Dose-response relationships for environmentally mediated infectious disease transmission models

Environmentally mediated infectious disease transmission models provide a mechanistic approach to examining environmental interventions for outbreaks, such as water treatment or surface decontamination. The shift from the classical SIR framework to one incorporating the environment requires codifying the relationship between exposure to environmental pathogens and infection, i.e. the dose–response relationship. Much of the work characterizing the functional forms of dose–response relationships has used statistical fit to experimental data. However, there has been little research examining the consequences of the choice of functional form in the context of transmission dynamics. To this end, we identify four properties of dose–response functions that should be considered when selecting a functional form: low-dose linearity, scalability, concavity, and whether it is a single-hit model. We find that i) middle- and high-dose data do not constrain the low-dose response, and different dose–response forms that are equally plausible given the data can lead to significant differences in simulated outbreak dynamics; ii) the choice of how to aggregate continuous exposure into discrete doses can impact the modeled force of infection; iii) low-dose linear, concave functions allow the basic reproduction number to control global dynamics; and iv) identifiability analysis offers a way to manage multiple sources of uncertainty and leverage environmental monitoring to make inference about infectivity. By applying an environmentally mediated infectious disease model to the 1993 Milwaukee Cryptosporidium outbreak, we demonstrate that environmental monitoring allows for inference regarding the infectivity of the pathogen and thus improves our ability to identify outbreak characteristics such as pathogen strain.


Alternate models
We consider several variations on the environmentally mediated infectious disease transmission model with dose-response (Eqs. (3)). In the first, we count the number W * rather than the concentration W of pathogens. This requires a slight redefinition of the shedding α and pick-up ρ parameters, because it shifts the implicit volume of the environment from incorporation in α to ρ.

Example details
In the example (Fig. 8) where a transmission model with each of the dose-response functional forms were fit to simulated data, the data were simulated in the following way. The model in Eqs.
Environmental concentration data were similarly simulated by drawing a number of oocysts per 10L from a Poisson distribution with mean determined from the model trajectory of the environment (modeled concentration times 10L). Computation was done in R (v3.3.1), and the seeds were set to 0 and 1 for the environmental monitoring and case data, respectively. Parameter combinations ακρ, ξ = µ + κρN/V , γ, and σ, as well as initial conditions E(0) and W (0)S(0)/α were estimated using the maximum likelihood approach described in the main text; the initial number of observed cases was used for I(0). We assumed S(0) = N − I(0) − E(0) (no prior immunity) and that N was known.
The model used for the Milwaukee cryptosporidosis outbreak (Fig. 9) uses the Cryptosporidium and turbidity data to estimate a time course of Cryptospordium concentration T (t) in the water supplied to homes. This model also uses two exposed compartments. Because the data is new onset of symptoms rather than infection, we only keep track of cumulative new cases Y (t). We assume new cases data K i on day t i comes from a binomial distribution with size N and probability S3

Dose-response model fits
Dose-response functions and corresponding dynamics for influenza, rotavirus, and Salmonella typhi are shown in Figs. S1, S2, and S3 respectively. Estimated parameters and negative log-likelihoods for the maximum-likelihood estimators of the seven dose-response functions and six pathogens considered are given in Table S1.

Stochastic basic reproduction number
Here we prove Proposition 2. A careful accounting of the transition events and rates for the stochastic analog of the model given in Eqs.
(3) is given in Table S2. Because pick-up and die-off are separate events, we do not use the parameterization ξ = κρN + µ here. Further, it is more intuitive to use number of pathogens W * in this derivation, although the concentration formulation is equivalent for a fixed environmental size V . Here, α and ρ are scaled as described in Section S1. (3) is

Proposition 2. The basic reproduction number for the stochastic analog of the model given in Eqs
Proof. Although there are three infected compartments (E, I, and W * ), because all exposed people necessarily become infectious, it is sufficient to consider I to be the "offspring" of W * without explicitly considering the intermediate E.
In the notation of [S11], we write the offspring probability generating function for I given I(0) = 1 and W * (0) = 0: Similarly, we write the offspring probability generating function for W * given I(0) = 0 and W (0) = 1: Then, the expectation matrix is Since f 1 and f 2 are not simple functions and M is irreducible, then spectral radius of M determines whether the probability of ultimate extinction is 1 or less than 1. We have The Jury conditions state that ρ(M) < 1 if and only if trace(M) < * 1 + det(M) < * * 2. The second ( * * ) Jury inequality is easily satisfied as det(M) < 0. The first ( * ) is satisfied if This condition can also be found by solving ρ(M) < 1 directly.
When R * 0 < 1, the branching process is subcritical, and the disease will die out with probability 1. If the disease is supercritical R * 0 > 1, then there are unique fixed points q 1 , and q 2 such that the probability of ultimate disease extinction is q i 1 i q i 2 2 given I(0) = i 1 and E(0) = i 2 .
The fixed points are found by solving This admits the following solution These fixed points have epidemiological interpretations. An infectious individual successfully transmits an infection with probability 1/R * 0 . A pathogen either dies with probability µ/(κρN + µ) or is picked up with probability κρN/(κρN + µ). If the pathogen is picked up, it either does not cause disease with probability 1 − f (1) or it does with probability f (1). If it causes disease, the probability of successfully transmitting an infection is 1/R * 0 .
S9 Table S2: Transition events and rates for the stochastic analog of the environmentally mediated infectious disease transmission model with dose-response relationship given in Eqs.
(3). Because ρW * will typically not be an integer, we assign an integer number in the following way: ρW *

Global dynamics results
We extend Theorem 1 to include person-to-person transmission.
Proof. First, we note that Ω = {(S, E, I, R, W ) : is a compact, invariant set for trajectories of Eqs. S2. In the notation of [S12], let x = (E, I, W ) be the disease compartments and y = (S, R) the non-disease compartments. Then, we may writeẋ (S16) Then we have new-infection and compartment transfer matrices Then, the next generation matrix is K = F V −1 , and R 0 is the spectral radius of K, namely Assume that f is concave down. Then ρEf (0) ≥ f (ρE), and h(x, y) ≥ 0.
The rest of the proof proceeds as in that of Theorem 1.