent host system. Further, coupling of the system with hosts of differing competencies will always reduce R₀, making it more difficult for the system to reach an endemic state.

Author Summary

An important development in the study of infectious diseases is the application of mathematical models to understand the interplay between various factors that determine epidemiological processes. Vector-borne diseases are additionally complex with interactions between multiple host and vector species. Understanding the transmission dynamics of vector-borne diseases is an important step towards controlling outbreaks and mitigating human infection risk. Identifying the biotic and abiotic interactions and mechanisms that may contribute to disease emergence, establishment and persistence is necessary for assessing current and future disease risk, as well as developing effective control strategies. Tick-borne relapsing fever (TBRF) is found around the world and is caused by several species of Borrelia spirochetes, which are vectored by soft ticks of the genus Ornithodoros. TBRF is a cryptic disease that still causes significant morbidity and mortality, especially in some African countries. Here, we develop and adapt a compartmentalized mathematical model (SIR) with a relapsing component to investigate the dynamics of TBRF.

Citation: Johnson TL, Landguth EL, Stone EF (2016) Modeling Relapsing Disease Dynamics in a Host-Vector Community. PLoS Negl Trop Dis 10(2): e0004428. https://doi.org/10.1371/journal.pntd.0004428

Editor: Janet Foley, UC Davis School of Medicine, UNITED STATES

Received: September 15, 2015; Accepted: January 13, 2016; Published: February 24, 2016

Copyright: © 2016 Johnson et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper.

Funding: TLJ and ELL received graduate stipend support from the Montana Ecology of Infectious Diseases IGERT Program (NSF IGERT Grant #0501628; William Holben, PI). TLJ also received graduate stipend support from the National Instituted of Health Pre-doctoral Intramural Research and Training Award (IRTA). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

An important development in the study of infectious diseases is the application of mathematical models to understand the interplay between various factors that determine epidemiological processes. Many systems show a rich variety of dynamics that arise from nonlinear interactions (due to the mixing of different infectious populations) or temporal forcing (caused by changes in the average contact rate) [1]. Vector-borne diseases are additionally complex with interactions between multiple host and vector species [2–4]. Compartmental models, such as susceptible, infectious, and removed models (SIR) [5], have been applied to many disease systems in an effort to examine system dynamics. In these epidemic models, susceptible individuals pass into the infective class, from which they transition to the removed class. For some diseases, recovered individuals may relapse with a reactivation of infection and revert back to an infective class. An example of such a system is found in van den Driessche et al. [6], which included a relapsing rate between the susceptible and the same infected compartment. Adding additional infected compartments simulates disease systems in which there is a relapsing component, leading to a prolonged infectious period, presumed to be important to disease persistence. To our knowledge, the addition of a relapsing component has not been applied to a host-vector system. Noteworthy vector-borne relapsing diseases include tick-borne relapsing fever (TBRF) and malaria.

An advantage of these types of models is the ability to vary parameters, while monitoring the overall effect on the disease system, allowing for the exploration of characteristics of the system that may not be well understood. Tick-borne relapsing fever (TBRF) is a cryptic disease that still causes significant morbidity and mortality worldwide, especially in African countries [7–10]. TBRF is a vector-borne zoonotic disease endemic to central Asia, Africa, and the Americas [11], and is caused by infection with Borrelia spirochetes. All but one species of relapsing fever spirochetes are vectored by soft ticks (Ornithodoros spp.) [12]. Relapsing fever is characterized by recurring febrile episodes and generalized symptoms including headache, chills, myalgia, nausea, and vomiting[13]. There is a rapid onset of disease symptoms, with a febrile episode lasting 3–6 days, after which symptoms subside, only to return in 7–10 days. Symptoms are associated with large number of spirochetes present in the bloodstream (spirochetemia), and subside when the host generates an antibody response against the variable major proteins (Vmps). The Vmps are involved with antigenic variation, and relapsing fever Borrelia produce a new variant during infection, subsequently attaining high densities [14, 15]. Little is known regarding the number of relapses in natural hosts, but studies have shown a range from 1 to 5 in experimentally infected animals [16]. In humans, there is an average relapse rate of three febrile episodes without treatment, but up to 13 relapses have been observed [17].

Ornithodoros spp. ticks are long-lived, fast feeding vectors that are known to live > 10 years, and have been shown to survive for up to five years without feeding [18]. Ornithodoros ticks are nidicolous ticks that rarely leave the confines of the host nest or burrow and are able to obtain a blood meal and detach from the host in < 90 min. Additionally, soft ticks only obtain a blood meal about once every 3 months; even when presented with the opportunity to feed daily. Ornithodoros ticks require several months between feedings and can survive years between feeding. The longevity of these ticks means that they outlive their rodent hosts, affording the potential to infect several cohorts of rodents over the course of the tick lifespan. Once infected ticks remain infected and infectious for the duration of their lifespan.

Here, we model TBRF caused by infection with B. hermsii and vectored by O. hermsi. We parameterize the model with field-derived values from hosts on Wild Horse Island in Montana and a single genomic group I (GGI) strain of B. hermsii. The overall goal of this study was to develop a SIR model using TBRF dynamics to describe a host-vector system with a relapsing class of host individuals. First, using specific information from a TBRF system located on Wild Horse Island, MT, a model for the dynamics of a single host-vector interaction was developed. For models with increasing numbers of relapses and multiple hosts, equilibrium analysis was performed and R₀ was generalized. Parameter values were considered in the model to provide theoretical criteria for population stability and to determine the parameters that would result in elimination of the disease. Finally, single and coupled host-vector systems were explored, focusing on the addition of less competent hosts and the number of relapses needed in order to maintain an endemic equilibrium. We use the model to ask several important biological questions pertaining to the TBRF system determining effect adding relapsing classes has on pathogen persistence and the effect of multiple host species with varying competency for acquiring and transmitting B. hermsii.

