Ted Cohen and Peter White
Why do some pathogens fail to spread effectively in a host community, while others increase in prevalence before eventual elimination? Why do some pathogens oscillate in frequency, and how can others become stably established at relatively constant levels over long periods of time? How is it possible that interventions can perversely increase the burden of disease in the community, even as they reduce the overall prevalence of infection?
This chapter introduces transmission-dynamic epidemic models as tools to help understand the patterns that arise from these complex interactions between pathogens and hosts. Like all models, transmission-dynamic models serve as simplified representations of more complex systems, but, in contrast to purely statistical models (see Chapter 13), this chapter focuses on models that explicitly include the transmission processes of communicable diseases. These transmission-dynamic models specify how the risk of infection among susceptible hosts depends on the current (and usually time-varying) prevalence of infectious individuals (or vectors or fomites), which has consequences for the distribution of infection and the (cost-) effectiveness of interventions (see Chapter 17).
Importantly, models should not be ‘black boxes’ but should be clearly described, so that non-modellers are able to assess the validity of the assumptions and parameters used in the model, as well as the model itself.
Although transmission-dynamic models have been used for many different purposes, it is useful to classify them into two categories:
• models to study how a system behaves and responds to interventions, particularly to inform economic analysis and policy-making
• models to better understand the system itself.
Models to study how a system behaves and responds to interventions
Is the prevalence of infection expected to increase or decrease in the absence of interventions? What fraction of the population needs to be immunized in order to prevent a pathogen from causing an outbreak? If vaccine supply is limited, should elderly individuals or school-aged children be prioritized to minimize the morbidity associated with infection? Questions about how pathogen prevalence and disease burden are expected to change over time and in response to available strategies for disease control are natural applications for transmission-dynamic models.
Models to better understand the system itself
To what extent does previous infection provide protection from reinfection? How much does the incidence of infection depend on the proportion of infectious individuals who are asymptomatic? Does uncertainty about the future trajectory of an epidemic depend more on lack of a precise estimate of the probability that an individual will progress to disease after exposure or the duration of infectiousness prior to symptom onset? By fitting transmission-dynamic models to available data, it is possible to study elements of the natural history of disease that may otherwise be difficult, expensive, or unethical to study through other means. Models can also be used to prioritize which features of the system must be better understood to improve future projections and reduce uncertainty.
This chapter focuses on models for studying the transmission of microparasities (pathogens multiplying directly within hosts) between humans. Approaches for modelling the transmission of macroparasites (pathogens for which important reproductive stages occur outside of hosts, especially worms) or vector-borne pathogens require additional considerations and are described in greater detail elsewhere.1
Before constructing a transmission-dynamic model for the spread of microparasitic organisms between hosts, the host health states to represent (e.g. susceptible to infection, infectious, or recovered from a previous infection) and the transition rules that govern how hosts move between these states must be specified.
Representing health states in the model
After becoming infected, does a host rapidly become infectious to others or is there an important latent period before that host becomes infectious to others? Do individuals recover from infection? If individuals do recover, are they then immune or do they remain at risk of reinfection if exposed to the pathogen again?
The structures of transmission-dynamic models are often named according to the core health and disease states represented. For example, a ‘susceptible–infectious–recovered (SIR)’ model has three health states: susceptible, infectious, and recovered. An SIR structure could be chosen if hosts become infectious soon after infection and retain at least partial immunity to reinfection on recovery. In contrast, a susceptible–exposed–infectious–recovered (SEIR) model would be appropriate for a pathogen with a period of latency between time of infection and time that an infected individual becomes infectious to others (by convention, ‘exposed’ is commonly used, instead of ‘latent’). For pathogens from which recovery from infectiousness is not associated with protection from reinfection, a susceptible–infectious–susceptible (SIS) model structure may be warranted, while a pathogen that causes a chronic infection may be represented by a susceptible–infectious (SI) structure. Box 16.1 shows the structures of several prototypical model structures and example disease prevalence trajectories associated with them (see also Figure 16.1).
Although knowledge of the natural history of the pathogen is always important, the decision about which health states should be included in the model is also strongly influenced by the question the model is being developed to address. For example, if the aim is the long-term dynamics of a disease with a short latent period, the E state could be omitted from the SEIR model. However, if understanding epidemic behaviour over a shorter period is important, exclusion of this latent period may not be reasonable.
Determining transition rules between modelled health states
After determining which health states should be represented, the rules by which hosts transition between these states must be determined. For example, in an SIR model structure, how hosts enter and exit each of these health states must be specified.
