Masking the Use of Masks

The principal messaging from health authorities in the COVID-19 outbreak to the general public has been that masks protect others from the wearer.  There are two important concepts in this simple message that need to be unpacked, and I want to put down my thoughts on them.  I believe the underlying miscommunication of concepts (and failure to update with new understandings, in some cases of old knowledge) has led to a great deal of the political debate about mask wearing (caveat — I am not a political scientist).

As a purely technical point, masks behave as filters.  Masks, especially those which are in use by the general public are not anisotropic, at least in the direction of airflow.  The key meaning of this is that particle removals are the same whether from the wearer to the surroundings (on talking, coughing, sneezing or exhalation), or from the surroundings to the wearer (on inhalation).  The following table, excerpted from a recent paper on testing cloth mask materials with simulated gaps showed that cloth masks of various types could achieve upwards of 54 % removal of small particles and 44% of large particles — and noteworthy was that with or without simulated gaps, most fabric masks performed as well or better than surgical masks.


Table excerpted from Konda et al. (2020)

Material <300 nm average ± error >300 nm average ± error
N95 (no gap) 85 ± 15 99.9 ± 0.1
N95 (with gap) 34 ± 15 12 ± 3
surgical mask (no gap) 76 ± 22 99.6 ± 0.1
surgical mask (with gap) 50 ± 7 44 ± 3
cotton quilt 96 ± 2 96.1 ± 0.3
flannel 57 ± 8 44 ± 2
cotton (600 TPI), 1 layer 79 ± 23 98.4 ± 0.2
cotton (600 TPI), 2 layers 82 ± 19 99.5 ± 0.1
natural silk, 1 layer 54 ± 8 56 ± 2
natural silk, 2 layers 65 ± 10 65 ± 2


In communicating about mask use, there is frequent use of the word “protect”.  The first definition of this word is: “to cover or shield from exposure, injury, damage, or destruction”.  To the lay public this is construed as “eliminate”,  There are two problems with use of this word in communicating about masks.  First is that no risk can be eliminated, but only reduced (perhaps to a level where the residual risk is regarded as acceptable).  Second, masks must be considered as part of a multiple barrier strategy (possibly a subject of a future post). It is becoming clear with COVID19 that transmission may occur not only via large particles (which the medical community has historically called droplets), but small particles.  

Total protection by any one intervention is not necessary.  In fact, concepts from industrial and occupational health recognize that the use of PPE such as masks are a last resort after interventions such as engineering design, administrative controls, etc. are employed.  The combination of all such interventions are what yields a needed risk reduction.

Complete elimination by controls, while an admiral goal, is not necessary.  In the current pandemic, the key desire is to reduce the reproduction number of cases below 1 so that there are diminishing numbers of cases and the case incidence rate is sufficiently low to permit testing, contact tracing and isolation as a final means of driving illness levels down.  Hence, for example, if the uncontrolled reproduction number (R0) is 4, then only an overall 75% reduction of risk is needed.

From the tests on homemade masks, it is clear that most materials can provide risk reductions to levels that would be desired, especially in conjunction with other measures such as improvement of indoor ventilation, social distancing, and prudence in gathering of large groups.

If I were designing a risk communication message about masks, I would phrase it along the lines that “wearing a mask reduces risk to you and those you interact with”.  Hopefully such changes will start to come about.


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.

#Zika Virus From a #RiskAssessment Point of View

Zika Virus From a Risk Assessment Point of View

Zika virus appears to have a transmission cycle of infected human host –> mosquito (via blood meal) –> susceptible host (in course of a second blood meal). To understand transmission via the route, the following need to be known:

  1. What are the levels in blood of an infected individual?
  2. What is the volume of blood ingested in feeding by a mosquito?
  3. What is the volume of disgorgement of blood by a mosquito upon a second blood meal?
  4. What is the die-off of Zika virus within a mosquito between blood meals?
  5. What is the dose-response in the human host for infection by Zika virus.

Questions (2) and (3) should be identifiable by a literature review and would not be expected to be a function of the pathogen (Zika). Question (1) may be obtainable from a review of case reports and deliberate trials in the literature, as well as on the ongoing primate trials at the University of WIsconsin, which are being done in an open science manner ( 

It is not anticipated that data on question (4) is available per se, however inferences may be drawn from persistence of other Flaviviridae in conditions analogous to carriage in the mosquito. A preliminary scan of the literature suggests prior data that could be useful in developing a dose-response relationship per question (5) for Zika. We have developed dose response relationships for many other organisms including several vector borne pathogens:

The assembly of this information can be useful, when embedded in a population transmission model, for projecting consequence and estimating the effectiveness of public health interventions. To my knowledge, such risk assessment approach has not been underway.

Unfortunately, serious risk analysis seems to be minimized as a tool to respond, right now.  Decision makers and funders need to be educated.