Methods

Study system

We sought to develop a model based on disease dynamics on Wild Horse Island (WHI), Flathead Lake, Lake County, MT. WHI is the largest island (~2100 acres) on Flathead Lake and like other islands on the lake has a limited diversity of rodent host species. WHI is almost exclusively inhabited by deer mice (Peromyscus maniculatus) and pine squirrels (Tamiasciurus hudsonicus) as the terrestrial rodents and provided an important opportunity to develop and parameterize a model including only two hosts. Although there are two genomic groups (GGI and GGII) of B. hermsii present on WHI, we parameterize the model using estimates for only GGI B. hermsii, as host competency experiments have primarily been performed with GGI B. hermsii [16].

Dynamical systems model

A key assumption for host-vector disease modeling is the definition of the transmission term, which represents the contact between hosts and vectors. The formulation of the transmission term affects the reproduction number, R₀, which is a central predictor of disease systems [19]. For host-vector disease models, the transmission term includes the vector biting rate. This rate controls the pathogen transmission both from the vector-to-host and from the host-to-vector. The TBRF model follows frequency-dependent transmission assumptions through the biting rate, since a blood meal is only required approximately once every three months regardless of the host population density. Following this framework, hosts would likely experience an increasing number of bites as the vector population increased.

Given a mathematical model for disease spread, R₀ is an essential summary parameter. It is defined as the average number of secondary infections produced when one infected individual is introduced into a completely susceptible host population [20]. When R₀ < 1, the disease free equilibrium (DFE) at which the population remains in the absence of disease is locally asymptotically stable. However, if R₀ > 1, then the DFE is unstable and invasion is always possible (see [21]) and a new endemic equilibrium (EE) exists. For this study, R₀ was extracted following the methodology developed in van den Driessche et al. [22] (see also [23, 24]) for general compartmental disease models, which can be extended to more complicated host-vector disease systems [25, 26].

Parameter estimates

Specific parameter values for this system have not yet been determined, but can be estimated from similar studies and from data collected on O. hermsi from laboratory experiments. The units of the rates are individuals per month. Table 1 summarizes the notation for all system parameters and variables. See Table 2 for specific model values used in all of the host-vector models. Note that parameters denoted with additional subscripts of ps and dm refers to values specific to the pine squirrel and deer mouse host-vector systems, respectively.

Download:

Table 1. Parameters and variable notation in the host-vector TBRF model for j—1 relapses between j infected compartments (rates are per month, competency values are probabilities (per bite)) and dimensionless forms (rescaled by γ or normalized by N(0)).

In the coupled system, additional subscripts with ps represent the pine squirrel host-vector system and dm represents the deer mouse host-vector system.

https://doi.org/10.1371/journal.pntd.0004428.t001

Download:

Table 2. Parameter values in the TBRF model for j-1 relapses between j infected compartments (rates are per month, competency values are probabilities per bite).

The subscripts ps and dm denote values used in the pine squirrel and deer mouse host-vector system, respectively. Note that if the subscripts do not appear, then the parameter is the same value in both systems.

https://doi.org/10.1371/journal.pntd.0004428.t002

The birth rates for host and vector are each set to a constant value (β and β_v, respectively) and the compartmental death rates (for host and vector) are identical and set equal to birth rate. Then the death rates must be (1) and (2)

The growth rate of pine squirrels (β_ps = 0.33 individuals per month) is an average of the rates found in the literature, i.e., four individuals per litter at 1 litter per year [27]. The growth rate of deer mice is also taken from average estimates from the literature; we estimate growth rate based on an average of three litters per year and four young per litter, (β_dm = 1 individual per month) [28]. The death rates are determined from Eq (1), which depends on the number of relapses in the system. For example, for a pine squirrel host-vector system with one relapse, all death rates would be 0.0825. Life history dynamics of O. hermsi are not well documented and virtually nothing is known about the reproductive behavior and survival of these ticks in nature. Conservative estimates from the laboratory show that soft-bodied ticks lay on average five clutches over their approximately 10 year lifespan with roughly 50 eggs per clutch [29] (T. Schwan personal communication). Thus, the vector birth rate is β_v = 2.08 individuals per month. Following Eq (2), we get death rates of μ_sv = μ_iv = 1.04 for the vector compartments.

The rate at which an individual transitions among infected compartments and to the removed compartment is fixed and is assumed to be the same for all compartments. As more infected compartments are added to the system, the corresponding constant rates are γ = α = α₁ = … = α_j-1, for j infected compartments. Field parameter estimates have not yet been made for these transition rates (i.e., relapse and recovery rates). Laboratory results from three pine squirrels indicate a transition rate of approximately 4.35 individuals per month for a single compartment (Burgdorfer and Mavros 1970). Then γ = α = α₁ = … = α_j-1 = 4.35.

Ticks are assumed to bite a host once every three months (i.e., f = 0.33). Competency values are between 0 and 1 and thus modify the transmission rate of the infection by multiplying the biting rate. Burgdorfer and Mavros [16] observed a high competency in pine squirrels successfully infecting 3/3 animals by tick bite or injecting them with triturated ticks. Using the same methods, they challenged deer mice with B. hermsii and were unsuccessful in establishing infection. Thus, we used competency values c_v = 0.95 for the probability of transmission for vectors, c_ps = 0.90 for pine squirrels, and c_dm = 0.10 for deer mice.

