Multi-scale dynamics of infectious diseases

Abstract

To address the challenge of multiscale dynamics of infectious diseases, the Mathematical Biosciences Institute organized a workshop at The Ohio State University to bring together scientists from a variety of disciplines to share expertise gained through looking at infectious diseases across different scales. The researchers at the workshop, held in April 2018, were specifically looking at three model systems: foot-and-mouth disease, vector-borne diseases and enteric diseases. Although every multiscale model must be necessarily derived from a multiscale system, not every multiscale system has to lead to multiscale models. These three model systems seem to have produced a variety of both multiscale and integrated single-scale mechanistic models that have developed their own strengths and particular challenges. Here, we present papers from some of the workshop participants to show the breadth of the field.

1. Introduction

Infectious disease systems are inherently multiscale complex systems: pathogens replicate within hosts, transmit between individuals and spread among host populations; consequently, selection may operate and trade-offs occur at different levels. Though not every question about disease dynamics will relate to every scale depicted in figure 1, the dimensions of physical size, population size and time exist within every question of disease dynamics. A deeper understanding of these multiscale systems requires a range of scientific tools that include data science methods, mathematical modelling, statistical analysis and theoretical biology principles supported by empirical approaches (experimental systems, surveillance systems, clinical trials, observational research, etc.) originating from such disciplines as biology, epidemiology, microbiology, population biology, ecology, pharmacology, medicine, veterinary science and many more. Unlike data mining methods (models for big data, artificial intelligence and machine learning) that seek solutions for predicting events and categories, mechanistic modelling seeks to describe the processes that cause observed events. As such, mechanistic modelling represents a fundamental shift in how modelling of disease dynamics and other complex systems is approached. And, unlike single-scale modelling, multiscale modelling involves developing models that represent at least two scales and how they interact with each other. To be useful, multiscale mechanistic models must provide some additional information about the system of interest that would not be available by studying the separate scales in isolation. Our ability to empirically analyse such systems has greatly improved over last several decades, owing to the advent of powerful computing platforms and data collection technologies. This newly acquired capacity for data collection and pooling across multiple scales (e.g. from tracing the interactions among viruses to those among humans) naturally leads to interest in building in silico models that can integrate such data in a meaningful way. Although the integration of data across spatial, temporal or physical scales is often a necessary aspect of developing meaningful mechanistic models, the aspect of data integration across scales is frequently the most confusing one [1]. The Mathematical Biosciences Institute (MBI) organized a workshop at The Ohio State University to bring together scientists from a variety of disciplines to share expertise gained through looking at infectious diseases across different scales. The researchers met at the workshop, held in April 2018. This workshop was the natural sequel to the workshop 'From Within Host Dynamics to the Epidemiology of Infectious Disease', held at MBI in April 2014 [1].

Figure 1.

Schematic illustrating the dimensions of physical size, population size and time scale that occur within an environmental context for all infectious diseases to produce a multiscale, complex system. (Online version in colour.)

Although infectious disease dynamics is often well understood at specific scales, a fundamental challenge for multiscale modelling is how to link or couple scales in order to translate the detailed knowledge of the individual scales into integrated knowledge informing the overall dynamics. For an infectious disease system, the hierarchical levels and their associated scales very often represent shifts in disease processes. The complete set of scales of infectious diseases (figure 1) range from those associated with the sub-cellular level to those associated with the macro-ecosystem level. Connections among any subset of physical and population sizes at different time scales either including or not including environmental factors may be relevant to a particular disease dynamics question. Because multiscale models are produced to serve a specific purpose or answer a specific question, they do not necessarily need to include this full range of scales, but rather focus on those scales and connections that are relevant for a specific study. However, the collection of papers in this thematic issue showcases some of the parallels that exist for all of these types of investigations.

2. Foot-and-mouth disease and high-impact viral diseases

Extremely contagious pathogens that cause acute disease are among the most important global public and animal health concerns, because of their high burden of morbidity and mortality, their violent outbreaks, and as potential threats to biosecurity. Understanding the conditions and mechanisms that allow continued circulation of such pathogens, and predicting when and where outbreaks are most likely to occur, is therefore of obvious interest. In addition, from a basic science perspective, fast-evolving contagious pathogens can provide study systems for observing evolutionary change, and its implications for the host–pathogen interactions and disease dynamics, in real time.

