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Informing policy via dynamic models: Cholera in Haiti [1]

['Jesse Wheeler', 'Statistics Department', 'University Of Michigan', 'Ann Arbor', 'Michigan', 'United States Of America', 'Annaelaine Rosengart', 'Statistics', 'Data Science', 'Carnegie Mellon University']

Date: 2024-07

Mechanistic models for disease modeling

Mechanistic models representing biological phenomena are valuable for epidemiology and consequently for public health policy [56, 57]. More broadly, they have useful roles throughout biology, especially when combined with statistical methods that properly account for stochasticity and nonlinearity [58]. In some situations, modern machine learning methods can outperform mechanistic models on epidemiological forecasting tasks [59, 60]. The predictive skill of non-mechanistic models can reveal limitations in mechanistic models, but cannot readily replace the scientific understanding obtained by describing the biological dynamics of the system in a mathematical model [60, 61].

In this article, we refer to models that focus on learning relationships between variables in a dataset as associative, whereas models that incorporate a known scientific property of the system we call causal or mechanistic. The danger in using forecasting techniques which rely on associative models to predict the consequence of interventions is called the Lucas critique in an econometric context. Lucas et al. [62] pointed out that it is naive to predict the effects of an intervention on a given system based entirely on historical associations. To successfully predict the effect of an intervention, a model should therefore both provide a quantitative explanation of existing data and should have a causal interpretation: a manipulation of the system should correspond quantitatively with the corresponding change to the model. This motivates the development of mechanistic models, which provides a statistical fit to the available data while also supporting a causal interpretation. Despite their limited ability to project the effect of interventions on a system, associative models can be effectively used to make inference on an certain features of a system. In our literature review, there were 14 studies that used associative models to describe various aspects of the cholera epidemic [9, 40, 63–74].

The four mechanistic models of Lee et al. [4] were deliberately developed with limited coordination. This allows us to treat the models as fairly independently developed expert approaches to understanding cholera transmission. However, it led to differences in notation, and in subsets of the data chosen for analysis, that hinder direct comparison. Here, we have created a common notational framework that facilitates model comparison, and put all comparable model parameters—including parameters that were estimated or held constant—into Fig 2. Translations back to the original notation of Lee et al. [4] are given in the supplement (S1 Table).

Each model describes the cholera dynamics as a partially observed Markov process (POMP) with a latent state vector X(t) for each continuous time point t. N observations on the system are collected at time points t 1 , …, t N , written as t 1:N . The observation at time t n is modeled by the random vector Y n . While the latent process exists between observation times, the value of the latent state at observations times is of particular interest. We therefore write X n = X(t n ) to denote the value of the latent process at the nth observation time, and X 1:N is the collection of latent state values for all observed time points. The observable random variables Y 1:N are assumed to be conditionally independent given X 0:N . Together, with the density for the initial value of the latent state X 0 = X(t 0 ), each model defines a joint density , where θ is a parameter vector that indexes the model. The observed data , along with the unobserved true value of the latent state, are modeled as a realization of this joint distribution.

Because of the probabilistic nature of both the unobserved latent state and the observable random variables, it is possible to consider various marginal and conditional densities of these two jointly random vectors. An important example is the marginal density of the observed random vector Y 1:N , evaluated at the observed data , as shown in Eq (1): (1) When treated as a function of the parameter vector θ, this marginal density is called the likelihood function, which is the basis of likelihood based statistical inference.

Using the conditional independence of Y 1:N given X 0:N and the Markov property of X 0:N , the joint density can be re-factored into the useful form given in Eq (2): (2) This factorization is useful because it demonstrates that POMP models may be completely described using three parts: the initialization model for the latent states ; the one-step transition density, or the process model ; and the measurement model . In the following subsections, we describe Models 1–3 in terms of these three components. The latent state vector X(t) for each model consists of individuals labeled as susceptible (S), infected (I), asymptomatically infected (A), vaccinated (V), and recovered (R), with various sub-divisions sometimes considered. The observable random vector Y 1:N represents the random vector of cholera incidence data for each model; Models 2 and 3 have metapopulation structure, meaning that each individual is a member of a spatial unit, denoted by a subscript u ∈ 1:U, in which case we denote the observed data for each unit using . Here, the spatial units are the U = 10 Haitian administrative départements (henceforth anglicized as departments).

While the complete model description is scientifically critical, as well as necessary for transparency and reproducibility, the model details are not essential to our methodological discussions of how to diagnose and address model misspecification with the purpose of informing policy. A first-time reader may choose to skim through the rest of this section, and return later. Additional details about the numeric implementation of these models are provided in a supplemental text (S1 Text). While each of the dynamic models considered in this manuscript can be fully described using the mathematical equations provided in the following section, diagrams of dynamic systems can be helpful to understand the equations. For this reason, we provide flow chart diagrams for Models 1–3 in supplement figures (S1–S3 Figs).