The carrying capacity for the pine squirrel and deer mouse system is determined specifically for WHI. On WHI there are approximately 425 ha of suitable habitat for pine squirrels with up to a maximum of 2 individuals per suitable habitat patch and approximately 850 ha of suitable deer mouse habitat with a conservative estimate of just less than 12 mice per ha [28]. Thus, the total number of pine squirrels (N_ps) is estimated at 850 and total number of deer mice (N_dm) is estimated at 10,000. The soft bodied tick population (N_v) is virtually unknown, however, we assume that they are limited to the nests of their hosts. Initial field collections have found as many as 14 ticks in one nest on the island [30]; other collection efforts show > 300 ticks can be collected from a single nest or snag [31]. Because the estimates of ticks per nest vary largely between our limited collection on WHI and the literature we chose a conservative number of ticks. We estimate that each squirrel has less than one nest (because of juveniles in the system), and each nest is inhabited by 14 ticks. We found no ticks in nest material collected from deer mice, however, nest material collected during the human outbreak in 2002 yielded 14 O. hermsi; the carcasses of two deer mice were found nearby and American Robins (Turdus migratorius) had been nesting there [30]. Thus it is nearly impossible to estimate the average number of ticks in a deer mouse nest, or if in fact they are coming in contact with ticks while visiting other nests. We used an estimate of 20,000 total ticks on the island split equally among host systems. We chose a conservative estimate of 1% of all ticks are infected, as none of 12 of 14 field collected ticks were found to be infected [30]. Thus, we used S_v(0) = 9,900 ticks for the single host-vector system and S_v(0) = 19,800 ticks for the coupled host-vector system.

Results

Single host-vector system

A model for the dynamics of TBRF in a single host-vector system is considered (see Fig 1A). The following assumptions are used to establish a model that is appropriate for the WHI TBRF system for the host pine squirrel and soft tick vector, O. hermsi. (1) The only sources of infection occur between the bite of an infective vector and susceptible host and between a bite of a susceptible vector and infective host (i.e. there are no horizontal or vertical transmission events). (2) The vector becomes infected and infectious for life immediately upon biting an infectious host. (3) The transmission terms are frequency-dependent through the biting rate, f. (4) The hosts relapse to different infected compartments (i.e. different serotypes within the hosts caused by antigenic variation) at rate α and recover from the disease at rate γ. (5) Though mortality rates are noted to differ for each compartment, we assume a constant total population for both hosts and vectors (N and N_v, respectively). Thus, recruitment (or birth) and the sum of the removal (or death) rates from each compartment must be equal (Eqs 1 and 2).

Download:

Fig 1. Conceptual models for the cross-infection dynamics between (a) a single host-vector system, which includes j—1 relapses between j infected compartments and (b) a coupled host-vector system with no relapses in either host.

Dashed lines are vital rates for each population, where solid lines refer to interaction rates between compartments. See Table 1 for a summary of notation.

https://doi.org/10.1371/journal.pntd.0004428.g001

The generalized system for the infection dynamics in a single host-vector system with j—1 relapsing rates for j = 1 infected compartments describes the number of susceptible hosts S(t), infectious hosts I_k(t), removed hosts R(t), susceptible vectors S_v(t), and infected vectors I_v(t), where the total host population is and the total vector population is N_v = S_v + I_v (see Fig 1A for a compartmental diagram and Table 1 for parameter definitions). The equations are

Host equations: (3)

Vector equations: (4)

To evaluate the invasiveness of the disease in this system, we extracted R₀ following the techniques developed by van den Driessche and Watmough [22] by sequentially adding infected compartments (see S1 for equilibrium analysis and derivations). The form of R₀ was then inferred for j—1 relapsing rates between j infected compartments as (5)

R₀ is directly proportional to the biting rate (f), competency values (c and c_v), and the ratio of initial vectors to initial hosts and inversely proportional to the vector death rate (μ_iv) and the rate that moves individuals out of the infected compartments (α α_1,…., α_j-1, μ_i1, …, μ_ij, and γ). In addition, a pattern emerges as more infected compartments are added: a nesting sequence of terms that increase the value of R₀ and potentially contribute to a change in stability of the DFE. To illustrate this concept, we used the pine squirrel host parameters (Table 2) for increasing number of infected compartments and plotted R₀. R₀ crosses 1 at between j = 4 and j = 5 infected compartments (i.e., four relapses; Fig 2).

Download:

Fig 2. Single host-vector system.

Increasing number of infected compartments are added to the single host-vector system and R₀ is plotted (Eq 5). R₀ becomes greater than one at four relapses.

https://doi.org/10.1371/journal.pntd.0004428.g002

Coupled host-vector system

Here, the single host-vector model is expanded to include two hosts, namely pine squirrels and deer mice. Fig 1B is a compartmental diagram for the two systems with no relapses. The first host-vector system (S_ps, I_1,ps, R_rs, S_v,ps, I_v,ps) is coupled with the second system (S_dm, I_1,dm, R_dm, S_v,dm, I_v,dm) through ticks biting either host species, with parameter f, and is further controlled by competency values of either the ticks (c_v) or hosts (c_ps or c_dm for pine squirrel and deer mice, respectively). Transmission occurs through three mechanisms: 1) fc_v, which is the biting rate modified by the tick competency through which an infected tick bites a host from each system, 2) fc_ps, which is the biting rate modified by the pine squirrel competency in that a susceptible tick bites an infected pine squirrel, and 3) fc_dm, which is the biting rate modified by the deer mouse competency, such that a susceptible tick bites an infected deer mouse. The parameters remain as in the single host vector system, denoted with additional subscripts to represent the respective host-vector system (either ps or dm), and are explained in Tables 1–2.

The generalized system for the infection dynamics in a coupled host-vector system with j—1 relapsing rates for j = 1 infected compartments describes the pine squirrel system with the number of susceptible hosts S_ps(t), infectious hosts I_k,ps(t), and removed hosts R_ps(t). The total pine squirrel host population is Likewise, the deer mouse host system consists of susceptible hosts S_dm(t), infectious hosts I_k,dm(t), and removed hosts R_dm(t) with a total deer mouse host population of The vector compartments are susceptible vectors S_v(t), infected vectors I_v(t) and a total vector population of N_v = S_v + I_v. The equations are

Pine squirrel host system: (6)

Deer mouse host system: (7)

Coupled vector system: (8)

As with the single host-vector system, we performed equilibrium analysis (S2) and the form of R₀ was inferred for j—1 relapsing rates between j infected compartments.

(9)

Where (10)

From the coupled host-vector system it is apparent that R₀ has the additional dependency for both the host competency values (c_ps and c_dm). Since competency values are probabilities between 0 and 1, then they will always decrease the value of R₀ as they decrease. Like the single host-vector system, a pattern emerges as more infected compartments are added to each host system (Eqs 9 and 10): a nested sequence of terms that increase the value of R₀ and potentially contribute to a change in stability of the DFE. To compare the results of the number of relapses needed for R₀ > 1 in the coupled host-vector system with the single host-vector, we added an incompetent deer mouse host system (c_dm = 0.2) and increased the number of relapses in a pine squirrel host system until R₀ reached 1. R₀ crosses 1 at between j = 7 and j = 8 infected compartments (seven relapses; Fig 3).

Download:

Fig 3. Coupled host-vector system.

An incompetent deer mouse (Peromyscus maniculatus) host system (c_dm = 0.2) is coupled with a competent pine squirrel (Tamiasciurus hudsonicus) host system (c_ps = 0.9). R₀ is plotted (Eqs 9 and 10) for the deer mouse host system that contained no relapses and the pine squirrel host system with increasing number of infected compartments. R₀ becomes greater than one at seven relapses.

https://doi.org/10.1371/journal.pntd.0004428.g003

Discussion

Incorporating a relapsing component into a host-vector SIR modeling framework represents a step towards a better understanding and representation of complex disease systems. We investigated the disease dynamics of TBRF and used the model to better understand the underlying dynamics and interactions among spirochetes, rodent hosts, and tick vectors that contribute to pathogen persistence. Disease models were presented that describes (1) a single host-vector system with a single relapsing class of host individuals, and generalized to j-1 relapsing host classes and (2) a coupled host-vector model generalized as above to j -1 relapsing host classes. Analytical techniques allowed for the generalization of R₀ with increasing numbers of relapses, and parameters were identified that affect the elimination or persistence of the pathogen (e.g., biting rates, competency values, and population numbers).

In the single host-vector system, R₀ is directly proportional to the biting rate (f), competency values (c and c_v), and the ratio of initial vectors to initial hosts An inverse relationship exists between R₀ and the vector death rate (μ_iv) and the rate that moves individuals out of the infected compartments (α_1,…., α_j-1, μ_i1, …, μ_ij, and γ). When additional relapsing classes are added to the system, R₀ always increases because of the addition of a nested sequence of terms that is always > 1 (Eq 5). The coupled host-vector system has similar dependencies with additional interesting dynamics that may be very important to understanding pathogen persistence and host diversity. Coupling of the system with hosts of lower competencies will always reduce R₀ (Eqs 9 and 10). As the number of incompetent hosts available as blood meals for infected ticks increases, an effect comparable to the dilution effect occurs and R₀ always decreases, leading to DFE. The dilution effect states that in the presence of a second, less competent species, competent host-vector encounters leading to transmission events may be replaced by incompetent host-vector encounters that do not end in a pathogen transmission event, thus decreasing R₀ [3, 4].

The model presented here addresses the presence of multiple hosts with varying competencies and a single pathogen, however, the model can be extended to address not only differences in host species diversity but also the presence of > 1 pathogen strain. The genetics of B. hermsii have been well characterized and isolates have been shown to fall into two distinct genomic groups, referred to as genomic group I and II (GGI and GGII) [32, 33]. The presence of both genomic groups of B. hermsii has been documented on WHI, while only GGII B. hermsii has been found to date on the mainland around Flathead Lake where host species diversity is greater than that of the WHI.

Field investigations of rodents on WHI confirmed infection in a single deer mouse (Peromyscus maniculatus) infected with GGII B. hermsii (Johnson et al. In. Prep.). This prompted a laboratory experiment in which we infected deer mice with both GGI and GGII B. hermsii and monitored them for infection. We challenged deer mice with infection via needle inoculation and infectious tick bite and observed that deer mice show no susceptibility to GGI but are highly susceptible to GGII spirochetes (Johnson et al. In. Prep.). These findings were in contrast with Burgdorfer and Mavros [16] who were unable to establish infection in deer mice, however, they used infected ticks from a TBRF outbreak near Spokane, WA, U.S.A., which resulted in isolation of GGI B. hermsii.

The coupled system presented here could be used to examine the effects of not only host species with varying competencies, but also diverse host communities in the presence of B. hermsii GGI and GGII. The presence of both genomic groups simultaneously may result in a dampening of the dilution effect if GGII is able to infect a diverse array of host species even though GGI is more species limited. Rodent trapping and tick collection on WHI showed one squirrel and one tick infected with GGI and three squirrels infected with GGII. On WHI, 95% of all pine squirrels captured were seropositive for relapsing fever spirochetes while only 4% of deer mice possessed antibodies (Johnson et al. In Prep.). All infected individuals at mainland sites with diverse host species were infected with GGII spirochetes (Johnson et al. In Prep.).

Although there are limitations to the model presented here, the model is an important first step in understanding a relapsing host-vector disease system. All known complexities of the system were not addressed at this time, including incorporation of GGII strains of B. hermsii which can infect deer mice and possibly a wide range of other potential hosts (Johnson et al. In Prep.). Although there is conflicting evidence at the rate which transovarial transmission of B. hermsii occurs in O. hermsi, we can see from the R₀ calculation that I_v does not appear in the equation and therefore will have little impact on disease persistence in the presence of hosts. However, the existence of transovarial transmission may provide insight into the implication of O. hermsi serving as the reservoir for B. hermsii, i.e., the ability to maintain infectious ticks in a prolonged absence of competent hosts and/or hosts in general. Additionally, the model could be used to explore drivers in the host and vector communities and prevention/intervention strategies may be explored to identify the effectiveness of host control versus vector control. Further, this may provide insight into human protective measures and the effectiveness of control strategies such as host vaccination; simulations could be run to assess the efficacy of control programs such as vaccination regimes and vector control.

Ecological factors including biotic and abiotic interactions may play a primary role in the emergence and persistence of infectious diseases [34–39]. Understanding the complete epidemiology of a disease is crucial to advancing the ability to predict and control outbreaks in human and wildlife populations, however, this is rarely an attainable goal. Sonenshine [40] outlines the sequence of steps typically undertaken when attempting to understand the epidemiology of a given system. The pathway typically begins with the identification of a clinical syndrome, followed by discovery of the causative disease agent, and then the identification of the source of the agent in nature. The final step includes investigating the often complex biology and ecology of the hosts and/or vectors involved. Given the difficulty frequently encountered when attempting to study a disease in nature, the last step is often the most difficult. The application of advanced modeling techniques to poorly understood systems is often the only way to begin to understand the drivers of these systems.

The ecological dynamics of relapsing fever systems around the world are poorly understood. Here we use a North American system of relapsing fever caused by B. hermsii; however, information gathered from this modeling exercise can be applied to TBRF systems around the world. TBRF remains a major public health threat in Africa [41]. In addition to other TBRF systems, the ideas presented here may provide the groundwork for relapsing components to be included in other disease systems with greater public health implications such as malaria.

Supporting Information

S1 Appendix. Single host-vector system: Equilibrium analysis.

https://doi.org/10.1371/journal.pntd.0004428.s001

(DOCX)

S2 Appendix. Coupled host-vector system: Equilibrium analysis.

https://doi.org/10.1371/journal.pntd.0004428.s002

(DOCX)

Acknowledgments

We thank Vivian Kirk and Tom Schwan for thoughtful discussion and guidance throughout this research endeavor.

Author Contributions

Conceived and designed the experiments: TLJ ELL. Performed the experiments: TLJ. Analyzed the data: ELL. Contributed reagents/materials/analysis tools: TLJ EFS ELL. Wrote the paper: TLJ EFS ELL.

References

1. Keeling MJ, Rohani P, Grenfell BT. Seasonally forced disease dynamics explored as switching between attractors. Physica D. 2001;148(3):317–35.
- View Article
- Google Scholar
2. Dobson A, Cattadori I, Holt RD, Ostfeld RS, Keesing F, Krichbaum K, et al. Sacred cows and sympathetic squirrels: the importance of biological diversity to human health. PLoS Med. 2006;3(6):e231. pmid:16729846
- View Article
- PubMed/NCBI
- Google Scholar
3. Ostfeld RS, Keesing F. Biodiversity and disease risk: the case of Lyme disease. Conserv Biol. 2000;14(3):722–8.
- View Article
- Google Scholar
4. Ostfeld RS, Keesing F. Biodiversity series: the function of biodiversity in the ecology of vector-borne zoonotic diseases. Can J Zool. 2000;78(12):2061–78.
- View Article
- Google Scholar
5. Kermack WM AG. A contribution to the mathematical theory of epidemics. Part 3. Further studies of the problem of endemicity. Proc R Soc Lond Ser A. 1927;115:700–21.
- View Article
- Google Scholar
6. van den Driessche P, Wang L, Zou X. Modeling diseases with latency and relapse. Math Biosci Eng. 2007;4(2):205. pmid:17658924
- View Article
- PubMed/NCBI
- Google Scholar
7. Cutler SJ. Relapsing fever—a forgotten disease revealed. J Appl Microbiol. 2010;108(4):1115–22. pmid:19886891.
- View Article
- PubMed/NCBI
- Google Scholar
8. Cutler SJ. Possibilities for relapsing fever reemergence. Emerg Infect Dis. 2006;12(3):369–74. pmid:16704771; PubMed Central PMCID: PMCPMC3291445.
- View Article
- PubMed/NCBI
- Google Scholar
9. Cutler SJ, Abdissa A, Trape JF. New concepts for the old challenge of African relapsing fever borreliosis. Clin Microbiol Infect. 2009;15(5):400–6. pmid:19489922.
- View Article
- PubMed/NCBI
- Google Scholar
10. Elbir H, Raoult D, Drancourt M. Relapsing fever borreliae in Africa. Am J Trop Med Hyg. 2013;89(2):288–92. pmid:23926141; PubMed Central PMCID: PMCPMC3741250.
- View Article
- PubMed/NCBI
- Google Scholar
11. Davis GE. Ticks and relapsing fever in the United States. Public Health Rep (1896–1970). 1940:2347–51.
- View Article
- Google Scholar
12. Felsenfeld O. Borrelia: strains, vectors, human and animal borreliosis. St. Louis, MO, USA: Warren H. Green, Inc.; 1971.
13. Dworkin MS, Schwan TG, Anderson DE Jr. Tick-borne relapsing fever in North America. Med Clin North Am. 2002;86(2):417–33, viii-ix. pmid:11982310.
- View Article
- PubMed/NCBI
- Google Scholar
14. Barbour AG. Antigenic variation of a relapsing fever Borrelia species. Annu Rev Microbiol. 1990;44(1):155–71.
- View Article
- Google Scholar
15. Barbour AG, Carter CJ, Sohaskey CD. Surface protein variation by expression site switching in the relapsing fever agent Borrelia hermsii. Infect Immun. 2000;68(12):7114–21. pmid:11083837
- View Article
- PubMed/NCBI
- Google Scholar
16. Burgdorfer W, Mavros AJ. Susceptibility of various species of rodents to the relapsing fever spirochete, Borrelia hermsii. Infect Immun. 1970;2(3):256–9. pmid:16557828
- View Article
- PubMed/NCBI
- Google Scholar
17. Trevejo RT, Schriefer ME, Gage KL, Safranek TJ, Orloski KA, Pape WJ, et al. An interstate outbreak of tick-borne relapsing fever among vacationers at a Rocky Mountain cabin. The Am J Trop Med Hyg. 1998;58(6):743–7. pmid:9660457
- View Article
- PubMed/NCBI
- Google Scholar
18. Francıs E. Longevity of the tick Ornithodoros turicata and of Spirochaeta recurrentis within this tick. Public Health Rep (1896–1970). 1938:2220–41.
- View Article
- Google Scholar
19. Wonham MJ, Lewis MA, Rencławowicz J, Van den Driessche P. Transmission assumptions generate conflicting predictions in host–vector disease models: a case study in West Nile virus. Ecol Lett. 2006;9(6):706–25. pmid:16706915
- View Article
- PubMed/NCBI
- Google Scholar
20. Dietz K. Transmission and control of arbovirus diseases. Epidemiology. 1975:104–21.
- View Article
- Google Scholar
21. Hethcote HW. The mathematics of infectious diseases. SIAM Rev Soc Ind Appl Math. 2000;42(4):599–653.
- View Article
- Google Scholar
22. Van den Driessche P, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci. 2002;180(1):29–48.
- View Article
- Google Scholar
23. Diekmann O, Heesterbeek J, Metz JA. On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J Math Biol. 1990;28(4):365–82. pmid:2117040
- View Article
- PubMed/NCBI
- Google Scholar
24. Roberts MG, Heesterbeek JA. A new method for estimating the effort required to control an infectious disease. Proc Biol Sci. 2003;270(1522):1359–64. pmid:12965026; PubMed Central PMCID: PMCPMC1691377.
- View Article
- PubMed/NCBI
- Google Scholar
25. Lewis MA, Rencławowicz J, van Den Driessche P, Wonham M. A comparison of continuous and discrete-time West Nile virus models. Bull Math Biol. 2006;68(3):491–509. pmid:16794942
- View Article
- PubMed/NCBI
- Google Scholar
26. Wonham MJ, de-Camino-Beck T, Lewis MA. An epidemiological model for West Nile virus: invasion analysis and control applications. Proc R Soc Lond B Biol Sci. 2004;271(1538):501–7.
- View Article
- Google Scholar
27. Steele MA. Tamiasciurus hudsonicus. Mammalian Species. 586. USA: American Society of Mammalogists; 1998. p. 1–9.
28. Sullivan J. Peromyscus maniculatus. Fire Effects Information System,[Online] US Department of Agriculture, Forest Service, Rocky Mountain Research Station, Fire Sciences Laboratory (Producer) Available: http://www.fs.fed.us/database/feis/animals/mammal/pema/all.html [2010 May 1]. 1995.
29. Herms WB. Medical Entomology, with special reference to the Health and Well-being of Man and Animals. 1956.
30. Schwan TG, Policastro PF, Miller Z, Thompson RL, Damrow T, Keirans JE. Tick-borne relapsing fever caused by Borrelia hermsii, Montana. Emerg Infect D. 2003;9(9):1151.
- View Article
- Google Scholar
31. Davis GE. Ornithodoros hermsi and relapsing fever in Oregon. Public Health Rep (1896–1970). 1941:2010–2.
- View Article
- Google Scholar
32. Porcella SF, Raffel SJ, Anderson DE, Gilk SD, Bono JL, Schrumpf ME, et al. Variable tick protein in two genomic groups of the relapsing fever spirochete Borrelia hermsii in western North America. Infect Immun. 2005;73(10):6647–58. pmid:16177341
- View Article
- PubMed/NCBI
- Google Scholar
33. Schwan TG, Raffel SJ, Schrumpf ME, Porcella SF. Diversity and distribution of Borrelia hermsii. Emerg Infect Dis. 2007;13(3):436. pmid:17552097
- View Article
- PubMed/NCBI
- Google Scholar
34. Daszak P, Cunningham AA, Hyatt AD. Emerging infectious diseases of wildlife—threats to biodiversity and human health. Science. 2000;287(5452):443–9. pmid:10642539
- View Article
- PubMed/NCBI
- Google Scholar
35. Gubler DJ. Resurgent vector-borne diseases as a global health problem. Emerg Infect Dis. 1998;4(3):442. pmid:9716967
- View Article
- PubMed/NCBI
- Google Scholar
36. Wilcox BA, Gubler DJ. Disease ecology and the global emergence of zoonotic pathogens. Environ Health Prev Med. 2005;10(5):263–72. pmid:21432130
- View Article
- PubMed/NCBI
- Google Scholar
37. Morse SS. Factors in the emergence of infectious diseases. Emerg Infect Dis. 1995;1(1):7. pmid:8903148
- View Article
- PubMed/NCBI
- Google Scholar
38. Patz JA, Graczyk TK, Geller N, Vittor AY. Effects of environmental change on emerging parasitic diseases. Int J Parasitol. 2000;30(12):1395–405.
- View Article
- Google Scholar
39. Schrag SJ, Wiener P. Emerging infectious disease: what are the relative roles of ecology and evolution? Trends Ecol Evol. 1995;10(8):319–24. pmid:21237055
- View Article
- PubMed/NCBI
- Google Scholar
40. Sonenshine DE. The biology of tick vectors of human disease. In: Goodman JL, David T. Dennis, and Daniel E. Sonenshine, editor. Tick-borne diseases of humans: American Society of Microbiology; 2005.
41. Cutler S. Relapsing fever–a forgotten disease revealed. J Appl Microbiol. 2010;108(4):1115–22. pmid:19886891
- View Article
- PubMed/NCBI
- Google Scholar

[ref1] 1. Keeling MJ, Rohani P, Grenfell BT. Seasonally forced disease dynamics explored as switching between attractors. Physica D. 2001;148(3):317–35.
View Article
Google Scholar

[2] View Article

[3] Google Scholar

[ref2] 2. Dobson A, Cattadori I, Holt RD, Ostfeld RS, Keesing F, Krichbaum K, et al. Sacred cows and sympathetic squirrels: the importance of biological diversity to human health. PLoS Med. 2006;3(6):e231. pmid:16729846
View Article
PubMed/NCBI
Google Scholar