Foot-and-mouth disease virus (FMDV) exemplifies fast-evolving, fast-transmitting pathogens that can cause explosive outbreaks in vulnerable populations. FMDV can infect over 70 ungulate species, including domestic livestock and numerous wildlife species [2], and is endemic across large parts of Africa, Asia and South America. Foot-and-mouth disease (FMD) inflicts severe economic losses in endemic countries and is arguably the most important trade-restricting livestock disease in the world [3]. Extreme contagiousness is one of the most striking features of FMDV biology. Minuscule amounts of virus, as little as 10 TCID50 (median tissue culture infectious dose) [4], are sufficient to cause infection in a naive host. The small amount of virus required to initiate infection, combined with the large quantities of virus excreted by affected animals and multiple direct or indirect routes of infection, allow for a fast rate of viral spread [5]. R0 in cattle has been estimated between 2.5 and 73 [6–9], despite a very brief mean infectious period of 1.7 days [10].

This capacity for explosive spread of FMD at multiple spatial scales has stimulated interest in multiscale disease dynamic models focused on theoretical FMD outbreaks in naive host populations [11]. These studies typically use data from past FMD outbreaks and laboratory experiments to parametrize models that predict disease spread at farm to regional and national scales, and evaluate interventions aimed at limiting transmission and the risk of major epidemics [7,12–15]. The paper by Tsao et al. [16] exemplifies this type of work, using simulation models to examine how size and spatial characteristics of cattle farms in the USA (location, clustering, connectivity with other farms via shipments) affect the likelihood of spread of FMDV from a given index farm. They find that the likelihood of a large outbreak is far more sensitive to spatial attributes of index farms and counties than to epidemiological parameters describing disease transmission. In the light of these findings, they examine the efficacy of different possible interventions aimed at containing FMD transmission. Concordant with the importance of spatial parameters, they find that the most effective control is achieved through interventions that identify and target farms that are tightly connected with infected farms (owing to traffic between premises, rather than mere spatial proximity). As such, Tsao et al.'s study [16] highlights the importance of network structure, including node characteristics (e.g. farm size, clustering) and edge distribution (e.g. movement of animals and goods among farms), for understanding disease transmission across spatial scales.

The other end of the epidemiological spectrum—pathogen persistence between outbreaks—has received less attention in FMD models, but poses questions of equal importance. For extremely contagious pathogens, persistence in their host populations poses a challenge because the pool of available susceptible hosts is rapidly depleted. As a result, very contagious pathogens risk fading out as transmission becomes inefficient because the majority of contacts occur with individuals that are no longer susceptible [17]. Some progress has been made in elucidating how rapidly spreading pathogens persist in large human populations [18]; however, persistence in animal populations—which are typically of moderate size, far below the tens or even hundreds of thousands required to maintain pathogens such as measles [19,20], polio [21] and pertussis [22,23]—remains enigmatic [24].

In FMD, transmission from carrier hosts may play a role [5,25–27], and rapid viral evolution may allow for antigenic shift at relevant ecological time scales ([28,29], but see [30]). In this theme issue, Orton et al. [31] report on work examining FMD viral transmission processes among different tissues in individual hosts, between hosts and between host populations. For each of these scales, they estimate the bottleneck size—the number of viral genomes passed between parent and daughter populations that contribute to the sequenced populations. Thus, they are explicitly estimating the level of connection between scales, although they do not create a multiscale model of the overall process. Interestingly, they find similar, moderate bottleneck sizes for within-host and between-host viral passage events, and show that bottleneck size is negatively associated with increases in variant frequency towards fixation. Bottleneck size during viral population propagation is fundamental to understanding the dynamics of these rapidly evolving systems because larger bottlenecks will more effectively maintain viral genomic diversity and allow for adaptive change, whereas tight bottlenecks will result in evolution dominated by genetic drift. The balance of selection versus drift underpins the adaptive potential of viral populations, and drives the mechanisms that these populations can exploit for their persistence in different host populations.

Understanding the mechanisms allowing FMD and other highly contagious pathogens to persist in their host population is, by its nature, a multiscale problem: processes occurring at molecular (e.g. viral genomic changes leading to antigenic shift) to cellular and individual host scales (e.g. pathogen retention on follicular dendritic cells, and shedding by carrier hosts) drive disease dynamics at population and community scales. Viral evolution and host–pathogen interactions are increasingly well understood in FMD, setting the stage for theoretical integration across scales from molecules to meta-populations. Currently, the field is most rich in the molecular (RNA) and meta-population scales depicted in figure 1. As research begins to link scales starting from the smallest as in Orton et al. [31] and largest as in Tsao et al. [16], we expect the two research processes to overlap and will be intrigued to see if they produce similar findings.

