Back in 2009 I was with my family in Poland and we were doing some hill walking in Beskidy, a beautiful mountain range in South Poland. Hills in that range are usually well covered by forests, both mixed and single species – mostly spruce. This is what I was used to at least.
However, in places, the mature spruce forest was giving way to patches of a rather sad looking dead trees. When the landscape opened even more, we were walking through miles and miles of a apocalyptic-looking dead tree forests; the result of a combined drought and bark beetle onslaught.
Ips typographus, known as an eight-toothed spruce bark beetle or the European spruce bark beetle, is a serious and destructive pest of trees. It usually attacks weakened trees, so its outbreaks often follows dry periods when spruce – with its rather shallow roots – becomes particularly stressed. But, if summers are long and warm, the beetle can fit several generations (from eggs through larvae to adults) into the year. As each generation represents an exponential increase compared to the previous one, a huge outbreak can follow. The resulting density of beetles can overcome the resistance even of healthy trees, resulting in a massive outbreak which devastated forest in southern Poland and nearby Czech Republic and Slovakia.
This pattern is now being repeated again and again in different locations and with different trees and pests. As the climate becomes warmer, with less snow and higher temperature in winters forest insects like Ips typographus are finding it easier to survive. Less snow in winter and hotter summers also means drought which stresses trees. This combination is likely to lead to more and more disappearing forests – at least forests as we know them.
What can we do? Ips typographus is a native in continental Europe, but not present in the UK. We can try to make sure it is not brought into the UK by accident, and if it arrives – as it happened in Kent in 2018 – to eradicate it as quickly as possible.
We can also replant the forests with other trees, recognising that perhaps the forests we are used to might need to look differently in 20, 50 or hundred years. This requires a lot of planning of how to adapt the forests to the climate change and make them useful as well as beautiful.
Having written about two very large and very important plant diseases, I want today to focus on a much less known fungal pathogen, Rhizoctonia solani.
There is a personal aspect to it. In 1993 I joined the Department of Plant Sciences at the University of Cambridge as a post-doctoral researcher. I already had been working on human disease modelling, but this was the first time I encountered plant pathogens. Colleagues who were biologists and mycologists introduced me to a wonderful world of soil-borne fungi. I have been fascinated with the fungi ever since.
Rhizoctonia is a group of fungi causing various symptoms in plants, including sheath blight (the second-most devastating disease of rice after rice blast) and damping-off which affects seedlings. In the field, it can form densely infected patches causing large yield losses.
As a modeller, I was not allowed to do experiments (in case I do something stupid which will spoil the work), but I enjoyed seeing the experiments set up. We grew Rhizoctonia solani in trays filled with sand in which seedlings of brassicas (radish, cauliflower) were planted.
Another fungus, Trichoderma viride was introduced on poppy seeds, in an attempt to control the disease. A poppy seed had to be placed in an exact position and if dropped by mistake, they would ruin the experiment. It was amazing that the poppy seeds could not really be distinguished from grains of sand which made placing them tricky – but Rhizoctonia knew exactly where they were. A race between Rhizoctonia and Trichoderma ensued to see which one will get to “food” earlier.
We also did field experiments and I really enjoyed going out, meeting farmers and getting my boots dirty. Talking to people who actually grow the plants – propagators, farmers and farm managers – made me realise that mathematical and statistical modelling of plant pests and diseases must be connected to economics and behavioural sciences – a line of research that I have been pursuing since.
There is now a lot of interest in testing for coronavirus. The ‘test-and-trace’ strategy in the UK has been hugely controversial with a lot of complaints about people missing tests. But there is also a lot of confusion about how reliable testing is – Does it overestimate the number of cases? Does it tell people wrongly about their status? Are there people who are told they are not infected but they actually go about and spread the disease? Conversely, how many people are told wrongly they have the disease and as a result, their life is completely turned upside down unnecessarily?
There are at least three broad types of tests for the coronavirus. The first one uses swabs from people’s throat and nose to detect the virus itself; they are based on a molecular technique called PCR. This detects viral RNA and uses a method of “amplification” by which small amounts of RNA which are difficult to detect are “amplified” by a lab procedure to reach the detection level. This test tells us who is currently infected.
The second one is based on blood tests and detects the presence of antibodies – molecules that our body uses to fight off the virus. This test gives information on how many people have been exposed to the disease in the past. The third group looks for symptoms like infection of lungs and concentrates on people who had a particularly bad case of the coronavirus infection.
This blog is about statistics, not about molecular biology, so it suffices to say that all tests are pretty accurate and are continuously being evaluated and improved. At the same time, none of the tests is 100% effective and so we need to look at the consequences of potential misreadings.