[5] View Article

[6] PubMed/NCBI

[7] Google Scholar

[ref3] 3. Ostfeld RS, Keesing F. Biodiversity and disease risk: the case of Lyme disease. Conserv Biol. 2000;14(3):722–8.
View Article
Google Scholar

[9] View Article

[10] Google Scholar

[ref4] 4. Ostfeld RS, Keesing F. Biodiversity series: the function of biodiversity in the ecology of vector-borne zoonotic diseases. Can J Zool. 2000;78(12):2061–78.
View Article
Google Scholar

[12] View Article

[13] Google Scholar

[ref5] 5. Kermack WM AG. A contribution to the mathematical theory of epidemics. Part 3. Further studies of the problem of endemicity. Proc R Soc Lond Ser A. 1927;115:700–21.
View Article
Google Scholar

[15] View Article

[16] Google Scholar

[ref6] 6. van den Driessche P, Wang L, Zou X. Modeling diseases with latency and relapse. Math Biosci Eng. 2007;4(2):205. pmid:17658924
View Article
PubMed/NCBI
Google Scholar

[18] View Article

[19] PubMed/NCBI

[20] Google Scholar

[ref7] 7. Cutler SJ. Relapsing fever—a forgotten disease revealed. J Appl Microbiol. 2010;108(4):1115–22. pmid:19886891.
View Article
PubMed/NCBI
Google Scholar

[22] View Article

[23] PubMed/NCBI

[24] Google Scholar

[ref8] 8. Cutler SJ. Possibilities for relapsing fever reemergence. Emerg Infect Dis. 2006;12(3):369–74. pmid:16704771; PubMed Central PMCID: PMCPMC3291445.
View Article
PubMed/NCBI
Google Scholar

[26] View Article

[27] PubMed/NCBI

[28] Google Scholar

[ref9] 9. Cutler SJ, Abdissa A, Trape JF. New concepts for the old challenge of African relapsing fever borreliosis. Clin Microbiol Infect. 2009;15(5):400–6. pmid:19489922.
View Article
PubMed/NCBI
Google Scholar

[30] View Article

[31] PubMed/NCBI

[32] Google Scholar

[ref10] 10. Elbir H, Raoult D, Drancourt M. Relapsing fever borreliae in Africa. Am J Trop Med Hyg. 2013;89(2):288–92. pmid:23926141; PubMed Central PMCID: PMCPMC3741250.
View Article
PubMed/NCBI
Google Scholar

[34] View Article

[35] PubMed/NCBI

[36] Google Scholar

[ref11] 11. Davis GE. Ticks and relapsing fever in the United States. Public Health Rep (1896–1970). 1940:2347–51.
View Article
Google Scholar

[38] View Article

[39] Google Scholar

[ref12] 12. Felsenfeld O. Borrelia: strains, vectors, human and animal borreliosis. St. Louis, MO, USA: Warren H. Green, Inc.; 1971.

[ref13] 13. Dworkin MS, Schwan TG, Anderson DE Jr. Tick-borne relapsing fever in North America. Med Clin North Am. 2002;86(2):417–33, viii-ix. pmid:11982310.
View Article
PubMed/NCBI
Google Scholar

[42] View Article

[43] PubMed/NCBI

[44] Google Scholar

[ref14] 14. Barbour AG. Antigenic variation of a relapsing fever Borrelia species. Annu Rev Microbiol. 1990;44(1):155–71.
View Article
Google Scholar

[46] View Article

[47] Google Scholar

[ref15] 15. Barbour AG, Carter CJ, Sohaskey CD. Surface protein variation by expression site switching in the relapsing fever agent Borrelia hermsii. Infect Immun. 2000;68(12):7114–21. pmid:11083837
View Article
PubMed/NCBI
Google Scholar

[49] View Article

[50] PubMed/NCBI

[51] Google Scholar

[ref16] 16. Burgdorfer W, Mavros AJ. Susceptibility of various species of rodents to the relapsing fever spirochete, Borrelia hermsii. Infect Immun. 1970;2(3):256–9. pmid:16557828
View Article
PubMed/NCBI
Google Scholar

[53] View Article

[54] PubMed/NCBI

[55] Google Scholar

[ref17] 17. Trevejo RT, Schriefer ME, Gage KL, Safranek TJ, Orloski KA, Pape WJ, et al. An interstate outbreak of tick-borne relapsing fever among vacationers at a Rocky Mountain cabin. The Am J Trop Med Hyg. 1998;58(6):743–7. pmid:9660457
View Article
PubMed/NCBI
Google Scholar

[57] View Article

[58] PubMed/NCBI

[59] Google Scholar

[ref18] 18. Francıs E. Longevity of the tick Ornithodoros turicata and of Spirochaeta recurrentis within this tick. Public Health Rep (1896–1970). 1938:2220–41.
View Article
Google Scholar

[61] View Article

[62] Google Scholar

[ref19] 19. Wonham MJ, Lewis MA, Rencławowicz J, Van den Driessche P. Transmission assumptions generate conflicting predictions in host–vector disease models: a case study in West Nile virus. Ecol Lett. 2006;9(6):706–25. pmid:16706915
View Article
PubMed/NCBI
Google Scholar

[64] View Article

[65] PubMed/NCBI

[66] Google Scholar

[ref20] 20. Dietz K. Transmission and control of arbovirus diseases. Epidemiology. 1975:104–21.
View Article
Google Scholar

[68] View Article

[69] Google Scholar

[ref21] 21. Hethcote HW. The mathematics of infectious diseases. SIAM Rev Soc Ind Appl Math. 2000;42(4):599–653.
View Article
Google Scholar

[71] View Article

[72] Google Scholar

[ref22] 22. Van den Driessche P, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci. 2002;180(1):29–48.
View Article
Google Scholar