An example of a simple transition is that between the infectious and recovered states, which may be governed in some models by a single time-invariant parameter (that is, the parameter that defines the rate of recovery from infectiousness). In contrast, the transition rule that governs the process of becoming infected—the transition between the susceptible and infectious states—will usually be more complicated, because the risk of a susceptible person becoming infected per unit time (the ‘force of infection’) depends on the prevalence of infectious individuals (or vectors/fomites), which typically varies with time.
Two distinct approaches can be used to translate health states and transition rules into an operable transmission-dynamic model. These approaches differ in the manner in which host populations are represented and have various attributes and limitations, as described below.
Models in which hosts are aggregated by health state
Most transmission-dynamic models consider populations of hosts aggregated by health state (e.g. susceptible, infectious, and recovered), such that, at any time, the model tracks the number or fraction of the population occupying each health state but does not track each individual independently. These models are often referred to as ‘compartmental’ models, because the population is divided into compartments representing aggregated groups of individuals sharing a health state (Box 16.1).
Compartmental models are encoded as systems of difference (discrete time-steps) or differential (continuous time) equations. The equations comprise parameters and variables. Variables ‘keep track’ of the numbers of individuals in each compartment, which can change over time. Parameters specify the per-capita rates of movement from one health state to another (except for the process of acquiring infection) and are usually time-invariant.
At the population level, the number of individuals flowing from one health state to another is determined by the per-capita rate (specified by the parameter) and the number of individuals subject to that process (specified by the variable). Changes in flow rates over time occur as a consequence of changes in the variables, not the parameters.
Importantly, the force of infection is a variable—it changes dynamically in response to changes in the number of infectious individuals. This means that the numbers of individuals becoming infected over time depends on both the number who are infectious and the number who are susceptible. The ‘transmission term’ also contains a parameter that specifies the rate of contact between individuals in the population and the probability of transmission occurring when a susceptible and infectious individual make contact.
One important attribute of compartmental models is that they can often be analysed mathematically. Simple, analytically tractable compartmental models of epidemics have facilitated fundamental insights into the dynamics and control of infectious diseases (Box 16.2), while more complicated models, which usually require numerical analysis, have also played an important role in advancing the understanding of epidemics. Other key attributes of compartmental models are that they are easy to communicate and to replicate or modify by others, as they can be written down as a series of equations that fully specify the system.
Models in which hosts are considered as individuals
Individual-based models (IBMs) are models in which individuals are tracked through health states over time. These models can generate similar output to compartment models by reporting the number or fraction of hosts in each health state over time.
Because these models follow individual hosts through their transitions between health states, they may be preferred over compartmental models for questions that require that the individual history of each host to be tracked through time. For example, an IBM might be needed if the specific contact patterns for individuals are important for the question under study (see Chapter 15) or if continuously variable host factors need to be recorded. The increased capacity of IBMs to track individual history comes at the cost of analytic tractability and usually results in diminished transparency and ease of communication, compared with compartmental models. They also typically are demanding computationally, which can limit the extent of analysis performed.
Determinism and stochasticity in models
Models can be either ‘deterministic’ (they omit the effects of randomness, so repeated simulations of the model with unchanged parameter values are identical) or ‘stochastic’ (they include the effects of randomness, so there is variation between repeated simulations, even when parameters are not changed). For questions about the expected long-term trends of pathogens circulating in large host populations, deterministic models often provide sufficient insight. However, consideration of stochastic behaviour of mathematical models is important for studying the dynamics of rare infections (e.g. when the pathogen is first introduced into a community, when a drug-resistant strain first appears through genetic change and is initially present in only one host, and when a pathogen is on the verge of local eradication) and when host population size is restricted (e.g. when modelling the spread HAIs).
Deterministic and stochastic versions of compartmental models and IBMs can be implemented, but, in practice, almost all IBMs are stochastic, as this feature is natural to include when programming the transition rules within an IBM framework.
To address all but the simplest questions, models usually require the inclusion of additional complexity to reflect important heterogeneities between hosts.
Age and sex of hosts
Age- or sex-structured transmission-dynamic models are used when there are different age- or sex-specific natural histories of disease (e.g. when natural history varies by sex or when morbidity is associated with age at infection; Box 16.3) and when there are differences in contact patterns between hosts of different ages or sexes (see Chapter 15).
Behavioural risk patterns
For specific types of transmission, host behaviour has an immense effect on individual risk of exposure. For example, for STIs and infections transmitted through well-defined risk behaviours, such as intravenous drug use, the risk of infection can vary by orders of magnitude between individual hosts in the same community, based on behavioural risk factors. Accordingly, models of these pathogens incorporate such behavioural heterogeneity by allowing hosts to differ by risk group or by explicitly modelling the relevant contact patterns for individuals in the population.
Once a model is built, the values for parameters governing the transition rules between health states must be specified, and the model will often be calibrated so that its behaviour fits observed data from a specific epidemic scenario. From which data sources can we estimate the model parameter values and calibrate epidemic behaviour?
