The metabolomic physics of complex diseases.

Human diseases involve metabolic alterations. Metabolomic profiles have served as a vital biomarker for the early identification of high-risk individuals and disease prevention. However, current approaches can only characterize individual key metabolites, without taking into account the reality that complex diseases are multifactorial, dynamic, heterogeneous, and interdependent. Here, we leverage a statistical physics model to combine all metabolites into bidirectional, signed, and weighted interaction networks and trace how the flow of information from one metabolite to the next causes changes in health state. Viewing a disease outcome as the consequence of complex interactions among its interconnected components (metabolites), we integrate concepts from ecosystem theory and evolutionary game theory to model how the health state-dependent alteration of a metabolite is shaped by its intrinsic properties and through extrinsic influences from its conspecifics. We code intrinsic contributions as nodes and extrinsic contributions as edges into quantitative networks and implement GLMY homology theory to analyze and interpret the topological change of health state from symbiosis to dysbiosis and vice versa. The application of this model to real data allows us to identify several hub metabolites and their interaction webs, which play a part in the formation of inflammatory bowel diseases. The findings by our model could provide important information on drug design to treat these diseases and beyond.

Ideal adaptive control in biological systems: an analysis of P -invariance and dynamical compensation properties.

BACKGROUND: Dynamical compensation (DC) provides robustness to parameter fluctuations. As an example, DC enables control of the functional mass of endocrine or neuronal tissue essential for controlling blood glucose by insulin through a nonlinear feedback loop. Researchers have shown that DC is related to the structural unidentifiability and the P -invariance property. The P -invariance property is a sufficient and necessary condition for the DC property. DC has been seen in systems with at least three dimensions. In this article, we discuss DC and P -invariance from an adaptive control perspective. An adaptive controller automatically adjusts its parameters to optimise performance, maintain stability, and deal with uncertainties in a system. RESULTS: We initiate our analysis by introducing a simplified two-dimensional dynamical model with DC, fostering experimentation and understanding of the system's behavior. We explore the system's behavior with time-varying input and disturbance signals, with a focus on illustrating the system's P -invariance properties in phase portraits and step-like response graphs. CONCLUSIONS: We show that DC can be seen as a case of ideal adaptive control since the system is invariant to the compensated parameter.

Mathematical models disentangle the role of IL-10 feedbacks in human monocytes upon proinflammatory activation.

Inflammation is one of the vital mechanisms through which the immune system responds to harmful stimuli. During inflammation, proinflammatory and anti-inflammatory cytokines interplay to orchestrate fine-tuned and dynamic immune responses. The cytokine interplay governs switches in the inflammatory response and dictates the propagation and development of the inflammatory response. Molecular pathways underlying the interplay are complex, and time-resolved monitoring of mediators and cytokines is necessary as a basis to study them in detail. Our understanding can be advanced by mathematical models that enable to analyze the system of interactions and their dynamical interplay in detail. We, therefore, used a mathematical modeling approach to study the interplay between prominent proinflammatory and anti-inflammatory cytokines with a focus on tumor necrosis factor and interleukin 10 (IL-10) in lipopolysaccharide-primed primary human monocytes. Relevant time-resolved data were generated by experimentally adding or blocking IL-10 at different time points. The model was successfully trained and could predict independent validation data and was further used to perform simulations to disentangle the role of IL-10 feedbacks during an acute inflammatory event. We used the insight to obtain a reduced predictive model including only the necessary IL-10-mediated feedbacks. Finally, the validated reduced model was used to predict early IL-10-tumor necrosis factor switches in the inflammatory response. Overall, we gained detailed insights into fine-tuning of inflammatory responses in human monocytes and present a model for further use in studying the complex and dynamic process of cytokine-regulated acute inflammation.

Fast and Accurate Maximum-Likelihood Estimation of Multi-Type Birth-Death Epidemiological Models from Phylogenetic Trees.