[74] View Article

[75] Google Scholar

[ref23] 23. Diekmann O, Heesterbeek J, Metz JA. On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J Math Biol. 1990;28(4):365–82. pmid:2117040
View Article
PubMed/NCBI
Google Scholar

[77] View Article

[78] PubMed/NCBI

[79] Google Scholar

[ref24] 24. Roberts MG, Heesterbeek JA. A new method for estimating the effort required to control an infectious disease. Proc Biol Sci. 2003;270(1522):1359–64. pmid:12965026; PubMed Central PMCID: PMCPMC1691377.
View Article
PubMed/NCBI
Google Scholar

[81] View Article

[82] PubMed/NCBI

[83] Google Scholar

[ref25] 25. Lewis MA, Rencławowicz J, van Den Driessche P, Wonham M. A comparison of continuous and discrete-time West Nile virus models. Bull Math Biol. 2006;68(3):491–509. pmid:16794942
View Article
PubMed/NCBI
Google Scholar

[85] View Article

[86] PubMed/NCBI

[87] Google Scholar

[ref26] 26. Wonham MJ, de-Camino-Beck T, Lewis MA. An epidemiological model for West Nile virus: invasion analysis and control applications. Proc R Soc Lond B Biol Sci. 2004;271(1538):501–7.
View Article
Google Scholar

[89] View Article

[90] Google Scholar

[ref27] 27. Steele MA. Tamiasciurus hudsonicus. Mammalian Species. 586. USA: American Society of Mammalogists; 1998. p. 1–9.

[ref28] 28. Sullivan J. Peromyscus maniculatus. Fire Effects Information System,[Online] US Department of Agriculture, Forest Service, Rocky Mountain Research Station, Fire Sciences Laboratory (Producer) Available: http://www.fs.fed.us/database/feis/animals/mammal/pema/all.html [2010 May 1]. 1995.

[ref29] 29. Herms WB. Medical Entomology, with special reference to the Health and Well-being of Man and Animals. 1956.

[ref30] 30. Schwan TG, Policastro PF, Miller Z, Thompson RL, Damrow T, Keirans JE. Tick-borne relapsing fever caused by Borrelia hermsii, Montana. Emerg Infect D. 2003;9(9):1151.
View Article
Google Scholar

[95] View Article

[96] Google Scholar

[ref31] 31. Davis GE. Ornithodoros hermsi and relapsing fever in Oregon. Public Health Rep (1896–1970). 1941:2010–2.
View Article
Google Scholar

[98] View Article

[99] Google Scholar

[ref32] 32. Porcella SF, Raffel SJ, Anderson DE, Gilk SD, Bono JL, Schrumpf ME, et al. Variable tick protein in two genomic groups of the relapsing fever spirochete Borrelia hermsii in western North America. Infect Immun. 2005;73(10):6647–58. pmid:16177341
View Article
PubMed/NCBI
Google Scholar

[101] View Article

[102] PubMed/NCBI

[103] Google Scholar

[ref33] 33. Schwan TG, Raffel SJ, Schrumpf ME, Porcella SF. Diversity and distribution of Borrelia hermsii. Emerg Infect Dis. 2007;13(3):436. pmid:17552097
View Article
PubMed/NCBI
Google Scholar

[105] View Article

[106] PubMed/NCBI

[107] Google Scholar

[ref34] 34. Daszak P, Cunningham AA, Hyatt AD. Emerging infectious diseases of wildlife—threats to biodiversity and human health. Science. 2000;287(5452):443–9. pmid:10642539
View Article
PubMed/NCBI
Google Scholar

[109] View Article

[110] PubMed/NCBI

[111] Google Scholar

[ref35] 35. Gubler DJ. Resurgent vector-borne diseases as a global health problem. Emerg Infect Dis. 1998;4(3):442. pmid:9716967
View Article
PubMed/NCBI
Google Scholar

[113] View Article

[114] PubMed/NCBI

[115] Google Scholar

[ref36] 36. Wilcox BA, Gubler DJ. Disease ecology and the global emergence of zoonotic pathogens. Environ Health Prev Med. 2005;10(5):263–72. pmid:21432130
View Article
PubMed/NCBI
Google Scholar

[117] View Article

[118] PubMed/NCBI

[119] Google Scholar

[ref37] 37. Morse SS. Factors in the emergence of infectious diseases. Emerg Infect Dis. 1995;1(1):7. pmid:8903148
View Article
PubMed/NCBI
Google Scholar

[121] View Article

[122] PubMed/NCBI

[123] Google Scholar

[ref38] 38. Patz JA, Graczyk TK, Geller N, Vittor AY. Effects of environmental change on emerging parasitic diseases. Int J Parasitol. 2000;30(12):1395–405.
View Article
Google Scholar

[125] View Article

[126] Google Scholar

[ref39] 39. Schrag SJ, Wiener P. Emerging infectious disease: what are the relative roles of ecology and evolution? Trends Ecol Evol. 1995;10(8):319–24. pmid:21237055
View Article
PubMed/NCBI
Google Scholar

[128] View Article

[129] PubMed/NCBI

[130] Google Scholar

[ref40] 40. Sonenshine DE. The biology of tick vectors of human disease. In: Goodman JL, David T. Dennis, and Daniel E. Sonenshine, editor. Tick-borne diseases of humans: American Society of Microbiology; 2005.

[ref41] 41. Cutler S. Relapsing fever–a forgotten disease revealed. J Appl Microbiol. 2010;108(4):1115–22. pmid:19886891
View Article
PubMed/NCBI
Google Scholar

[133] View Article

[134] PubMed/NCBI

[135] Google Scholar

Figures

Abstract

Author Summary

Introduction

Methods

Study system

Dynamical systems model

Parameter estimates

Results

Single host-vector system

Coupled host-vector system

Discussion

Supporting Information

S1 Appendix. Single host-vector system: Equilibrium analysis.

S2 Appendix. Coupled host-vector system: Equilibrium analysis.

Acknowledgments

Author Contributions

References