Disease dynamics in wild populations: modeling and estimation: a review

Abstract

Models of infectious disease dynamics focus on describing the temporal and spatial variations in disease prevalence, and on understanding the factors that affect how many cases will occur in each time period and which individuals are likely to become infected. Classical methods for selecting and fitting models, mostly motivated by human diseases, are almost always based solely on raw counts of infected and uninfected individuals. We begin by reviewing the main classical approaches to parameter estimation, and some of their applications. We then review recently developed methods which enable representation of component processes such as infection and recovery, with observation models that acknowledge the complexities of the sampling and detection processes. We demonstrate the need to account for detectability in modeling disease dynamics, and explore a number of mark–recapture and occupancy study designs for estimating disease parameters while simultaneously accounting for variation in detectability. We highlight the utility of different modeling approaches and also consider the typically strong assumptions that may actually serve to limit their utility in general application to the study of disease dynamics (e.g., assignment of individuals to discrete disease states when underlying state space is more generally continuous; transitions assumed to be simple first-order Markov; temporal separation of hazard and transition events).

Appendices

Appendices

Appendix A. Classical disease models: a (very) short review

“It is utterly implausible that a mathematical formula should make the future known to us, and those who think it can would once have believed in witchcraft…”—B. de Jouvenel

The literature on the mathematical modeling of disease dynamics is enormous—here, we provide only a brief introduction (very brief) to provide some of the necessary terminology and classical model structures. The reader is referred to Daley and Gani (2001) for a general introduction; see Diekmann and Heesterbeek (2000) and Keeling and Rohani (2007) for more mathematically advanced treatments.

Most disease models are constructed by subdividing the population into discrete divisions (‘compartments’) reflecting the underlying disease ‘state’ of the individual. Frequently, these states include individuals that may be susceptible to infection (S), those that are infected (I), and those who are ‘removed’ (R), either by virtue of having died, or potentially recovered with some degree of immunity. The dynamics of such an SIR-type system are governed by the rate of transition between states (Fig. 4).

Fig. 4

Classical SIR compartmental model. Nodes refer to susceptible (S), infected (or infectious) (I), and ‘removed’ or recovered (R) disease states. Transition parameters β and γ reflect rate of movement between successive disease states over some time scale

The classical Ross–Kermack–McKendrick SIR model represented in Fig. 5 is a sequential model with a single absorbing state (implying permanent removal from the susceptible class, typically by death, or permanent immunity). Although there are a very large number of elaborations of this simple model (e.g., the SIRS model, where removal from the susceptible class is not permanent, as might be expected if immunity to the pathogen is temporary), we focus here on the classical deterministic SIR model since it is very general.

Fig. 5

Life cycle graph for a SIR-type disease (modified following Oli et al. 2006), assuming a post-breeding census. Nodes represent susceptible (S), infective (I) and removed (R) disease states. Model parameters are p _k = survival probability of individuals in disease state k (k = S, I or R), β is the probability that a susceptible individual becomes infective between time (t) and (t + 1), γ = the probability that an infective individual recovers between time (t) and (t + 1), B _k is the fertility rate of individuals in disease state k

Classical SIR models: deterministic, continuous time

The SIR model (Fig. 4) in continuous time is traditionally represented by a set of coupled ordinary differential equations:

$$ \begin{aligned} \frac{dS}{dt}&=-\beta{IS}\\ \frac{dI}{dt}&=\beta{IS}-\gamma{I}\\ \frac{dR}{dt}&=\gamma{I} \\ \end{aligned} $$

(11)

Analysis of such models generally focuses on two things: the conditions under which the system described by these equations is at equilibrium, and the sensitivity of this equilibrium (or any other point in plausible state space) to perturbation of one or more parameters. Classical sensitivity analysis uses partial derivatives of model output with respect to parameters in the model—the most common approach is based on finalization analysis, where equations for the partial derivatives of the solution of a particular model with respect to parameters are found by differentiating the model

$$ \frac{\partial}{\partial{t}}\left(\frac{\partial{x}} {\partial{\theta}}\right)=D_xf(x(t);\theta)\frac{\partial{x}} {\partial{\theta}}+D_{\theta}f(x(t);\theta) $$

where ∂x/∂θ is the gradient of x with respect to parameter θ, and D _x f and D _θ f are the Jacobians of f (model functions) with respect to x and θ, respectively. The stability and local transient dynamics of equilibria (and other fixed points) defined by the system of equations is determined by the eigenspectrum of the Jacobian matrix. See Keeling and Gilligan (2000) and Buzby et al. (2008) for a general discussion, with application to analysis of a vector-transmitted disease system.

Analysis of the equilibrium condition(s) often provide qualitative insights to the conditions necessary for an emerging pathogen to increase in a population of susceptible hosts. For example, if we apply the constraint that N = S + I + R is a constant (which implies that dS/dt + dI/dt + dR/dt = 0), we can look at the dynamics of the system simply by focusing on two of the three variables. If we consider S and I, we see that the invasion of a population that initially consists entirely of susceptible individuals requires that dI/dt > 0, which is clearly possible only if (βS − γ > 0); unless the rate of increase in the number of infective individuals is greater than the rate at which they are removed, then dI/dt cannot be greater than 0.

We can re-arrange this condition as

$$ R_o=\frac{\beta{S}}{\gamma}>1 $$

(12)

Traditionally, R _o is known as the reproductive number for the disease (occasionally also referred to as the basic reproduction ratio). We see that βS is the rate at which an infective causes new infections. Since 1/γ is the mean time an individual is infective, then R _o is the mean number of new infections caused by a single infective individual. The incidence of the disease will increase if R _o > 1, and decline if R _o < 1.

The reproductive number R _o is fundamental for analysis of disease dynamics. For example, if S _max = N (which is the null initial condition where all individuals in the population are susceptible), then dI/dt > 0 if and only if N > γ/β (which is equivalent to R _o > 1), indicating there is a minimum population size necessary for the pathogen to invade (i.e., for dI/dt > 0).

In the classical SIR model, the function F = βI models the transition from the susceptible compartment to the infectious compartment. Generally, this function F = βI is referred to as the force of infection.

Classical SIR models: deterministic, discrete time

Compared to continuous time models, discrete time forms of classic epidemiological models have received little attention (see van Boven and Weissing 2004; Oli et al. 2006; Korobeinikov et al. 2008 as notable exceptions). One possible discrete parameterization of the SIR equation, based on a post-breeding census is shown in Fig. 5

Here, we allow for state-specific survival and fertility (production of neonates). This model can be reduced to the equivalent constant N model discussed above, by fixing survival at 1 and fertility to 0.

Following Oli et al. (2006), we can decompose the projection matrix A corresponding to life cycle graph as the sum of a transition matrix T, where the elements (t _ij) represent the probability than an individual in state j at time (t) will be in state i at time (t + 1), and a fecundity matrix F, where the elements f _ij are the expected number of type i offspring produced at (t + 1) by an individual in state j at time (t); A = T+F (Cochran and Ellner 1992; Caswell 2001). The fundamental matrix N is defined as ${\bf I}+{\bf T}+{\bf T}^2+\dots=({\bf I}-{\bf T})^{-1}\equiv {\bf N}$ where I is an identity matrix with dimension equal to the number of disease states. The fundamental matrix N gives the expected number of time steps in each state. Since the fertility matrix F gives the expected number of offspring produced by each stage per time step, then the matrix R = FN has elements r _ij which quantify the expected lifetime production of offspring of type i by an individual in state j. The dominant eigenvalue of R is the net reproduction ratio R _o. Further, evaluation of the sensitivity of key population parameters (e.g., λ,R ₀) is straightforward in the projection matrix framework.

However, while discrete-time analogues of many disease model parameters are available (see Oli et al. 2006 for a general review, and discussion of important differences in calculating and interpreting several key disease paramors), we note that the discretization of both disease state and time have important implications for the modeling of disease dynamics, and the estimation of model parameters. Most interactions in disease systems are truly continuous (at least temporally)—discrete-time models generally represent approximations to continuous systems, necessitated in many instances by the discrete timing of sampling (or, equivalently, the aggregation of disease data into discrete time frames). The statistical challenge in the estimation of disease parameters concerns the need for methods to accommodate discretized systems, which are typically only partially observable, non-stationary and (generally) nonlinear.