There are two kinds of errors that are associated with testing. Firstly, the test might produce a negative result when a person is in fact infected. In other words, a person who in reality is infected by the virus is told “all clear”. There are many reasons for this. The swab test, for example, only can pick up the virus during a certain period following infection – not too early and not too late. Also, it is not a pleasant test – the swab spatula need to go far into the throat or nose. If not inserted properly (particularly if you are doing it yourself!), it will not pick up the viral particles.
In statistics, such an error is called a “false negative” result.
Secondly, the test can produce a positive result when a person is not infected. They perhaps went to get tested because they had fever and cough, but it was caused by an “ordinary” common cold. However, the test swab was for some reason contaminated with the viral RNA, or there was a problem with the procedure. This is usually less likely, but not impossible, as discussed recently in The New York Times.
We call it a “false positive” result.
It is important to stress at this point that the problems do not in any way undercut the importance of testing. However, it is important to understand the limitations and think about ways to deal with them.
Let’s look at some numbers. Suppose that on a certain day 200,000 people were tested and 3,000 positive results returned. These numbers are not too dissimilar to what is happening in the UK at present (18th September 2020: 233,199 tests were carried out with 3,395 positives).
False positives: ?
False negatives: ?
Our first go at the table.
We are now asking, how many people who went to get tested were “true” positives – not how many people in the whole population carried the virus which is a different question.
To calculate this we also need to have an idea of how good the tests are and so how likely the “false positives” and “false negatives” are. We currently do not have firm estimates, but we know that the “false positives” are not very likely, but “false negatives” could be more probable. Let’s assume first that the odds for both “false positives” and “false negatives” are 1:999. In other words that the (conditional) probability of the test to produce a wrong result is 0.1% in each case – very low indeed.
Then, a simple calculation shows that:
False positives: 197
False negatives: 3
Table for 1:999 odds for both “false positives” and “false negatives”.
What does this mean? Firstly, the number of people “really” infected (2,806) is similar to but slightly lower than the 3,000 reported – good so far. Secondly, the number of “false negatives” is very low (3/2806=0.1%). However, there are quite a few “false positives” – in fact, 6% (197/3000) of all positive tests are “false”.
We have made quite optimistic assumptions about the accuracy of the test, particularly for the “false negative”. Changing the odds to 2:8 just for these entries (there is some evidence that the best rate at which we can detect the virus is about 80% if tested 3-4 days after infection and 1-2 days after symptoms), we get:
False positives: 197
False negatives: 350
Table for 2:8 odds for “false positives” and 1:999 for “false negatives”.
The number of “false positives” stays the same (roughly, as we are doing some rounding off), but the number of “false negatives” shoots up. We now expect to have more “true” cases (3,153) than detected and to include a sizeable proportion of “false negatives”.
Why do we bother? The consequences of these two “false” test results are a bit different. If a person is told they are COVID-positive, they will need to self-isolate. It might also mean that their family, or friends, or contacts, will need to isolate as well. Or, if this happens at school, or at a care home, then the whole class or a home population will be affected, albeit for a relatively short time. If this is based on a wrong diagnosis, it can be a nuisance and it can lead to economic and social hardship, but there are hardly any epidemiological consequences.
In fact, if at all, they seem to be positive rather than negative. Most people who go to get tested have some symptoms or have been in contact with those who have symptoms. Flu symptoms are similar to the ones for COVID-19 and so the self-isolation might break the flu transmission as well. Maybe this is one reason why Australia and New Zealand apparently have been recording fewer influenza cases in their Winter season (June-September 2020) than in other years (masks and social distancing contribute as well).
The consequences of “false negatives” are more serious, as a person that carries the coronavirus infection will get a green light to carry on normal life. This could lead to further infections and, if a person is involved in a superspreader event, could result in a large outbreak.
In this post I only looked at the efficiency of testing, thinking exclusively about people who decided or were told to be tested. Thus, I only really looked at the PCR/swab tests.
There is a whole group of people who are infected – and infectious – but who for one reason or another are not tested. Some of them will self-isolate if they have symptoms. But many will simply carry on, unwillingly – perhaps sometimes carelessly – spreading the virus. In the first wave (February-June) we think there were about 8 such asymptomatic and untested people for every positive tested case. It is believed that with better testing this multiplier is now 2-3. As with the “false negatives” talked about above, it is very important to “catch” as many of them and to reduce the potential for transmission. But, this is a story for another blog post.
For the mathematically curious, this is the set of equations involved in the calculations: