It’s the Dose Response, Stupid

During the first campaign of Bill Clinton, James Carville hung a sign in the campaign office, “It’s the Economy, Stupid”.  Amidst the discussions on COVID19 and SARS-COV-2, the concept of dose-response seems to have been lost.  Too many, especially from medical backgrounds, continue to use terms such as “infectious dose” or “minimal infectious dose”.  Ever since my first paper on microbial dose response modeling in 1983 [1], this concept has been outmoded.  But it is timely for me to restate the evidence and implications for the current pandemic. 

More than 60 years ago, Meynell and Stocker [2] outlined two prevailing views for microbial infection:

“The relationship between inoculated bacteria might take one of two extreme
forms. Either they could be assumed to be acting co-operatively, death being
a consequence of their joint action, or they could be regarded as acting
independently, more than one bacterium usually being needed because the probability
of a given bacterium being lethal is less than unity …. The situation when the LD50 dose contains many organisms
is analogous to that of a poor marksman firing at a bottle. Since his aim is
poor, the bottle is unlikely to have been broken after a small number of shots
has been fired but if he persists he will probably hit the bottle eventually.
A local observer might be aware that the bottle was broken by the action of
one bullet. On the other hand, a distant observer, informed only of the total
number of shots fired before the bottle broke, would not be able to exclude
the hypothesis that the breakage was due to the accumulated stresses produced
by all the bullets fired.

We describe (1) the hypothesis of independent action, and (2) hypotheses of
synergistic action,”

In their work, by experimentation, Meynell and Stocker provided strong evidence that the hypothesis of independent action governed bacterial infections.

In work over the last 37 years, I, my students, and colleagues around the world have analyzed numerous dose-response studies of bacteria, virus, protozoa and fungi, in both animal and human hosts, and found ALL to fit dose-response models that are consistent with the independent action dose response model.  Many of these have been compiled in a wiki that started when US EPA and Department of Homeland Security co-funded the Center for Advancing Microbial Risk Assessment (CAMRA) which Joan Rose and I co-led.  The wiki is still available.  

In dose-response modeling, the dose frequently used is the population-average dose.  So for example, an average dose of 0.1 might be that less than 10% of individuals might experience an actual dose of a single organism, a smaller fraction would experience a dose of more than one organism, and the vast bulk of individuals would experience a dose of zero organisms.

It is emphasized that a successful infection and disease results from the growth of progeny of successful exogenous organism(s) from the exposure.  There is a significant literature on describing the dynamics of this in vivo birth-death process, and a useful starting point is the work of Williams [3, 4, 5].

The two dose response models that have been found to fit ALL such data are the exponential and the beta-Poisson.  The exponential is derived from a random (Poisson) distribution of organisms between individuals, a probability that a single organism in vivo survives to colonize and initiate infection, and a binomial variation between multiple exogenous organisms within a host of surviving.  Furthermore, it is assumed that only one survivor initiating infection is sufficient.  The combination of these assumptions yields the following equation:

           UntitledImage                                    (1)

where p is the proportion of individuals experiencing the average dose d who are affected, and k is the survival probability of an individual organism in vivo. 

There is variability in the propensity of microorganisms to survive in a host.  It is reasonable to describe “k” by a probability distribution.  Furumoto and Mickey [6] were the first to describe this and used a beta distribution to describe this variability.  The exact result is given by a confluent hypergeometric distribution, which can be approximated by:

       UntitledImage                 (2)

where UntitledImage is the average dose eliciting 50% response, and UntitledImage is the dispersion parameter (as this approaches infinity, the best-Poisson model approaches the exponential).  

The figure below illustrates the behavior of both the exponential and beta-Poisson models (at different values of alpha).  There are several salient things to observe:

  • The beta-Poisson model is never steeper than the exponential
  • At low dose, the dose-response slope is linear (a straight line on a log-log plot)
  • There is no dose at which the response probability is zero



The last point is critical.  The phrase “minimal infectious dose” is often thrown around in common parlance, and even by the medical community.  The exponential and beta-Poisson relationships show that even at very low average doses, there is still a non-zero proportion of individuals who may become affected by the pathogen (ultimately since the Poisson distribution predicts that some proportion of people will become exposed to one or more organisms).  This often discomfits those making decisions, since it indicates the impossibility of assuring certitude of zero risk, but rather explicitly or implicitly some acceptance of a level of residual risk (e.g. after a cleanup) needs to occur.

