Category Archives: Math biology

A National Face Mask Law Could End the Pandemic

How do I know this? Because I read an article in the April 2020 issue of the Atlantic Monthly explained the “real reason to wear a mask.”

Medical workers use them and other PPE to avoid ingress, transmission of outside particle to the wearer. However, individuals should wear masks to prevent egress. A key transmission route of COVID-19 is via droplets that fly out of our mouths when we cough, sneeze or even just speak. The purpose of wearing a mask is to avoid you transmitting the virus to others around us.

To develop this article, the magazine assembled an interdisciplinary team of 19 experts and looked at a range of mathematical models and other research. They wrote a scientific paper that was published online

The conclusion was that if 80% of people wore masks that were 60% efficient (easily achievable with cloth masks) the basic reproduction number R0 for the epidemic would be < 1 and the epidemic would die out. A graphic shows that possible combinations of mask wearing percentages and mask efficiencies that would achieve this goal.

I admit that the time scale over which things will happen is somewhat of a guess. Not much reduction will be seen in the first week since many infected people have yet to show symptoms. As the graphic shows the reduction will depend on the percent of people complying with the order and the quality of face masks, which should be much better now than when the article was initially published. On the other hand large numbers of people congregating in bars without wearing face masks could negate the effort.

The effectiveness of masks in containing the virus is not just a theoretical result. There are a number of spectacular examples of success. In Hong Kong only four deaths due to COVID-19 have been recorded since the beginning of the pandemic. Hong Kong health authorities credit their citizens’ near universal mask-wearing as a key factor. Similarly, Taiwan ramped up mask production early on and distributed masks to the population, mandating their use in public transit and recommending their use in public places, a suggestion that was been widely complied with. Their death toll has been 6, and the schools have been open since early February.

While other countries have been smart, the US has not. Thanks in no small part to Trump’s decision to not wear a mask and to have large rallies where very few people wore them, the issue has become politicized. Recently the governor of Georgia sued the mayor of Atlanta to stop her from imposing a mask order. Each weekend in Raleigh, hundreds of young people crowd into restaurants and bars on Glendale South and there is not a mask in sight, a situation that occurs in many parts of the country. This behavior occurs because of the perception that young people rarely get sick and if they do get infected the symptoms are mild. However, in recent weeks 1/3 of the new infected have been under the age of 30.

As Dr. Fauci has Said when Trump has allowed him to be on TV, large gatherings in which face masks are not worn can lead to transmission of the virus from one asymptomatic person to another. It is difficult to determine the extent to which this occurs, but contact tracing data from North Carolina shows that 50% of symptomatic cases are caused by contact with an asymptomatic individual. Another sign of the invisible epidemic is that the CDC estimates that there have been 10 times as many cases as those that have been verified by a COVID-19 test.

Trump has recently worn a mask, and at his corona virus briefing on Tuesday July 21, uttered the words that everyone should wear a mask when they are in a situation where social distancing is impossible. The history of pandemic in America shows that people will not voluntarily do the right thing. It must be mandatory. The president could dramatically improve his chances of being re-elected by signing an executive order to make mask mandatory.

I hate to point the president to a road to re-election, but I do not want to see 90,000 more people die. The IHME web site

projects 224,500 deaths by election day, while the CDC data shows that 140,000 have occurred as of July 21. To get re-elected, Trump must first admit stop lying about the pandemic. The US has 5% of the world’s population but the fraction of deaths that have occurred here is 140,000/617,000 = 22.7%, more than 4 times as many as a typical country. It does not have the lowest death rate in the world.

The US cannot reopen its economy or send students back to school five days a week with a pandemic raging in the streets. The crisis needs to be stopped now. It seems unlikely that sttes will go back into lockdown, so making masks mandatory is our only hope. If hospitalizations continue to spiral out of control (and they are NOT caused by our high level of testing) then the death toll could easily go higher than projected. In April when stay at home orders and other control measures were in place, the IHME projected death toll was roughly 70,000. This means that the premature re-opening of the economy has cost 150,000 lives. If we had followed the lead of Europe and dramatically reduce the number of cases before opening up the country things would be much better now but that opportunity is gone. We need to act now to prevent a complete disaster.


Pooled Tests for COVID-19

When one is dealing with a disease that is at a low frequency in the population and one has a large number of people to test, it is natural to do group testing. A fixed number of samples, say 10, are mixed together. If the combined sample is negative, we know all the individuals are. But if a group tests positive then all the samples in the group have to be retested individually.

If the groups are too small then not much work is saved. If the groups are too large then there are too many positive group tests. To find the optimal group size, suppose there are a total of  N individuals, the group size is k, and 1% of the population has the disease. The number of group tests that must be performed is N/k. The probability a group tests positive is k/100. If this happens then we need k more tests. Thus we want to minimize

(N/k)( 1 + k2/100) = N/k + Nk/100

Differentiating we want –N/k2 + N/100=0 or k = 10. In the concrete case N = 1000, the number of tests is 200.

Note: the probability a group test is positive is p = 1 – (1 – 1/100)k but this makes the optimization very messy. When k=10, 1 + kp = 1.956, so the answer does not change by very much.

Recent work reported on in Nature on July 10, 2020 shows that the number of tests needed can be reduced substantially if the individuals are divided into groups in two different ways for group testing before one has to begin testing individuals. To visualize the set-up consider a k by k matrix with one individual in each cell. We will group test the rows and group test the columns . An individual who tests negative in either test can be eliminated. The number of k by k squares is N/k2. For each square there are 2k tests that are always performed. Each of the k2 individuals in the square have their group test positive twice with probability (k/100)2. These events are NOT independent, but that does not matter in computing the expected number of tests

(N/ k2)(2k + k4/10,000) = 2N/k + N k2/10,000

Differentiating we want –2N/k2 + 2Nk/10,000 = 0 or k = (10,000)1/3 = 21.54. In the concrete case N=1000 the expected number of tests is 139.