Model 1. The latent state vector X(t) = (S z (t), E z (t), I z (t), A z (t), R z (t), z ∈ 0 : Z) describes susceptible, latent (exposed), infected (and symptomatic), asymptomatic, and recovered individuals in vaccine cohort z at time t. Here, z = 0 corresponds to unvaccinated individuals, and z ∈ 1:Z describes hypothetical vaccination programs. Each program z indexes differences in both the number of doses administered (one versus two doses per individual) and the round of vaccine administration, separating individuals into compartments with distinct dynamics based on vaccination status. The force of infection is (3) where β(t) is a periodic cubic spline representation of seasonality, given in terms of a B-spline basis {s j (t), j ∈ 1:6} and parameters β 1:6 as (4) where (wk)-1 is a dimensionality constant. The process noise dΓ(t)/dt is multiplicative Gamma-distributed white noise, with infinitesimal variance parameter . Lee et al. [4] included process noise in Model 3 but not in Model 1, i.e., they fixed . Gamma white noise in the transmission rate gives rise to an over-dispersed latent Markov process [75] which has been found to improve the statistical fit of disease transmission models [2, 76]. For any time point in t 1:N , the process model is defined by describing how individuals move from one latent state compartment to another. Per-capita transition rates are given in Eqs (5)–(12): (5) (6) (7) (8) (9) (10) (11) (12) where z ∈ 0:Z. Here, μ AB is a transition rate from compartment A to B. We have an additional demographic source and sink compartment • modeling entry into the study population due to birth or immigration, and exit from the study population due to death or immigration. Thus, μ A• is a rate of exiting the study population from compartment A and μ •B is a rate of entering the study population into compartment B. In Model 1, the advantage afforded to vaccinated individuals is an increased probability that an infection is asymptomatic. Conditional on infection status, vaccinated individuals are also less infectious than their non-vaccinated counterparts by a rate of ϵ = 0.05 in Eq (3). In Eqs (7) and (6) the asymptomatic ratio for non-vaccinated individuals is set f 0 (t) = 0, so that the asymptomatic route is reserved for vaccinated individuals. For z ∈ 1:Z, the vaccination cohort z is assigned a time τ z , and we take f z (t) = cϑ*(t − τ z ) where ϑ*(t) is efficacy at time t since vaccination for adults, a step-function represented in Table S4 of [4], and c = (1 − (1 − 0.4688) × 0.11) is a correction to allow for reduced efficacy in the 11% of the population aged under 5 years. Single and double vaccine doses were modeled by changing the waning of protection; protection was modeled as equal between single and double dose until 52 weeks after vaccination, at which point the single dose becomes ineffective. The latent state vector X(t) is initialized by setting the counts for each compartment and vaccination scenario z ≠ 0 as zero, and introducing initial-value parameters I 0,0 and E 0,0 such that R 0 (0) = 0, I 0 (0) = Pop × I 0,0 , E 0 (0) = Pop × E 0,0 and S 0 (0) = Pop × (1 − I 0,0 − E 0,0 ), where Pop is the total population of Haiti. The measurement model describes reported cholera cases at time point n come from a negative binomial distribution, where only a fraction (ρ) of new weekly cases are reported. More details about the initialization model and the measurement model for Models 1–3 are provided in a supplement text (S2 and S3 Text).

Model 2. Susceptible individuals are in compartments S uz (t), where u ∈ 1:U corresponds to the U = 10 departments, and z ∈ 0:4 describes vaccination status: z = 0: Unvaccinated or waned vaccination protection. z = 1: One dose at age under five years. z = 2: Two doses at age under five years. z = 3: One dose at age over five years. z = 4: Two doses at age over five years. Like Model 1, the process model is primarily defined via the description of movement of individuals between compartments, however Model 2 also includes a dynamic description of a latent bacterial compartment as well. Individuals can progress to a latent infection E uz followed by symptomatic infection I uz with recovery to R uz or asymptomatic infection A uz with recovery to . The force of infection depends on both direct transmission and an aquatic reservoir, W u (t), and is given by (13) The latent state is therefore described by the vector . The cosine term in Eq (13) accounts for annual seasonality, with a phase parameter ϕ. The Lee et al. [4] implementation of Model 2 fixes ϕ = 0. Individuals move from department u to v at rate T uv , and aquatic cholera moves at rate . The nonzero transition rates are (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) In Eq (18) the spatial coupling is specified by a gravity model, (24) where Pop u is the mean population for department u, D uv is a distance measure estimating average road distance between randomly chosen members of each population, and v rate = 10−12 km2yr−1 was fixed at the value used in [4]. In Eq (23), is a measure of river flow between departments. The unit of W u (t) is cells per ml, with dose response modeled via a saturation constant of W sat in Eq (13). In Eq (14), ϑ z denotes the vaccine efficacy for each vaccination campaign z ∈ Z, with ϑ 0 = 0, ϑ 1 = 0.429q, ϑ 2 = 0.519q, ϑ 3 = 0.429, and ϑ 4 = 0.519 Here, q = 0.4688 represents the reduced efficacy of the vaccination for children under the age of five years, and the values 0.429 and 0.519 are the median effectiveness of one and two doses over their effective period respectively, according to Table S4 in the supplement material of Lee et al. [4]. Because vaccine efficacy remains constant, individuals in this model transition from a vaccinated compartment to the susceptible compartment at the end of the vaccine coverage period. The starting value for each element of the latent state vector X(0) are set to zero except for and R u0 (0) = Pop u − I u0 (0), where is the reported number of cholera cases in department u at time t = 0. Reported cases are described using a log-normal distribution, with the log-scale mean equal to the reporting rate ρ times the number of newly infected individuals. See the supplement material on model initializations for more details (S2 Text).

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[1] Url: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1012032

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