Appendix B. HMM models: a (very) short introduction

A system can be modeled by a HMM if the sequence of hidden states is Markov, and if the sequence of observations are either independent, or Markov, given the hidden state.

A HMM is specified by:

• Q, the set of possible states $\{q_1,q_2,\ldots,q_n\}$

• O, the output ‘alphabet’ (often referred to as ‘emission’ distributions’); the set of observed ‘event’ states (sensu Pradel 2005); $\{o_1,o_2,\ldots,o_m\}$

• π_i, the probability of being in state q _i at time t = 0 (i.e., the initial states)

• $\varvec{\Upphi}$, the matrix of transition probabilities ϕ_ij, where ϕ_ij = Pr(entering state q _j at time $t+1 \mid$ being in state q _i at time t). Assuming the system is Markov, then the state transitions do not depend on the previous states at earlier times.

• B, the matrix of output probabilities {b _j(k)} (i.e., the ‘observed event’ array), where b _j(k) = Pr(producing ‘event’ v _k at time $t \mid$ being in state q _j at time t)

Formally, we define a HMM model M as

$$ M=(\Upphi,{\bf B},\pi) $$

The key point is that the data consist of observations of the events V, which are mapped to the underlying Markovian transitions between unobserved states Q by the transition matrices $\varvec{\Upphi}$ and B. A Markov chain has a strict one-to-one mapping between observations and underlying states. This is not a requirement for HMM, where an observation can typically be generated by several different states, and the probabilities of generating an observation given a state differ.

From above, we define Q to be an unobserved fixed state sequence (path through possible states Q) of length T (i.e., T sampling events), and corresponding observations O made at discrete time intervals:

$$ \begin{aligned} Q &= q_1, q_2, \ldots , q_T \\ O &= o_1, o_2, \ldots , o_T \\ \end{aligned} $$

Given O, there are 3 primary questions of interest:

• 1. What is the probability of the observed ‘event history’ O, given the model M? In other words, we wish to calculate Pr(O|M). This is usually referred to as evaluation.

• 2. At each time step, what state is most likely? In other words, what is the hidden state sequence Q that was most likely to have produced a given observation sequence O? It is important to note that the sequence of states computed by this criterion might be impossible. Thus more often we are interested in what single sequence of states has the largest probability of occurrence. That is, find the state sequence $q_1,q_2,\ldots,q_T$ such that $Pr(o_1,o_2,\ldots,o_T | M)$ is maximized. This is usually referred to as decoding.

• 3. Given some data, how do we “learn” a good hidden Markov model to describe the data? That is, given the structure of a HMM, and observed event data, how do we parameterize the model which maximizes Pr(O|M)? This is referred to as learning.

Due to the large number of possible sample paths, even for low dimension HMM with small time horizon T, the need for an efficient algorithm to satisfy the objectives (noted above) should be apparent. Consider, for example, the evaluation problem—trying to find the probability of observations $O=\{o_1,o_2,\ldots,o_T\}$ by means of considering all possible hidden state sequences is clearly impractical (for even simple problems, the number of possible state sequences R is astronomically large). Generally, dynamic programming approaches are used to minimize the computational burden. Forward–backward and Viterbi algorithms are frequently applied to the evaluation and decoding problems, respectively. For the learning problem (estimation of parameters for the HMM), the Baum–Welch algorithm is frequently used (the Baum–Welch algorithm is a generalized expectation–maximization routine, and can compute maximum likelihood estimates and posterior mode estimates for the parameters—both transition and event probabilities—of a HMM, when given only ‘events’ as training data), but analysis with complete data (sensu Schofield and Barker 2008) is certainly also possible. The Baum–Welch and related estimation approaches do not estimate the number of states—that must be specified and fixed. For a more comprehensive introduction to general inference from HMM, see Cappé et al. (2005).