The effect of ongoing exposure dynamics in dose response relationships

Abstract

Characterizing infectivity as a function of pathogen dose is integral to microbial risk assessment. Dose-response experiments usually administer doses to subjects at one time. Phenomenological models of the resulting data, such as the exponential and the Beta-Poisson models, ignore dose timing and assume independent risks from each pathogen. Real world exposure to pathogens, however, is a sequence of discrete events where concurrent or prior pathogen arrival affects the capacity of immune effectors to engage and kill newly arriving pathogens. We model immune effector and pathogen interactions during the period before infection becomes established in order to capture the dynamics generating dose timing effects. Model analysis reveals an inverse relationship between the time over which exposures accumulate and the risk of infection. Data from one time dose experiments will thus overestimate per pathogen infection risks of real world exposures. For instance, fitting our model to one time dosing data reveals a risk of 0.66 from 313 Cryptosporidium parvum pathogens. When the temporal exposure window is increased 100-fold using the same parameters fitted by our model to the one time dose data, the risk of infection is reduced to 0.09. Confirmation of this risk prediction requires data from experiments administering doses with different timings. Our model demonstrates that dose timing could markedly alter the risks generated by airborne versus fomite transmitted pathogens.

Figure 1. Evolution of the number of pathogens over time for a characteristic parameter set .

Each line represents an individual replicate with the same parameter set (100 in total). The fraction of replicates in which the number of pathogens diverge towards infinity, as opposed to going extinct, is equivalent to the probability of infection (p_inf) for the dose (main graph, and for the insets a) and b) respectively). Temporal exposure length is fixed at T_e = 1 hour. Probability of infection is 0.67, 0.02 and 0.98 for the main graph, the inset a), and the inset b) respectively.

Figure 2. State probability distribution at the end of inoculation () for a dose of and temporal exposure length of ) 0.1 h, B) 1.0 h, C) 10.0 h and D) 50.0 h.

The distribution of probabilities if would be given the parameters of the system are . The dashed white line is the separatrix of the deterministic version of the model (see subsection Deterministic Analysis); if the system were deterministic once inoculation has been completed, the states that fall below the separatrix would end up in no infection, and the states above would end up in infection.

Figure 3. Vector field plot of the deterministic cumulative dose model for a characteristic parameter set .

To avoid overlaps of the vectors they have been normalized. The solid red lines are the nullclines, the intersections of the nullclines are the fixed points (stable pathogen elimination equilibrium) and (unstable saddle point equilibrium). The dash black line is the separatrix that separates those configurations that will go to non-infection equilibrium, , and those that will diverge in the number of pathogens resulting on infection. The separatrix has been calculated numerically.

Figure 4. Dose-response curves based on the Exponential Model () and the Cumulative Dose model () compared to the experimental dataset for Poliovirus type 1 (squares).

The estimated parameters are for the Exponential model and for the Cumulative Dose model.

Figure 5. Dose-response curves based on the Exponential Model () and the Cumulative Dose model () compared to the experimental dataset for Cryptosporidium parvum (squares).

The estimated parameters are for the Exponential model [30] and for the Cumulative Dose model.

Figure 6. Dose-response curves based on the Exponential Model (), the Beta-Poisson model () and the Cumulative Dose model () compared to the experimental dataset for Rotavirus (squares).

The estimated parameters are for the Exponential model, for the Beta-Poisson model [31] and for the Cumulative Dose model.

Figure 7. Predicted effects of varying exposure times () when inoculated with Poliovirus type 1.

Parameters are defined as stated in Figure 4.

Figure 8. Predicted effects of varying exposure times () when inoculated with Cryptosporidium parvum.

The top graph comes from simulations using the parameter set defined in Figure 5. The bottom graph comes simulations using the parameter set .

Figure 9. Predicted effects of different temporal patterns of exposure when inoculated with Cryptosporidium parvum.

The main figure displays the probability of infection as function of the number of inoculation events. The line with circular markers comes from simulation results using the parameter set defined in Figure 5, and the line with square markers comes from simulation results using the parameter set The insets below demonstrate three temporal patterns for three different patterns of inoculation events: A = 1, B = 4 and C = 50 events respectively. The solid line represents one instance of the 5000 replicas used in the experiment. The dashed line represents the average of dose inoculated over time.