Specifying model parameter values
Observational, and occasionally experimental, studies provide the commonest sources of data from which to estimate parameters governing the natural history of infection. Numerous challenges related to the use of such data sources for making inference on these fundamental parameters are worth considering. For example, as it is often difficult to identify the time that an exposed individual is infected with a pathogen, determining the rate of progression from infection to infectiousness can be challenging. For diseases for which treatments are available, studies to estimate the average duration of infectiousness for untreated disease are not ethical, so studies often rely on data gathered in an era before treatment was available. Although these data may exist, their relevance for the duration of infectiousness at other times or in different settings may be limited. In addition, there is often substantial between-host heterogeneity in the values of these natural history parameters, and understanding the importance of this variance for the behaviour of pathogen dynamics is an area of substantial current research interest.
The most difficult model parameter to measure directly is the transmission parameter, defined earlier as the product of the probability of infection conditional on contact between an infectious and a susceptible individual and the rate of contacts between individuals in the population. In most modelling studies, this parameter must be estimated by fitting models to available data on the burden of disease in the community.
Specifying epidemiological measures of disease burden over time
Data informing the burden of disease in communities over time include cross-sectional sources (e.g. seroprevalence surveys) and time-series sources (e.g. notifications data, incidence, prevalence, and disease trends). Availability of these types of data allows models to be calibrated against observed or estimated trends in disease burden such as infections, disease, mortality, case notifications, and hospitalizations.
In using such data to calibrate models, it is important to consider how observable measures may be biased representations of underlying disease trends and may thus challenge the validity of model-fitting exercises. For example, if the probability of a patient seeking care or the probability that a physician will report or notify the occurrence of an additional case changes over the course of an epidemic, notification data may be an a unreliable metric of the actual trend in incidence.
There may also be underlying bias related to the selection of which epidemics are recognized and used to calibrate models. If only large outbreaks of a novel pathogen are recognized (smaller, stuttering outbreaks that fail to propagate successfully will be more difficult to detect), models fit to these data will systematically overestimate the reproductive number, which is a key measure of how effectively the pathogen spreads.
Sensitivity and uncertainty analyses
Many methods are used to calibrate models to existing data on disease trends; although the technical details of such methods are beyond the scope of this chapter, these fitting methods aim to minimize the difference between modelled and observed trajectories given information on the credible values (or ranges of values) for prespecified model parameters.
The relationship between parameter values and the behaviour of models may be investigated in at least two ways.
• Sensitivity analyses can be performed, in which the behaviour of the model is evaluated as single or multiple parameters which are varied from baseline values. Accordingly, the sensitivity of the model behaviour to changes in specific parameters can be evaluated and compared. Identifying the parameters to which the model behaviour is most sensitive can provide insight into the role that these parameters play in determining the epidemic trajectory.
• Uncertainty analyses can be undertaken, in which selected outcomes of the model are evaluated across plausible or credible ranges of input parameters. Uncertainty analyses provide insight into the variability of outcomes that may arise from a model conditional on the precision (or lack thereof) in estimated model parameters. By identifying parameters that drive uncertainty in models, it is possible to prioritize future studies that, by increasing precision of relevant parameter estimates, would best reduce uncertainty in the model’s behaviour and better inform public health policy.
Several examples of phenomena that are well explained, even by the simplest models, are used to illustrate the types of insights made possible by transmission-dynamic models.
Vaccination coverage and herd immunity
Herd immunity is a property of communities that contain sufficient numbers of immune individuals to provide protection for individuals that are not immune (see Chapter 12). From a mechanistic perspective, the presence of immune individuals effectively ‘blocks’ the transmission of pathogens and thus prevents the sustained transmission of a pathogen that would otherwise lead to epidemic spread.
Simple models provide insight into the fraction of the population that must be immune to generate such herd immunity. The effective reproductive number (Reffective) is defined as the number of expected secondary infections attributable to a single infectious individual in the population (see Chapters 1 and 13). The presence of individuals with immunity limits transmission. When there are no immune individuals in the population, Reffective is equal to the basic reproductive number R0 (Box 16.1). Assuming homogeneous mixing of individuals, the general relationship is Reffective = R0 × (1 − the fraction of the population that is immune). Algebraic manipulation of the relationship provides insight that epidemics are not expected to occur when the fraction of the population that is immune exceeds the quantity 1 – (1/R0). Thus, 1 − (1/R0) defines the critical fraction of the population to immunize to prevent an epidemic. For example, if R0 = 3, at least two-thirds of the population must be immune to guard against epidemic spread. For pathogens with higher R0, a larger fraction of the population must be successfully immunized to prevent such epidemics; importantly, if a vaccine is not 100% protective (see Chapter 12), it may not be feasible to deliver the vaccine to enough individuals to provide herd immunity.