Practical Considerations:

One could do fewer tests by eliminating the negative rows before testing the columns, but the  algorithm used here allows all the tests to be done at once, avoiding the need to wait for the first round results to come back before  the second round is done.

Larger group sizes will make it harder to detect the virus if only one individual in the group. The Nature article, Sigrum Smola of the Saarland University Medical Center in Homburg has been is quoted as saying he doesn’t recommend grouping more than 30 individuals in one test. Others claim that it is possible to identify the virus when there is one positive individual out of 100.

Ignoring the extra work in creating the group samples, the method described above reduces the cost of test by 86%. The price of $9 per test quoted in the article would be reduced to $1.26, so this could save a considerable amount of money for a university that has to test 6000 undergraduates several times in one semester.

In May, officials in Wuhan used a method of this type to test 2.3 million samples in two weeks.


Mutesa, L et al (2020) A strategy for finding people infected with SARS-CoV-2: optimizing pooled testing at low prevalence arXiv: 2004.14934

Malliapaty, Smriti (2020) The mathematical strategy that could transform coronavirus testing. Nature News July 10. https://www-nature-com/articles/d41586-020-02053-6


China is NOT to blame for the COVID-19 epidemic in the US

As President, Donald has told tens of thousands of lies. In many cases, he can hide behind the silence of his loyal supporters. However, when it comes to the coronavirus epidemic the details are on TV, in the press, and in publicly available databases for all the world to see.

One of his most egregious lies is that China is to blame for the epidemic. A May 20 story in USA today says “As the political rhetoric blaming China for the pandemic escalates, law enforcement officials and human rights advocates have seen an increasing number of hate crimes and incidents of harassment and discrimination against Asian Americans.” Trump has fanned these flames in his rallies, referring to the virus as the “Kung flu.”

One of the most incredible lies (i.e., too extraordinary and improbable to be believed) is that the corona virus was made in a laboratory in Wuhan. To protect this lie, the White House directed the National Institutes of Health to cancel funding for a project studying how coronaviruses spread from bats to people. The NIH typically only cancels active grant when there is scientific misconduct or improper financial behavior, neither of which it has occurred in this case. The PI on the grant, Dr. Peter Daszak, is President of EcoHealth Alliance, a US-based organization that conducts research and outreach programs on global health, conservation and international development. His research has been instrumental in identifying and predicting the origins and impact of emerging disease, which is very important for avoiding future pandemics..

Early Spread.  A special report published on July 5 in the New York Times gives new information about the early days of the epidemic. In mid-February the official case count was 15 but there is evidence of 2000 other infections. Given what we now know about the spread of the disease, it is natural to guess that many of these cases were asymptomatic. However, as explained in the paper cited in the next paragraph, part of the discrepancy was due to the fact that testing done before March 4, 2020 was only done for symptomatic patients who had recently traveled internationally.

This idea that the corona virus was widespread in the US in January 2020 was discussed in news stories about Alessandro Vepignani’s work. These appeared on the Northeastern web site in April, but the paper has only recently appeared on the medRxiv: Jessica T. Davis et al. Estimating the establishment of local transmission and the cryptic phase of the COVID-19 pandemic in the US. Their conclusions are based on the use of a rather complicated individual-based, stochastic and spatial epidemic model called  GLEAM (GLobal Epidemic and Mobility Model) that divides the global population 3200 subpopulations. See PNAS 166 (2009), 21484-21489 and J. Computational Science 1 (2010), 132-145 for more details.

Origins of the virus in the US.  Recently two genetic sequencing studies published online in Science, have investigated the origins of the corona virus in US. A.S. Gonzalez-Reiche et al, published on May 29, 2020 studied Introductions and early spread of SARS-CoV-2 in the New York City area. Phylogenetic analysis of 84 distinct SARS-CoV-2 genomes from samples taken February 29 – March 18 provided evidence for multiple, independent introduction . Phylogenetic analysis of full length genome sequences suggested that the majority of the introductions came from Europe and other parts of the United States.

A medRxiv preprint by M.T. Maurano et al which reports on the analysis of 864 SARS-CoV-2 sequences, reached the same conclusion: comparisons to global viral sequences showed that early strains was most likely linked to cases from Europe

Deng et al, published on June 8, 2020 studied the introduction of Sars-CoV-2 into Northern California. They studied 36 patients spanning 9 counties and the Grand Princess cruise ship using a method they called MSSPE (Metagenomic Sequencing with Spiked Primer Enrichment) to assemble genomes directly from clinical samples. Phylogenetic analysis described in detail in the paper indicated that 14 were associated with the Washington State WA1 lineage, 10 associated with a Santa Clara outbreak cluster (SCC1), 3 from a Solano County cluster, 5 related to lineages circulating in Europe but only 4 related to lineages from Wuhan. This precision comes from the fact that as of March 20, 2020 when this work was done, there were 789 worldwide genomes in the GISAID database. This wealth of data is possible because coronaviruses are unsegmented single-stranded RNA viruses that are about 30 kilobases in length.

The results in the last three paragraphs demonstrate that most lineages came from Europe not China. In hindsight the fact that Europe is the primary source of coronavirus is the US. Travel from China was banned February 2, but travel from Europe was only ended on March 13.

I have concentrated on the science.  I’ll leave it to you to decide if you want the Senate to vote for Thom Tillis’ 18-point plan in May “to hold China accountable” for what he says is its role in the coronavirus pandemic.