Impact of heterogeneity on infection probability: Insights from single-hit dose-response models

TL;DR

This paper analyzes how heterogeneity in dose and microbial infectivity shapes infection probability within the single-hit dose-response framework. By deriving general results and exploring multiple concrete models, it shows that within-host infectivity heterogeneity increases the probability of infection for a fixed mean infectivity, whereas between-host infectivity heterogeneity and dose heterogeneity tend to decrease the expected probability of infection in the small-infectivity regime. The study unifies top-down dose-response analysis with a mechanistic within-host growth perspective, illustrating hierarchies among models (A, C, C′) and connecting beta-Poisson parameters to underlying variability. The results offer mathematical predictions and suggest laboratory experiments to validate how heterogeneity types influence infection outcomes, potentially guiding risk assessment and mechanistic understanding of infections.

Abstract

The process of infection of a host is complex, influenced by factors such as microbial variation within and between hosts as well as differences in dose across hosts. This study uses dose-response and within-host microbial infection models to delve into the impact of these factors on infection probability. It is rigorously demonstrated that within-host heterogeneity in microbial infectivity enhances the probability of infection. The effect of infectivity and dose variation between hosts is studied in terms of the expected value of the probability of infection. General analytical findings, derived under the assumption of small infectivity, reveal that both types of heterogeneity reduce the expected infection probability. Interestingly, this trend appears consistent across specific dose-response models, suggesting a limited role for the small infectivity condition. Additionally, the vital dynamics behind heterogeneous infectivity are investigated with a within-host microbial growth model which enhances the biological significance of single-hit dose-response models. Testing these mathematical predictions inspire new and challenging laboratory experiments that could deepen our understanding of infections.

Figures (7)

• Figure 1: Single-hit models for the expected probability of infection. Each axis represents a different source of heterogeneity: (I) Randomly distributed infectivity within hosts, (II) randomly distributed infectivity for different hosts and (III) randomly distributed dose for different hosts. Model $A$ is located at the origin where both infectivity and dose are homogeneous. Models of type $A^\prime$ are located along the axis corresponding to heterogeneous doses. Models of type $B$ are located along the axis corresponding to random infectivity within hosts and homogeneous between-hosts infectivity and dose. Models of type $C$ are located on the horizontal plane with homogeneous dose but heterogeneous infectivity within and between hosts. Models of type $C^\prime$ incorporate all three types of heterogeneity and are located in the bulk of the model space.
• Figure 2: Graphical illustration of the inequalities given in Proposition \ref{['pro:Inequalities_PA']}. The dependence of the probability of infection on the infectivity $x$ for a system with homogeneous infectivity (Model $A$, continuous black line) is compared to the dependence on $x$ of the expected probability of infection of models $A^{\prime}_1$ (dashed red line) and $A^{\prime}_2$ (dot-dashed blue line). The dose was set to $n=5$ for model $A$. The mean and variance of the dose were set to $\mu_n=5$ and $v_n=20$, respectively, for models $A^{\prime}_1$ and $A^{\prime}_2$.
• Figure 3: Probability density function of the infectivity for model $C_1$ (dashed line, Eq. \ref{['eq:rho_beta']}) and the within-host microbial infection model proposed in Sec. \ref{['sec:Birth-Death_Particular']} (continous line, Eq. \ref{['eq:rhox_ExpGamma']}). Different panels illustrate the dependence of $\rho_h^{(C_1)}(x;\alpha,\beta)$ on infectivity in each of the regimes described in the text: (a) Regime (i) with $(\alpha,\beta)=(0.5,0.25)$, (b) regime (ii) with $(\alpha,\beta)=(1.2,0.36)$, (c) regime (iii) with $(\alpha,\beta)=(0.89,1.32)$, and (d) regime (iv) with $(\alpha,\beta)=(1.5,1.8)$. When parametrised by the mean and variance of the effective microbe mortality considered in the microbial infection model of Sec. \ref{['sec:Birth-Death_Particular']}, the four panels correspond to: (a) $(\mu_\lambda,v_\lambda)=(0.5,1.0)$, (b) $(\mu_\lambda,v_\lambda)=(0.3,0.25)$, (c) $(\mu_\lambda,v_\lambda)=(1.5,1.7)$ and (d) $(\mu_\lambda,v_\lambda)=(1.2,0.8)$.
• Figure 4: Location of the four regimes for the shape of the infectivity PDF $\rho_h^{(C_1)}(x;\alpha,\beta)$ in the space spanned by the mean and variance of the infectivity.
• Figure 5: Expected probability of infection for examples $C_1$ with dose $n=5$ (continuous lines, Eq. \ref{['eq:Pinf_C1']}) and $\tilde{C_1}$ (dashed lines, Eq. \ref{['eq:Pinf_tildeC1']}). Panel (a) shows the dependence of $P^{(C_1)}$ and $P^{(\tilde{C_1})}$ on the mean infectivity, $\mu_x$, for two values of the variance, $v_x$, as marked by the legend. The dotted diagonal line indicates the bounds for $P^{(C_1)}$ given by Eq. \ref{['eq:lim_PC1_mu+-']}. Panel (b) shows the dependence of $P^{(C_1)}$ and $P^{(\tilde{C_1})}$ on the infectivity variance $v_x$. The dotted line shows the limit of $P^{(C_1)}$ as $v_x \nearrow v_{\max}(\mu_x)$ (cf. Eqs. \ref{['eq:vmax']} and \ref{['eq:limPC1_vxvmax']}).

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 4.1
• proof
• Corollary 4.1.1
• proof
• Proposition 4.1
• proof