Multi-type birth-death (MTBD) models are phylodynamic analogies of compartmental models in classical epidemiology. They serve to infer such epidemiological parameters as the average number of secondary infections Re and the infectious time from a phylogenetic tree (a genealogy of pathogen sequences). The representatives of this model family focus on various aspects of pathogen epidemics. For instance, the birth-death exposed-infectious (BDEI) model describes the transmission of pathogens featuring an incubation period (when there is a delay between the moment of infection and becoming infectious, as for Ebola and SARS-CoV-2), and permits its estimation along with other parameters. With constantly growing sequencing data, MTBD models should be extremely useful for unravelling information on pathogen epidemics. However, existing implementations of these models in a phylodynamic framework have not yet caught up with the sequencing speed. Computing time and numerical instability issues limit their applicability to medium data sets (≤ 500 samples), while the accuracy of estimations should increase with more data. We propose a new highly parallelizable formulation of ordinary differential equations for MTBD models. We also extend them to forests to represent situations when a (sub-)epidemic started from several cases (e.g., multiple introductions to a country). We implemented it for the BDEI model in a maximum likelihood framework using a combination of numerical analysis methods for efficient equation resolution. Our implementation estimates epidemiological parameter values and their confidence intervals in two minutes on a phylogenetic tree of 10,000 samples. Comparison to the existing implementations on simulated data shows that it is not only much faster but also more accurate. An application of our tool to the 2014 Ebola epidemic in Sierra-Leone is also convincing, with very fast calculation and precise estimates. As MTBD models are closely related to Cladogenetic State Speciation and Extinction (ClaSSE)-like models, our findings could also be easily transferred to the macroevolution domain.

Choice of landscape discretisation method affects the inferred rate of spread in wildlife disease spread models.

Disease modelling at the livestock-wildlife interface is an important topic for which discrete-space models are used for the wildlife component. One prominent example is African Swine Fever, where wild boar play an influential role as reservoirs of disease spillover into domestic pig farms. In this paper, we present a simulation study that demonstrates the impact of seemingly arbitrary choices of landscape discretisation method on the inferred rate of spread within the model. We use an ordinary differential equation model to implement a simplified model of disease transmission between discrete groups of wild boar with spillover into domestic pig farms contained within a homogeneous landscape. We examine a range of scenarios whereby the landscape is discretised into wild boar patches of varying size and shape, and compare the rate of spread between domestic pig farms placed at fixed points on the landscape. Our results demonstrate a non-monotonic relationship between patch size and rate of spread, which is particularly unstable and unpredictable for square and triangular shaped patches. Discretisation of the landscape into hexagons appears to produce a more stable relationship between patch size and rate of spread for the three types of transmission kernel we investigated. Although the rate of disease spread does converge to a stable value, this occurs at patch sizes that are much smaller than would be used in practice for wild boar. We conclude that outputs of disease models containing a wildlife component should not be considered to be robust to arbitrary choices for patch size and placement, but rather as a source of uncertainty to be examined using sensitivity analysis. Furthermore, we strongly recommend the use of hexagons rather than squares or right triangles for landscape discretisation.

Parameter estimation and forecasting with quantified uncertainty for ordinary differential equation models using QuantDiffForecast: A MATLAB toolbox and tutorial.

Mathematical models based on systems of ordinary differential equations (ODEs) are frequently applied in various scientific fields to assess hypotheses, estimate key model parameters, and generate predictions about the system's state. To support their application, we present a comprehensive, easy-to-use, and flexible MATLAB toolbox, QuantDiffForecast, and associated tutorial to estimate parameters and generate short-term forecasts with quantified uncertainty from dynamical models based on systems of ODEs. We provide software ( https://github.com/gchowell/paramEstimation_forecasting_ODEmodels/) and detailed guidance on estimating parameters and forecasting time-series trajectories that are characterized using ODEs with quantified uncertainty through a parametric bootstrapping approach. It includes functions that allow the user to infer model parameters and assess forecasting performance for different ODE models specified by the user, using different estimation methods and error structures in the data. The tutorial is intended for a diverse audience, including students training in dynamic systems, and will be broadly applicable to estimate parameters and generate forecasts from models based on ODEs. The functions included in the toolbox are illustrated using epidemic models with varying levels of complexity applied to data from the 1918 influenza pandemic in San Francisco. A tutorial video that demonstrates the functionality of the toolbox is included.

Minimal Mechanisms of Microtubule Length Regulation in Living Cells.