Costs and Benefits of Quarantine and Isolation


Clearly, at least in the US, ebolaphobia has been contagious.  But lets look at the concepts of quarantine and isolation.  According to CDC:

  • “Isolation separates sick people with a contagious disease from people who are not sick.
  • Quarantine separates and restricts the movement of people who were exposed to a contagious disease to see if they become sick.”

I want to focus on the concept of quarantine.  The operative phrase in the CDC definition is “who were exposed to a contagious disease”.  What precisely does that mean — is it that they are almost certain to progress to illness (i.e., the level of exposure was sufficiently high to give a near 100% probability), or that they were in circumstances where they could have received a dose (but they have perhaps a less, or much less probability of progressing to illness)?

There are clear costs and potential benefits to quarantine.  The obvious costs include the following:

  • lost wages for individuals unable to work during the quarantine period
  • room and board if the quarantine is not home quarantine
  • medical monitoring
  • costs associated with enforcement
  • there is the less tangible, but nonetheless real, cost of reducing civil liberties of the affected persons
  • for quarantine of health care workers after they have cared for patients (either in Africa or domestically) there is the cost that is also not well quantifiable in deterring others from giving such vital services in the future.

There are also potential benefits, which may be more difficult to calculate.  For the fraction of individuals who will succumb to disease, placing them in quarantine may reduce the spread of disease in others.  But to adequately quantify this one needs to employ a disease transmission model, which will require estimation of the underlying parameters, and also the underlying baseline disease prevalence.

Right now rather than such rational decision making, the mad rush towards quarantine seems to be political.  The general consensus seems to be that a signal early symptom of Ebola is a rapid onset of fever.  Therefore, if a person is deemed to be responsible (and presumably a default ought to be that health care workers are regarded as such), self monitoring and reporting is sufficient.

It seems to be unfortunate that decision making now has a strong element of science denial.

Quantifying Ebola – I

With the ongoing Ebola outbreak, what seems to me missing in the discourse is some quantification.  There have been a number of detailed epidemiologic models in the literature and on the web.  For example:

These models all use some version of the SEIR model, depicted in the graphic below 






In this approach, a human population is considered to be either susceptible, exposed, infectious or recovered, with a progression as indicated by the arrows (with each arrow representing a rate of “flow” between one state and another.  There may be other paths as well (for example, conversion of some of the recovered to once again susceptible).

If we focus on the first two boxes, susceptible (S) and exposed (E), then using a deterministic form and homogenous populations, two differential equations for S and E can be written following Daley and Gani (1)








The initial conversion to exposed is therefore the term “beta” x S x I, where “beta” is the rate at which new exposures occur from interactions between infectious (I) and susceptible (S) populations.

It seems to me that it is useful and interesting to regard “beta” as being the product of two terms:

beta = b1 x b2

where b1 is the frequency (#/unit time) of contacts between susceptible and infectious individuals and b2 is the probability that a single contact would result in conversion to an “exposed” state.  Note that in standard epidemiological parlance, the term “exposed” refers to an individual who will become but is not yet infectious.  The term b1 would presumably be driven principally by the nature and velocity of population mixing (which could be reduced by isolation of the exposed from the susceptible).

In this terminology, b2 is essentially the risk probability (frequently termed the probability of infection) which is widely used in quantitative microbial risk assessment (2).  To obtain b2, we essentially need two sets of information:

  • the average dose (d) of infectious agent transferred in the interaction between susceptible and infectious individuals during an interaction
  • a dose-response relationship giving the relation between dose and probability of infection.  the following two forms are widely used (see reference 2):







The first equation is the exponential dose response relationship (with an unknown parameter k) and the second is the approximate beta Poisson with unknown parameters N50 and alpha.  N50 is the median infectious dose.  As alpha in the second equation goes to infinity, the beta Poisson equation becomes the exponential.  

This analysis points to two types of data that are in need of quantification in the current Ebola outbreak.  First, what is the average dose transferred between an infectious person and a susceptible person during their contact?  This clearly will be a distribution depending on the nature and extent of contact.  Risk assessors are accustomed to incorporating and modeling various sources of uncertainty and variability (3).

Second, the dose response curve (parameters k or alpha and N50) need to be known.  I have not yet seen data (either in animal systems or humans) needed to construct such a curve, although we have many relationships for a variety of pathogens (4) .  In the context of weaponized aerosolized Ebola, an infectious dose (my interpretation of this is the median infectious dose) of 1-10 organisms is widely reported, e.g. (5), however the infectivity by what are thought to be the most relevant large droplet routes (6) (which might have different portals of entry) does not appear to have been established.

So it seems to me that there are two important research needs identified for quantification of the outbreak: (1) quantification or estimation of the transferred dose upon different types of contacts between infectious and susceptible individuals; and (2) estimation of the dose response parameters for the most important routes of exposure.


1.Daley, D. and J. Gani, Epidemic Modeling: An Introduction NY, New York1999: Cambridge University Press.

2.Haas, C.N., J.B. Rose and C.P. Gerba, Quantitative Microbial Risk Assessment. 2nd ed2014, New York: John Wiley.