3. Vector-borne diseases

Vector-borne disease involves, by definition, the interaction between multiple host species, in addition to the pathogen. From an ecological point of view, a pathogen simply transitions between different ecological niches (vectors and hosts) many times in a multi-generational manner. The duration of the life cycle and the size of vectors and pathogens tend to be orders of magnitude smaller than those of a vertebrate host (although exceptions are notable such as some helminths [32] and some chronic infections). In the field, particular attention has been given to the size and traits of pathogen populations within a host or vector as in Yan et al. [33] and in ecological influence on vector populations as reviewed in Reiner et al. [34].

Given the large diversity in vectors, hosts, parasites and all biologically viable permutations, any quantitative attempt to analyse a vector-borne disease necessarily passes through the fundamental question of categories. That is, can we find commonalities among a configuration of vectors and pathogens such that it makes sense to group them in a single identifiable and unambiguous category? How many fundamental categories would it be possible to identify? These are the questions addressed by Garira & Chirove [35].

Garira & Chirove [35] describe a modelling approach to characterize multiple pathogen populations within the host and within the vector. This framework is then applied to the case of onchocerciasis, a devastating disease that causes blindness in humans as a result of the damage caused by the parasitic worm Onchocerca volvulus. The results of the modelling exercise inform a number of control points, which in turn can be easily translated into clinical and environmental interventions. This investigation is quite similar, methodologically, to the one used by Tsao et al. [16], where estimates of parameters from a variety of sources are extrapolated across scales using a complex mechanistic compartmental model. In both cases, a sensitivity analysis is used to express limitations in certainty created by assumptions about parameter values at individual scales. As those parameters are varied in the sensitivity analyses, some picture is created of how estimates at different scales interact with one another.

While the study by Garira & Chirove [35] provides a paradigm for ecological studies, it does not address molecular and behavioural mechanisms that might prove very influential in the outcome of an epidemic and are currently under investigation in the field [36,37]. Vector-borne multiscale models are most rich in the host–pathogen and host (vector)–environment scales depicted in figure 1. There is a gap, left for future research that could examine how variability in host, vector and pathogen genetic traits, immune history and behaviour influence disease dynamics.

4. Enteric diseases

The life cycle of enteric pathogens takes place in two distinct habitats: the gut and the outside environment. The need to survive in both habitats shapes the pathogen ecology and evolution—resulting in diverse pathogen life histories and fitness differences with numerous implications for disease transmission and control. With the important role of food and water contamination for transmission of enteric pathogens, susceptible–infected–recovered-type differential equation models have been used much less in this system than in vector-borne and high-impact viral disease systems. Exposure dose and subsequent interactions among pathogen, host and associated gut microbiome can lead to differences in infection outcomes. Similarly, differences in pathogen survival in the outside environment can shape transmission. To investigate these within-host and out-of-host dynamics, the literature on enteric pathogens is very rich in experimental data in model systems [38] as well as explorations of environmental processes [39,40]. However, multiscale models are needed to capture the cross-scale influences that shape enteric pathogen dynamics as these systems are linked in natural environments. However, linking pathogen and host processes at different hierarchical scales is challenging because of the divergence of temporal and spatial scales and the lack of cross-scale data.

The use of digital telemetry to monitor physiological parameters and animal movement is rapidly expanding our ability to quantify and describe the time course of an infection and host behaviour with an unparalleled resolution. These data can inform infection outcomes and provide a way of collecting data to use in cross-scale models much like Orton et al. [31] provide a way to link scales in the FMDV system. However, the methods used by Aminian et al. [41] also need extensive processing and data analysis. In this theme issue, Aminian et al. [41] show how techniques for modelling time series on high-dimensional domains can be applied to identify patterns in temperature time-series data. The researchers gathered temperature data for mice challenged with Salmonella enterica. Anomaly detection methods were able to detect first ‘off-pattern’ anomalies within the first days post infection, providing an early estimate of disease onset.

Multiscale approaches are needed to elucidate how immune responses at the individual level influence population-level disease dynamics. Stout et al. [42] demonstrate that exposure to virulence-attenuated Listeria monocytogenes strains confers cross-immunity to full virulent strains in a dose-dependent manner. This cross-protection has some counterintuitive population-level implications: control aiming at reducing exposure to L. monocytogenes may decrease population immunity level against L. monocytogenes, increasing the incidence of listeriosis, the clinical manifestation of L. monocytogenes infection. By using mechanistic models, Stout et al. [42] explore the implications of pathogen–host interactions at the population scale much like Tsao et al. [16] and Garira & Chirove [35].