There are a number of extensions and embellishments of the exponential and beta-Poisson models that I have reviewed elsewhere.[7].

So what does this mean with respect to SARS-COV-2.  As of this writing, there do not appear to be any animal data sets suitable for dose response modeling.  In all of our prior work, we have found that data on inhaled dose of pathogens in a competent animal species is suitable without inter-species correction factors, for direct use in human health risk assessment.  

In my group, after the SARS outbreaks, we reviewed coronavirus data that would be suitable for dose response modeling.  We were able to fit human CoV 229E data (primarily associated with common cold) and mouse data for several coronaviruses to exponential relationships [8]. The plots of the best fit models (in the original paper we provide fitting statistics and uncertainties) are shown below.  There is about a 40 fold difference between the two curves.  The mouse data seemed to provide plausible estimates based on the attack rates for the SARS-1 Amoy Gardens cluster [9]/



This graph shows, for example, that if we want to keep the risk to a population below 0.0001, we would need to have an inhaled dose less than 0.004 based on the mouse data, and considerably less (about 0.0001) for the human data.  The tolerable risk level is a risk management decision.  Conventionally we would use this for a daily exposure.  Given a dose criterion, based on a breathing rate and an hours at exposure, we could develop concentration limits.

Undoubtedly as we gain more information on SARS-COV-2, we will be able to ascertain the more exact positioning of the dose-response relationship.  However there is no reason to believe that it would differ from all other pathogens that have been investigated with respect to functional forms of the dose response relationship.

Parenthetically, I note that in indoor air, microbial risk has frequently been estimated using the approach of Wells and Riley [10].  This is essentially an exponential in form as well, however it conflates the exposure assessment and dose response assessment components of modern risk assessment, and therefore the use of separate dose-response relationships should be preferred.  

The bottom line, in assessing risks from SARS-COV-2, in the paraphrase of James Carville, remember: “It’s the dose-response, stupid”.



[1] Haas, C. N. “Estimation of Risk Due to Low Doses of Microorganisms: A Comparison of Alternative Methodologies.” American Journal of Epidemiology 118 (1983): 573–82.

[2] Meynell, G. G., and B. A. D. Stocker. “Some Hypotheses on the Aetiology of Fatal Infections in Partially Resistant Hosts and Their Application to Mice Challenged with Salmonella Paratyphi-B or Salmonella Typhimurium by Intraperitoneal Injection.” Journal of General Microbiology 16 (1957): 38–58.

[3] Trevor Williams, and G. G. Meynell. “Time-Dependence and Count-Dependence in Microbial Infection.” Nature 214 (1967): 473–75.

[4] Williams, T. “The Basic Birth-Death Model for Microbial Infections.” Journal of the Royal Statistical Society Part B 27 (1965): 338–60.

[5] Williams, Trevor. “The Distribution of Response Times in a Birth-Death Process.” Biometrika 52, no. 3/4 (December 1965): 581.

[6] Furumoto, W. A., and R. Mickey. “A Mathematical Model for the Infectivity-Dilution Curve of Tobacco Mosaic Virus: Theoretical Considerations.” Virology 32 (1967): 216.

[7] Haas, Charles N. “Microbial Dose Response Modeling: Past, Present, and Future.” Environmental Science & Technology 49 (February 3, 2015): 1245–59.

[8] Watanabe, Toru, Timothy A. Bartrand, Mark H. Weir, Tatsuo Omura, and Charles N. Haas. “Development of a Dose-Response Model for SARS Coronavirus.” Risk Analysis 30 (2010): 1129–38.

[9] Li, Y., S. Duan, I. T. Yu, and T. W. Wong. “Multi-Zone Modeling of Probable SARS Virus Transmission by Airflow between Flats in Block E, Amoy Gardens.” Indoor Air 15 (April 2005): 96–111.

[10] Sze To, G. N., and C. Y. Chao. “Review and Comparison between the Wells-Riley and Dose-Response Approaches to Risk Assessment of Infectious Respiratory Diseases.” Indoor Air 20 (February 2010): 2–16.