Using Disease Dynamics

Infectious diseases are inherently dynamic; that is, they are continually changing in time and space.  While summary characteristics (e.g. prevalence) may remain stationary over time, this arises from the balance of rates that govern the continual processes of transmission, infection, and recovery (or removal).  More commonly, these dynamic processes generate fluctuations in time and space, and non-linear phenomena that imply that the future state of the system cannot necessarily be predicted with a simple extrapolation from the current state.  These non-linear dynamics present some challenges in terms of estimation and prediction, but may also take advantage of these dynamics to design effective strategies for control.  Mathematical and computational models have been critical tools in exploring the interaction of dynamics and control.  Here we review various applications of disease dynamics to the monitoring and evaluation of infectious disease management. We first discuss the general applications of disease dynamics and then present lessons from specific classes of dynamic phenomena.

There are three key applications of dynamical models to control and management of infectious diseases: surveillance, evaluation, and prediction.

Surveillance: Quantifying both the absolute incidence or prevalence of disease and trends in incidence or prevalence over time is critical to the design and evaluation of control.  In this section we discuss the role of dynamic models in the estimation of disease burden (e.g. estimating reporting rates, modern state-space methods) and disease trends (i.e. the relative change in burden over time or space).

Here I refer to disease surveillance as the repeated measurement of a disease system in terms of its states, where the states are the specific quantifiable biological properties that change through time, such as the population size, the number of susceptible, infected, and recovered individuals in different classes, or its behavior, where the behavior can be summarized in terms of some aggregate phenomena, such as cycles .

The incidence or prevalence of disease (I will use the “incidence” as a catchall term for the burden of disease and infection in a population respectively) in a population is a straightforward, interpretable, and operational quantity. These quantities can be estimated using classic survey design (e.g. Morris 2011), however the utility of such estimates is limited to the scope of the survey.  Routine Cauchemez 2013  or syndromic surveillance Mandl 2003 provides an alternative for assessing trends in disease incidence from measures collected through an existing health system.  Translating these measured quantities into absolute disease incidence presents challenges because these measures are often imperfect.  For example, many cases of illness occur within the community Harpaz 2004, but are not recorded in the health system, thus leading to under-reporting. Case records for diseases like polio and Zika virus, for which rates of symptomatic disease are low relative to infection, may reasonably inform disease rates but will be necessarily insensitive to small changes in the burden of infection.  In the absence of diagnostic tests, many infections are monitored through clinical symptoms which have low specificity Gary 1997Hutchins 2004.  Thus many measures could lead to over-reporting of infection (e.g. ILI, fever and rash, diarrheal disease).  The bias (in either direction) due to imperfect reporting can be estimated using classic survey methodology or a combination of clinical diagnosis and lab-confirmation Page 2014.  However, these methods are again limited the scope and scale of these specific efforts. The problem of imperfect measurement and observation is common to many dynamical systems and the tools to address this have been increasingly adopted in epidemiology.

Though reported incidence of disease may reflect temporal trends, they rarely reflect total burden due to under-reporting.  Thus, while a true, unobservable dynamic process of transmission exists, we are only able to view it through the imperfect lens of the public health. State-space models are a general class of statistical models that are designed for such partially observable dynamic processes Bretó 2009 Chen 2011.  The basic structure of a state-space model consists of a process model that describes the evolution of the states of the system (where states are measureable biological quantities, such as the numbers of susceptibles and infecteds) from one time point to the next (formally, these must adhere to the Markov property that the state in time t+1 depends only on the parameters and the states at time t), and an observation model that governs the likelihood of observing a reported level of disease incidence conditional on the states (i.e. the true incidence).  In the absence of a dynamic link between the states of the system at time t and time t+1, the absolute value of the states and the reporting process would be confounded.  However, the temporal autocorrelation in sequential observations induced by the dynamic process allows these methods to generate simultaneous estimates of both the st