Finally, Lanzas et al. [43] brings attention to the challenges of explicitly modelling environmental transmission. The authors provide a conceptual framework to organize pathogens based on the role that non-host habitats play in the pathogen life cycle. In addition, they show that bottom-up and top-down approaches to model environmental transmission can yield divergent model predictions under certain conditions. They conclude with recommendations on appropriate approaches to simultaneously consider pathogen and host dynamics in modelling environmental transmission.

5. Analytical approaches to modelling multiscale systems

For the purpose of building and validating a multiscale model, one typically needs to consider two issues: (i) what are the scales of interest and (ii) how to bridge the data collected across these scales. The first issue is sometimes resolved simply by considering the operating principles of the particular system. For instance, in the analysis of the dynamics of malaria or other vector-borne diseases, the data collected on the status of a host and a vector are often naturally separated by their physical and temporal scales. However, in many other types of biological systems, like those describing host–pathogen interactions or inter- and intra-cellular dynamics, the identification of appropriate model scales may not be obvious and may need to be derived empirically based on some observables of the system. When the effects of scales are not well understood, there is a tendency in all systems to mechanistically model all those scales that may have a role in the system and use data available at those different scales to parametrize theoretical models and explore the implications of assumptions [16,35,42]. To inform these complex models, new methods of data collection and analysis [31,41] are needed to explain how different scales are connected and modelled. Modelling frameworks [35,43] as well as looking at both top-down and bottom-up approaches to multiscale modelling of disease systems are helpful in harmonizing findings from the expanding literature within these different disease systems. Beyond these methods for piecing together models across scales, new analytical techniques hold promise for integrating models and data across scales.

To fit mechanistic models to observed data, analytical methods that have been successfully applied range from the static structural equation methods to nested regression models and hierarchical Bayesian analysis [44]. An example of the latter technique in the context of genomic data is given in the paper by Panchal & Linder [45]. The second modelling issue, that is, the bridging of the scales of data collection, requires an explicit way of relating various model observables. In many of the biological systems of interest, we may do so with the so-called quasi-steady-state approximation. This approach explores the fact that often one part of a system associated with a particular physical scale (typically but not always it is the smallest, shortest or most local one) reaches its equilibrium faster than the rest of the system. This allows for replacing the equilibrated portion of the system with its steady state. The examples of application of various versions of quasi-steady-state approximations may be seen in the papers on FMD, vector-borne infections and enteric infections [16,35,42] in this theme issue. These approaches often employ sensitivity analyses to explore the limitations of their steady-state assumptions.

Another approach to consistently bridge the system scales is the method of averaging the dynamics, also known as the mean-field approximation [46]. This technique is especially useful for bridging temporal and physical scales and in general takes advantage of the fact that the average dynamics of a collection of individuals (large-scale or macro-dynamics) may often be approximated by the dynamics of an average of individuals (host-scale or micro-dynamics). As shown in the paper by KhudaBukhsh et al. [47], this approach may sometimes lead to very substantial simplification and efficiency gains, particularly if the model of interest consists of a large number of homogeneous components.

While the different disease systems presented here all explore multiscale dynamics of infectious diseases through mechanistic models, they each emphasize different scales and connections between scales. The literature in FMD emphasizes both meta-population [11,16] and molecular [31] scales. Vector-borne research tends to emphasize within-host [33] and host–environment [34] scales. Enteric disease research tends to emphasize the within-host [38] as well as explicit modelling of processes external to the host, which we could classify as modelling the environment [39,40]. While emphasis on these particular scales may be appropriate to these disease systems, new empirical methods of linking scales [31,41] as well as modelling frameworks [35,43] and analytical methods [45,47] may allow researchers to explore connections among the full system of scales in physical size, population size, time and environmental context (figure 1) to determine whether the scales that have been emphasized are truly the drivers of dynamics for these diseases.

Data accessibility

This article has no additional data.

Authors' contributions

All authors contributed to writing this manuscript. A.J. contributed the section on FMD modelling. J.G. contributed the section on vector-borne diseases. C.L. contributed the section on enteric diseases. G.R. and W.G. contributed the section on methods. W.G. also provided assistance in framing the overall paper. R.B.G. drafted the introduction, assembled the content, edited for overall themes and flow, and added the conclusions and references. All authors discussed the development of the manuscript and reviewed the final submission.

Competing interests

We declare we have no competing interests.

Funding

We received no funding for this study.