The microtubule cytoskeleton is responsible for sustained, long-range intracellular transport of mRNAs, proteins, and organelles in neurons. Neuronal microtubules must be stable enough to ensure reliable transport, but they also undergo dynamic instability, as their plus and minus ends continuously switch between growth and shrinking. This process allows for continuous rebuilding of the cytoskeleton and for flexibility in injury settings. Motivated by in vivo experimental data on microtubule behavior in Drosophila neurons, we propose a mathematical model of dendritic microtubule dynamics, with a focus on understanding microtubule length, velocity, and state-duration distributions. We find that limitations on microtubule growth phases are needed for realistic dynamics, but the type of limiting mechanism leads to qualitatively different responses to plausible experimental perturbations. We therefore propose and investigate two minimally-complex length-limiting factors: limitation due to resource (tubulin) constraints and limitation due to catastrophe of large-length microtubules. We combine simulations of a detailed stochastic model with steady-state analysis of a mean-field ordinary differential equations model to map out qualitatively distinct parameter regimes. This provides a basis for predicting changes in microtubule dynamics, tubulin allocation, and the turnover rate of tubulin within microtubules in different experimental environments.

Fitting Epidemic Models to Data: A Tutorial in Memory of Fred Brauer.

Fred Brauer was an eminent mathematician who studied dynamical systems, especially differential equations. He made many contributions to mathematical epidemiology, a field that is strongly connected to data, but he always chose to avoid data analysis. Nevertheless, he recognized that fitting models to data is usually necessary when attempting to apply infectious disease transmission models to real public health problems. He was curious to know how one goes about fitting dynamical models to data, and why it can be hard. Initially in response to Fred's questions, we developed a user-friendly R package, fitode, that facilitates fitting ordinary differential equations to observed time series. Here, we use this package to provide a brief tutorial introduction to fitting compartmental epidemic models to a single observed time series. We assume that, like Fred, the reader is familiar with dynamical systems from a mathematical perspective, but has limited experience with statistical methodology or optimization techniques.

A Role of Effector CD 8 + T Cells Against Circulating Tumor Cells Cloaked with Platelets: Insights from a Mathematical Model.

Cancer metastasis accounts for a majority of cancer-related deaths worldwide. Metastasis occurs when the primary tumor sheds cells into the blood and lymphatic circulation, thereby becoming circulating tumor cells (CTCs) that transverse through the circulatory system, extravasate the circulation and establish a secondary distant tumor. Accumulating evidence suggests that circulating effector CD 8 + T cells are able to recognize and attack arrested or extravasating CTCs, but this important antitumoral effect remains largely undefined. Recent studies highlighted the supporting role of activated platelets in CTCs's extravasation from the bloodstream, contributing to metastatic progression. In this work, a simple mathematical model describes how the primary tumor, CTCs, activated platelets and effector CD 8 + T cells participate in metastasis. The stability analysis reveals that for early dissemination of CTCs, effector CD 8 + T cells can present or keep secondary metastatic tumor burden at low equilibrium state. In contrast, for late dissemination of CTCs, effector CD 8 + T cells are unlikely to inhibit secondary tumor growth. Moreover, global sensitivity analysis demonstrates that the rate of the primary tumor growth, intravascular CTC proliferation, as well as the CD 8 + T cell proliferation, strongly affects the number of the secondary tumor cells. Additionally, model simulations indicate that an increase in CTC proliferation greatly contributes to tumor metastasis. Our simulations further illustrate that the higher the number of activated platelets on CTCs, the higher the probability of secondary tumor establishment. Intriguingly, from a mathematical immunology perspective, our simulations indicate that if the rate of effector CD 8 + T cell proliferation is high, then the secondary tumor formation can be considerably delayed, providing a window for adjuvant tumor control strategies. Collectively, our results suggest that the earlier the effector CD 8 + T cell response is enhanced the higher is the probability of preventing or delaying secondary tumor metastases.

Inference of dynamic systems from noisy and sparse data via manifold-constrained Gaussian processes.

Parameter estimation for nonlinear dynamic system models, represented by ordinary differential equations (ODEs), using noisy and sparse data, is a vital task in many fields. We propose a fast and accurate method, manifold-constrained Gaussian process inference (MAGI), for this task. MAGI uses a Gaussian process model over time series data, explicitly conditioned on the manifold constraint that derivatives of the Gaussian process must satisfy the ODE system. By doing so, we completely bypass the need for numerical integration and achieve substantial savings in computational time. MAGI is also suitable for inference with unobserved system components, which often occur in real experiments. MAGI is distinct from existing approaches as we provide a principled statistical construction under a Bayesian framework, which incorporates the ODE system through the manifold constraint. We demonstrate the accuracy and speed of MAGI using realistic examples based on physical experiments.

