Dose-response relationships for environmentally mediated infectious disease transmission models

<div><p>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 <i>Cryptosporidium</i> 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.</p></div

Model variables and parameters.

<p>Compartments and parameterizations of an environmentally mediated infectious disease transmission model with dose–response and a latency period, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.e009" target="_blank">Eq (3)</a>.</p

Attack ratios and basic reproduction numbers by dose–response function.

<p>The attack ratio, or cumulative incidence, is the fraction of at-risk (susceptible) individuals in the population who become infected during the outbreak. The simulations of a <i>Cryptosporidium</i> outbreak are pictured in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.g002" target="_blank">Fig 2b</a>. Because some transmission model parameters are uncertain, these values are for comparison between functions only and do not necessarily reflect real-world dynamics.</p

<i>Cryptosporidium</i> dose–response and dynamics with the fixed stochastic .

<p>Modeled number of infected people under different dose–response relationships, where <i>α</i> is adjusted so that for each simulation. The exponential, exact and approximate beta–Poisson, and Hill-1 functions lie nearly on top of each other, and the Hill-<i>n</i> and lognormal functions have no outbreak.</p

Common dose–response functions and selected properties.

<p>Equations give a probability of infection <i>f</i>(<i>x</i>) (response) for a number of pathogens <i>x</i> (dose). Parameter <i>π</i> = <i>f</i>′(0) for the functions with finite, non-zero slope at the origin. Some functions have been given non-standard parameterizations in order facilitate comparison between functions. In particular, the beta–Poisson functions are often parameterized in terms of <i>α</i> = <i>πβ</i> and <i>N</i>₅₀ = <i>β</i>(2^1/<i>α</i> − 1). Some functions are known by different names with different parameterizations. Hill-<i>n</i> is called log-logistic when written in terms of <i>β</i> = −<i>n</i> and <i>α</i> = 1/<i>π</i>, and log-normal is called log-probit when written in terms of <i>β</i>₀ = −<i>μ</i>/<i>σ</i> and <i>β</i>₁ = 1/<i>σ</i>.</p

Schematic of an environmentally mediated infectious disease transmission model with dose–response and exposed compartment.

<p>Solid lines represent people and dashed lines represent pathogens. Variables and parameters are defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.t002" target="_blank">Table 2</a>.</p

<i>Cryptosporidium</i> dose–response and dynamics.

<p>a) Maximum-likelihood estimates of dose–response functions for <i>Cryptosporidium</i>. Data from [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref045" target="_blank">45</a>]; sizes of data points correspond to sample size. Best-fit parameters are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.t003" target="_blank">Table 3</a>. b) Modeled fraction of infected people under different dose–response relationships. Model parameters are <i>N</i> = 1000, <i>S</i>₀ = 999, <i>I</i>₀ = 1, <i>W</i>₀ = 0, <i>σ</i> = 1/7 [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref046" target="_blank">46</a>], <i>γ</i> = 1/10 [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref046" target="_blank">46</a>], <i>κ</i> = 8 and <i>ρ</i> = 0.15 so that <i>κρ</i> = 1.2 L [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref047" target="_blank">47</a>], <i>V</i> = 4<i>E</i>8 L, <i>α</i> = 1E6/<i>V</i> (taken from a range [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref048" target="_blank">48</a>]), <i>μ</i> = 0.069 (taken from a range [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref049" target="_blank">49</a>]). Model results using the exact and approximate beta–Poisson functions lie nearly on top of each other, and those using the Hill-<i>n</i> and log-normal functions have no outbreak. c) Average pathogen dose over time for different dose–response relationships. Model parameters are as in Fig 2b. d) Low dose behavior of the dose–response functions given in Fig 2a. Confidence intervals are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.g003" target="_blank">Fig 3</a>.</p

<i>Vibrio cholerae</i> dose–response and dynamics.

<p>a) Maximum-likelihood estimates of dose–response functions for buffered Inaba strain of <i>Vibrio cholerae</i>. Buffering was used in this experiment to approximate eating contaminated food, as food buffers stomach acid; because <i>Vibrio cholerae</i> does not tolerate gastric acidity, a buffered strain is more infectious. Data from [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref050" target="_blank">50</a>]; sizes of data points correspond to sample size. Best-fit parameters are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.t003" target="_blank">Table 3</a>. b) Modeled fraction of infected people under different <i>Vibrio cholerae</i> dose–response relationships. Model parameters are <i>N</i> = 1000, <i>S</i>₀ = 999, <i>I</i>₀ = 1, <i>W</i>₀ = 0, <i>σ</i> = 5/2 [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref051" target="_blank">51</a>], <i>γ</i> = 1/5 [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref009" target="_blank">9</a>], <i>κ</i> = 8 and <i>ρ</i> = 0.15 so that <i>κρ</i> = 1.2 L [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref047" target="_blank">47</a>], <i>V</i> = 4<i>E</i>8, <i>α</i> = 2E6/<i>V</i>, <i>μ</i> = 0.23 [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref009" target="_blank">9</a>]. Model results using the exact and approximate beta–Poisson functions lie nearly on top of each other, and those using the Hill-1 and exponential functions have no outbreak.</p

<i>Shigella flexneri</i> dose–response and dynamics.

<p>a) Maximum-likelihood estimates of dose–response functions for <i>Shigella flexneri</i>. Data from [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref052" target="_blank">52</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref053" target="_blank">53</a>]; sizes of data points correspond to sample size. Best-fit parameters are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.t003" target="_blank">Table 3</a>. b) Modeled fraction of infected people under different <i>Shigella flexneri</i> dose–response relationships. Model parameters are <i>N</i> = 1000, <i>S</i>₀ = 999, <i>I</i>₀ = 1, <i>W</i>₀ = 0, <i>σ</i> = 2/3 [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref054" target="_blank">54</a>], <i>γ</i> = 1/6 [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref054" target="_blank">54</a>], <i>κ</i> = 8 and <i>ρ</i> = 0.15 so that <i>κρ</i> = 1.2 L [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref047" target="_blank">47</a>], <i>V</i> = 4<i>E</i>8, <i>α</i> = 4E7/<i>V</i>, <i>μ</i> = 5 [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref055" target="_blank">55</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.ref056" target="_blank">56</a>]. Model results using the Weibull, log-normal, and Hill-<i>n</i> functions lie nearly on top of each other, and those using the Hill-1 and exponential functions have no outbreak.</p

Force of infection changes with dose aggregation.

<p>Force of infection <i>κf</i>(<i>ρW</i>) relative to that when the contact rate is <i>κ</i> = 8 for constant daily dose <i>κρE</i>, i.e. <i>κf</i>(<i>ρW</i>)/(8<i>f</i>(<i>κρW</i>/8)), for each dose–response model parameterized to have an ID₅₀ of 1E6 (comparable to influenza) at a) high, b) medium, and c) low total doses. Any parameters not constrained by the median dose were chosen from the best-fit influenza parameters (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005481#pcbi.1005481.s001" target="_blank">S1 Appendix</a>).</p