Controllability of infections and contribution of transmission from asymptomatic cases
The simplest models predict that if Reffective of a pathogen can be reduced to <1, the pathogen will eventually be eliminated from the host population. Calculations of R0 and Reffective for directly transmitted pathogens depend directly on the duration of infectiousness; for otherwise identical pathogens, one with twice the duration of infectiousness will also be expected to produce twice the number of secondary cases in an exponentially growing epidemic.
Accordingly, reducing the average duration of infectiousness may allow for effective control of an emerging pathogen by lowering Reffective below this critical threshold. From a practical perspective, reductions in the duration of infectiousness usually require an infectious individual effectively to be removed from the population, so they do not further expose susceptible individuals; this may be achieved through treatment or isolation. However, in most cases, early identification of these infectious cases requires that symptom onset either precedes or is synchronous with the onset of infectiousness (see Chapter 1). If infectiousness precedes the onset of symptoms (or asymptomatic infections are infectious), as is the case for many important pathogens, the feasibility and degree to which the average duration of potential transmission may be reduced are themselves reduced.
Simple models provide insight into how transmission from asymptomatic cases would make control of infections impossible through earlier detection of symptomatic cases. Using the notation of Fraser et al.5, if θ = the proportion of transmission that occurs before the onset of symptoms, then when θ > 1/R0, it is not feasible to control an epidemic by early identification of symptomatic individuals. This insight follows from the fact that, under the optimistic assumption of immediate detection and removal of symptomatic cases, R0 can be reduced to θ × R0, and when the value of this product is <1, epidemic spread is obstructed. Algebraic manipulation reveals that when θ > 1/R0, it is not possible to bring the reproductive number below 1 in the absence of additional public health measures.
New sources of data about host and pathogen population structure and dynamics improve the structure and parameterization of transmission-dynamic models and expand the types of questions that may be answered by these models.
New technologies for measuring an individual’s location, such as mobile phones equipped with GPS trackers and sensors for measuring proximity and contacts between individuals in a community, offer opportunities for more detailed monitoring of contact networks in populations. These tools are increasingly important for pathogens, for which such detailed accounting of host contact patterns is most important for explaining existing epidemiological patterns or predicting future disease trajectories.
Molecular epidemiological tools for identification of pathogens have advanced our ability to identify linkages between infected hosts and thus to make inferences on which disease events are related through possible transmission. Such data have been used routinely to inform the structure and parameterization of dynamic mathematical models. The rapidly expanding access to technologies for WGS of pathogens has greatly increased the resolution of such approaches and offers new hope for clarifying and determining directionality of transmission between individuals and resolving clear chains of transmission within clusters of cases. Many current developments within the field of transmission-dynamic modelling are focused around the opportunities and challenges associated with use of the increasing volumes of genomic data to inform the development of new models, particularly individual-based models.
Novel approaches and expanding use of existing methods to inform the selection of model structures (i.e. which health states are included and how transition rules link these states) and to understand the contribution of model structural uncertainty to overall uncertainty are likely to play a larger role in future modelling studies.
Transmission-dynamic models are powerful tools that, when properly developed and analysed, can provide important insights into the complex behaviour of pathogens, as they spread through a host population. In all cases, such models formalize decisions about which details of the complicated systems of relationships between hosts in a population and of pathogens with hosts can be greatly simplified or completely ignored. Well-informed decisions about which details can be ‘stripped away’ from the system provide opportunities to understand better the fundamental relationships within that system and to predict how it will change over time or in response to external perturbations. This is the aim of a transmission-dynamic modeller.
At the same time, judgements that inform the development of dynamic epidemiological models are often based on incomplete (and potentially erroneous) understanding of the natural history of disease or the behaviours of host populations. In the best of cases, such uncertainty will be made explicit in the model and incorporated through sensitivity and uncertainty analyses. However, in other cases, deficiencies in the understanding of the system may not be recognized and ultimately may undermine the usefulness of the model. An attribute of a transmission-dynamic model is that it requires explicit identification of key assumptions within the model equations or code, and best practices dictate that the researcher must highlight such key assumptions when communicating the model-based findings.
The discipline of transmission-dynamic modelling is young but rapidly maturing. Transmission-dynamic models are increasingly linked with economic analysis (see Chapter 17) and are being used to inform policy and to help design and analyse studies. Improving modelling methods, coupled with increasing computing power, better sources of data with which to calibrate models, and sober consideration of the particular strengths and challenges associated with the use of such models, will ensure that these tools will continue to be used to improve insight into the behaviour of epidemics.
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