Low-dimensional neural ODEs and their application in pharmacokinetics.

Machine Learning (ML) is a fast-evolving field, integrated in many of today's scientific disciplines. With the recent development of neural ordinary differential equations (NODEs), ML provides a new tool to model dynamical systems in the field of pharmacology and pharmacometrics, such as pharmacokinetics (PK) or pharmacodynamics. The novel and conceptionally different approach of NODEs compared to classical PK modeling creates challenges but also provides opportunities for its application. In this manuscript, we introduce the functionality of NODEs and develop specific low-dimensional NODE structures based on PK principles. We discuss two challenges of NODEs, overfitting and extrapolation to unseen data, and provide practical solutions to these problems. We illustrate concept and application of our proposed low-dimensional NODE approach with several PK modeling examples, including multi-compartmental, target-mediated drug disposition, and delayed absorption behavior. In all investigated scenarios, the NODEs were able to describe the data well and simulate data for new subjects within the observed dosing range. Finally, we briefly demonstrate how NODEs can be combined with mechanistic models. This research work enhances understanding of how NODEs can be applied in PK analyses and illustrates the potential for NODEs in the field of pharmacology and pharmacometrics.

Continuous patient state attention model for addressing irregularity in electronic health records.

BACKGROUND: Irregular time series (ITS) are common in healthcare as patient data is recorded in an electronic health record (EHR) system as per clinical guidelines/requirements but not for research and depends on a patient's health status. Due to irregularity, it is challenging to develop machine learning techniques to uncover vast intelligence hidden in EHR big data, without losing performance on downstream patient outcome prediction tasks. METHODS: In this paper, we propose Perceiver, a cross-attention-based transformer variant that is computationally efficient and can handle long sequences of time series in healthcare. We further develop continuous patient state attention models, using Perceiver and transformer to deal with ITS in EHR. The continuous patient state models utilise neural ordinary differential equations to learn patient health dynamics, i.e., patient health trajectory from observed irregular time steps, which enables them to sample patient state at any time. RESULTS: The proposed models' performance on in-hospital mortality prediction task on PhysioNet-2012 challenge and MIMIC-III datasets is examined. Perceiver model either outperforms or performs at par with baselines, and reduces computations by about nine times when compared to the transformer model, with no significant loss of performance. Experiments to examine irregularity in healthcare reveal that continuous patient state models outperform baselines. Moreover, the predictive uncertainty of the model is used to refer extremely uncertain cases to clinicians, which enhances the model's performance. Code is publicly available and verified at https://codeocean.com/capsule/4587224 . CONCLUSIONS: Perceiver presents a computationally efficient potential alternative for processing long sequences of time series in healthcare, and the continuous patient state attention models outperform the traditional and advanced techniques to handle irregularity in the time series. Moreover, the predictive uncertainty of the model helps in the development of transparent and trustworthy systems, which can be utilised as per the availability of clinicians.

What Influence Could the Acceptance of Visitors Cause on the Epidemic Dynamics of a Reinfectious Disease?: A Mathematical Model.

The globalization in business and tourism becomes crucial more and more for the economical sustainability of local communities. In the presence of an epidemic outbreak, there must be such a decision on the policy by the host community as whether to accept visitors or not, the number of acceptable visitors, or the condition for acceptable visitors. Making use of an SIRI type of mathematical model, we consider the influence of visitors on the spread of a reinfectious disease in a community, especially assuming that a certain proportion of accepted visitors are immune. The reinfectivity of disease here means that the immunity gained by either vaccination or recovery is imperfect. With the mathematical results obtained by our analysis on the model for such an epidemic dynamics of resident and visitor populations, we find that the acceptance of visitors could have a significant influence on the disease's endemicity in the community, either suppressive or supportive.

Mathematical complexities in radionuclide metabolic modelling: a review of ordinary differential equation kinetics solvers in biokinetic modelling.

Biokinetic models have been employed in internal dosimetry (ID) to model the human body's time-dependent retention and excretion of radionuclides. Consequently, biokinetic models have become instrumental in modelling the body burden from biological processes from internalized radionuclides for prospective and retrospective dose assessment. Solutions to biokinetic equations have been modelled as a system of coupled ordinary differential equations (ODEs) representing the time-dependent distribution of materials deposited within the body. In parallel, several mathematical algorithms were developed for solving general kinetic problems, upon which biokinetic solution tools were constructed. This paper provides a comprehensive review of mathematical solving methods adopted by some known internal dose computer codes for modelling the distribution and dosimetry for internal emitters, highlighting the mathematical frameworks, capabilities, and limitations. Further discussion details the mathematical underpinnings of biokinetic solutions in a unique approach paralleling advancements in ID. The capabilities of available mathematical solvers in computational systems were also emphasized. A survey of ODE forms, methods, and solvers was conducted to highlight capabilities for advancing the utilization of modern toolkits in ID. This review is the first of its kind in framing the development of biokinetic solving methods as the juxtaposition of mathematical solving schemes and computational capabilities, highlighting the evolution in biokinetic solving for radiation dose assessment.

Yeast cells depleted of the frataxin homolog Yfh1 redistribute cellular iron: Studies using Mössbauer spectroscopy and mathematical modeling.

The neurodegenerative disease Friedreich's ataxia arises from a deficiency of frataxin, a protein that promotes iron-sulfur cluster (ISC) assembly in mitochondria. Here, primarily using Mössbauer spectroscopy, we investigated the iron content of a yeast strain in which expression of yeast frataxin homolog 1 (Yfh1), oxygenation conditions, iron concentrations, and metabolic modes were varied. We found that aerobic fermenting Yfh1-depleted cells grew slowly and accumulated FeIII nanoparticles, unlike WT cells. Under hypoxic conditions, the same mutant cells grew at rates similar to WT cells, had similar iron content, and were dominated by FeII rather than FeIII nanoparticles. Furthermore, mitochondria from mutant hypoxic cells contained approximately the same levels of ISCs as WT cells, confirming that Yfh1 is not required for ISC assembly. These cells also did not accumulate excessive iron, indicating that iron accumulation into yfh1-deficient mitochondria is stimulated by O2. In addition, in aerobic WT cells, we found that vacuoles stored FeIII, whereas under hypoxic fermenting conditions, vacuolar iron was reduced to FeII. Under respiring conditions, vacuoles of Yfh1-deficient cells contained FeIII, and nanoparticles accumulated only under aerobic conditions. Taken together, these results informed a mathematical model of iron trafficking and regulation in cells that could semiquantitatively simulate the Yfh1-deficiency phenotype. Simulations suggested partially independent regulation in which cellular iron import is regulated by ISC activity in mitochondria, mitochondrial iron import is regulated by a mitochondrial FeII pool, and vacuolar iron import is regulated by cytosolic FeII and mitochondrial ISC activity.

A multiscale mathematical model describing the growth and development of bambara groundnut.

A principal objective in agriculture is to maximise food production; this is particularly relevant with the added demands of an ever increasing population, coupled with the unpredictability that climate change brings. Further improvements in productivity can only be achieved with an increased understanding of plant and crop processes. In this respect, mathematical modelling of plants and crops plays an important role. In this paper we present a two-scale mathematical model of crop yield that accounts for plant growth and canopy interactions. A system of nonlinear ordinary differential equations (ODEs) is formulated to describe the growth of each individual plant, where equations are coupled via a term that describes plant competition via canopy-canopy interactions. A crop of greenhouse plants is then modelled via an agent based modelling approach in which the growth of each plant is described via our system of ODEs. The model is formulated for the African drought tolerant legume bambara groundnut (Vigna subterranea), which is currently being investigated as a food source in light of climate change and food insecurity challenges. Our model allows us to account for plant diversity and also investigate the effect of individual plant traits (e.g. plant canopy size and planting distance) on the yield of the overall crop. Informed with greenhouse data, model results show that plant positioning relative to other plants has a large impact on individual plant yield. Variation in physiological plant traits from genetic diversity and the environmental effects lead to experimentally observed variations in crop yield. These traits include plant height, plant carrying capacity, leaf accumulation rate and canopy spread. Of these traits plant height and ground cover growth rates are found to have the greatest impact on crop yield. We also consider a range of different planting arrangements (uniform grid, staggered grid, circular rings and random allocation) and find that the staggered grid leads to the greatest crop yield (6% more compared to uniform grid). Whilst formulated specifically for bambara groundnut, the generic formulation of our model means that with changes to certain parameter's, it may be extended to other crop species that form a canopy.

Age-related model for estimating the symptomatic and asymptomatic transmissibility of COVID-19 patients.

Estimation of age-dependent transmissibility of COVID-19 patients is critical for effective policymaking. Although the transmissibility of symptomatic cases has been extensively studied, asymptomatic infection is understudied due to limited data. Using a dataset with reliably distinguished symptomatic and asymptomatic statuses of COVID-19 cases, we propose an ordinary differential equation model that considers age-dependent transmissibility in transmission dynamics. Under a Bayesian framework, multi-source information is synthesized in our model for identifying transmissibility. A shrinkage prior among age groups is also adopted to improve the estimation behavior of transmissibility from age-structured data. The added values of accounting for age-dependent transmissibility are further evaluated through simulation studies. In real-data analysis, we compare our approach with two basic models using the deviance information criterion (DIC) and its extension. We find that the proposed model is more flexible for our epidemic data. Our results also suggest that the transmissibility of asymptomatic infections is significantly lower (on average, 76.45% with a credible interval (27.38%, 88.65%)) than that of symptomatic cases. In both symptomatic and asymptomatic patients, the transmissibility mainly increases with age. Patients older than 30 years are more likely to develop symptoms with higher transmissibility. We also find that the transmission burden of asymptomatic cases is lower than that of symptomatic patients.

Investigating the Influence of Growth Arrest Mechanisms on Tumour Responses to Radiotherapy.

Cancer is a heterogeneous disease and tumours of the same type can differ greatly at the genetic and phenotypic levels. Understanding how these differences impact sensitivity to treatment is an essential step towards patient-specific treatment design. In this paper, we investigate how two different mechanisms for growth control may affect tumour cell responses to fractionated radiotherapy (RT) by extending an existing ordinary differential equation model of tumour growth. In the absence of treatment, this model distinguishes between growth arrest due to nutrient insufficiency and competition for space and exhibits three growth regimes: nutrient limited, space limited (SL) and bistable (BS), where both mechanisms for growth arrest coexist. We study the effect of RT for tumours in each regime, finding that tumours in the SL regime typically respond best to RT, while tumours in the BS regime typically respond worst to RT. For tumours in each regime, we also identify the biological processes that may explain positive and negative treatment outcomes and the dosing regimen which maximises the reduction in tumour burden.

Asymptotic behavior of an epidemic model with infinitely many variants.

We investigate the long-time dynamics of a SIR epidemic model with infinitely many pathogen variants infecting a homogeneous host population. We show that the basic reproduction number [Formula: see text] of the pathogen can be defined in that case and corresponds to a threshold between the persistence ([Formula: see text]) and the extinction ([Formula: see text]) of the pathogen population. When [Formula: see text] and the maximal fitness is attained by at least one variant, we show that the systems reaches an endemic equilibrium state that can be explicitly determined from the initial data. When [Formula: see text] but none of the variants attain the maximal fitness, the situation is more intricate. We show that, in general, the pathogen is uniformly persistent and any family of variants that have a fitness which is uniformly lower than the optimal fitness, eventually gets extinct. We derive a condition under which the total pathogen population converges to a limit which can be computed explicitly. We also find counterexamples that show that, when our condition is not met, the total pathogen population may converge to an unexpected value, or the system can even reach an eternally transient behavior where the total pathogen population between several values. We illustrate our results with numerical simulations that emphasize the wide variety of possible dynamics.

A perturbative approach to the analysis of many-compartment models characterized by the presence of waning immunity.

The waning of immunity after recovery or vaccination is a major factor accounting for the severity and prolonged duration of an array of epidemics, ranging from COVID-19 to diphtheria and pertussis. To study the effectiveness of different immunity level-based vaccination schemes in mitigating the impact of waning immunity, we construct epidemiological models that mimic the latter's effect. The total susceptible population is divided into an arbitrarily large number of discrete compartments with varying levels of disease immunity. We then vaccinate various compartments within this framework, comparing the value of [Formula: see text] and the equilibria locations for our systems to determine an optimal immunization scheme under natural constraints. Relying on perturbative analysis, we establish a number of results concerning the location, existence, and uniqueness of the system's endemic equilibria, as well as results on disease-free equilibria. We use numerical techniques to supplement our analytical ones, applying our model to waning immunity dynamics in pertussis, among other diseases. Our analytical results are applicable to a wide range of systems composed of arbitrarily many ODEs.